Advanced Functional Evolution Equations and Inclusions

Developments in Mathematics Mouffak Benchohra Saïd Abbas Advanced Functional Evolution Equations and Inclusions Deve...

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Developments in Mathematics

Mouffak Benchohra Saïd Abbas

Advanced Functional Evolution Equations and Inclusions

Developments in Mathematics VOLUME 39 Series Editors: Krishnaswami Alladi, University of Florida, Gainesville, FL, USA Hershel M. Farkas, Hebrew University of Jerusalem, Jerusalem, Israel

More information about this series at http://www.springer.com/series/5834

Mouffak Benchohra • Saïd Abbas

Advanced Functional Evolution Equations and Inclusions

123

Mouffak Benchohra Department of Mathematics University of Sidi Bel Abbes Sidi Bel Abbes, Algeria

Saïd Abbas Laboratoire de Mathématiques Université de Saïda Saïda, Algeria

ISSN 1389-2177 ISSN 2197-795X (electronic) Developments in Mathematics ISBN 978-3-319-17767-0 ISBN 978-3-319-17768-7 (eBook) DOI 10.1007/978-3-319-17768-7 Library of Congress Control Number: 2015936330 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www. springer.com)

We dedicate this book to our family members. In particular, Saïd Abbas dedicates to the memory of his father Abdelkader Abbas; and Mouffak Benchohra makes his dedication to the memory of his father Yahia Benchohra

Preface

Functional differential equations and inclusions occur in a variety of areas of biological, physical, and engineering applications, and such equations have received much attention in recent years. This book is devoted to the existence of local and global mild solutions for some classes of functional differential evolution equations and inclusions, and other densely and non-densely defined functional differential equations and inclusions in separable Banach spaces or in Fréchet spaces. Some of these equations and inclusions present delay which may be finite, infinite, or statedependent. Other equations are subject to impulses effect. The tools used include classical fixed point theorems and the measure of non-compactness (MNC). Each chapter concludes with a section devoted to notes and bibliographical remarks. All the presented abstract results are illustrated by examples. The content of the book is new and complements the existing literature devoted to functional differential equations and inclusions. It is useful for researchers and graduate students for research, seminars, and advanced graduate courses, in pure and applied mathematics, engineering, biology, and all other applied sciences. We are grateful to our colleagues and friends N. Abada, E. Alaidarous, S. Baghli, A. Baliki, M. Belmekki, K. Ezzinbi, H. Hammouche, J. Henderson, I. Medjedj, J.J. Nieto, and M. Ziane for their collaboration in research related to the problems considered in this book. Last but not least, we are grateful to Elizabeth Loew and Dahlia Fisch for their support and to Jeffin Thomas Varghese for his help during the production of the book. Sidi Bel Abbes, Algeria Saïda, Algeria

Mouffak Benchohra Saïd Abbas

vii

Introduction

Nonlinear evolution equations, i.e., partial differential equations with time t as one of the independent variables, arise not only from many fields of mathematics, but also from other branches of science such as physics, mechanics, and material science. For example, Navier–Stokes and Euler equations from fluid mechanics, nonlinear reaction-diffusion equations from heat transfers and biological sciences, nonlinear Klein–Gordon equations and nonlinear Schrödinger equations from quantum mechanics, and Cahn–Hilliard equations from material science, to name just a few, are special examples of nonlinear evolution equations. See the books [174, 176–178]. Functional differential equations and inclusions arise in a variety of areas of biological, physical, and engineering applications, and such equations have received much attention in recent years. A good guide to the literature for functional differential equations is the books by Hale [131], Hale and Verduyn Lunel [133], Kolmanovskii and Myshkis [148], and the references therein. During the last decades, existence and uniqueness of mild, strong, classical, almost periodic, almost automorphic solutions of semi-linear functional differential equations and inclusions has been studied extensively by many authors using the semigroup theory, fixed point argument, degree theory, and measures of non-compactness. We mention, for instance, the books by Ahmed [16], Diagana [103], Engel and Nagel [106], Kamenskii et al. [144], Pazy [168], Wu [184], Zheng [187], and the references therein. In recent years, there has been a significant development in evolution equations and inclusions; see the monograph of Perestyuk et al. [169], the papers of Baliki and Benchohra [33, 37], Benchohra and Medjedj [55, 56], Benchohra et al. [82], and the references therein. Neutral functional differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years. A good guide to the literature for neutral functional differential equations is the books by Hale [131], Hale and Verduyn Lunel [133], Kolmanovskii and Myshkis [148], and the references therein. Hernandez in [137] proved the existence of mild, strong, and periodic solutions for neutral equations. Fu in [117, 118] studies the controllability on a bounded interval of a class of neutral functional differential equations. Fu and ix

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Introduction

Ezzinbi [119] considered the existence of mild and classical solutions for a class of neutral partial functional differential equations with nonlocal conditions. Various classes of partial functional and neutral functional differential equations with infinite delay are studied by Adimy et al. [10–12], Belmekki et al. [52], and Ezzinbi [108]. Henriquez [136] and Hernandez [137, 138] studied the existence and regularity of solutions to functional and neutral functional differential equations with unbounded delay. Balachandran and Dauer have considered various classes of first and second order semi-linear ordinary, functional and neutral functional differential equations on Banach spaces in [43]. By means of fixed point arguments, Benchohra et al. have studied various classes of functional differential equations and inclusions and proposed some controllability results in [28, 33, 33, 37, 58, 72, 73, 75, 76, 80]. See also the works by Gatsori [120], Li et al. [155], Li and Xue [156], and Li and Yong [157]. Impulsive differential equations and inclusions appear frequently in applications such as physics, aeronautic, economics, engineering, and population dynamics; see the monographs of Bainov and Simeonov [39, 40], Benchohra et al. [81], Erbe and Krawcewicz [107], Graef et al. [127], Samoilenko and Perestyuk [172], and Perestyuk et al. [169], and the paper of Coldbeter et al. [95] where numerous properties of their solutions are studied. In this way, they make changes of states at certain moments of time between intervals of continuous evolution such changes can be reasonably well approximated as being instantaneous changes of this state which we will represent by impulses and then these processes are modeled by impulsive differential equations and for this reason the study of this type of equations has received great attention in the last years. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments. See, for instance, the monographs by Benchohra et al. [81], Lakshmikantham et al. [150], and Samoilenko and Perestyuk [172]. There exists an extensive literature devoted to the case where the impulses are absent (i.e., Ik D 0; k D 1; : : : ; m), see, for instance, the monograph by Liang and Xiao [158] and the paper by Schumacher [158]. We mention here also the use of impulsive differential equations in the study of oscillation and non-oscillation of impulsive dynamic equations, see, for instance, the papers of Graef et al. [124, 125], oscillation of dynamic equations with delay was considered in [13, 14]. During the last 10 years impulsive ordinary differential inclusions and functional differential equations and inclusions have attracted the attention of many mathematicians and are intensively studied. At present the foundations of the general theory and such kind of problems are already laid and many of them are investigated in detail in [58, 59, 63, 79, 81, 107] and the references therein. It is well known that the issue of controllability plays an important role in control theory and engineering because they have close connections to pole assignment, structural decomposition, quadratic optimal control, observer design, etc. In recent years, the problem of controllability for various kinds of differential and impulsive differential systems has been extensively studied by many authors [71, 155–157, 186] using different approaches. Several authors have extended the controllability concept to infinite dimensional systems in Banach space with

Introduction

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unbounded operators, see the monographs [85, 98, 157, 186] and the references therein. Sufficient conditions for controllability are established by Lasiecka and Triggiani [153]. Fu in [117, 118] studied the controllability on a bounded interval of a class of neutral functional differential equations. Fu and Ezzinbi [119] considered the existence of mild and classical solutions for a class of neutral partial functional differential equations with nonlocal conditions. Adimy et al. [10–12] studied some classes of partial functional and neutral functional differential equations with infinite delay. When the delay is infinite, the notion of the phase space B plays an important role in the study of both qualitative and quantitative theory. A usual choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato in [132], see also Corduneanu and Lakshmikantham [97] and Kappel and Schappacher [145]. The literature related to ordinary and partial functional differential equations with delay is very extensive. On the other hand, functional differential equations with state-dependent delay appear frequently in applications as model of equations, and for this reason the study of this type of equations has received great attention in the last year, see, for instance [31, 183] and the references therein. The literature related to partial functional differential equations with state-dependent delay is limited; see [139, 171]. Several authors have considered extensively the problem x0 .t/ D A.t/x.t/ C f .t; xt / when A.t/ D A: Existence of mild solutions is developed by Heikkila and Lakshmikantham [134], Kamenski et al. [144], and the pioneer Hino and Murakami paper [141] for some semi-linear functional differential equations with finite delay. By means of fixed point arguments, Benchohra and his collaborators have studied many classes of first and second order functional differential inclusions on a bounded interval with local and nonlocal conditions in [59, 60, 62, 64, 65, 77, 78, 121]. Extension to the semi-infinite interval is given by Benchohra and Ntouyas in [58, 61]. When A depends on time, Arara et al. [26, 28] considered a control multi-valued problem on a bounded interval. Uniqueness results of mild solutions for some classes of partial functional and neutral functional differential evolution equations on the semi-infinite interval J D RC for a finite delay with local and nonlocal conditions were given in [33, 37]. When the delay is infinite, existence and uniqueness results for evolution problems are proposed in [33], and controllability result of mild solutions for the evolution equations are given in [15, 36]. The case when A is non-densely defined and generates an integrated semigroup was done by Benchohra et al. [80]. Some global existence results for impulsive differential equations and inclusions were obtained by Guo [129], Graef and Ouahab [126], and the references therein. Partial functional evolution equations and inclusions with infinite and statedependent delay, controllability on finite interval are our concerns. Our approach is based upon the fixed point theory for multi-valued condensing maps under assumptions expressed in terms of the MNC [144].

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In the last three decades, the theory of C0 -semigroup has been developed extensively, and the achieved results have found many applications in the theory of partial differential equations, for instance see [106, 122, 168] and the papers of Arara et al. [26, 27] and Benchohra et al. [74]. Recently, increasing interest has been observed in applications to impulsive differential equations and inclusions, see Liu [69, 159]. The case where the generator of the semigroup is non-densely defined, the existence of integral solutions on compact intervals for differential equations and inclusions were studied by Adimy et al. [8–10], Arendt [29, 30], Ezzinbi and Liu [109, 110], and Henderson and Ouahab [135]. The model with multi-valued jump sizes arises in a control problem where we want to control the jump sizes in order to achieve given objectives. There are very few results for impulsive evolution inclusions with multi-valued jump operator, see [161]. We present the existence of solutions for both densely or non-densely defined impulsive functional differential inclusions. ˇ The multi-valued jumps (i.e., the difference operator xˇtDtk 2 Ik .x.tk //) is a natural model of an impulsive system where the jump sizes are not deterministic as in [17–19, 161] but rather they are uncertain. However given the state x and time ti ; the set of possible jump sizes at this state is determined by the set Ik .x/: The set-valued maps Ik may be given by the sub-differential of a lower semi-continuous convex functional i : In this case, the system is governed by evolution inequations at the points of time tk : Another situation that may give rise to such a dynamic model originates from the parametric uncertainty such as Ik .x/ D fIk .t; x/I t 2 Ig; where fIk g is a suitable family of functions I E ! E: To our knowledge, there are very few results for impulsive evolution inclusions with multi-valued jump operators; see [5, 19, 36]. The results of this book extend and complement those obtained in the absence of the impulse functions Ik ; and for those with single-valued impulse functions Ik : This book is arranged and organized as follows: In Chap. 1, we introduce notations, definitions, and some preliminary notions. In Sect. 1.1, we give some notations from the theory of Banach spaces. Section 1.2 is concerned to recall some basic definitions and some properties in Fréchet spaces. In Sect. 1.3, we recall some basic definitions and give some examples of Phase spaces. Section 1.4 contains some properties of set-valued maps. In Sect. 1.5, we give some preliminaries about evolution systems. Some definitions and properties of the theory of semigroups are presented in Sect. 1.6. In Sect. 1.6.3, we give some properties of the extrapolation method. The last section (Sect. 1.7) contains some fixed point theorems. In Chap. 2, we study some first order classes of partial functional, neutral functional, integro-differential, and neutral integro-differential evolution equations with finite delay on the positive real line. Section 2.2 deals with the existence and uniqueness of mild solutions for some classes of partial evolution equations with local and nonlocal conditions. We give some results based on the fixed point theorem of Frigon in Fréchet spaces. An example will be presented at the last illustrating the abstract theory. In Sect. 2.3, we study some neutral differential evolution equations in Fréchet spaces. In Sect. 2.4, we give existence results for other

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classes of partial functional integro-differential evolution equations. Section 2.5 deals with uniqueness results of neutral functional integro-differential evolution equations. In Chap. 3, we provide sufficient conditions for the existence of the unique mild solution on the positive half-line RC for some classes of first order partial functional and neutral functional differential evolution equations with infinite delay. In Sect. 3.2, we study the existence and uniqueness of mild solutions for partial functional evolution equations in Fréchet spaces. Section 3.3 deals with the controllability of mild solutions on finite interval for partial evolution equations. In Sect. 3.4, we study the controllability of mild solutions on semi-infinite interval for partial evolution equations. Section 3.5 deals with the existence of the unique mild solution of neutral functional evolution equations. In Sect. 3.6, we study the controllability of mild solutions on finite interval for neutral evolution equations. Section 3.7 deals with the controllability of mild solutions on semi-infinite interval for neutral evolution equations. In Chap. 4, we shall be concerned by perturbed partial functional and neutral functional evolution equations with finite and infinite delay on the semi-infinite interval RC : Our main tool is the nonlinear alternative proved by Avramescu (1.30) for the sum of contractions and completely continuous maps in Fréchet spaces [32], combined with semigroup theory. In Sect. 4.2, we study the existence of mild solutions for perturbed partial functional evolution equations with finite delay. Section 4.3 deals with perturbed neutral functional evolution equations with finite delay. In Sect. 4.4, we study the existence of mild solutions for perturbed partial evolution equations with infinite delay. In Chap. 5, we provide sufficient conditions for the existence of mild solutions on the semi-infinite interval RC for some classes of first order partial functional and neutral functional differential evolution inclusions with finite delay. In Sect. 5.2, we study the existence of mild solutions for a class of functional partial evolution equations. Section 5.3 deals with neutral partial evolution equations. In Chap. 6, we study the existence of mild solutions on the semi-infinite interval RC for some classes of first order partial functional and neutral functional differential evolution inclusions with infinite delay. In Sect. 6.2, we study functional partial evolution equations. Section 6.3 deals with neutral partial evolution equations. In Chap. 7, we are concerned by the existence of mild and extremal solutions of some first order classes of impulsive semi-linear functional differential inclusions with local and nonlocal conditions when the delay is finite in separable Banach spaces. Using a recent theorem due to Dhage combined with the semigroup theory, the existence of the mild and extremal mild solution are assured. The nonlocal case is studied too. In Sect. 7.2, we study the existence of mild solutions with local conditions. Section 7.3 deals with the existence of mild solutions with nonlocal conditions. In Sect. 7.4, we give an application to the control theory. In Chap. 8, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some non-densely defined impulsive semi-linear functional differential inclusions in separable Banach spaces with local and nonlocal conditions. In Sect. 8.2, we give some results for integral solutions

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of non-densely defined functional differential inclusions with local conditions. Section 8.3 deals with extremal integral solutions with local conditions, and Sect. 8.4 deals with extremal integral solutions with nonlocal conditions. In Sect. 8.5, we give an application to the control theory. In Chap. 9, we study the existence of mild solutions impulsive semi-linear functional differential equations. In Sect. 9.2, we study some existence results for semi-linear differential evolution equations with impulses and delay. Section 9.3 is devoted to some classes of impulsive semi-linear functional differential equations with non-densely defined operators. In Sect. 9.4, we study impulsive semi-linear neutral functional differential equations with infinite delay. Section 9.5 deals with integral solutions of non-densely defined impulsive semi-linear functional differential equations with state-dependent delay. In Chap. 10, we shall establish sufficient conditions for the existence of mild, extremal mild, integral, and extremal integral solutions for some impulsive semilinear neutral functional differential inclusions in separable Banach spaces. In Sect. 10.2, we study some densely defined impulsive functional differential inclusions. Section 10.3 deals with the existence of mild solutions for non-densely defined impulsive neutral functional differential inclusions. In Sect. 10.4, we study the controllability of impulsive semi-linear differential inclusions in Fréchet spaces. In Chap. 11, we study functional differential inclusions with multi-valued jumps. In Sect. 11.2, we study some existence of integral solutions for semi-linear functional differential inclusions with state-dependent delay and multi-valued jump. Section 11.3 deals with impulsive evolution inclusions with infinite delay and multi-valued jumps. Section 11.4 deals with impulsive semi-linear differential evolution inclusions with non-convex right-hand side. In Sect. 11.5, we study some impulsive evolution inclusions with state-dependent delay and multi-valued jumps. Section 11.6 deals with the controllability of impulsive differential evolution inclusions with infinite delay. In Chap. 12, we study functional differential equations and inclusions with delay. In Sect. 12.2, we prove some global existence for functional differential equations with state-dependent delay. Section 12.3 deals with global existence results for neutral functional differential equations with state-dependent delay. In Sect. 12.4, we give some global existence results for functional differential inclusions with delay. Section 12.4.1 deals with global existence results for functional differential inclusions with state-dependent delay. In Chap. 13, we shall establish sufficient conditions for global existence results of second order functional differential equations with delay. In Sect. 13.2, we give some global existence results of second order functional differential equations with delay. Keywords and Phrases: Evolution differential equations and inclusions, integrodifferential equations, densely and non-densely defined differential equations, convex and non-convex valued multi-valued, mild solution, weak solution, initial value problem, nonlocal conditions, contraction, existence, uniqueness, measure of noncompactness, Banach space, Fréchet space, phase space, impulses, time delay, state-dependent delay, fixed point.

Contents

1

Preliminary Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Some Properties in Fréchet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Phase Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Set-Valued Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Evolution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 C0 -Semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Integrated Semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Extrapolated Semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Some Fixed Point Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 7 7 7 9 10 10

2

Partial Functional Evolution Equations with Finite Delay . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Partial Functional Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Nonlocal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Neutral Functional Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Nonlocal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Partial Functional Integro-Differential Evolution Equations . . . . . . . 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Nonlocal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Neutral Functional Integro-Differential Evolution Equations . . . . . . 2.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 17 17 18 21 22 24 24 24 29 30 32 32 32 36 37 38 38 xv

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2.5.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Nonlocal Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 42 44 45

Partial Functional Evolution Equations with Infinite Delay . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Partial Functional Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Existence and Uniqueness of Mild Solution . . . . . . . . . . . . . . 3.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Controllability on Finite Interval for Partial Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Controllability of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Controllability on Semi-infinite Interval for Partial Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Controllability of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Neutral Functional Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Existence and Uniqueness of Mild Solution . . . . . . . . . . . . . . 3.5.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Controllability on Finite Interval for Neutral Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Controllability of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Controllability on Semi-infinite Interval for Neutral Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.2 Controllability of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 47 47 48 52

Perturbed Partial Functional Evolution Equations . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Perturbed Partial Functional Evolution Equations with Finite Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93 93

2.6 3

4

53 53 54 61 62 62 63 67 68 68 69 74 75 75 76 82 84 84 84 91 92

93 93 94 99

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4.3

4.4

4.5

xvii

Perturbed Neutral Functional Evolution Equations with Finite Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbed Partial Functional Evolution Equations with Infinite Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

100 100 100 105 106 106 106 111 112

5

Partial Functional Evolution Inclusions with Finite Delay . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Partial Functional Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Neutral Functional Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 113 113 114 118 119 119 120 124 126

6

Partial Functional Evolution Inclusions with Infinite Delay . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Partial Functional Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Neutral Functional Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 127 127 128 133 135 135 135 141 142

7

Densely Defined Functional Differential Inclusions with Finite Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Existence of Mild Solutions with Local Conditions . . . . . . . . . . . . . . . . 7.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Existence of Extremal Mild Solutions . . . . . . . . . . . . . . . . . . . . . 7.3 Existence of Mild Solutions with Nonlocal Conditions . . . . . . . . . . . . 7.3.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 143 143 144 153 155 155

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7.4

7.5 8

9

Application to the Control Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Non-densely Defined Functional Differential Inclusions with Finite Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Integral Solutions of Non-densely Defined Functional Differential Inclusions with Local Conditions . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Extremal Integral Solutions with Local Conditions . . . . . . . . . . . . . . . . 8.4 Integral Solutions with Nonlocal Conditions . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Application to the Control Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impulsive Semi-linear Functional Differential Equations . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Semi-linear Differential Evolution Equations with Impulses and Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Existence of Extremal Mild Solutions . . . . . . . . . . . . . . . . . . . . . 9.2.4 Impulsive Differential Equations with Nonlocal Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Impulsive Semi-linear Functional Differential Equations with Non-densely Defined Operators . . . . . . . . . . . . . . . . . . . . 9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Examples of Operators with Non-dense Domain . . . . . . . . . 9.3.3 Existence of Integral Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Existence of Extremal Integral Solutions . . . . . . . . . . . . . . . . . . 9.3.5 Impulsive Differential Equations with Nonlocal Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.6 Applications to Control Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.7 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Impulsive Semi-linear Neutral Functional Differential Equations with Infinite Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Existence of Integral Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

156 156 162 163 165 165 165 166 179 180 181 182 183 187 189 191 191 191 191 192 199 200 201 203 203 204 205 215 216 218 223 225 225 226 234 243

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9.5

9.6 10

11

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Non-densely Defined Impulsive Semi-linear Functional Differential Equations with State-Dependent Delay . . . . . . . . . . . . . . . . 9.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Existence of Integral Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Impulsive Functional Differential Inclusions with Unbounded Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Densely Defined Impulsive Functional Differential Inclusions . . . . 10.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Extremal Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Extremal Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Controllability of Impulsive Semi-linear Differential Inclusions in Fréchet Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Differential Inclusions with Multi-valued Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Semi-linear Functional Differential Inclusions with State-Dependent Delay and Multi-valued Jump . . . . . . . . . . . . . . . . . . . . 11.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Existence of Integral Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Impulsive Evolution Inclusions with Infinite Delay and Multi-valued Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Impulsive Semi-linear Differential Evolution Inclusions with Non-convex Right-Hand Side . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Existence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245 245 246 259 261 261 261 261 263 274 276 278 279 289 291 293 293 303 304 305 305 305 305 306 306 316 317 317 318 323 324 324 325 329

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11.5

11.6

11.7 12

13

Impulsive Evolution Inclusions with State-Dependent Delay and Multi-valued Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Existence Results for the Convex Case . . . . . . . . . . . . . . . . . . . . 11.5.3 Existence Results for the Non-convex Case . . . . . . . . . . . . . . . 11.5.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Controllability of Impulsive Differential Evolution Inclusions with Infinite Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.2 Existence and Controllability Results . . . . . . . . . . . . . . . . . . . . . 11.6.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Functional Differential Equations and Inclusions with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Global Existence for Functional Differential Equations with State-Dependent Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Global Existence Results for Neutral Functional Differential Equations with State-Dependent Delay . . . . . . . . . . . . . . . . 12.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Global Existence Results for Functional Differential Inclusions with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Global Existence Results for Functional Differential Inclusions with State-Dependent Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Existence of Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second Order Functional Differential Equations with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Global Existence Results of Second Order Functional Differential Equations with Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Existing Result for the Finite Delay Case . . . . . . . . . . . . . . . . .

330 330 331 337 341 342 342 343 351 352 353 353 353 353 354 359 361 361 361 368 369 369 370 374 375 375 376 382 383 385 385 385 385 386

Contents

xxi

13.2.3

13.3

Existing Results for the State-Dependent Delay Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 13.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

Chapter 1

Preliminary Background

In this chapter, we introduce notations, definitions, and preliminary facts which are used throughout this book.

1.1 Notations and Definitions Let RC D Œ0; C1/ be the positive real line, H D Œr; 0 be an interval with r > 0, and .E; j j/ be a real Banach space. By C.H; E/ we denote the Banach space of continuous functions from H into E; with the norm kyk D sup jy.t/j: t2H

Let B.E/ be the space of all bounded linear operators from E into E, with the norm kNkB.E/ D sup jN.y/j: jyjD1

A measurable function y W RC ! E is Bochner integrable if and only if jyj is Lebesgue integrable. For properties of the Bochner integral, see for instance, Yosida [185].

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_1

1

2

1 Preliminary Background

As usual, by L1 .RC ; E/ we denote the Banach space of measurable functions y W RC ! E which are Bochner integrable normed by Z kykL1 D

C1

jy.t/jdt: 0

1 Let Lloc .RC ; E/ be the space of measurable functions which are locally Bochner integrable. For any continuous function y defined on Œr; C1/ and any t 2 RC ; we denote by yt the element of C.H; E/ defined by

yt . / D y.t C /

for 2 H:

Here yt ./ represents the history of the state from time t r up to the present time t: Definition 1.1. A function f W RC E ! E is said to be an L1 -Carathéodory function if it satisfy: (i) for each t 2 RC the function f .t; :/ W E ! E is continuous; (ii) for each y 2 E the function f .:; y/ W RC ! E is measurable; (iii) for every positive integer k there exists hk 2 L1 .RC ; RC / such that jf .t; y/j hk .t/ for all jyj k and almost each t 2 RC :

1.2 Some Properties in Fréchet Spaces Let X be a Fréchet space with a family of semi-norms fk kn gn2N : Let Y X; we say that F is bounded if for every n 2 N; there exists Mn > 0 such that kykn Mn

for all y 2 Y:

To X we associate a sequence of Banach spaces f.X n ; k kn /g as follows: For every n 2 N; we consider the equivalence relation n defined by: x n y if and only if kx ykn D 0 for all x; y 2 X: We denote X n D .Xjn ; k kn / the quotient space, the completion of X n with respect to k kn : To every Y X, we associate a sequence fY n g of subsets Y n X n as follows: For every x 2 X; we denote Œxn the equivalence class of x of subset X n and we define Y n D fŒxn W x 2 Yg: We denote Y n ; intn .Y n / and @n Y n ; respectively, the closure, the interior, and the boundary of Y n with respect to k k in X n : We assume that the family of semi-norms fk kn g verifies: kxk1 kxk2 kxk3 : : : for every x 2 X:

1.3 Phase Spaces

3

Definition 1.2 ([116]). A function f W X ! X is said to be a contraction if for each n 2 N there exists kn 2 Œ0; 1/ such that: kf .x/ f .y/kn kn kx ykn for all x; y 2 X:

1.3 Phase Spaces In this section, we will define the phase space B axiomatically, using ideas and notations developed by Hale and Kato [132] and follow the terminology used in [142], (see also Kapper and Schappacher [145] and Schumacher [173]). More precisely, .B; k kB / will denote the vector space of functions defined from .1; 0 into E endowed with a semi norm denoted k:kB and satisfying the following axioms: (A1 ) If y W .1; b/ ! E; b > 0; is continuous on Œ0; b and y0 2 B, then for every t 2 Œ0; b/ the following conditions hold: (i) yt 2 B; (ii) There exists a positive constant H such that jy.t/j Hkyt kB ; (iii) There exist two functions K./; M./ W RC ! RC independent of y.t/ with K continuous and M locally bounded such that: kyt kB K.t/ supf jy.s/j W 0 s tg C M.t/ky0 kB : Denote Kb D supfK.t/ W t 2 Œ0; bg and Mb D supfM.t/ W t 2 Œ0; bg: (A2 ) (A3 )

For the function y.:/ in .A1 /, yt is a B-valued continuous function on Œ0; b. The space B is complete.

Remark 1.3. 1. (ii) is equivalent to j.0/j HkkB for every 2 B. 2. Since k kB is a seminorm, two elements ; 2 B can verify k kB D 0 without necessarily ./ D . / for all 0. 3. From the equivalence of (ii), we can see that for all ; 2 B such that k kB D 0: This implies necessarily that .0/ D .0/: For any continuous function y and any t 0; we denote by yt the element of B defined by yt . / D y.t C /

for 2 .1; 0:

We assume that the histories yt belong to some abstract phase space B:

4

1 Preliminary Background

Consider the following space ˚ BC1 D y W R ! E W yjRC 2 C.RC ; E/; y0 2 B ; where yjRC / is the restriction of y to Œ0; C1/. Hereafter are some examples of phase spaces. For other details we refer, for instance, to the book by Hino et al. [142]. Example 1.4. The spaces BC; BUC; C1 , and C0 : Let: BC the space of bounded continuous functions defined from .1; 0 to EI BUC the space of bounded uniformly continuous functions defined from .1; 0to EI C1 C0

WD 2 BC W lim ./ exists in E I !1 WD 2 BC W lim ./ D 0 ; endowed with the uniform norm !1

kk D supfj./j W 0g: We have that the spaces BUC; C1 , and C0 satisfy conditions .A1 /–.A3 /: BC satisfies .A2 /; .A3 / but .A1 / is not satisfied. Example 1.5. The spaces Cg ; UCg ; Cg1 , and Cg0 . Let g be a positive continuous function on .1; 0. We define: ./ is bounded on .1; 0 I Cg WD 2 C..1; 0; E/ W g. / ./ 0 D 0 ; endowed with the uniform norm Cg WD 2 Cg W lim !1 g. / kk D sup

j./j W 0 : g. /

We consider the following condition on the function g. g.t C / W 1 < t < 1: .g1 / For all a > 0; sup sup g. / 0ta Then we have that the spaces Cg and Cg0 satisfy conditions .A3 /. They satisfy conditions .A1 / and .A2 / if g1 holds. Example 1.6. The space C . For any real constant , we define the functional space C by C WD 2 C..1; 0; E/ W lim e ./ exist in E !1

1.4 Set-Valued Maps

5

endowed with the following norm kk D supfe j./j W 0g: Then in the space C the axioms .A1 /–.A3 / are satisfied.

1.4 Set-Valued Maps Let .X; d/ be a metric space. We use the following notations: Pcl .X/ D fY 2 P.X/ W Y closedg;

Pb .X/ D fY 2 P.X/ W Y boundedg;

Pcv .X/ D fY 2 P.X/ W Y convexg;

Pcp .X/ D fY 2 P.X/ W Y compactg:

Consider Hd W P.X/ P.X/ ! RC [ f1g, given by Hd .A; B/ D max

sup d.a; B/; sup d.A; b/ a2A

;

b2B

where d.A; b/ D inf d.a; b/, d.a; B/ D inf d.a; b/. Then .Pb;cl .X/; Hd / is a metric a2A

b2B

space and .Pcl .X/; Hd / is a generalized (complete) metric space (see [147]). Lemma 1.7. If A and B are compact, then there exists either an a 2 A with d.a; B/ D Hd .A; B/ or a b 2 B with d.A; b/ D Hd .A; B/ Definition 1.8. A multi-valued map G W RC ! Pcl .X/ is said to be measurable if for each x 2 E, the function Y W RC ! X defined by Y.t/ D d.x; G.t// D inffjx zj W z 2 G.t/g is measurable where d is the metric induced by the normed Banach space X. 1 Definition 1.9. A function F W RC X ! P.X/ is said to be an Lloc -Carathéodory multi-valued map if it satisfies:

(i) x 7! F.t; y/ is continuous for almost all t 2 RC ; (ii) t ! 7 F.t; y/ is measurable for each y 2 X; 1 (iii) for every positive constant k there exists hk 2 Lloc .RC ; RC / such that kF.t; y/k hk .t/ for all kykB k and for almost all t 2 RC : Let .X; k k/ be a Banach space. A multi-valued map G W X ! P.X/ has convex (closed) values if G.x/ is convex (closed) for all x 2 X. We say that G is bounded on bounded sets if G.B/ is bounded in X for each bounded set B of X, i.e.,

6

1 Preliminary Background

sup fsupfkyk W y 2 G.x/gg < 1: x2B

Finally, we say that G has a fixed point if there exists x 2 X such that x 2 G.x/: For each y 2 C.RC ; E/ let the set SF;y known as the set of selectors from F defined by SF;y D fv 2 L1 .RC ; E/ W v.t/ 2 F.t; y.t//; a:e: t 2 RC g: For more details on multi-valued maps we refer to the books of Deimling [101], Djebali et al. [104], Górniewicz [123], Hu and Papageorgiou [143], and Tolstonogov [181]. Definition 1.10. A multi-valued map F W X ! P.X/ is called an admissible contraction with constant fkn gn2N if for each n 2 N there exists kn 2 Œ0; 1/ such that i) Hd .F.x/; F.y// kn kx ykn for all x; y 2 X: ii) for every x 2 X and every 2 .0; 1/n , there exists y 2 F.x/ such that kx ykn kx F.x/kn C n for every n 2 N Lemma 1.11 ([154]). Let X be a Banach space. Let F W Œa; b X ! Pcp;c .X/ be an L1 -Carathéodory multi-valued map with SF;y 6D ; and let be a linear continuous mapping from L1 .Œa; b; X/ into C.Œa; b; X/, then the operator ı SF W C.Œa; b; X/ ! Pcp;c .C.Œa; b; X//; y 7! . ı SF /.y/ WD .SF;y / is a closed graph operator in C.Œa; b; X/ C.Œa; b; X/: Proposition 1.12 ([167]). Let the space E be separable and the multi-function ˚ W Œ0; b ! P.E/ be integrable bounded and .˚.t// q.t/ for a.a t 2 Œ0; b where q.:/ 2 L1 .Œ0; b; RC /. Then Z

0

Z ˚.s/ds

q.s/ds;

0

for all 2 Œ0; b:

In particular, if the multi-function ˚ W Œ0; b ! Pcl .E/ is measurable and integrable bounded, then the function .˚.:// is integrable and Z

0

Z ˚.s/ds

0

.˚.s//ds;

for all 2 Œ0; b:

1.6 Semigroups

7

1.5 Evolution System In what follows, for the family fA.t/; t 0g of closed densely defined linear unbounded operators on the Banach space E we assume that it satisfies the following assumptions (see [16], p. 158). (P1) The domain D.A.t// is independent of t and is dense in E: (P2) For t 0; the resolvent R.; A.t// D .I A.t//1 exists for all with Re 0, and there is a constant M independent of and t such that kR.t; A.t//k M.1 C jj/1 ; for Re 0: (P3)

There exist constants L > 0 and 0 < ˛ 1 such that k.A.t/ A. //A1 . /k Ljt j˛ ; for t; ; 2 J:

Lemma 1.13 ([16], p. 159). Under assumptions (P1)–(P3), the Cauchy problem y0 .t/ A.t/y.t/ D 0; t 2 J; y.0/ D y0 ; has a unique evolution system U.t; s/, .t; s/ 2 WD f.t; s/ 2 J J W 0 s t < C1g satisfying the following properties: 1. U.t; t/ D I where I is the identity operator in E, 2. U.t; s/ U.s; / D U.t; / for 0 s t < C1, 3. U.t; s/ 2 B.E/ the space of bounded linear operators on E, where for every .t; s/ 2 and for each y 2 E, the mapping .t; s/ ! U.t; s/ y is continuous. More details on evolution systems and their properties can be found in the books of Ahmed [16], Engel and Nagel [106], and Pazy [168].

1.6 Semigroups 1.6.1 C0 -Semigroups Let E be a Banach space and B.E/ be the Banach space of linear bounded operators on E. Definition 1.14. A semigroup of class .C0 / is a one parameter family fT.t/ j t 0g B.E/ satisfying the conditions: (i) T.0/ D I, (ii) T.t/T.s/ D T.t C s/, for t; s 0, (iii) the map t ! T.t/.x/ is strongly continuous, for each x 2 E, i.e˙, lim T.t/x D x; 8x 2 E:

t!0

8

1 Preliminary Background

A semigroup of bounded linear operators T.t/; is uniformly continuous if lim kT.t/ Ik D 0:

t!0

Here I denotes the identity operator in E: We note that if a semigroup T.t/ is class .C0 /, then satisfies the growth condition kT.t/kB.E/ Meˇt ; for 0 t < 1 with some constants M > 0 and ˇ: If, in particular M D 1 and ˇ D 0, i.e˙, kT.t/kB.E/ 1; for t 0; then the semigroup T.t/ is called a contraction semigroup .C0 /. Definition 1.15. Let T.t/ be a semigroup of class .C0 / defined on E. The infinitesimal generator A of T.t/ is the linear operator defined by T.h/x x ; h!0 h

A.x/ D lim where D.A/ D fx 2 E j limh!0

T.h/.x/x h

for x 2 D.A/;

exists in Eg.

Let us recall the following property: Proposition 1.16. The infinitesimal generator A is a closed, linear, and densely defined operator in E. If x 2 D.A/, then T.t/.x/ is a C1 -map and d T.t/.x/ D A.T.t/.x// D T.t/.A.x// dt

on Œ0; 1/:

Theorem 1.17 (Hille and Yosida [168]). Let A be a densely defined linear operator with domain and range in a Banach space E. Then A is the infinitesimal generator of uniquely determined semigroup T.t/ of class .C0 / satisfying kT.t/kB.E/ Me!t ;

t 0;

where M > 0 and ! 2 R if and only if .I A/1 2 B.E/ and k.I A/n k M=. !/n ; n D 1; 2; : : : ; for all 2 R: For more details on strongly continuous operators, we refer the reader to the books of Ahmed [16], Goldstein [122], Fattorini [111], Pazy [168], and the papers of Travis and Webb [179, 180].

1.6 Semigroups

9

1.6.2 Integrated Semigroups Definition 1.18 ([29]). Let E be a Banach space. An integrated semigroup is a family of bounded linear operators .S.t//t0 on E with the following properties: (i) S.0/ D 0I (ii) t ! S.t/ is Z strongly continuous; s

(iii) S.s/S.t/ D 0

.S.t C r/ S.r//dr; for all t; s 0:

Definition 1.19 ([146]). An operator A is called a generator of an integrated semigroup if there exists ! 2 R such that .!; 1/ .A/ . .A/; is the resolvent set of A) and there exists a strongly continuous exponentially bounded family 1 .S.t// Z t0 of bounded operators such that S.0/ D 0 and R.; A/ WD .I A/ D 1

0

et S.t/dt exists for all with > !:

Proposition 1.20 ([29]). Let A be the generator of an integrated semigroup .S.t//t0 : Then for all x 2 E and t 0; Z

t

Z

t

S.s/xds 2 D.A/ and S.t/x D A

0

S.s/xds C tx:

0

Definition 1.21 ([146]). (i) An integrated semigroup .S.t//t0 is called locally Lipschitz continuous if, for all > 0 there exists a constant L such that jS.t/ S.s/j Ljt sj; t; s 2 Œ0; : (ii) An integrated semigroup .S.t//t0 is called nondegenerate if S.t/x D 0; for all t 0 implies that x D 0: Definition 1.22. We say that the linear operator A satisfies the Hille–Yosida condition if there exists M 0 and ! 2 R such that .!; 1/ .A/ and supf. !/n j.I A/n j W n 2 N; > !g M: Theorem 1.23 ([146]). The following assertions are equivalent: (i) A is the generator of a nondegenerate, locally Lipschitz continuous integrated semigroup; (ii) A satisfies the Hille–Yosida condition. If A is the generator of an integrated semigroup .S.t//t0 which is locally Lipschitz, then from [29], S./x is continuously differentiable if and only if x 2 D.A/ and .S0 .t//t0 is a C0 semigroup on D.A/:

10

1 Preliminary Background

1.6.3 Extrapolated Semigroups Let A0 be the part of A in X0 D D.A/ which is defined by D.A0 / D fx 2 D.A/ W Ax 2 D.A/g; and A0 x D Ax; for x 2 D.A0 /: Lemma 1.24 ([106]). A0 generates a strongly continuous semigroup .T0 .t//t0 on X0 and jT0 .t/j N0 e!t ; for t 0: Moreover .A/ .A0 / and R.; A0 / D R .; A/=X0 ; for 2 .A/: For a fixed 0 2 .A/; we introduce on X0 a new norm defined by kxk1 D jR.0 ; A0 /xj for x 2 D.A0 /: The completion X1 of .X0 ; kk1 / is called the extrapolation space of X associated with A: Note that k k1 and the norm on X0 given by jR.; A0 /xj, for 2 .A/, are extensions T1 .t/ to the Banach space X1 , and .T1 .t//t0 is a strongly continuous semigroup on X1 . .T1 .t//t0 is called the extrapolated semigroup of .T0 .t//t0 , and we denote its generator by .A1 ; D.A1 //: Lemma 1.25 ([130]). The following properties hold: (i) jT.t/jB.X1 / D jT0 .t/jL.X0 / : (ii) D.A1 / D X0 : (iii) A1 W X0 ! X1 is the unique continuous extension of A0 W D.A0 / .X0 ; j:j/ ! .X0 ; k:k1 /; and . A1 /1 is an isometry from .X0 ; j:j/ into .X0 ; k:k1 /: (iv) If 2 .A0 /; then . A1 / is invertible and . A1 /1 2 B.X1 /: In particular 2 .A1 / and R.; A1 /=X0 D R.; A0 / (v) The space X0 D D.A/ is dense in .X1 ; k k1 /: Hence the extrapolation space X1 is also the completion of .X; k:k1 / and X ,! X1 . (vi) The operator A1 is an extension of A: In particular if 2 .A/; then R.; A1 /=X D R.; A/ and . A1 /X D D.A/: Abstract extrapolated spaces have been introduced by Da Prato and Grisvard [99] and Engel and Nagel [106] and used for various purposes [23–25, 160, 163, 164].

1.7 Some Fixed Point Theorems First we will introduce the following compactness criteria in the space of continuous and bounded functions defined on the positive half line.

1.7 Some Fixed Point Theorems

11

Lemma 1.26 (Corduneanu [96]). Let D BC.Œ0; C1/; E/: Then D is relatively compact if the following conditions hold: (a) D is bounded in BC. (b) The function belonging to D is almost equi-continuous on Œ0; C1/, i.e., equicontinuous on every compact of Œ0; C1/: (c) The set D.t/ WD fy.t/ W y 2 Dg is relatively compact on every compact of Œ0; C1/: (d) The function from D is equiconvergent, that is, given > 0; responds T./ > 0 such that ju.t/ lim u.t/j < ; for any t T./ and u 2 D: t!C1

Lemma 1.27 (Nonlinear Alternative [105]). Let X be a Banach space with C X closed and convex. Assume U is a relatively open subset of C with 0 2 U and G W U ! C is a compact map. Then either, (i) G has a fixed point in U; or (ii) there is a point u 2 @U and 2 .0; 1/ with u D G.u/. The multi-valued version of Nonlinear Alternative Lemma 1.28 ([105]). Let X be a Banach space with C X a convex. Assume U is a relatively open subset of C with 0 2 U and G W X ! Pcp;c .X/ be an upper semi-continuous and compact map. Then either, (a) there is a point u 2 @U and 2 .0; 1/ with u 2 G.u/, or (b) G has a fixed point in U. Theorem 1.29 (Nonlinear Alternative of Frigon and Granas [116]). Let X be a Fréchet space and Y X a closed subset in Y and let N W Y ! X be a contraction such that N.Y/ is bounded. Then one of the following statements holds: .S1/ .S2/

N has a unique fixed point; There exists 2 Œ0; 1/, n 2 N and x 2 @n Y n such that kx N .x/kn D 0.

The following nonlinear alternative is given by Avramescu in Fréchet spaces which is an extension of the same version given by Burton [87] and Burton and Kirk [88] in Banach spaces. Theorem 1.30 (Nonlinear Alternative of Avramescu [32]). Let X be a Fréchet space and let A; B W X ! X be two operators satisfying: (1) A is a compact operator, (2) B is a contraction. Then either one of the following statements holds: (S1)

The operator A C B has a fixed point;

12

1 Preliminary Background

(S2)

The set n

x 2 X; x D A.x/ C B

x o

is unbounded for 2 .0; 1/: Theorem 1.31 (Nonlinear Alternative of Frigon [114, 115]). Let X be a Fréchet space and U an open neighborhood of the origin in X and let N W U ! P.X/ be an admissible multi-valued contraction. Assume that N is bounded. Then one of the following statements holds: .S1/ .S2/

N has a fixed point; There exists 2 Œ0; 1/ and x 2 @U such that x 2 N.x/.

The following fixed point theorem is due to Burton and Kirk. Theorem 1.32 ([88]). Let X be a Banach space, and A; B two operators satisfying: (i) A is a contraction, and (ii) B is completely. Then either (a) the operator equation y D A.y/ C B.y/ has a solution, or (b) the set " D fu 2 X W A. u / C B.u/g is unbounded for 2 .0; 1/: We need the following definitions in the sequel. Definition 1.33. A nonempty closed subset C of a Banach space X is said to be a cone if (i) C C C C; (ii) C C for > 0, and, (iii) C \ C D f0g: A cone C is called normal if the norm k k is semi-monotone on C, i.e., there exists a constant N > 0 such that kxk Nkyk, whenever x y. We equip the space X D C.J; E/ with the order relation induced by a regular cone C in E, that is for all y; y 2 X W y y if and only if y.t/ y.t/ 2 C; 8t 2 J: In what follows will assume that the cone C is normal. Cones and their properties are detailed in [134]. Let a; b 2 X be such that a b. Then, by an order interval Œa; b we mean a set of points in X given by Œa; b D fx 2 X j a x bg: Definition 1.34. Let X be an ordered Banach space. A mapping T W X ! X is called isotone increasing if T.x/ T.y/ for any x; y 2 X with x < y. Similarly, T is called isotone decreasing if T.x/ T.y/ whenever x < y:

1.7 Some Fixed Point Theorems

13

Definition 1.35. We say that x 2 X is the least fixed point of G in X if x D Gx and x y whenever y 2 X and y D Gy. The greatest fixed point of G in X is defined similarly by reversing the inequality. If both least and greatest fixed point of G in X exist, we call them extremal fixed point of G in X. The following fixed point theorem is due to Heikkila and Lakshmikantham. Theorem 1.36. Let Œa; b be an order interval in an order Banach space X and let Q W Œa; b ! Œa; b be a nondecreasing mapping. If each sequence .Qxn / Q.Œa; b/ converges, whenever .xn / is a monotone sequence in Œa; b, then the sequence of Q-iteration of a converges to the least fixed point x of Q and the sequence of Qiteration of a converges to the greatest fixed point x of Q. Moreover x D minfy 2 Œa; b; y Qyg and x D maxfy 2 Œa; b; y Qyg: As a consequence, Dhage and Henderson have proved the following fixed point theorem, which will be used to prove the existence of extremal solutions. Theorem 1.37 ([102]). Let Œa; b be an order interval in a Banach space X and let B1 ; B2 W Œa; b ! X be two functions satisfying: (a) (b) (c) (d)

B1 is a contraction, B2 is completely continuous, B1 and B2 are strictly monotone increasing, and B1 .x/ C B2 .x/ 2 Œa; b; 8 x 2 Œa; b. Further if the cone C in X is normal, then the equation x D B1 .x/ C B2 .x/ has a least fixed point x and a greatest fixed point x 2 Œa; b. Moreover x D lim xn n!1

and x D lim yn , where fxn g and fyn g are the sequences in Œa; b defined by n!1

xnC1 D B1 .xn / C B2 .xn /; x0 D a and ynC1 D B1 .yn / C B2 .yn /; y0 D b: V Given a space X and metrics d˛ ; ˛ 2 ; denote P.X/ D fY X W Y 6D ;g, Pcl .X/ D fY 2 P.X/ V W Y closedg, Pb .X/ D fY 2 P.X/ W Y boundedg: We denote by D˛ ; ˛ 2 ; the Hausdorff pseudo-metric induced by d˛ I that is, for V; W 2 P.X/; n D˛ .V; W/ D inf " > 0 W 8x 2 V; 8y 2 W; 9Nx 2 V; yN 2 W such that o d˛ .x; yN / < "; d˛ .Nx; y/ < " ; with inf ; D 1: In the particular case where X is a complete locally convex V space, we say that a subset V X is bounded if D˛ .f0g; V/ < 1 for every ˛ 2 : Definition 1.38. A multi-valued map F W X ! P.E/ V is called an admissible contraction with constant fk˛ g˛2V if for each ˛ 2 there exists k˛ 2 .0; 1/ such that

14

1 Preliminary Background

i) D˛ .F.x/; F.y// k˛ d˛ .x; y/ for all x;Vy 2 X: ii) for every x 2 X and every " 2 .0; 1/ ; there exists y 2 F.x/ such that d˛ .x; y/ d˛ .x; F.x// C "˛ for every ˛ 2

^

:

Lemma 1.39 (Nonlinear Alternative, [113]). Let E be a Fréchet space and U an open neighborhood of the origin in E, and let N W U ! P.E/ be an admissible multi-valued contraction. Assume that N is bounded. Then one of the following statements holds: (C1) (C2)

N has at least one fixed point; there exists 2 Œ0; 1/ and x 2 @U such that x 2 N.x/:

Lemma 1.40 ([144]). If U is a closed convex subset of a Banach space E and R W U ! Pcv;k .E/ is a closed ˇ-condensing multi-function, where ˇ is a nonsingular MNC defined on the subsets of U. Then R has a fixed point. The next results are concerned with the structure of solution sets for ˇ-condensing u.s.c. multi-valued maps. Lemma 1.41 ([144]). Let W be a closed subset of a Banach space E and R W W ! Pcv;k .E/ be a closed multi-function which is ˇ-condensing on every bounded subset of W, where ˇ is a monotone MNC. If the fixed points set FixR is bounded, then it is compact. The following theorem is due to Mönch. Theorem 1.42 ([162]). Let E be a Banach space, U an open subset of E and 0 2 U. Suppose that N W U ! E is a continuous map which satisfies Mönch’s condition (i.e., if D U is countable and D co.f0g [ N.D//, then D is compact) and assume that x ¤ N.x/;

for x 2 @U and 2 .0; 1/

holds. Then Nhas a fixed point in U. Lemma 1.43 ([144, Theorem 2]). The generalized Cauchy operator G satisfies the properties (G1)

there exists 0 such that Z kGf .t/ Gg.t/k

0

t

kf .s/ g.s/kds; for every f ; g 2 L1 .J; E/; t 2 J:

(G2) for any compact K E and any sequence .fn /n1 L1 .J; E/ such that for all n 1, fn .t/ 2 K, a. e. t 2 J, the weak convergence fn * f0 in L1 .J; E/ implies the convergence Gfn ! Gf0 in C.J; E/.

1.7 Some Fixed Point Theorems

15

Lemma 1.44 ([144]). Let S W L1 .J; E/ ! C.J; E/ be an operator satisfying condition (G2) and the following Lipschitz condition (weaker than (G1)). (G1’) kSf SgkC.J;E/ kf gkL1 .J;E/ : C1 1 Then for every semi-compact set ffn gC1 nD1 L .J; E/ the set fSfn gnD1 is relatively compact in C.J; E/. Moreover, if .fn /n1 converges weakly to f0 in L1 .J; E/ then Sfn ! Sf0 in C.J; E/.

Lemma 1.45 ([144]). Let S W L1 .J; E/ ! C.J; E/ be an operator satisfying conditions (G1), (G2) and let the set ffn g1 nD1 be integrable bounded with the property .ffn .t/ W n 1g/ .t/, for a.e. t 2 J, where .:/ 2 L1 .J; RC / and is the Hausdorff MNC. Then Z .fSfn .t/ W n 1g/ 2

t 0

.s/ds; for all t 2 J;

where 0 is the constant in condition (G1). Let us recall the following result that will be used in the sequel. Lemma 1.46 ([86]). Let E be a separable metric space and let G W E ! P .L1 .Œ0; b; E// be a multi-valued operator which is lower semi-continuous and has nonempty closed and decomposable values. Then G has a continuous selection, i.e., there exists a continuous function f W E ! L1 .Œ0; b; E/ such that f .y/ 2 G.y/ for every y 2 E.

Chapter 2

Partial Functional Evolution Equations with Finite Delay

2.1 Introduction In this chapter, we study some first order classes of partial functional, neutral functional, integro-differential, and neutral integro-differential evolution equations on a positive line RC with local and nonlocal conditions when the historical interval H is bounded, i.e., when the delay is finite. In the literature devoted to equations with finite delay, the phase space is much of time the space of all continuous functions on H for r > 0; endowed with the uniform norm topology. Using a recent nonlinear alternative of Leray–Schauder type for contractions in Fréchet spaces due to Frigon and Granas combined with the semigroup theory, the existence and uniqueness of the mild solution will be obtained. The method we are going to use is to reduce the existence of the unique mild solution to the search for the existence of the unique fixed point of an appropriate contraction operator in a Fréchet space. The nonlocal Cauchy problem has been studied first by Byszewski in 1991 [90] (see also [89, 91, 92]). Then, Balachandran and his collaborators have considered various classes of nonlinear integro-differential systems [44].

2.2 Partial Functional Evolution Equations 2.2.1 Introduction In this section, we consider partial functional evolution equations with local and nonlocal conditions where the existence of the unique mild solution is assured. Firstly, in Sect. 2.2.2 we consider the following partial functional evolution system

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_2

17

18

2 Partial Functional Evolution Equations with Finite Delay

y0 .t/ D A.t/y.t/ C f .t; yt /; a.e. t 2 J D RC

(2.1)

y.t/ D '.t/; t 2 H;

(2.2)

where r > 0, f W J C.H; E/ ! E and ' 2 C.H; E/ are given functions and fA.t/gt0 is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of bounded linear operators fU.t; s/g.t;s/2JJ for 0 s t < C1 from E into E: Later, we consider the functional evolution problem with a nonlocal condition of the form y0 .t/ D A.t/y.t/ C f .t; yt /; a.e. t 2 J D RC

(2.3)

y.t/ C ht .y/ D '.t/; t 2 H;

(2.4)

where A./, f and ' are as in evolution problem (2.1)–(2.2) and ht W C.H; E/ ! E is a given function. Using the fixed point argument, Frigon applied its own alternative to some differential and integral equations in [113]. In the literature devoted to equations with A./ D A on a bounded interval, we can found the recent works by Benchohra and Ntouyas for semi-linear equations and inclusions [58, 59, 65], controllability results are established by Benchohra et al. in [26, 75, 76] and Li et al. in [156].

2.2.2 Main Result Let us introduce the definition of the mild solution of the partial functional evolution system (2.1)–(2.2). Definition 2.1. We say that the continuous function y./ W Œr; C1/ ! E is a mild solution of (2.1)–(2.2) if y.t/ D '.t/ for all t 2 H and y satisfies the following integral equation Z y.t/ D U.t; 0/ '.0/ C

0

t

U.t; s/ f .s; ys / ds;

for each t 2 Œ0; C1/:

We will need the following hypotheses which are assumed hereafter: b 1 such that (2.1.1) There exists a constant M b kU.t; s/kB.E/ M for every .t; s/ 2 WD f.t; s/ 2 J J W 0 s t < C1g;

2.2 Partial Functional Evolution Equations

19

(2.1.2) There exist a continuous nondecreasing function 1 p 2 Lloc .Œ0; C1/; RC / such that jf .t; u/j p.t/

W RC ! .0; C1/ and

.kuk/;

for a.e. t 2 Œ0; C1/ and each u 2 C.H; E/; 1 .Œr; C1/; RC / such that (2.1.3) For all R > 0, there exists lR 2 Lloc jf .t; u/ f .t; v/j lR .t/ ku vk for all u; v 2 C.H; E/ with kuk R and kvk R. For every n 2 N, we define in C.Œr; C1/; E/ the semi-norms by: kykn WD sup f e

where Ln .t/ D

Z 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

b ln .t/ and ln is the function from .2:1:3/: ln .s/ ds; ln .t/ D M

Then C.Œr; C1/; E/ is a Fréchet space with the family of semi-norms fk kn gn2N . In what follows we will choose > 1. Theorem 2.2 ([33]). Suppose that hypotheses (2.1.1)–(2.1.3) are satisfied and moreover for n > 0 Z

C1

c1

ds b >M .s/

Z

n

p.s/ ds;

(2.5)

0

b k'k. Then the problem (2.1)–(2.2) has a unique mild solution. where c1 D M Proof. Transform the problem (2.1)–(2.2) into a fixed point problem. Consider the operator N W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by:

.N1 y/.t/ D

8 ˆ < '.t/; ˆ : U.t; 0/ '.0/ C

if t 2 H; Z

t 0

U.t; s/ f .s; ys / ds;

if t 2 RC :

Clearly, the fixed points of the operator N1 are mild solutions of the problem (2.1)–(2.2). Let y be a possible solution of the problem (2.1)–(2.2). Given n 2 N and t n, then from .2:1:1/ and .2:1:2/ we have: Z jy.t/j jU.t; 0/j j'.0/j C

0

t

kU.t; s/kB.E/ jf .s; ys /jds

20

2 Partial Functional Evolution Equations with Finite Delay

b k'k C M b M

Z

t

.kys k/ds:

p.s/ 0

We consider the function defined by .t/ WD supf jy.s/j W 0 s t g; 0 t < C1: Let t 2 Œr; t be such that .t/ D jy.t /j: If t 2 Œ0; n, by the previous inequality we get b k'k C M b .t/ M

Z

t

..s// ds; t 2 Œ0; n:

p.s/ 0

If t 2 H; then .t/ D k'k and the previous inequality holds. Let us take the right-hand side of the above inequality as v.t/: Then we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we get b c1 WD v.0/ D Mk'k

and

b p.t/ v 0 .t/ D M

Using the nondecreasing character of b p.t/ v 0 .t/ M

..t// a.e. t 2 Œ0; n:

, we have .v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using (2.5) we get Z

v.t/

c1

Z

ds b M .s/

p.s/ ds 0

Z

b M Z

t

n

p.s/ ds 0 C1

< c1

ds : .s/

Thus there exists a constant n such that v.t/ n ; t 2 Œ0; n and hence .t/ n ; t 2 Œ0; n. Since for every t 2 Œ0; n; kyt k .t/, we have kykn maxfk'k; n g WD n : Set Y D f y 2 C.Œr; C1/; E/ W supfjy.t/j W 0 t ng n C 1 for all n 2 N g:

2.2 Partial Functional Evolution Equations

21

Clearly, Y is a closed subset of C.Œr; C1/; E/. We shall show that N1 W Y ! C.Œr; C1/; E/ is a contraction operator. Indeed, consider y; y 2 C.Œr; C1/; E/; thus using .2; 1; 1/ and .2:1:3/ for each t 2 Œ0; n and n 2 N we get Z j.N1 y/.t/ .N1 y/.t/j

t

0

Z

t

0

Z

t

0

Z

t

kU.t; s/kB.E/ jf .s; ys / f .s; ys /j ds b ln .s/ kys ys k ds M Œln .s/ e

"

0

1

e

e

Ln .s/

Ln .s/

Ln .t/

Œe

Ln .s/

kys ys k ds

#0 ds ky ykn

ky ykn :

Therefore, k.N1 y/ .N1 y/kn

1 ky ykn :

So, for > 1, the operator N1 is a contraction for all n 2 N: From the choice of Y there is no y 2 @Y n such that y D N1 .y/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.29 does not hold. A consequence of the nonlinear alternative of Frigon and Granas that .S1/ holds, we deduce that the operator N1 has a unique fixed point y which is the unique mild solution of the problem (2.1)–(2.2). t u

2.2.3 An Example As an application of Theorem 2.2, we consider the following partial functional differential equation 8 @z @2 z ˆ ˆ .t; x/ D a.t; x/ 2 .t; x/ C Q.t; z.t r; x// ˆ ˆ ˆ @t @x ˆ ˆ ˆ < z.t; 0/ D z.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : z.t; x/ D ˚.t; x/

t 2 Œ0; C1/;

x 2 Œ0;

t 2 Œ0; C1/ t 2 H;

x 2 Œ0; ; (2.6)

22

2 Partial Functional Evolution Equations with Finite Delay

where r > 0; a.t; x/ W Œ0; 1/Œ0; ! R is a continuous function and is uniformly Hölder continuous in t, Q W Œ0; C1/ R ! R and ˚ W H Œ0; ! R are continuous functions. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0g: Then A.t/ generates an evolution system U.t; s/ satisfying assumption .2:1:1/ (see [112, 149]). For x 2 Œ0; , we set y.t/.x/ D z.t; x/

t 2 RC ;

f .t; yt /.x/ D Q.t; z.t r; x//

t 2 RC

and '.t/.x/ D ˚.t; x/

r t 0:

Thus, under the above definitions of f , ' and A./, the system (2.6) can be represented by the abstract partial functional evolution problem (2.1)–(2.2). Furthermore, more appropriate conditions on Q ensure the existence of unique mild solution for (2.6) by Theorems 2.2 and 1.29.

2.2.4 Nonlocal Case In this section, we extend the above results about the existence and uniqueness of mild solution to the partial functional evolution equations with nonlocal conditions (2.3)–(2.4). The nonlocal condition can be applied in physics with better effect than the classical initial condition y.0/ D y0 . For example, ht .y/ may be given by ht .y/ D

p X

ci y.ti C t/;

t2H

iD1

where ci ; i D 1; : : :; p are given constants and 0 < t1 < < tp < C1. At time t D 0, we have h0 .y/ D

p X

ci y.ti /:

iD1

Nonlocal conditions were initiated by Byszewski [90] to which we refer for motivation and other references.

2.2 Partial Functional Evolution Equations

23

Before giving the main result, we give first the definition of mild solution of the nonlocal partial functional evolution problem (2.3)–(2.4). Definition 2.3. A function y 2 C.Œr; C1/; E/ is said to be a mild solution of (2.3)–(2.4) if y.t/ D '.t/ ht .y/ for all t 2 H and y satisfies the following integral equation Z y.t/ D U.t; 0/ Œ'.0/ h0 .y/ C

t 0

U.t; s/ f .s; ys / ds;

for each t 2 Œ0; C1/:

We will need the following hypotheses on ht ./ in the proof of the main result of this section. (2.3.1) For all n 0, there exists a constant n > 0 such that jht .u/ ht .v/j n ku vk for all t 2 H, u; v 2 C.Œr; 1/; E/ with kuk n and kvk n; (2.3.2) there exists > 0 such that jht .u/j for each u 2 C.H; E/; and t 2 J: Theorem 2.4 ([33]). Assume that the hypotheses (2.1.1)–(2.1.3), (2.3.1), and (2.3.2) hold and moreover for n > 0 Z

C1

c2

Z

ds b >M .s/

n

p.s/ ds;

(2.7)

0

b where c2 D M.k'k C /. Then the nonlocal evolution problem (2.3)–(2.4) has a unique mild solution. Proof. Transform the problem (2.3)–(2.4) into a fixed point problem. Consider the operator N2 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by:

.N2 y/.t/ D

8 ˆ < '.t/ ht .y/; ˆ : U.t; 0/ Œ'.0/ h0 .y/ C

if t 2 H; Z 0

t

U.t; s/ f .s; ys / ds;

if t 2 RC :

Clearly, the fixed points of the operator N2 are mild solutions of the problem (2.3)– (2.4). Then, by parallel steps of Theorem 2.2’s proof, we can easily show that the operator N2 is a contraction which have a unique fixed point by statement .S1/ in Theorem 1.29. The details are left to the reader. t u

24

2 Partial Functional Evolution Equations with Finite Delay

2.3 Neutral Functional Evolution Equations 2.3.1 Introduction In this section, we investigate neutral functional evolution equations with local and nonlocal conditions. First, we study in Sect. 2.3.2 the following neutral functional evolution equations d Œy.t/ g.t; yt / D A.t/y.t/ C f .t; yt /I a.e. t 2 RC ; dt y.t/ D '.t/I t 2 H;

(2.8) (2.9)

where r > 0; f ; g W J C.H; E/ ! E and ' 2 C.H; E/ are given functions and fA.t/gt0 is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators fU.t; s/g.t;s/2JJ for 0 s t < C1: An extension of these existence results will be given in Sect. 2.3.4 for the following neutral functional evolution equation with nonlocal conditions d Œy.t/ g.t; yt / D A.t/y.t/ C f .t; yt /; a.e. t 2 J D RC dt y.t/ C ht .y/ D '.t/; t 2 H;

(2.10) (2.11)

where A./, f , g, and ' are as in problem (2.8)–(2.9) and ht W C.Œr; 1/; E/ ! E is a given function. Neutral equations have received much attention in recent years: existence and uniqueness of mild, strong, and classical solutions for semi-linear functional differential equations and inclusions has been studied extensively by many authors. Hernandez in [138] proved the existence of mild, strong, and periodic solutions for neutral equations. Fu in [117] studied the controllability on a bounded interval of a class of neutral functional differential equations. Fu and Ezzinbi [119] considered the existence of mild and classical solutions for a class of neutral partial functional differential equations with nonlocal conditions. Here we are interesting to give existence and uniqueness of the mild solution for the neutral functional evolution equations (2.8)–(2.9) and the corresponding nonlocal problem (2.10)–(2.11).

2.3.2 Main Result We give first the definition of the mild solution of (2.8)–(2.9).

2.3 Neutral Functional Evolution Equations

25

Definition 2.5. We say that the continuous function y./ W Œr; C1/ ! E is a mild solution of (2.8)–(2.9) if y.t/ D '.t/ for all t 2 H and y satisfies the following integral equation Z y.t/ D U.t; 0/ Œ'.0/ g.0; '/ C g.t; yt / C Z

t

C 0

U.t; s/ f .s; ys / ds;

t 0

U.t; s/ A.s/ g.s; ys / ds

for each t 2 Œ0; C1/:

We will need to introduce the following assumptions: .G1/

There exists a constant M 0 > 0 such that: kA1 .t/k M 0

.G2/

There exists a constant 0 < L <

1 M0

for all t 0: , such that:

jA.t/ g.t; '/j L .k'k C 1/ for all t > 0 and ' 2 C.H; E/: There exists a constant L > 0 such that:

.G3/

jA.s/ g.s; '/ A.s/ g.s; '/j L .js sj C k' 'k/ for all s; s > 0 and '; ' 2 C.H; E/. For every n 2 N, we define in C.Œr; C1/; E/ the semi-norms by: kykn WD sup f e

where Ln .t/ D

Z 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

b Œ L C ln .t/ and ln is the function from ln .s/ ds; ln .t/ D M

.2:1:3/: Then C.Œr; C1/; E/ is a Fréchet space with the family of semi-norms fk 1 < 1. kn gn2N . Let us fix > 0 and assume M 0 L C

Theorem 2.6 ([37]). Suppose that hypotheses (2.1.1)–(2.1.3) and the assumptions .G1/–.G3/ are satisfied. If Z

C1

c3;n

Z n b ds M max.L; p.s//ds; for each n > 0 > s C .s/ 1 M0L 0

(2.12)

26

2 Partial Functional Evolution Equations with Finite Delay

with c3;n D

b C 1/ C MLn b b C M 0 L/k'k C M 0 L.M M.1 1 M0L

:

Then the problem (2.8)–(2.9) has a unique mild solution. Proof. Transform the problem (2.8)–(2.9) into a fixed point problem. Consider the operator N3 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by: 8 '.t/; if t 2 H ˆ ˆ ˆ < Œ'.0/ g.0; '/ C g.t;Zyt / .N3 y/.t/ D U.t; Z 0/ ˆ t t ˆ ˆ :C U.t; s/A.s/g.s; ys /ds C U.t; s/f .s; ys /ds; if t 2 RC : 0

0

Clearly, the fixed points of the operator N3 are mild solutions of the problem (2.8)–(2.9). We are going to use Theorem 1.29 in the following way. (i) Define a set Y such that (2.8)–(2.9) does’nt have any solution out of Y. (ii) Show that (S2) of Theorem 1.29 doesn’t hold under the above choice of Y, hence (S1) takes place. Let y be such that y D N3 .y/ for 2 Œ0; 1. Given n 2 N and t n, then from .2:1:1/, .2:1:2/, .G1/ and .G2/ we have jy.t/j jU.t; 0/j j'.0/ g.0; '/j C jg.t; yt /j Z t Z t C jU.t; s/j jA.s/ g.s; ys /j ds C jU.t; s/j jf .s; ys /j ds 0

0

b k'k C M b kA .0/k jA.0/ g.0; '/j C kA1 .t/kjA.t/ g.t; yt /j M Z t Z t b b CM jA.s/ g.s; ys /j ds C M p.s/ .kys k/ ds 1

0

0

b k'k C M b M 0 L.k'k C 1/ C M 0 L.kyt k C 1/ M Z t Z t bL b CM .kys k C 1/ ds C M p.s/ .kys k/ ds: 0

0

We consider the function defined by .t/ WD supf jy.s/j W 0 s t g; 0 t < C1: Let t 2 Œr; t be such that .t/ D jy.t /j: If t 2 Œ0; n, by the previous inequality we get for t 2 Œ0; n b k'k C M b M 0 L.k'k C 1/ C M 0 L..t/ C 1/ .t/ M Z t Z t b b CM L ..s/ C 1/ ds C M p.s/ ..s// ds: 0

0

2.3 Neutral Functional Evolution Equations

27

So b k'k C M b M 0 L k'k C M b M0 L C M0 L .1 M 0 L/ .t/ M Z t Z t b b LnCM b L.s/ds C M p.s/ ..s// ds: CM 0

0

Set c3;n WD

b C 1/ C MLn b b C M 0 L/k'k C M 0 L.M M.1 1 M0L

;

then .t/

1 1 M0L C

i h b C 1/ C MLn b b C M 0 L/k'k C M 0 L.M M.1 Z

b M 1 M0L

D c3;n C

t

0

b M

Z L.s/ds C 0

Z

1 M0L

t

t

p.s/ ..s//ds Z

t

L.s/ds C

0

0

p.s/ ..s//ds :

If t 2 H, then .t/ D k'k and the previous inequality holds. Let us take the right-hand side of the above inequality as v.t/. Then we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we get v.0/ D c3;n

v 0 .t/ D

and

b M 1 M0L

Using the nondecreasing character of v 0 .t/

b M 1 M0L

ŒL.t/ C p.t/ ..t// a.e. t 2 Œ0; n:

, we have for a.e. t 2 Œ0; n ŒLv.t/ C p.t/ .v.t//:

This implies that for each t 2 Œ0; n and using (2.12) we get Z

v.t/

c3;n

Z t b M ds max.L; p.s//ds s C .s/ 1 M0L 0 Z n b M max.L; p.s//ds 1 M0L 0 Z C1 ds : < s C .s/ c3;n

28

2 Partial Functional Evolution Equations with Finite Delay

Thus, there exists a constant n such that v.t/ n ; t 2 Œ0; n and hence .t/ n ; t 2 Œ0; n. Since for every t 2 Œ0; n; kyt k .t/, we have kykn maxfk'k; n g WD n : Set Y D f y 2 C.Œr; C1/; E/ W kykn n C 1 for each n 2 Ng: Clearly, Y is an open subset of C.Œr; C1/; E/. We shall show that N3 W Y ! C.Œr; C1/; E/ is a contraction operator. Indeed, consider y; y 2 C.Œr; C1/; E/, thus using (2.1.1), (2.1.3), (G1) and (G3) for each t 2 Œ0; n and n 2 N we get j.N3 y/.t/ .N3 y/.t/j jg.t; yt / g.t; yt /j Z t C jU.t; s/j jA.s/ .g.s; ys / g.s; ys //j ds Z

0

t

C 0

jU.t; s/j jf .s; ys / f .s; ys /j ds

kA1 .t/k jA.t/ g.t; yt / A.t/ g.t; yt /j Z t b jA.s/ g.s; ys / A.s/ g.s; ys /j ds C M Z

0

t

C 0

b jf .s; ys / f .s; ys /j ds M Z

M 0 L kyt yt k C Z

t

C 0

t 0

b L kys ys k ds M

b ln .s/ kys ys k ds M

M 0 L kyt yt k C

Z th i b ln .s/ kys ys k ds b L C M M 0

i h i h M 0 L e Ln .t/ e Ln .t/ kyt yt k Z th i h i C ln .s/ e Ln .s/ e Ln .s/ kys ys k ds 0

h M 0 L e

Ln .t/

i

ky ykn C

Z t" 0

e

Ln .s/

#0 ds ky ykn :

2.3 Neutral Functional Evolution Equations

29

Then i h 1 j.N3 y/.t/ .N3 y/.t/j M 0 L e Ln .t/ ky ykn C e

1 e Ln .t/ ky ykn : M 0 L C

Ln .t/

ky ykn

Therefore,

1 ky ykn : kN3 .y/ N3 .y/kn M 0 L C

1 < 1, the operator N3 is a contraction for all n 2 N. From So, for M 0 L C

the choice of Y there is no y 2 @Y n such that y D N3 .y/ for some 2 .0; 1/: Then the statement .S2/ in Theorem 1.29 does’nt hold. Thus statement .S1/ holds, and hence the operator N3 has a unique fixed point y in Y, which is the unique mild solution of the neutral functional evolution problem (2.8)–(2.9). t u

2.3.3 An Example As an application of our results we consider the following model 8

Z t Z @ ˆ ˆ z.t; x/ b.s t; u; x/ z.s; u/ du ds ˆ ˆ ˆ @t r 0 ˆ ˆ ˆ @z @2 z ˆ ˆ ˆ < D a.t; x/ @x2 .t; x/ C Q.t; z.t r; x/; @x .t r; x//; t 2 Œ0; C1/; x 2 Œ0; ˆ ˆ ˆ z.t; 0/ D z.t; / D 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : z.t; x/ D ˚.t; x/;

t 2 Œ0; C1/ t 2 H;

x 2 Œ0; (2.13)

where r > 0; a.t; x/ is a continuous function and is uniformly Hölder continuous in t, Q W Œ0; C1/ R R ! R and ˚ W H Œ0; ! R are continuous functions. Let y.t/.x/ D z.t; x/; t 2 Œ0; 1/; x 2 Œ0; ; './.x/ D ˚.; x/; 2 H; x 2 Œ0; ; Z t Z g.t; yt /.x/ D b.s t; u; x/z.s; u/duds; x 2 Œ0; r

0

30

2 Partial Functional Evolution Equations with Finite Delay

and f .t; yt /.x/ D Q.t; z.; x/;

@z .; x//; 2 H; x 2 Œ0; : @x

Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0g: Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .2:1:1/ and .G1/ (see [112, 149]). Here we consider that ' W H ! E such that ' is Lebesgue measurable and h.s/j'.s/j2 is Lebesgue integrable on H where h W H ! R is a positive integrable function. The norm is defined here by: Z k'k D j˚.0/j C

0

2

12

h.s/j'.s/j ds r

:

The function b is measurable on Œ0; 1/ Œ0; Œ0; ; b.s; u; 0/ D b.s; u; / D 0; .s; u/ 2 Œ0; 1/ Œ0; ; Z

Z

0

t

Z

r

0

b2 .s; u; x/ dsdudx < 1 h.s/

and sup N .t/ < 1; where t2Œ0;1/

Z N .t/ D

0

Z

t

r

Z

0

2 @2 1 a.s; x/ 2 b.s; u; x/ dsdudx: h.s/ @x

Thus, under the above definitions of f , g, and A./, the system (2.13) can be represented by the abstract neutral functional evolution problem (2.8)–(2.9). Furthermore, more appropriate conditions on Q ensure the existence of at least one mild solution for (2.13) by Theorems 2.6 and 1.29.

2.3.4 Nonlocal Case In this section we give existence and uniqueness results for the neutral functional evolution equation with nonlocal conditions (2.10)–(2.11). Nonlocal conditions were initiated by Byszewski [89]. Before giving the main result, we give first the definition of the mild solution.

2.3 Neutral Functional Evolution Equations

31

Definition 2.7. A function y 2 C.Œr; C1/; E/ is said to be a mild solution of (2.10)–(2.11) if y.t/ D '.t/ ht .y/ for all t 2 H and y satisfies the following integral equation y.t/ D U.t; Z 0/ Œ'.0/ h0 .y/ g.0; '/ZC g.t; yt /C t

C 0

U.t; s/ A.s/ g.s; ys / ds C

t

0

U.t; s/ f .s; ys / ds;

for t 2 RC :

We take here the same assumptions in Sect. 2.2.4 for the function ht ./: Theorem 2.8 ([37]). Assume that the hypotheses (2.1.1)–(2.1.3), .G1/–.G3/, .D1/, and .D2/ hold. If Z

C1

c4;n

Z n b M ds > max.L; p.s//ds; for each n > 0 s C .s/ 1 M0L 0

(2.14)

with c4;n D

b M

b C 1/ C MLn b 1 C M 0 L k'k C C M 0 L.M 1 M0L

I

Then the nonlocal neutral functional evolution problem (2.10)–(2.11) has a unique mild solution. Proof. Consider the operator N4 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by: 8 '.t/ ht .y/; ˆ ˆ ˆ < Œ'.0/ h0 .y/ g.0; '/ .N4 y/.t/ D U.t; Z 0/ Z tC g.t; yt / ˆ t ˆ ˆ :C U.t; s/A.s/g.s; ys /ds C U.t; s/f .s; ys / ds; 0

0

if t 2 H;

if t 2 RC :

Clearly, the fixed points of the operator N4 are mild solutions of the problem (2.10)–(2.11). Then, by parallel steps of the Theorem 2.6’s proof, we can prove that the operator N4 is a contraction which have a fixed point by statement .S1/ in Theorem 1.29. The details are left to the reader. t u

32

2 Partial Functional Evolution Equations with Finite Delay

2.4 Partial Functional Integro-Differential Evolution Equations 2.4.1 Introduction In this section, we are interested by partial functional integro-differential evolution equations with local and nonlocal conditions. First, we look for the class of partial functional integro-differential evolution equations of the form Z

0

y .t/ D A.t/y.t/ C

t 0

K.t; s/ f .s; ys / ds; a.e. t 2 J D RC

y.t/ D '.t/; t 2 H;

(2.15) (2.16)

where K W J J ! E, f W J C.H; E/ ! E and ' 2 C.H; E/ are given functions and fA.t/gt0 is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators fU.t; s/g.t;s/2JJ for 0 s t < C1. Also, an extension of these existence results is given for the following partial functional integro-differential evolution problem with nonlocal conditions 0

Z

y .t/ D A.t/y.t/ C

t 0

K.t; s/ f .s; ys / ds; a.e. t 2 J D RC

y.t/ C ht .y/ D '.t/; t 2 H;

(2.17) (2.18)

where A./, f , K, and ' are as in evolution problem (2.15)–(2.16) and ht W C.H; E/ ! E is a given function. The problem of proving the existence of mild solutions for integro-differential equations and inclusions in abstract spaces has been studied by several authors; see Balachandran and Anandhi [41, 42], Balachandran and Leelamani [45] and Benchohra et al. [72, 77], Benchohra and Ntouyas [60, 61, 66], Ntouyas [165]. Here we are interested to study the existence and uniqueness of the mild solution for the partial functional integro-differential evolution equations (2.15)– (2.16) and the corresponding nonlocal problem (2.17)–(2.18). The motivation of these problems is to look for the integro-differential equations considered in [33].

2.4.2 Main Result Before stating and proving the main result, we give first the definition of mild solution of the partial functional integro-differential evolution problem (2.15)–(2.16).

2.4 Partial Functional Integro-Differential Evolution Equations

33

Definition 2.9. We say that the function y./ W Œr; C1/ ! E is a mild solution of (2.15)–(2.16) if y.t/ D '.t/ for all t 2 H and y satisfies the following integral equation Z y.t/ D U.t; 0/ '.0/ C

Z

t

s

U.t; s/ 0

0

K.s; / f . ; y / d ds;

for each t 2 RC :

We will need to add the following assumption: (2.9.1) For each t 2 J, K.t; s/ is measurable on Œ0; t and K.t/ D ess supfjK.t; s/jI 0 s tg is bounded on Œ0; nI let Sn WD sup K.t/: t2Œ0;n

For every n 2 N, we define in C.Œr; C1/; E/ the semi-norms by: kykn WD sup f e

where

Ln .t/

Z D 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

b n Sn ln .t/ and ln is the function from .2:1:3/: ln .s/ ds, ln .t/ D M

Then C.Œr; C1/; E/ is a Fréchet space with the family of semi-norms fk kn gn2N . In what follows we will choose > 1. Theorem 2.10. Suppose that hypotheses (2.1.1)–(2.1.3) are satisfied and the assumption .2:9:1/ holds. If Z

C1

c5

Z

ds b n Sn >M .s/

n

p.s/ ds; 0

for each n > 0

(2.19)

b k'k. Then the problem (2.15)–(2.16) has a unique mild solution. with c5 D M Proof. Transform the problem (2.15)–(2.16) into a fixed point problem. Consider the operator N5 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by:

.N5 y/.t/ D

8 ˆ < '.t/; ˆ : U.t; 0/ '.0/ C

if t 2 H; Z

Z

t

U.t; s/ 0

0

s

K.s; / f . ; y / d ds;

if t 2 RC :

Clearly, the fixed points of the operator N5 are mild solutions of the problem (2.15)–(2.16).

34

2 Partial Functional Evolution Equations with Finite Delay

Let y be a possible solution of the problem (2.15)–(2.16). Given n 2 N and t n, then from .2:1:1/, .2:1:2/ and .2:9:1/ we have: Z jy.t/j jU.t; 0/j j'.0/j C b j'.0/j C M b M

t

0

0

b k'k C M b n Sn M

s

0

Z tZ

b k'k C M b M

0

Z tZ 0

s 0

Z

ˇZ s ˇ ˇ ˇ kU.t; s/kB.E/ ˇˇ K.s; / f . ; y / d ˇˇ ds jK.s; /j jf . ; y /j d ds

jK.s; /j p. / t

.ky k/ d ds

.kys k/ ds:

p.s/ 0

We consider the function defined by .t/ WD supf jy.s/j W 0 s t g; 0 t < C1: Let t 2 Œr; t be such that .t/ D jy.t /j: If t 2 Œ0; n, by the previous inequality we get Z t b k'k C M b n Sn .t/ M p.s/ ..s// ds; t 2 Œ0; n: 0

If t 2 H, then .t/ D k'k and the previous inequality holds. Let us take the right-hand side of the above inequality as v.t/. Then we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we get b c5 WD v.0/ D Mk'k

and

b n Sn p.t/ v 0 .t/ D M

Using the nondecreasing character of

..t// a.e. t 2 Œ0; n:

, we have

b n Sn p.t/ v 0 .t/ M

.v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using (2.19) we get Z

v.t/

c5

ds b n Sn M .s/ b n Sn M Z

C1

< c5

Z

t

p.s/ ds 0

Z

n

p.s/ ds 0

ds : .s/

2.4 Partial Functional Integro-Differential Evolution Equations

35

Thus, there exists a constant n such that v.t/ n ; t 2 Œ0; n and hence .t/ n ; t 2 Œ0; n. Since for every t 2 Œ0; n; kyt k .t/, we have kykn maxfk'k; n g WD n : Set Y D f y 2 C.Œr; C1/; E/ W supfjy.t/j W 0 t ng n C 1 for all n 2 N g: Clearly, Y is a closed subset of C.Œr; C1/; E/. We shall show that N5 W Y ! C.Œr; C1/; E/ is a contraction operator. Indeed, consider y; y 2 C.Œr; C1/; E/, thus using .2:1:1/, .2:1:3/, and .2:9:1/ for each t 2 Œ0; n and n 2 N we get ˇZ t ˇ Z s ˇ ˇ j.N5 y/.t/ .N5 y/.t/j D ˇˇ U.t; s/ K.s; / Œf . ; y / f . ; y / d dsˇˇ 0 0 Z t Z s kU.t; s/kB.E/ jK.s; /j jf . ; y / f . ; y /j d ds 0

Z

t

b M

0

Z

t

t

0

Z

t

jK.s; /j jf . ; y / f . ; y /j d ds

b n Sn ln .s/ kys ys k ds M Œln .s/ e

"

0

s

0

0

Z

0

Z

1

e

e

Ln .s/

Ln .s/

Ln .t/

Œe

Ln .s/

kys ys k ds

#0 ds ky ykn

ky ykn :

Therefore, kN5 .y/ N5 .y/kn

1 ky ykn :

So, for > 1, the operator N5 is a contraction for all n 2 N. From the choice of Y there is no y 2 @Y n such that y D N5 .y/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.29 does not hold. A consequence of the nonlinear alternative of Frigon and Granas [116] that .S1/ holds, we deduce that the operator N5 has a unique fixed point y in Y, which is the unique mild solution of the problem (2.15)– (2.16). t u

36

2 Partial Functional Evolution Equations with Finite Delay

2.4.3 An Example As an application of our results we consider the following partial functional integrodifferential equation 8 @z.t; x/ @2 z.t; x/ ˆ ˆ D a.t; x/ ˆ ˆ ˆ @t Z t @x2 ˆ ˆ ˆ ˆ ˆ ˛.t; s/Q.s; z.s r; x//ds; t 0; x 2 Œ0; < C r

ˆ ˆ ˆ ˆ z.t; 0/ D z.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ : z.t; x/ D ˚.t; x/

(2.20) t0 t 2 H; x 2 Œ0;

where a.t; x/ is a continuous function and is uniformly Hölder continuous in t, ˛ W Œ0; C1/ Œ0; C1/ ! R, Q W Œ0; C1/ R ! R and ˚ W H Œ0; ! R are continuous functions. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumption .2:1:1/ (see [112, 149]). For x 2 Œ0; , we have y.t/.x/ D z.t; x/

t 2 RC ;

K.t; s/ D ˛.t; s/ t; s 2 RC ; f .t; yt /.x/ D Q.t; z.t r; x//

t 2 RC

and '.t/.x/ D ˚.t; x/

r t 0:

Thus, under the above definitions of f , K, and A./, the system (2.20) can be represented by the abstract partial functional integro-differential evolution problem (2.15)–(2.16). Furthermore, more appropriate conditions on Q ensure the existence of unique mild solution for (2.20) by Theorems 2.10 and 1.29.

2.4 Partial Functional Integro-Differential Evolution Equations

37

2.4.4 Nonlocal Case In this section, we extend the above results of existence and uniqueness of mild solution to the partial functional integro-differential evolution equations with nonlocal conditions (2.17)–(2.18). Nonlocal conditions were initiated by Byszewski [89]. First, we define the mild solution. Definition 2.11. A function y 2 C.Œr; C1/; E/ is said to be a mild solution of (2.17)–(2.18) if y.t/ D '.t/ ht .y/ for all t 2 H and y satisfies the following integral equation Z y.t/ D U.t; 0/ Œ'.0/ h0 .y/ C

Z

t

s

U.t; s/ 0

0

K.s; / f . ; y / d ds;

for t 2 RC :

Under the same assumptions in Sect. 2.2.4 for the function ht ./, we establish that: Theorem 2.12. Assume that the hypotheses (2.1.1)–(2.1.3), (2.9.1), .D1/, and .D2/ hold and moreover Z

C1

c6

ds b n Sn >M .s/

Z

n

p.s/ ds; 0

for each n > 0

(2.21)

b where c6 D M.k'k C /. Then the nonlocal integro-differential evolution problem (2.17)–(2.18) has a unique mild solution. Proof. Transform the problem (2.17)–(2.18) into a fixed point problem. Consider the operator N6 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by: 8 '.t/ ht .y/; ˆ ˆ ˆ < .N6 y/.t/ D U.t; 0/ Z tŒ'.0/ Zh0 .y/ ˆ s ˆ ˆ : C U.t; s/ K.s; / f . ; y / d ds; 0

0

if t 2 H;

if t 0:

Clearly, the fixed points of the operator N6 are mild solutions of the problem (2.17)–(2.18). Then, by parallel steps of the Theorem 2.10’s proof, we can easily show that the operator N6 is a contraction which have a unique fixed point by statement .S1/ in Theorem 1.29. The details are left to the reader. t u

38

2 Partial Functional Evolution Equations with Finite Delay

2.5 Neutral Functional Integro-Differential Evolution Equations 2.5.1 Introduction In this section, we investigate neutral functional integro-differential evolution equations with local and nonlocal conditions where the existence of the unique mild solution is assured. Firstly, we study in Sect. 2.5.2 the neutral functional integrodifferential evolution equations of the form d Œy.t/ g.t; yt / D A.t/y.t/ C dt

Z

t 0

K.t; s/ f .s; ys / ds; a.e. t 2 J D RC (2.22)

y.t/ D '.t/; t 2 H;

(2.23)

where K W J J ! E; f ; g W J C.H; E/ ! E and ' 2 C.H; E/ are given functions and fA.t/gt0 is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators fU.t; s/g.t;s/2JJ : An extension of these existence results, we consider the following neutral functional evolution equation with nonlocal conditions d Œy.t/ g.t; yt / D A.t/y.t/ C dt

Z

t 0

K.t; s/ f .s; ys / ds; a.e. t 2 J D RC (2.24)

y.t/ C ht .y/ D '.t/; t 2 H;

(2.25)

where A./, K, f , g, and ' are as in problem (2.22)–(2.23) and ht W C.Œr; 1/; E/ ! E is a given function. The problem of proving the existence of mild solutions for integro-differential equations and inclusions in abstract spaces has been studied by several authors; see Balachandran and Anandhi [41, 42], Balachandran and Leelamani [45], Benchohra et al. [72, 77], Benchohra and Ntouyas [60, 61, 66], Ntouyas [165]. We are motivated by the mixed problems in [42, 64, 66]. Here we are interested to study of the existence and uniqueness of the mild solution for the neutral functional integro-differential evolution equations (2.22)– (2.23) and the corresponding nonlocal problem (2.24)–(2.25). These results are an extension of [37] for the neutral case.

2.5.2 Main Result We give first the definition of the mild solution of the neutral functional integrodifferential evolution problem (2.22)–(2.23).

2.5 Neutral Functional Integro-Differential Evolution Equations

39

Definition 2.13. We say that the continuous function y./ W Œr; C1/ ! E is a mild solution of (2.22)–(2.23) if y.t/ D '.t/ for all t 2 H and y satisfies the following integral equation Z t y.t/ D U.t; 0/ Œ'.0/ g.0; '/ C g.t; yt / C U.t; s/ A.s/ g.s; ys / ds 0 Z t Z s C U.t; s/ K.s; / f . ; y / d ds for each t 2 RC : 0

0

For every n 2 N, we define in C.Œr; C1/; E/ the semi-norms by: kykn WD sup f e

where

Ln .t/

Z D 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

b .L C nSn ln .t// and ln is the function from ln .s/ ds, ln .t/ D M

(2.1.3). Then C.Œr; C1/; E/ is a Fréchet space with the family of semi-norms 1 < 1. fk kn gn2N . Let us fix > 0 and assume M 0 L C

Theorem 2.14. Suppose that hypotheses (2.1.1)–(2.1.3), (2.9.1), and .G1/–.G3/ are satisfied. If Z

C1 c7;n

Z n b M ds > max.L; nSn p.s//ds; s C .s/ 1 M0L 0

for each n > 0

(2.26)

with c7;n D

b C 1/ C MLn b b Mk'k.1 C M 0 L/ C M 0 L.M 1 M0L

I

Then the neutral functional integro-differential evolution problem (2.22)–(2.23) has a unique mild solution. Proof. Transform the problem (2.22)–(2.23) into a fixed point problem. Consider the operator N7 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by: 8 '.t/; if t 2 H ˆ ˆ ˆ ˆ ˆ Z t < U.t; s/A.s/g.s; ys /ds .N7 y/.t/ D U.t; 0/ Œ'.0/ g.0; '/ C g.t; yt / C ˆ 0 Z t Z s ˆ ˆ ˆ ˆ :C U.t; s/ K.s; / f . ; y / d ds; if t 2 RC . 0

0

Clearly, the fixed points of the operator N7 are mild solutions of the problem (2.22)–(2.23).

40

2 Partial Functional Evolution Equations with Finite Delay

Let y be a possible solution of the problem (2.22)–(2.23). Given n 2 N and t n, then from (2.1.1), (2.9.1), .G1/, and .G2/, we have Z jy.t/j jU.t; 0/j j'.0/ g.0; '/j C jg.t; yt /j C Z

t

C 0

t

0

kU.t; s/kB.E/ jA.s/ g.s; ys /j ds

ˇZ s ˇ ˇ ˇ ˇ kU.t; s/kB.E/ ˇ K.s; / f . ; y / d dsˇˇ 0

b k'k C M b kA .0/k kA.0/ g.0; '/k C kA1 .t/k kA.t/ g.t; yt /k M Z t Z tZ s b b CM kA.s/ g.s; ys /k ds C M jK.s; /j jf . ; y /j d ds 1

0

0

0

b k'k C M b M 0 L .k'k C 1/ C M 0 L .kyt /k C 1/ M Z t Z tZ s b b CM L .kys k C 1/ ds C M jK.s; /j p. / .ky k/ d ds 0

0

0

b C 1/ C M bLn b k'k.1 C M 0 L/ C M 0 L.M M Z t Z t bL b n Sn CM 0 Lkyt k C M kys k ds C M p.s/ .kys k/ ds: 0

0

We consider the function defined by .t/ WD supf jy.s/j W 0 s t g; 0 t < C1: Let t 2 Œr; t be such that .t/ D jy.t /j: If t 2 Œ0; n, by the previous inequality we get for t 2 Œ0; n b C 1/ C MLn b b .t/ Mk'k.1 C M 0 L/ C M 0 L.M Z t Z t b b n CM 0 L.t/ C ML .s/ds C MnS p.s/ ..s//ds: 0

0

Then b b C 1/ C MLn b .1 M 0 L/.t/ Mk'k.1 C M 0 L/ C M 0 L.M Z t Z t b b n CML .s/ds C MnS p.s/ ..s//ds: 0

0

Then .t/

1 1 M0L C

i h b C 1/ C MLn b b Mk'k.1 C M 0 L/ C M 0 L.M

b ML 1 M0L

Z

t 0

.s/ds C

b n Z MnS 1 M0L

t 0

p.s/ ..s//ds

2.5 Neutral Functional Integro-Differential Evolution Equations

Z

b M

D c7;n C

1 M0L

t

Z L.s/ds C

0

0

t

41

nSn p.s/ ..s//ds :

If t 2 H, then .t/ D k'k and the previous inequality holds. Let us take the right-hand side of the above inequality as v.t/. Then we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we get c WD v.0/ D c7;n

and

v 0 .t/ D

b M 1 M0 L

Using the nondecreasing character of v 0 .t/

b M 1 M0L

ŒL.t/ C nSn p.t/ ..t//

a.e. t 2 Œ0; n:

, we have for a.e. t 2 Œ0; n

ŒLv.t/ C nSn p.t/ .v.t// :

This implies that for each t 2 Œ0; n and using (2.26) we get Z

v.t/

c7;n

Z t b ds M max.L; nSn p.s//ds s C .s/ 1 M0L 0 Z n b M max.L; nSn p.s//ds 1 M0L 0 Z C1 ds < : s C .s/ c7;n

Thus, there exists a constant n such that v.t/ n ; t 2 Œ0; n and hence .t/ n ; t 2 Œ0; n. Since for every t 2 Œ0; n; kyt k .t/, we have kykn maxfk'k; n g WD n : Set Y D f y 2 C.Œr; C1/; E/ W kyk1 n C 1 for all n 2 N g: Clearly, Y is a closed subset of C.Œr; C1/; E/: We shall show that N7 W Y ! C.Œr; C1/; E/ is a contraction operator. Indeed, consider y; y 2 C.Œr; C1/; E/, thus using .2:1:1/; .2:1:3/; .2:9:1/; .G1/ and .G3/ for each t 2 Œ0; n and n 2 N, we get j.N7 y/.t/ .N7 y/.t/j jg.t; yt / g.t; yt /j Z t C kU.t; s/kB.E/ jA.s/.g.s; ys / g.s; ys //jds 0

ˇZ t ˇ Z s ˇ ˇ C ˇˇ U.t; s/ K.s; /Œf . ; y / f . ; y /d dsˇˇ 0

1

0

kA .t/kkA.t/g.t; yt / A.t/g.t; yt /k

42

2 Partial Functional Evolution Equations with Finite Delay

b CM

Z

t 0

Z

t

C 0

kA.s/g.s; ys / A.s/g.s; ys /kds Z

kU.t; s/kB.E/

b 0 L kyt yt k C M Z

t

C 0

0

Z

t 0

s

jK.s; /jjf . ; y / f . ; y /jd ds

b kys ys kds ML

b n ln .s/kys ys kds MnS

b 0 L kyt yt k C M

Ln .t/

Z th i b C MnS b n ln .s/ kys ys kds ML 0

Ln .t/

M 0 L e Œe kyt yt k Z t C ln .s/e Ln .s/ Œe Ln .s/ kys ys kds 0

Ln .t/

M 0 L e

ky ykn C

Z t" 0

e Ln .s/

#0 dsky ykn

1 M 0 L e Ln .t/ ky ykn C e Ln .t/ ky ykn

1 Ln .t/ e M 0 L C ky ykn :

Therefore,

1 ky ykn : kN7 .y/ N7 .y/kn M 0 L C

1 < 1, the operator N7 is a contraction for all n 2 N. From So, for M 0 L C

the choice of Y there is no y 2 @Y n such that y D N7 .y/ for some 2 .0; 1/: Then the statement .S2/ in Theorem 1.29 does not hold. A consequence of the nonlinear alternative of Frigon and Granas [116] that .S1/ holds, we deduce that the operator N7 has a unique fixed point y in Y, which is the unique mild solution of the problem (2.22)–(2.23). t u

2.5.3 An Example As an application of our results we consider the following neutral functional integrodifferential evolution equation

2.5 Neutral Functional Integro-Differential Evolution Equations

43

8 Rt R @2 z ˆ llc @t@ z.t; x/ r 0 b.s t; u; x/ z.s; u/ du ds D a.t; x/ @x ˆ 2 .t; x/ ˆ

Rt ˆ ˆ @z ˆ C 0 ˛.t; s/Q s; z.s r; x/; @x .s r; x/ ds; t 0; x 2 Œ0; ˆ ˆ ˆ ˆ < ˆ z.t; 0/ D z.t; / D 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :z.t; x/ D ˚.t; x/;

t0 t 2 H; x 2 Œ0; (2.27)

where a.t; x/ is a continuous function and is uniformly Hölder continuous in t, ˛ W Œ0; C1/ Œ0; C1/ ! R; Q W Œ0; C1/ R R ! R and ˚ W H Œ0; ! R are continuous functions. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .2:1:1/ and .G1/ (see [112, 149]). For x 2 Œ0; , we have t 2 RC ;

y.t/.x/ D z.t; x/

K.t; s/ D ˛.t; s/ t; s 2 RC ; f .t; yt /.x/ D Q.t; z.t r; x// t 2 RC ; Z t Z g.t; yt /.x/ D b.s t; u; x/z.s; u/duds; x 2 Œ0; r

0

and '.t/.x/ D ˚.t; x/

t 2 H:

Here we consider that ' W H ! E is Lebesgue measurable and h k'k2 is Lebesgue integrable on H where h W H ! R is a positive integrable function. The norm is defined here by: Z k'k D k˚.0/k C

0

r

2

h.s/ k'k ds

12

:

(i) The function b is measurable and Z

0

Z

t

r

Z

0

b2 .s; u; x/ dsdudx < 1: h.s/

44

2 Partial Functional Evolution Equations with Finite Delay

@2 b @b .s; u; x// and . 2 .s; u; x// are measurable, b.s; u; 0/ D @x @x b.s; u; // D 0 and sup N .t/ < 1; where

(ii) The function .

t2Œ0;b

Z N .t/ D

0

Z

t

r

Z

0

2 @2 1 a.s; x/ 2 b.s; u; x/ dsdudx: h.s/ @x

Thus, under the above definitions of f , g, K, and A./; the system (2.27) can be represented by the abstract neutral functional integro-differential evolution problem (2.22)–(2.23). Furthermore, more appropriate conditions on Q ensure the existence of a least one mild solution for (2.27) by Theorems (2.14) and (1.29).

2.5.4 Nonlocal Case An extension of these results is given here for the neutral functional integrodifferential evolution equation with nonlocal conditions (2.24)–(2.25). Nonlocal conditions were initiated by Byszewski [89]. Before giving the main result, we give first the definition of the mild solution. Definition 2.15. A function y 2 C.Œr; C1/; E/ is said to be a mild solution of (2.24)–(2.25) if y.t/ D '.t/ ht .y/ for all t 2 H and y satisfies the following integral equation Z t y.t/ D U.t; 0/ Œ'.0/ h0 .y/ g.0; '/ C g.t; yt / C U.t; s/ A.s/ g.s; ys / ds 0 Z t Z s C U.t; s/ K.s; / f . ; y / d ds; for each t 2 RC : 0

0

Under the same assumptions in Sect. 2.2.4 for the function ht ./, we establish that Theorem 2.16. Assume that the hypotheses (2.1.1)–(2.1.3), (2.9.1), .G1/–.G3/, .D1/, and .D2/ hold. If Z

C1

c8;n

Z n b M ds > max .L; nSn p.s// ds; .s/ 1 M0L 0

for each n > 0

(2.28)

with c8;n D

b C 1/ C MLn b b k'k.1 C M 0 L/ C C M 0 L.M M 1 M0L

:

Then the nonlocal neutral functional integro-differential evolution problem (2.24)–(2.25) has a unique mild solution.

2.6 Notes and Remarks

45

Proof. Transform the problem (2.24)–(2.25) into a fixed point problem. Consider the operator N8 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by: 8 ˆ '.t/ ht .y/; if t 2 HI ˆ Z t ˆ ˆ < U.t; 0/ Œ'.0/ h0 .y/ g.0; '/ C U.t; s/ A.s/ g.s; ys / ds .N8 y/.t/ D 0 Z Z ˆ t s ˆ ˆ ˆ : C U.t; s/ K.s; / f . ; y / d ds; for each t 0: 0

0

Clearly, the fixed points of the operator N8 are mild solutions of the problem (2.24)–(2.25). Then, by parallel steps of the Theorem 2.14’s proof, we can prove that the operator N8 is a contraction which have a fixed point by statement .S1/ in Theorem 1.29. The details are left to the reader. t u

2.6 Notes and Remarks The results of Chap. 2 are taken from Baghli and Benchohra [33, 33, 37]. Other results may be found in [41, 42, 44].

Chapter 3

Partial Functional Evolution Equations with Infinite Delay

3.1 Introduction In this chapter, we provide sufficient conditions for the existence of the unique mild solution on the positive half-line RC for some classes of first order partial functional and neutral functional differential evolution equations with infinite delay.

3.2 Partial Functional Evolution Equations 3.2.1 Introduction In this section, we consider the following partial functional evolution equations with infinite delay y0 .t/ D A.t/y.t/ C f .t; yt /; a.e. t 2 J D RC

(3.1)

y0 D 2 B;

(3.2)

where f W J B ! E and 2 B are given functions and fA.t/g0t
© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_3

47

48

3 Partial Functional Evolution Equations with Infinite Delay

Here we are interested to give existence and uniqueness results of the mild solution for the partial functional evolution equations (3.1)–(3.2) by using Theorem 1.29 due to Frigon and Granas [116].

3.2.2 Existence and Uniqueness of Mild Solution Before stating and proving the main result, we give first the definition of mild solution of the partial functional evolution problem (3.1)–(3.2). Definition 3.1. We say that the continuous function y./ W R ! E is a mild solution of (3.1)–(3.2) if y.t/ D .t/ for all t 2 .1; 0 and y satisfies the following integral equation Z t y.t/ D U.t; 0/ .0/ C U.t; s/ f .s; ys / ds; for each t 2 RC : 0

We will need to introduce the following hypotheses which are assumed hereafter: b 1 such that: (3.1.1) There exists a constant M b for every .t; s/ 2 : kU.t; s/kB.E/ M 1 (3.1.2) There exist a function p 2 Lloc .J; RC / and a continuous nondecreasing function W RC ! .0; 1/ such that :

jf .t; u/j p.t/

.kukB / for a.e. t 2 J and each u 2 B:

1 (3.1.3) For all R > 0, there exists lR 2 Lloc .R; RC / such that:

jf .t; u/ f .t; v/j lR .t/ ku vkB for all u; v 2 B with kukB R and kvkB R. Consider the following space ˚ BC1 D y W R ! E W yjŒ0;T 2 C.Œ0; T; E/; y0 2 B ; where yjŒ0;T is the restriction of y to any real compact interval Œ0; T. For every n 2 N, we define in BC1 the semi-norms by: kykn WD sup f e

where Ln .t/ D

Z 0

t

Ln .t/

jy.t/j W t 2 Œ0; ng

b n .t/ and ln is the function from .3:1:3/: ln .s/ ds ; ln .t/ D Kn Ml

Then BC1 is a Fréchet space with the family of semi-norms k kn2N . In what follows let us fix > 1.

3.2 Partial Functional Evolution Equations

49

Theorem 3.2 ([33]). Suppose that hypotheses (3.1.1)–(3.1.3) are satisfied and moreover Z

C1

c9;n

ds b > Kn M .s/

Z

n 0

p.s/ ds; for each n > 0

(3.3)

b C Mn /kkB : Then the problem (3.1)–(3.2) has a unique mild with c9;n D .Kn MH solution. Proof. Consider the operator N9 W BC1 ! BC1 defined by:

.N9 y/.t/ D

8 ˆ < .t/;

if t 0;

ˆ : U.t; 0/ .0/ C

Z

t

0

U.t; s/ f .s; ys / ds;

if t 0:

Clearly, fixed points of the operator N9 are mild solutions of the problem (3.1)–(3.2). For 2 B, we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 2 .1; 0I

: U.t; 0/ .0/;

if t 2 J:

Then x0 D . For each function z 2 C.J; E/, set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 3.1 if and only if z satisfies z0 D 0 and Z

t

z.t/ D

U.t; s/ f .s; zs C xs / ds;

0

for t 2 J:

Let B0C1 D fz 2 BC1 W z0 D 0g : Define the operator F W B0C1 ! B0C1 by: Z F.z/.t/ D 0

t

U.t; s/ f .s; zs C xs / ds;

for t 2 J:

Obviously the operator N9 has a fixed point is equivalent to F has one, so it turns to prove that F has a fixed point.

50

3 Partial Functional Evolution Equations with Infinite Delay

Let z 2 B0C1 be a possible fixed point of the operator F. By the hypotheses .3:1:1/ and .3:1:2/; we have for each t 2 Œ0; n Z

t

jz.t/j 0

b M

kU.t; s/kB.E/ jf .s; zs C xs /j ds Z

t

p.s/ 0

.kzs C xs kB / ds:

Assumption .A1 / gives kzs C xs kB kzs kB C kxs kB K.s/jz.s/j C M.s/kz0 kB C K.s/jx.s/j C M.s/kx0 kB Kn jz.s/j C Kn kU.s; 0/kB.E/ j.0/j C Mn kkB b Kn jz.s/j C Kn Mj.0/j C Mn kkB b Kn jz.s/j C Kn MHkk B C Mn kkB b C Mn /kkB : Kn jz.s/j C .Kn MH b C Mn /kkB , then we have Set ˛n WD c9;n D .Kn MH kzs C xs kB Kn jz.s/j C ˛n Using the nondecreasing character of b jz.t/j M

Z

(3.4)

, we get

t

p.s/ 0

.Kn jz.s/j C ˛n / ds:

Then b Kn jz.t/j C ˛n Kn M

Z

t 0

p.s/ .Kn jz.s/j C ˛n /ds C c9;n :

Consider the function defined by .t/ WD sup f Kn jz.s/j C ˛n W 0 s t g; 0 t < C1: Let t 2 Œ0; t be such that .t/ D Kn jz.t /j C ˛n : By the previous inequality, we have Z t b p.s/ ..s// ds C c9;n ; for t 2 Œ0; n: .t/ Kn M 0

3.2 Partial Functional Evolution Equations

51

Let us take the right-hand side of the above inequality as v.t/. Then, we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we have b v.0/ D c9;n and v 0 .t/ D Kn Mp.t/ Using the nondecreasing character of

..t// a.e. t 2 Œ0; n:

, we get

b p.t/ v 0 .t/ Kn M

.v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using the condition (3.3), we get Z

v.t/ c9;n

ds b Kn M .s/ b Kn M Z

Z

p.s/ ds Z

C1

< c9;n

t 0 n

p.s/ ds 0

ds : .s/

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kzkn .t/, we have kzkn n : Set Z D f z 2 B0C1 W supf jz.t/j W 0 t ng n C 1 for all n 2 Ng: Clearly, Z is a closed subset of B0C1 : We shall show that F W Z ! B0C1 is a contraction operator. Indeed, consider z; z 2 B0C1 ; thus using .3:1:1/ and .3:1:3/ for each t 2 Œ0; n and n 2 N ˇZ t ˇ ˇ ˇ ˇ j.Fz/.t/ .Fz/.t/j D ˇ U.t; s/ Œf .s; zs C xs / f .s; zs C xs / dsˇˇ 0 Z t kU.t; s/kB.E/ jf .s; zs C xs / f .s; zs C xs /j ds 0

Z

t

0

Z

0

t

b ln .s/ kzs C xs zs xs kB ds M b ln .s/ kzs zs kB ds: M

52

3 Partial Functional Evolution Equations with Infinite Delay

Using .A1 /, we obtain Z j.Fz/.t/ .Fz/.t/j

t

0

Z

t

0

Z

t

0

Z

t

b ln .s/ .K.s/ jz.s/ z.s/j C M.s/ kz0 z0 kB / ds M b Kn ln .s/ jz.s/ z.s/j ds M Œln .s/ e

"

0

1

e

e

Ln .s/

Ln .s/

Ln .t/

Œe

Ln .s/

jz.s/ z.s/j ds

#0 ds kz zkn

kz zkn :

Therefore, kF.z/ F.z/kn

1 kz zkn :

So, for > 1, the operator F is a contraction for all n 2 N. From the choice of Z there is no z 2 @Z n such that z D F.z/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.29 does not hold. A consequence of the nonlinear alternative of Frigon and Granas that .S1/ holds, we deduce that the operator F has a unique fixed point z . Then y .t/ D z .t/ C x.t/; t 2 R is a fixed point of the operator N9 , which is the unique mild solution of the problem (3.1)–(3.2). t u

3.2.3 An Example Consider the following partial functional differential equation 8 @z @2 z ˆ ˆ .t; x/ D a.t; x/ 2 .t; x/ C Q.t; z.t r; x// t 0; x 2 Œ0; ˆ ˆ ˆ @t @x ˆ < z.t; 0/ D z.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ : z.t; x/ D ˚.t; x/

t0

(3.5)

t 0; x 2 Œ0; ;

where a.t; x/ is a continuous function and is uniformly Hölder continuous in t; Q W RC R ! R and ˚ W B Œ0; ! R are continuous functions.

3.3 Controllability on Finite Interval for Partial Evolution Equations

53

Let y.t/.x/ D z.t; x/; t 2 Œ0; 1/; x 2 Œ0; ; ./.x/ D ˚.; x/; 0; x 2 Œ0; and f .t; /.x/ D Q.t; .; x//; 0; x 2 Œ0; : Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g: Then A.t/ generates an evolution system U.t; s/ satisfying assumption .3:1:1/ (see [112, 149]). Thus, under the above definitions of f and A./, the system (3.5) can be represented by the abstract evolution problem (3.1)–(3.2). Furthermore, more appropriate conditions on Q ensure the existence of the unique mild solution of (3.5) by Theorems 3.2 and 1.29.

3.3 Controllability on Finite Interval for Partial Evolution Equations 3.3.1 Introduction In this section, we give sufficient conditions to ensure the controllability of mild solutions on a bounded interval JT WD Œ0; T for T > 0 for the partial functional evolution equations with infinite delay of the form y0 .t/ D A.t/y.t/ C Cu.t/ C f .t; yt /; a.e. t 2 JT

(3.6)

y0 D 2 B;

(3.7)

where f W JB ! E and 2 B are given functions, the control function u.:/ is given in L2 .Œ0; T; E/; the Banach space of admissible control functions with E be a real separable Banach space with the norm jj, C is a bounded linear operator from E into E, and fA.t/g0tT is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators fU.t; s/g.t;s/2JJ for 0 s t T:

54

3 Partial Functional Evolution Equations with Infinite Delay

3.3.2 Controllability of Mild Solutions Before stating and proving the main result, we give first the definition of mild solution of problem (3.6)–(3.7) and the definition of controllability of the mild solution. Definition 3.3. We say that the continuous function y./ W R ! E is a mild solution of (3.6)–(3.7) if y.t/ D .t/ for all t 2 .1; 0 and y satisfies the following integral equation Z y.t/ D U.t; 0/.0/ C

t

Z

t

U.t; s/Cu.s/ds C

0

0

U.t; s/f .s; ys /ds; for each t 2 Œ0; T:

Definition 3.4. The problem (3.6)–(3.7) is said to be controllable on the interval Œ0; T if for every initial function 2 B and yQ 2 E there exists a control u 2 L2 .Œ0; T; E/ such that the mild solution y./ of (3.6)–(3.7) satisfies y.T/ D yQ . We will need to introduce the following hypotheses which are assumed hereafter: b 1 such that: (3.4.1) U.t; s/ is compact for t s > 0 and there exists a constant M b for every 0 s t T: kU.t; s/kB.E/ M (3.4.2) There exists a function p 2 L1 .JT ; RC / and a continuous nondecreasing function W RC ! .0; 1/ such that: jf .t; u/j p.t/

.kukB / for a.e. t 2 JT and each u 2 B:

(3.4.3) The linear operator W W L2 .Œ0; T; E/ ! C.Œ0; T; E/ is defined by Z Wu D

T

U.T; s/Cu.s/ds; 0

has a bounded inverse operator W 1 which takes values in L2 .Œ0; T; E/= Q and M Q 1 such that: ker W and there exists positive constants M Q 1: Q and kW 1 k M kCk M Remark 3.5. For the construction of W see the book of Carmichael and Quinn [93]. Consider the following space BT D fy W .1; T ! E W yjJ 2 C.J; E/; y0 2 Bg ; where yjJ is the restriction of y to J.

3.3 Controllability on Finite Interval for Partial Evolution Equations

55

Theorem 3.6. Suppose that hypotheses (3.4.1)–(3.4.3) are satisfied and moreover there exists a constant M > 0 such that

c10;T

M bM QM Q 1T C 1 b M C KT M

.M / kpkL1

> 1;

(3.8)

with i h bM QM Q 1 T C 1 C MT kkB C KT M b bM QM Q 1 T jQyj : c10;T D c10 .; yQ ; T/ D KT MH M Then the problem (3.6)–(3.7) is controllable on .1; T: Proof. Transform the problem (3.6)–(3.7) into a fixed point problem. Consider the operator N10 W BT ! BT defined by: 8 ˆ .t/; if t 2 .1; 0; ˆ ˆ ˆ ˆ Z t ˆ < U.t; s/ C uy .s/ ds .N10 y/.t/ D U.t; 0/ .0/ C ˆ 0 ˆ Z ˆ t ˆ ˆ ˆ :C U.t; s/ f .s; ys / ds; if t 2 Œ0; T: 0

Clearly, fixed points of the operator N10 are mild solutions of the problem (3.6)–(3.7). Using assumption .3:4:3/; for arbitrary function y./, we define the control Z uy .t/ D W 1 yQ U.T; 0/ .0/

0

T

U.T; s/ f .s; ys / ds .t/:

Noting that, we have Z juy .t/j kW 1 k jQyj C kU.T; 0/kB.E/ j.0/j C Z b Q 1 jQyj C MHkk b C M M B Z b b Q 1 jQyj C MHkk C M M B

T

0

T 0 T 0

kU.T; /kB.E/ jf . ; y /jd

jf . ; y /jd

p. /

.ky kB / d :

For 2 B; we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 2 .1; 0I

: U.t; 0/ .0/;

if t 2 JT :

56

3 Partial Functional Evolution Equations with Infinite Delay

Then x0 D . For each function z 2 BT , set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 3.3 if and only if z satisfies z0 D 0 and Z z.t/ D 0

t

Z U.t; s/ C uz .s/ ds C

t

0

U.t; s/ f .s; zs C xs / ds;

for t 2 JT :

Let B0T D fz 2 BT W z0 D 0g : For any z 2 B0T we have kzkT D supf jz.t/j W t 2 Œ0; T g C kz0 kB D supf jz.t/j W t 2 Œ0; T g: Thus .B0T ; k kT / is a Banach space. Define the operator F W B0T ! B0T by: Z Z t U.t; s/ C uz .s/ ds C .Fz/.t/ D 0

t

U.t; s/ f .s; zs C xs / ds;

0

for t 2 JT :

Obviously the operator N10 has a fixed point is equivalent to F has one, so it turns to prove that F has a fixed point. The proof will be given in several steps. Let us first show that the operator F is continuous and compact. Step 1: F is continuous. Let .zn /n be a sequence in B0T such that zn ! z in B0T . Then, we get Z j.Fzn /.t/ .Fz/.t/j

0

t

kU.t; s/kB.E/ kCk juzn .s/ uz .s/j ds

Z

t

C 0

bM Q M b CM

kU.t; s/kB.E/ jf .s; zns C xs / f .s; zs C xs /j ds

Z

Z

t 0 t

0

Q 1M b M

Z

T 0

T 0

jf . ; zn C x / f . ; z C x /j d ds

jf .s; zns C xs / f .s; zs C xs /j ds

b 2M QM Q 1T M b CM

Z

Z

T 0

jf .s; zns C xs / f .s; zs C xs /j ds

jf .s; zns C xs / f .s; zs C xs /j ds

bM QM Q 1 T C 1 kf .; zn C x / f .; z C x /kL1 : b M M Since f is continuous, we obtain by the Lebesgue dominated convergence theorem jF.zn /.t/ F.z/.t/j ! 0 as n ! C1: Thus F is continuous.

3.3 Controllability on Finite Interval for Partial Evolution Equations

57

Step 2: F maps bounded sets of B0T into bounded sets. For any d > 0, there exists a positive constant ` such that for each z 2 Bd D fz 2 B0T W kzkT dg we have kF.z/kT `. Let z 2 Bd , for each t 2 Œ0; T, we have Z t Z t j.Fz/.t/j kU.t; s/kB.E/ kCk juz .s/j ds C kU.t; s/kB.E/ jf .s; zs C xs /j ds 0

bM Q M

Z

Z

t

0

t

b juz .s/j ds C M jf .s; zs C xs /j ds 0

Z t Z T bM Q b Q 1 jQyj C MHkk b M C M p.

/ .kz C x k / d

ds M B

B 0 0 Z t b CM jf .s; zs C xs /j ds 0

Z T b Q Q b b M M M1 T jQyj C MHkkB C M p.s/ .kzs C xs kB / ds 0 Z t b p.s/ .kzs C xs kB / ds CM 0 i h bM QM Q 1 T jQyj C MHkk b M B Z T bM QM Q 1T C 1 b M CM p.s/ .kzs C xs /kB / ds: 0

0

By (3.4) on JT , we get for each z 2 Bd kzs C xs kB KT d C ˛T WD ıT :

(3.9)

Then, using the nondecreasing character of , we get for each t 2 Œ0; T i h b bQ Q bM QM Q 1 T jQyj C MHkk b j.Fz/.t/j M .ıT / kpkL1 WD `: B C M M M M1 T C 1 Thus there exists a positive number ` such that kF.z/kT `: Hence F.Bd / Bd . Step 3: F maps bounded sets into equi-continuous sets of B0T . We consider Bd as in Step 2 and we show that F.Bd / is equi-continuous. Let 1 ; 2 2 JT with 2 > 1 and z 2 Bd . Then Z 1 kU. 2 ; s/ U. 1 ; s/kB.E/ kCk juz .s/j ds j.Fz/. 2 / .Fz/. 1 /j 0 Z 1 kU. 2 ; s/ U. 1 ; s/kB.E/ jf .s; zs C xs /j ds C Z0 2 kU. 2 ; s/kB.E/ kCk juz .s/j ds C

1 Z 2 kU. 2 ; s/kB.E/ jf .s; zs C xs /j ds: C

1

58

3 Partial Functional Evolution Equations with Infinite Delay

In the property of uy , we use (3.9) and the nondecreasing character of h Q 1 jQyj C MHkk b b juz .t/j M B CM Then

Z

j.Fz/. 2 / .Fz/. 1 /j kCkB.E/ ! Z C .ıT /

1

0

kU. 2 ; s/ U. 1 ; s/kB.E/ p.s/ ds Z

CkCkB.E/ ! Z C .ıT /

i .ıT / kpkL1 WD !:

kU. 2 ; s/ U. 1 ; s/kB.E/ ds

0

1

to get

2

1

2

1

kU. 2 ; s/kB.E/ ds

kU. 2 ; s/kB.E/ p.s/ ds:

Noting that j.Fz/. 2 / .Fz/. 1 /j tends to zero as 2 1 ! 0 independently of z 2 Bd . The right-hand side of the above inequality tends to zero as 2 1 ! 0. Since U.t; s/ is a strongly continuous operator and the compactness of U.t; s/ for t > s implies the continuity in the uniform operator topology (see [16, 168]). As a consequence of Steps 1 to 3 together with the Arzelá–Ascoli theorem it suffices to show that the operator F maps Bd into a precompact set in E. Let t 2 JT be fixed and let be a real number satisfying 0 < < t. For z 2 Bd we define Z t .F z/.t/ D U.t; t / U.t ; s/ C uz .s/ ds 0

Z

CU.t; t /

t

0

U.t ; s/ f .s; zs C xs / ds:

Since U.t; s/ is a compact operator, the set Z .t/ D fF .z/.t/ W z 2 Bd g is precompact in E for every sufficiently small, 0 < < t. Moreover using the definition of !, we have Z t j.Fz/.t/ .F z/.t/j kU.t; s/kB.E/ kCk juz .s/j ds t

Z

C

t

kU.t; s/kB.E/ jf .s; zs C xs /j ds

t

Z

kCkB.E/ ! Z C .ıT /

t

kU.t; s/kB.E/ ds

t t t

kU.t; s/kB.E/ p.s/ ds:

3.3 Controllability on Finite Interval for Partial Evolution Equations

59

Therefore there are precompact sets arbitrary close to the set fF.z/.t/ W z 2 Bd g. Hence the set fF.z/.t/ W z 2 Bd g is precompact in E. So we deduce from Steps 1, 2, and 3 that F is a compact operator. Step 4: For applying Theorem 1.27, we must check .S2/: i.e., it remains to show that the set ˚ E D z 2 B0T W z D F.z/ for some 0 < < 1 is bounded. Let z 2 E, for each t 2 Œ0; T we have Z

t

jz.t/j 0

Z kU.t; s/kB.E/ kCk juz .s/j ds C

bM Q M

Z Z

t

p.s/

t

p.s/ 0

p. /

.kz C x kB / d

T

p.s/ 0

.kzs C xs kB / ds

.kzs C xs kB / ds:

Using the inequality (3.4) over JT and the nondecreasing character of Z bM QM Q 1 T jQyj C MHkk b b jz.t/j M C M B b CM

Z

p.s/ 0

, we get

T

p.s/

0

t

ds

.kzs C xs kB / ds

Z b b Q Q b M M M1 T jQyj C MHkkB C M Z

kU.t; s/kB.E/ jf .s; zs C xs /j ds

T

0

0

b CM

0

Z b Q 1 jQyj C MHkk b C M M B

t

0

b CM

t

.KT jz.s/j C ˛T / ds

.KT jz.s/j C ˛T / ds:

Then bM QM Q 1T KT jz.t/j C ˛T ˛T C KT M Z b b M jQyj C MHkk B b CKT M

Z

t

p.s/ 0

T

p.s/ 0

.KT jz.s/j C ˛T / ds

.KT jz.s/j C ˛T / ds:

60

3 Partial Functional Evolution Equations with Infinite Delay

i h bM QM Q 1 T jQyj C MHkk b Set c10;T WD ˛T C KT M B , thus KT jz.t/j C ˛T c10;T

b 2 KT M QM Q 1T CM

b CKT M

Z

t

p.s/ 0

Z

T

p.s/ 0

.KT jz.s/j C ˛T / ds

.KT jz.s/j C ˛T / ds:

We consider the function defined by .t/ WD sup f KT jz.s/j C ˛T W 0 s t g; 0 t T: Let t 2 Œ0; t be such that .t/ D KT jz.t /j C ˛T : If t 2 Œ0; T, by the previous inequality, we have for t 2 Œ0; T b 2 KT M QM Q 1T .t/ c10;T C M

Z

T

p.s/ 0

b ..s// ds C KT M

Z

t

p.s/ 0

..s// ds:

Then, we have .t/ c10;T

Z b b Q Q C KT M M M M1 T C 1

T

p.s/ 0

..s// ds:

Consequently,

c10;T

kzkT b M bM QM Q 1T C 1 C KT M

.kzkT / kpkL1

1:

Then by (3.8), there exists a constant M such that kzkT ¤ M . Set Z D f z 2 B0T W kzkT M C 1 g: Clearly, Z is a closed subset of B0T . From the choice of Z there is no z 2 @Z such that z D F.z/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.27 does not hold. As a consequence of the nonlinear alternative of Leray–Schauder type [128], we deduce that .S1/ holds: i.e., the operator F has a fixed point z . Then y .t/ D z .t/ C x.t/, t 2 .1; T is a fixed point of the operator N10 , which is a mild solution of the problem (3.6)–(3.7). Thus the evolution system (3.6)–(3.7) is controllable on .1; T. t u

3.3 Controllability on Finite Interval for Partial Evolution Equations

61

3.3.3 An Example As an application of Theorem 3.6, we present the following control problem 8 @v @2 v ˆ ˆ ˆ .t; / D a.t; / 2 .t; / C d./u.t/ ˆ ˆ @t ˆ Z 0 @ ˆ ˆ ˆ ˆ ˆ C P. /r.t; v.t C ; //d <

t 2 Œ0; T

1

2 Œ0; (3.10)

ˆ ˆ ˆ ˆ v.t; 0/ D v.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ : v.; / D v0 .; /

t 2 Œ0; T 1 < 0; 2 Œ0; ;

where a.t; / is a continuous function and is uniformly Hölder continuous in t ; P W .1; 0 ! R ; r W Œ0; T R ! R ; v0 W .1; 0 Œ0; ! R and d W Œ0; ! E are continuous functions. u./ W Œ0; T ! E is a given control. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumption .3:5:1/ (see [112, 149]). For the phase space B, we choose the well-known space BUC.R ; E/: the space of uniformly bounded continuous functions endowed with the following norm k'k D sup j'./j for 0

' 2 B:

If we put for ' 2 BUC.R ; E/ and 2 Œ0; y.t/./ D v.t; /; t 2 Œ0; T; 2 Œ0; ; . /./ D v0 .; /; 1 < 0; 2 Œ0; ; and Z f .t; '/./ D

0

1

P. /r.t; '. /.//d; 1 < 0; 2 Œ0;

Finally let C 2 B.R; E/ be defined as Cu.t/./ D d./u.t/; t 2 Œ0; T; 2 Œ0; ; u 2 R; d./ 2 E:

62

3 Partial Functional Evolution Equations with Infinite Delay

Then, problem (3.10) takes the abstract evolution form (3.6)–(3.7). In order to show the controllability of mild solutions of system (3.10), we suppose the following assumptions: – There exists a continuous function p 2 L1 .JT ; RC / and a nondecreasing continuous function W Œ0; 1/ ! Œ0; 1/ such that jr.t; u/j p.t/ .juj/; for t 2 JT ; and u 2 R: – P is integrable on .1; 0: By the dominated convergence theorem, one can show that f is a continuous function mapping B into E: In fact, we have for ' 2 B and 2 Œ0; Z jf .t; '/./j Since the function

0

1

jp.t/P. /j .j.'. //./j/d:

is nondecreasing, it follows that Z

jf .t; '/j p.t/

0 1

jP. /j d .j'j/; for ' 2 B:

Proposition 3.7. Under the above assumptions, if we assume that condition (3.8) in Theorem 3.6 is true, ' 2 B, then the problem (3.10) is controllable on .1; T.

3.4 Controllability on Semi-infinite Interval for Partial Evolution Equations 3.4.1 Introduction We obtain in this section the controllability of mild solutions on the semi-infinite interval J D RC for the partial functional evolution equations with infinite delay of the form y0 .t/ D A.t/y.t/ C Cu.t/ C f .t; yt /; a.e. t 2 J D RC y0 D 2 .1; 0;

(3.11) (3.12)

where f W J B ! E and 2 B are given functions, the control function u.:/ is given in L2 .Œ0; 1/; E/; the Banach space of admissible control function with E is a real separable Banach space with the norm j j for some n > 0, C is a bounded linear operator from E into E, and fA.t/g0t
3.4 Controllability on Semi-infinite Interval for Partial Evolution Equations

63

3.4.2 Controllability of Mild Solutions In this section, we give controllability result for the system (3.11)–(3.12). Before this, we introduce the the following type of solutions for the problem (3.11)–(3.12). Definition 3.8. We say that the continuous function y./ W R ! E is a mild solution of (3.11)–(3.12) if y.t/ D .t/ for all t 2 .1; 0 and y satisfies the following integral equation Z y.t/ D U.t; 0/.0/ C

t

Z

t

U.t; s/Cu.s/ds C

0

0

U.t; s/f .s; ys /ds; for each t 2 RC

Definition 3.9. The evolution problem (3.11)–(3.12) is said to be controllable if for every initial function 2 B and yO 2 E, there is some control u 2 L2 .Œ0; n; E/ such that the mild solution y./ of (3.11)–(3.12) satisfies the terminal condition y.n/ D yO : We will consider the hypotheses (3.1.1)–(3.1.3) and we will need to introduce the following one which is assumed hereafter: (3.9.1) For each n 2 N, the linear operator W W L2 .Œ0; n; E/ ! E is defined by Z Wu D

n

U.n; s/Cu.s/ds; 0

has a bounded inverse operator W 1 which takes values in L2 .Œ0; n; E/= ker W Q and M Q 1 such that: and there exists positive constants M Q 1: Q and kW 1 k M kCk M Remark 3.10. For the construction of W see [93]. Consider the following space ˚ BC1 D y W R ! E W yjŒ0;T 2 C.Œ0; T; E/; y0 2 B ; where yjŒ0;T is the restriction of y to any real compact interval Œ0; T. For every n 2 N, we define in BC1 the semi-norms by kykn WD sup f e

where

Ln .t/

Z D 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

b ln .t/ and ln is the function from .3:1:3/: ln .s/ ds ; ln .t/ D Kn M

Then BC1 is a Fréchet space with the family of semi-norms fk kn gn2N . In what follows let us fix > 1.

64

3 Partial Functional Evolution Equations with Infinite Delay

Theorem 3.11. Suppose that hypotheses (3.1.1)–(3.1.3), (3.9.1) are satisfied and moreover there exists a constant M > 0 M b M bM QM Q 1 n C 1/ c11;n C Kn M.

.M / kpkL1

> 1;

(3.13)

with i h b bM QM Q 1 n jOyj : bM QM Q 1 n C 1 C Mn kkB C Kn M c11;n D c11 .Oy; ; n/ WD Kn MH M Then the evolution problem (3.11)–(3.12) is controllable on R: Proof. Consider the operator N11 W BC1 ! BC1 defined by: 8 ˆ .t/; if t 0; ˆ ˆ ˆ ˆ Z ˆ t < U.t; s/ C uy .s/ ds .N11 y/.t/ D U.t; 0/ .0/ C ˆ 0 ˆ Z t ˆ ˆ ˆ ˆ :C U.t; s/ f .s; ys / ds; if t 0: 0

Using assumption .3:9:1/; for arbitrary function y./, we define the control

Z n uy .t/ D W 1 yO U.n; 0/ .0/ U.n; s/ f .s; ys / ds .t/: 0

Noting that, we have Z juy .t/j kW 1 k jOyj C kU.t; 0/kB.E/ j.0/j C

0

Z Q 1 jOyj C MHkk b b M B CM

n

Z Q b b M1 jOyj C MHkkB C M

n

0

0

n

kU.n; /kB.E/ jf . ; y /jd

jf . ; y /jd

p. /

.ky kB / d :

We shall show that using this control the operator N11 has a fixed point y./. Then y./ is a mild solution of the evolution system (3.11)–(3.12). For 2 B, we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 0I

: U.t; 0/ .0/;

if t 0:

3.4 Controllability on Semi-infinite Interval for Partial Evolution Equations

65

Then x0 D . For each function z 2 BC1 , set y.t/ D z.t/ C x.t/: Then z satisfies z0 D 0 and Z

t

z.t/ D 0

Z U.t; s/ C uzCx .s/ ds C

t 0

U.t; s/ f .s; zs C xs / ds;

for t 0:

Let B0C1 D fz 2 BC1 W z0 D 0g : Define the operators F; G W B0C1 ! B0C1 by: Z

t

F.z/.t/ D 0

U.t; s/ C uzCx .s/ ds;

for t 0:

and Z

t

G.z/.t/ D

U.t; s/ f .s; zs C xs / ds;

0

for t 0:

Obviously the operator N11 has a fixed point is equivalent to F C G has one, so it turns to prove that F C G has a fixed point. The proof will be given in several steps. We can show as in Sect. 3.3.2 that the operator F is continuous and compact. We can prove also that the operator G is a contraction as in the proof of Theorem 2.2). For applying Theorem 1.30, it remains to show that .S2/ doesn’t hold: i.e., we will prove that the following set is bounded o n z for some 0 < < 1 : E D z 2 B0C1 W z D F.z/ C G Let z 2 E, for each t 2 Œ0; n, we have Z jz.t/j

t

0

kU.t; s/kB.E/ kCk juzCx .s/j ds

Z

t

C 0

ˇ z ˇ ˇ ˇ s kU.t; s/kB.E/ ˇf s; C xs ˇ ds:

Then 1 bM Q jz.t/j M

Z

Z b Q 1 jOyj C MHkk b C M M B

t

0

Z

0

n

p. /

z s ds C xs B 0 Z n b b bM QM Q 1 t jOyj C MHkk C M p.s/ M B b CM

b CM

.kz C x kB / d

t

p.s/

Z

0

t

p.s/ 0

z s ds C xs B

.kzs C xs kB / ds

ds

66

3 Partial Functional Evolution Equations with Infinite Delay

Z i h 2 Q Q b b Q Q b M M M1 n jOyj C MHkkB C M M M1 n b CM

Z

t

p.s/ 0

z s ds: C xs B

n

.kzs C xs kB / ds

p.s/ 0

Using the inequality (3.4) and the nondecreasing character of , we get Z n i h 1 2 Q Q b Q Q b b jz.t/j M M M1 n jOyj C MHkkB C M M M1 n p.s/ .Kn jz.s/j C ˛n / ds 0 Z t Kn b CM jz.s/j C ˛n ds: p.s/ 0 Then, we get i h Kn bM QM Q 1 n jOyj C MHkk b jz.t/j C ˛n ˛n C Kn M B Z n b 2M QM Q 1n CKn M p.s/ .Kn jz.s/j C ˛n / ds b CKn M

0

Z

t

p.s/ 0

Kn jz.s/j C ˛n

ds:

i h bM QM Q 1 n jOyj C MHkk b Set c11;n WD ˛n C Kn M B . By the nondecreasing character of and for < 1, we obtain Z n Kn Kn 2 Q Q b jz.t/j C ˛n c11;n C Kn M M M1 n jz.s/j C ˛n ds p.s/ 0 Z t Kn b C Kn M jz.s/j C ˛n ds: p.s/ 0 We consider the function defined by Kn .t/ WD sup jz.s/j C ˛n W 0 s t ; 0 t n: Let t 2 Œ0; t be such that .t/ D inequality, we have for t 2 Œ0; n b2M QM Q 1n .t/ c11;n C Kn M

Z

Kn jz.t /j C ˛n : If t 2 Œ0; n, by the previous

n

p.s/ 0

b ..s// ds C Kn M

Z

t

p.s/ 0

..s// ds:

3.4 Controllability on Semi-infinite Interval for Partial Evolution Equations

67

Then, we have b M bM QM Q 1 n C 1/ .t/ c11;n C Kn M.

Z

n

p.s/ 0

..s// ds:

Consequently, kzkn b b Q Q c11;n C Kn M.M M M1 n C 1/

.kzkn / kpkL1

1:

Then by the condition (3.13), there exists a constant M such that .t/ M . Since kzkn .t/, we have kzkn M : This shows that the set E is bounded, i.e., the statement .S2/ in Theorem 1.30 does not hold. Then the nonlinear alternative of Avramescu [32] implies that .S1/ holds, i.e., the operator F C G has a fixed-point z . Then y .t/ D z .t/ C x.t/, t 2 R is a fixed point of the operator N11 , which is a mild solution of the problem (3.11)–(3.12). Thus the evolution system (3.11)–(3.12) is controllable. t u

3.4.3 An Example As an application of Theorem 3.11, we present the following control problem 8 @v @2 v ˆ ˆ ˆ .t; / D a.t; / 2 .t; / C d./u.t/ ˆ ˆ @t ˆ Z 0 @ ˆ ˆ ˆ ˆ ˆ C P. /r.t; v.t C ; //d < 1

ˆ ˆ ˆ ˆ v.t; 0/ D v.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ : v.; / D v0 .; /

t0

2 Œ0; (3.14)

t0 1 < 0; 2 Œ0; ;

where a.t; / is a continuous function and is uniformly Hölder continuous in t ; P W .1; 0 ! R ; r W RC R ! R ; v0 W .1; 0Œ0; ! R and d W Œ0; ! E are continuous functions. u./ W RC ! E is a given control. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumption .3:1:1/ (see [112, 149]).

68

3 Partial Functional Evolution Equations with Infinite Delay

For the phase space B, we choose the well-known space BUC.R ; E/: the space of uniformly bounded continuous functions endowed with the following norm k'k D sup j'./j for 0

' 2 B:

If we put for ' 2 BUC.R ; E/ and 2 Œ0; y.t/./ D v.t; /; t 0; 2 Œ0; ; . /./ D v0 .; /; 1 < 0; 2 Œ0; ; and Z f .t; '/./ D

0

1

P. /r.t; '. /.//d; 1 < 0; 2 Œ0; :

Finally let C 2 B.R; E/ be defined as Cu.t/./ D d./u.t/; t 0; 2 Œ0; ; u 2 R; d./ 2 E: Then, problem (3.14) takes the abstract evolution form (3.11)–(3.12). Furthermore, more appropriate conditions on P and r ensure the controllability of mild solutions on .1; C1/ of the system (3.14) by Theorems 3.11 and 1.30.

3.5 Neutral Functional Evolution Equations 3.5.1 Introduction In this section, we investigate the following neutral functional differential evolution equation with infinite delay d Œy.t/ g.t; yt / D A.t/y.t/ C f .t; yt /; a.e. t 2 J D RC dt y0 D 2 B;

(3.15) (3.16)

where A./, f , and are as in problem (3.1)–(3.2) and g W J B ! E is a given function.

3.5 Neutral Functional Evolution Equations

69

3.5.2 Existence and Uniqueness of Mild Solution We give first the definition of the mild solution of our neutral functional evolution problem (3.15)–(3.16) before stating our main result and proving it. Definition 3.12. We say that the continuous function y./ W R ! E is a mild solution of (3.15)–(3.16) if y.t/ D .t/ for all t 2 .1; 0 and y satisfies the following integral equation Z t y.t/ D U.t; 0/Œ.0/ g.0; / C g.t; yt / C U.t; s/A.s/g.s; ys /ds 0 Z t C U.t; s/f .s; ys / ds; for each t 2 RC : 0

We will need to introduce the following assumptions which are assumed hereafter: .G1/

There exists a constant M 0 > 0 such that: kA1 .t/kB.E/ M 0

.G2/

There exists a constant 0 < L <

1 M 0 Kn

for all t 2 J: such that:

jA.t/ g.t; /j L .kkB C 1/ for all t 2 J and 2 B: .G3/

There exists a constant L > 0 such that: jA.s/ g.s; / A.s/ g.s; /j L .js sj C k kB /

for all s; s 2 J and ; 2 B. Consider the following space ˚ BC1 D y W R ! E W yjŒ0;T 2 C.Œ0; T; E/; y0 2 B ; where yjŒ0;T is the restriction of y to any real compact interval Œ0; T. For every n 2 N, we define in BC1 the semi-norms by: kykn WD sup f e

where Ln .t/ D

Z 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

b C ln .t/ and ln is the function from ln .s/ ds, ln .t/ D Kn MŒL

(3.1.3). Then BC1 is a Fréchet space with the family of semi-norms k kn2N . Let us fix 1 < 1.

> 0 and assume that M 0 L Kn C

70

3 Partial Functional Evolution Equations with Infinite Delay

Theorem 3.13. Suppose that hypotheses .G1/–.G3/ are satisfied and moreover Z

C1 c12;n

(3.1.1)–(3.1.3)

Z n b ds Kn M max.L; p.s//ds; > s C .s/ 1 M 0 LKn 0

and

assumptions

for each n > 0

(3.17)

with " c12;n D

b M 0 LKn M 1 M 0 LKn

C

Kn 1 M 0 LKn

M 0 LKn

b C Mn / 1 C C .Kn MH

1 M 0 LKn

!# kkB

h i b C 1/ C MLn b M 0 L.M ;

then the problem (3.15)–(3.16) has a unique mild solution. Proof. Let the operator N12 W BC1 ! BC1 be defined by: 8 .t/; if t 0I ˆ ˆ ˆ < Œ.0/ g.0; / C g.t;Zyt / .N12 y/.t/ D U.t; Z 0/ ˆ t t ˆ ˆ :C U.t; s/A.s/g.s; ys /ds C U.t; s/f .s; ys /ds; if t 0: 0

0

Then, fixed points of the operator N12 are mild solutions of the problem (3.15)– (3.16). For 2 B; we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 2 .1; 0I

: U.t; 0/ .0/;

if t 2 J:

Then x0 D . For each function z 2 BC1 , set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 3.12 if and only if z satisfies z0 D 0 and for t 2 J, we get z.t/ D g.t; zt C xt / U.t; 0/g.0; / Z t Z t C U.t; s/A.s/g.s; zs C xs /ds C U.t; s/f .s; zs C xs /ds: 0

0

Let B0C1 D fz 2 BC1 W z0 D 0g : Define the operator F W B0C1 ! B0C1 by: .Fz/.t/ D g.t; Z zt C xt / U.t; 0/g.0; / t

C 0

U.t; s/A.s/g.s; zs C xs /ds C

Z

t 0

U.t; s/f .s; zs C xs /ds:

3.5 Neutral Functional Evolution Equations

71

Obviously the operator N12 has a fixed point is equivalent to F has one, so it turns to prove that F has a fixed point. Let z 2 B0C1 be a possible F fixed point of the operator . Then, using .3:1:1/, .3:1:2/, .G1/ and .G2/, we have for each t 2 Œ0; n Z jz.t/j jg.t; zt C xt /j C jU.t; 0/g.0; /j C j Z

t

Cj 0

t 0

U.t; s/A.s/g.s; zs C xs /dsj

U.t; s/f .s; zs C xs /dsj

kA1 .t/kB.E/ kA.t/g.t; zt C xt /k C kU.t; 0/kB.E/ kA1 .0/kkA.0/ g.0; /k Z t Z t C kU.t; s/kB.E/ kA.s/g.s; zs C xs /kds C kU.t; s/kB.E/ jf .s; zs C xs /jds 0

0

b 0 L.kkB C 1/ M 0 L.kzt C xt kB C 1/ C MM Z t Z t b b CM L.kzs C xs kB C 1/ds C M p.s/ .kzs C xs kB /ds 0

0

b C 1/ C MLn b C MM b 0 LkkB M 0 Lkzt C xt kB C M 0 L.M Z t Z t b b CM Lkzs C xs kB ds C M p.s/ .kzs C xs kB /ds: 0

0

Using the inequality (3.4) and the nondecreasing character of

, we get

b C 1/ C MLn b C MM b 0 LkkB jz.t/j M 0 L.Kn jz.t/j C ˛n / C M 0 L.M Z t Z t b b CM L.Kn jz.s/j C ˛n /ds C M p.s/ .Kn jz.s/j C ˛n /ds 0

0

b 0 LkkB C M 0 L.M b C 1/ C MLn b M 0 LKn jz.t/j C M 0 L˛n C MM Z t Z t b b CM L.Kn jz.s/j C ˛n /ds C M p.s/ .Kn jz.s/j C ˛n /ds: 0

0

Then b 0 LkkB C M 0 L.M b C 1/ C MLn b .1 M 0 LKn /jz.t/j M 0 L˛n C MM Z t Z t b b CM L.Kn jz.s/j C ˛n /ds C M p.s/ .Kn jz.s/j C ˛n /ds: 0

0

72

3 Partial Functional Evolution Equations with Infinite Delay

Set c12;n WD ˛n C

Kn 1 M 0 LKn

h i b b b M 0 L ˛n C Mkk B C M 0 L.M C 1/ C MLn . Thus

Kn jz.t/j C ˛n c12;n C

Z

b Kn M

t

L.Kn jz.s/j C ˛n /ds 1 M 0 LKn 0 Z t b Kn M p.s/ .Kn jz.s/j C ˛n /ds: C 1 M 0 LKn 0

We consider the function defined by .t/ WD sup f Kn jz.s/j C ˛n W 0 s t g; 0 t < C1: Let t 2 Œ0; t be such that .t/ D Kn jz.t /j C ˛n : By the previous inequality, we have .t/ c12;n C

Z

b Kn M 1 M 0 LKn

t

Z L.s/ds C

0

t

0

p.s/ ..s//ds;

for t 2 Œ0; n:

Let us take the right-hand side of the above inequality as v.t/. Then, we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we have v.0/ D c12;n and v 0 .t/ D

b Kn M 1 M 0 LKn

ŒL.t/ C p.t/ ..t// a.e. t 2 Œ0; n:

Using the nondecreasing character of v 0 .t/

b Kn M 1 M 0 LKn

, we get

ŒLv.t/ C p.t/ .v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using the condition (3.17), we get Z

v.t/

c12;n

Z t b Kn M ds max.L; p.s//ds s C .s/ 1 M 0 LKn 0 Z n b Kn M max.L; p.s//ds 1 M 0 LKn 0 Z C1 ds : < c12;n s C .s/

3.5 Neutral Functional Evolution Equations

73

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kzkn .t/, we have kzkn n : Set Z D fz 2 B0C1 W supfjz.t/j 0 t ng n C 1 for all n 2 Ng: Clearly, Z is a closed subset of B0C1 . Now, we shall show that F W Z ! B0C1 is a contraction operator. Indeed, consider z; z 2 Z, thus for each t 2 Œ0; n and n 2 N and using .3:1:1/, .3:1:3/; G1 and .G3/, we get jF.z/.t/ F.z/.t/j jg.t; zt C xt / g.t; zt C xt /j Z t C kU.t; s/kB.E/ jA.s/Œg.s; zs C xs / g.s; zs C xs /jds Z

0

t

C 0

kU.t; s/kB.E/ jf .s; zs C xs / f .s; zs C xs /jds

kA1 .t/kB.E/ jA.t/g.t; zt C xt / A.t/g.t; zt C xt /j Z t C kU.t; s/kB.E/ jA.s/g.s; zs C xs / A.s/g.s; zs C xs /jds Z

0

t

C 0

kU.t; s/kB.E/ jf .s; zs C xs / f .s; zs C xs /jds Z

M 0 L kzt zt kB C Z

t

C 0

t

0

b kzs zs kB ds ML

b n .s/kzs zs kB ds: Ml

Using .A1 /, we obtain jF.z/.t/ F.z/.t/j M 0 L .K.t/ jz.t/ z.t/j C M.t/ kz0 z0 kB / Z t b C ln .s/.K.s/ jz.s/ z.s/j C M.s/ kz0 z0 kB /ds C MŒL 0

Z

M 0 L Kn jz.t/ z.t/j C Z M 0 L Kn jz.t/ z.t/j C

t 0 t 0

b C ln .s/ jz.s/ z.s/jds Kn MŒL ln .s/ jz.s/ z.s/jds

i h i h M 0 L Kn e Ln .t/ e Ln .t/ jz.t/ z.t/j Z th i h i C ln .s/ e Ln .s/ e Ln .s/ jz.s/ z.s/j ds 0

74

3 Partial Functional Evolution Equations with Infinite Delay

M 0 L Kn e

Ln .t/

kz zkn C

Z t"

e

Ln .s/

#0

0

ds kz zkn

1 M 0 L Kn e Ln .t/ kz zkn C e Ln .t/ kz zkn

1 e Ln .t/ kz zkn : M 0 L Kn C

Therefore,

1 kF.z/ F.z/kn M 0 L Kn C

kz zkn :

1 < 1, the operator F is a contraction for all n 2 N. So, for M 0 L Kn C

From the choice of Z there is no z 2 @Z n such that z D F.z/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.29 does not hold. We deduce that the operator F has a unique fixed point z . Then y .t/ D z .t/ C x.t/, t 2 R is a fixed point of the operator N12 , which is the unique mild solution of the problem (3.15)–(3.16). t u

3.5.3 An Example As an application we consider the following neutral functional evolution equation

Z t Z 8 @ ˆ ˆ z.t; x/ b.s t; u; x/ z.s; u/ du ds ˆ ˆ @t ˆ 1 0 ˆ ˆ ˆ @z @2 z ˆ ˆ < D a.t; x/ 2 .t; x/ C Q.t; z.t r; x/; .t r; x//; t 0; x 2 Œ0; @x @x ˆ ˆ ˆ ˆ z.t; 0/ D z.t; / D 0; t0 ˆ ˆ ˆ ˆ ˆ ˆ : z.t; x/ D ˚.t; x/; t 0; x 2 Œ0;

(3.18)

where r > 0, a.t; x/ is a continuous function and is uniformly Hölder continuous in t, Q W RC R R ! R and ˚ W B Œ0; ! R are continuous functions. Let y.t/.x/ D z.t; x/; t 2 Œ0; 1/; x 2 Œ0; ; ./.x/ D ˚.; x/; 0; x 2 Œ0; ; Z g.t; /.x/ D

t

1

Z

0

b.s t; u; x/.s; u/duds; x 2 Œ0;

3.6 Controllability on Finite Interval for Neutral Evolution Equations

75

and @ .; x/ ; 0; x 2 Œ0; : f .t; /.x/ D Q t; .; x/; @x

Consider E D L2 Œ0; and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D f w 2 E = w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g: Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .3:1:1/ and .G1/ (see [112, 149]). Here we consider that ' W .1; 0 ! E such that ' is Lebesgue measurable and h.s/j'.s/j2 is Lebesgue integrable on H where h W .1; 0 ! R is a positive integrable function. The norm is defined here by: Z k'k D j˚.0/j C

0

2

h.s/ j'.s/j ds

12

1

:

The function b is measurable on RC Œ0; Œ0; ; b.s; u; 0/ D b.s; u; / D 0; .s; u/ 2 RC Œ0; ; Z

Z

0

Z

t

1

0

b2 .s; u; x/ dsdudx < 1; h.s/

and sup N .t/ < 1; where t2RC

Z N .t/ D

0

Z

t

1

Z

0

2 @2 1 a.s; x/ 2 b.s; u; x/ dsdudx: h.s/ @x

Thus, under the above definitions of f , g, and A./, the system (3.18) can be represented by the abstract neutral functional evolution problem (3.15)–(3.16). Furthermore, more appropriate conditions on Q ensure the existence of the unique mild solution of (3.18) by Theorem 3.13 and 1.29.

3.6 Controllability on Finite Interval for Neutral Evolution Equations 3.6.1 Introduction In this section, we give sufficient conditions ensuring the controllability of mild solutions on a bounded interval JT WD Œ0; T for T > 0 for the neutral functional differential evolution equation with infinite delay of the form

76

3 Partial Functional Evolution Equations with Infinite Delay

d Œy.t/ g.t; yt / D A.t/y.t/ C Cu.t/ C f .t; yt /; a.e. t 2 JT D Œ0; T dt y0 D 2 B;

(3.19) (3.20)

where A./, f , u, C, and are as in problem (3.6)–(3.7) and g W JT B ! E is a given function.

3.6.2 Controllability of Mild Solutions Before stating and proving the controllability result, we give first the definition of mild solution of our evolution problem (3.19)–(3.20). Definition 3.14. We say that the continuous function y./ W .1; T ! E is a mild solution of (3.19)–(3.20) if y.t/ D .t/ for all t 2 .1; 0 and y satisfies the following integral equation Z t y.t/ D U.t; 0/Œ.0/ g.0; / C g.t; yt / C U.t; s/A.s/g.s; ys /ds 0 Z t Z t C U.t; s/Cu.s/ds C U.t; s/f .s; ys / ds; for each t 2 Œ0; T: 0

0

Definition 3.15. The neutral functional evolution problem (3.19)–(3.20) is said to be controllable on the interval Œ0; T if for every initial function 2 B and yQ 2 E there exists a control u 2 L2 .Œ0; T; E/ such that the mild solution y./ of (3.19)– (3.20) satisfies y.T/ D yQ : We consider the hypotheses (3.4.1)–(3.4.3) and we will need to introduce the following assumptions which are assumed hereafter: f .G1/

There exists a constant M 0 > 0 such that: kA1 .t/kB.E/ M 0

f .G2/

There exists a constant 0 < L <

1 M 0 KT

for all t 2 JT : such that:

jA.t/ g.t; /j L .kkB C 1/ for all t 2 JT and 2 B: f .G3/

There exists a constant L > 0 such that: jA.t/ g.s; / A.t/ g.s; /j L .js sj C k kB /

for all 0 t; s; s T and ; 2 B.

3.6 Controllability on Finite Interval for Neutral Evolution Equations

77

f .G4/ The function g is completely continuous and for any bounded set Q BT the set ft ! g.t; xt / W x 2 Qg is equi-continuous in C.Œ0; T; E/. Consider the following space BT D fy W .1; T ! E W yjJ 2 C.J; E/; y0 2 Bg ; where yjJ is the restriction of y to J. f Theorem 3.16. Suppose that hypotheses (3.4.1)–(3.4.3) and assumptions .G1/– f are satisfied and moreover there exists a constant M > 0 with .G4/

c13;T

M b Q Q b M M M1 T C 1 ŒM C C KT M 1 M 0 LKT

> 1;

(3.21)

.M / k kL1

where .t/ D max.L; p.t// and c13;T D c13 .; yQ ; T/ D " C

b KT M

i bM QM Q 1 T C 1/ h KT .M b C 1/ C MLT b M 0 L.M 1 M 0 LKT bM QM Q 1 T C 1/ C M QM Q 1 T.MH b C M 0 LMT / M 0 L.M

1 M 0 LKT

b C MT / 1 C C.KT MH bM QM Q 1T CKT M

1 C M 0 LKT 1 M 0 LKT

M 0 LKT

!#

1 M 0 LKT

kkB

jQyj

then the neutral functional evolution problem (3.19)–(3.20) is controllable on .1; T. Proof. Consider the operator N13 W BT ! BT defined by: 8 .t/; if t 2 .1; 0I ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ U.t; 0/ Œ.0/ g.0; / C g.t; yt / ˆ < Z t N13 .y/.t/ D ˆ C U.t; s/A.s/g.s; ys /ds ˆ ˆ ˆ 0 ˆ ˆ Z Z t t ˆ ˆ ˆ :C U.t; s/Cuy .s/ds C U.t; s/f .s; ys /ds; if t 2 JT : 0

0

78

3 Partial Functional Evolution Equations with Infinite Delay

Using assumption .3:4:3/; for arbitrary function y./; we define the control uy .t/ D W 1 ŒQy U.T; 0/ ..0/ g.0; // g.T; yT /

Z T Z T U.T; s/A.s/g.s; ys /ds U.T; s/f .s; ys /ds .t/: 0

0

Noting that

juy .t/j kW 1 k jQyj C kU.t; 0/kB.E/ j.0/j C kA1 .0/kjA.0/g.0; /j Z T kU.T; /kB.E/ jA. /g. ; y /jd

CkA1 .T/kjA.T/g.T; yT /j C 0

Z

T

C 0

kU.T; /kB.E/ jf . ; y /jd

h i b Q 1 jQyj C MHkk b M B C MM 0 L.kkB C 1/ C M 0 L.kyT kB C 1/ b Q 1 ML CM

Z

T 0

Q 1M b .ky kB C 1/d C M

Z

T 0

jf . ; y /jd :

From .3:4:2/, we get h i Q 1 jQyj C M.H b C 1/ C MLT b b C M 0 L/kkB C M 0 L.M juy .t/j M Z T Z T Q 1 ML Q 1M b b Q 1 M 0 LkyT kB C M ky kB d C M jf . ; y /j d

CM 0 0 h i Q 1 jQyj C M.H b C 1/ C MLT b b C M 0 L/kkB C M 0 L.M M Z T Z T Q 1 ML Q 1M b b Q 1 M 0 LkyT kB C M ky kB d C M p. / .ky kB /d : CM 0

0

It shall be shown that using this control the operator N13 has a fixed point y./. Then y./ is a mild solution of the neutral functional evolution system (3.19)–(3.20). For 2 B, we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 2 .1; 0I

: U.t; 0/ .0/;

if t 2 JT :

Then x0 D . For each function z 2 BT , set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 3.15 if and only if z satisfies z0 D 0 and for t 2 JT , we get Z z.t/ D g.t; zt C xt / U.t; 0/g.0; / C Z

t

C 0

Z U.t; s/Cuz .s/ds C

0

t

0

t

U.t; s/A.s/g.s; zs C xs /ds

U.t; s/f .s; zs C xs /ds:

3.6 Controllability on Finite Interval for Neutral Evolution Equations

79

Define the operator F W B0T ! B0T by: Z

t

F.z/.t/ D g.t; zt C xt / U.t; 0/ g.0; / C U.t; s/ A.s/ g.s; zs C xs / ds 0 Z t Z t C U.t; s/ C uz .s/ ds C U.t; s/ f .s; zs C xs / ds: 0

0

Obviously the operator N13 has a fixed point is equivalent to F has one, so it turns to prove that F has a fixed point. The proof will be given in several steps. We can show that the operator F is continuous and compact. For applying Theorem 1.27, we must check .S2/: i.e., it remains to show that the set ˚ E D z 2 B0T W z D F.z/ for some 0 < < 1 is bounded. f and .G2/, f we have for each t 2 Œ0; T Let z 2 E. By (3.4.1)–(3.4.3), .G1/ jz.t/j kA1 .t/k jA.t/g.t; zt C xt /j C kU.t; 0/kB.E/ kA1 .0/k jA.0/g.0; /j Z t Z t C kU.t; s/kB.E/ jA.s/g.s; zs C xs /j ds C kU.t; s/kB.E/ kCk juz .s/j ds Z

0

0

t

C

kU.t; s/kB.E/ jf .s; zs C xs /j ds

0

Z

b 0 L .kkB C 1/ C ML b M 0 L .kzt C xt kB C 1/ C MM bM Q CM

Z

t 0

Z

0

.kzs C xs kB C 1/ ds

b C 1/ C MLT b Q b C M 0 L/kkB C M 0 L.M M1 jQyj C M.H

b C M 0 LkzT C xT kB C ML b CM

t

t

Z

T 0

b kz C x kB d C M

Z 0

T

p. / .kz C x kB /d ds

.kzs C xs kB / ds

p.s/ 0

i h b C 1/ C MLT b bM QM Q 1 T C 1/ C M bM QM Q 1 T jQyj .M M 0 L.M h i b M 0 L.M bM QM Q 1 T C 1/ C M bM QM Q 1 TH kkB C M bM QM Q 1 M 0 LTkzT C xT kB CM b CM 0 Lkzt C xt kB C ML b2M QM Q 1T CM

Z 0

T

Z 0

t

b 2M QM Q 1 LT kzs C xs kB ds C M

b p.s/ .kzs C xs kB / ds C M

Z 0

t

Z

T 0

kzs C xs kB ds

p.s/ .kzs C xs kB / ds:

80

3 Partial Functional Evolution Equations with Infinite Delay

Noting that we have kzT C xT kB KT jQyj C MT kkB and using (3.4) and by the nondecreasing character of , we obtain i h

b C 1/ C MLT b bM QM Q 1 T C 1/ C M bM QM Q 1 T 1 C M 0 LKT jQyj .M jz.t/j M 0 L.M h i b M 0 L.M bM QM Q 1 T C 1/ C M QM Q 1 T MH b C M 0 LMT kkB CM CM 0 L .KT jz.t/j C ˛T / Z t Z 2 Q Q b b .KT jz.s/j C ˛T / ds C M M M1 LT CML 0

b 2M QM Q 1T CM b CM

Z 0

t

Z

T 0

T 0

.KT jz.s/j C ˛T / ds

p.s/ .KT jz.s/j C ˛T / ds

p.s/ .KT jz.s/j C ˛T / ds:

Then h i

b C 1/ C MLT b bM QM Q 1 T C 1/ 1 M 0 LKT jz.t/j M 0 L.M .M

bM QM Q 1 T 1 C M 0 LKT jQyj C M 0 L˛T CM h b M 0 L.M bM QM Q 1 T C 1/ CM i QM Q 1 T MH b C M 0 LMT kkB CM Z t b .KT jz.s/j C ˛T / ds CML 0

b 2M QM Q 1 LT CM b 2M QM Q 1T CM b CM

Z 0

t

Z 0

Z

T 0

T

.KT jz.s/j C ˛T / ds

p.s/ .KT jz.s/j C ˛T / ds

p.s/ .KT jz.s/j C ˛T / ds:

Set c13;T WD ˛T C

KT

1 M 0 LKT nh i

b C 1/ C MLT b bM QM Q 1 T C 1/ C M bM QM Q 1 T 1 C M 0 LKT jQyj M 0 L.M .M i o h b C M 0 LMT kkB b M 0 L.M bM QM Q 1 T C 1/ C M QM Q 1 T MH C M 0 L˛T C M

3.6 Controllability on Finite Interval for Neutral Evolution Equations

81

thus b KT M

KT jz.t/j C ˛T c13;T C

1 M 0 LKT Z t Z bM QM Q 1 LT .KT jz.s/j C ˛T / ds C M L 0

bM QM Q 1T CM

Z

T

0

T 0

Z p.s/ .KT jz.s/j C ˛T / ds C

0

t

.KT jz.s/j C ˛T / ds

p.s/ .KT jz.s/j C ˛T / ds :

We consider the function defined by .t/ WD sup f KT jz.s/j C ˛T W 0 s t g; 0 t T: Let t 2 Œ0; t be such that .t/ D KT jz.t /j C ˛T : If t 2 Œ0; T, by the previous inequality, we have b KT M

.t/ c13;T C

Z t Z bM QM Q 1 LT L .s/ ds C M

T

.s/ ds 1 M 0 LKT 0 0

Z T Z t bM QM Q 1T CM p.s/ ..s// ds C p.s/ ..s// ds : 0

0

Then, we have b .t/ c13;T C KT M

Z T bM QM Q 1T C 1 Z T M L .s/ ds C p.s/ ..s// ds : 1 M 0 LKT 0 0

Set .t/ WD max.L; p.t// for t 2 Œ0; T .t/ c13;T

bM QM Q 1T C 1 Z T M b C KT M

.s/ Œ.s/ C 1 M 0 LKT 0

..s// ds:

Consequently,

c13;T

kzkT b Q Q b M M M1 T C 1 ŒkzkT C C KT M 1 M 0 LKT

1: .kzkT / k kL1

Then by (3.21), there exists a constant M such that kzkT ¤ M . Set ZQ D f z 2 B0T W kzkT M C 1 g:

82

3 Partial Functional Evolution Equations with Infinite Delay

Clearly, ZQ is a closed subset of B0T . From the choice of ZQ there is no z 2 @ZQ such that z D F.z/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.27 does not hold. As a consequence of the nonlinear alternative of Leray–Schauder type [128], we deduce that .S1/ holds: i.e., the operator F has a fixed point z . Then y .t/ D z .t/ C x.t/, t 2 .1; T is a fixed point of the operator N13 , which is a mild solution of the problem (3.19)–(3.20). Thus the evolution system (3.19)–(3.20) is controllable on .1; T. t u

3.6.3 An Example As an application of Theorem 3.16, we present the following control problem 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <

Z 0 @ v.t; / T. /w.t; v.t C ; //d @t 1 2 @v D a.t; / 2 .t; / C d./u.t/ Z @ C

0

P. /r.t; v.t C ; //d ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ v.t; 0/ D v.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ : v.; / D v0 .; /

t 2 Œ0; T

2 Œ0;

t 2 Œ0; T 1 < 0; 2 Œ0; ; (3.22)

where a.t; / is a continuous function and is uniformly Hölder continuous in t ; T; P W .1; 0 ! R ; w; r W Œ0; T R ! R ; v0 W .1; 0 Œ0; ! R and d W Œ0; ! E are continuous functions. u./ W Œ0; T ! E is a given control. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .3:6:1/ f (see [112, 149]). and .G1/ For the phase space B, we choose the well-known space BUC.R ; E/: the space of uniformly bounded continuous functions endowed with the following norm k'k D sup j'./j for 0

' 2 B:

If we put for ' 2 BUC.R ; E/ and 2 Œ0; y.t/./ D v.t; /; t 2 Œ0; T; 2 Œ0; ;

3.6 Controllability on Finite Interval for Neutral Evolution Equations

83

. /./ D v0 .; /; 1 < 0; 2 Œ0; ; Z g.t; '/./ D

0

1

T. /w.t; '. /.//d; 1 < 0; 2 Œ0; ;

and Z f .t; '/./ D

0

1

P. /r.t; '. /.//d; 1 < 0; 2 Œ0;

Finally let C 2 B.R; E/ be defined as Cu.t/./ D d./u.t/; t 2 Œ0; T; 2 Œ0; ; u 2 R; d./ 2 E: Then, problem (3.22) takes the abstract neutral functional evolution form (3.19)– (3.20). In order to show the controllability of mild solutions of system (3.22), we suppose the following assumptions: – w is Lipschitz with respect to its second argument. Let lip.w/ denotes the Lipschitz constant of w. – There exist a function p 2 L1 .JT ; RC / and a nondecreasing continuous function W Œ0; 1/ ! Œ0; 1/ such that jr.t; u/j p.t/ .juj/; for t 2 JT ; and u 2 R: – T and P are integrable on .1; 0: By the dominated convergence theorem, one can show that f is a continuous function from B to E. Moreover the mapping g is Lipschitz continuous in its second argument, in fact, we have Z jg.t; '1 / g.t; '2 /j M 0 L lip.w/

0 1

jT. /j d j'1 '2 j ; for '1 ; '2 2 B:

On the other hand, we have for ' 2 B and 2 Œ0; Z jf .t; '/./j Since the function

0

1

jp.t/P. /j .j.'. //./j/d:

is nondecreasing, it follows that Z

jf .t; '/j p.t/

0 1

jP. /j d .j'j/; for ' 2 B:

Proposition 3.17. Under the above assumptions, if we assume that condition (3.21) in Theorem 3.16 is true, ' 2 B, then the problem (3.22) is controllable on .1; T.

84

3 Partial Functional Evolution Equations with Infinite Delay

3.7 Controllability on Semi-infinite Interval for Neutral Evolution Equations 3.7.1 Introduction We investigate in this section the controllability of mild solutions on the semiinfinite interval J D RC for the following neutral functional evolution equations with infinite delay d Œy.t/ g.t; yt / D A.t/y.t/ C Cu.t/ C f .t; yt /; a.e. t 2 J D RC dt y0 D 2 .1; 0;

(3.23) (3.24)

where A./, f , u, C, and are as in problem (3.11)–(3.12) (Sect. 3.4) and g W J B ! E is a given function. Here we are interested to give an application of (3.15) in [33] to control theory on the semi-infinite interval J D RC for the partial functional evolution equations (3.23)–(3.24) by Theorem 1.30 due to Avramescu in [32] for sum of compact and contraction operators in Fréchet spaces, combined with the semigroup theory [16, 168].

3.7.2 Controllability of Mild Solutions Before stating and proving the controllability result, we give first the definition of mild solution of the evolution problem (3.23)–(3.24). Definition 3.18. We say that the function y./ W R ! E is a mild solution of (3.23)– (3.24) if y.t/ D .t/ for all t 2 .1; 0 and y satisfies the following integral equation Z t y.t/ D U.t; 0/Œ.0/ g.0; / C g.t; yt / C U.t; s/A.s/g.s; ys /ds 0 Z t Z t C U.t; s/Cu.s/ds C U.t; s/f .s; ys / ds; for each t 2 RC : 0

0

Definition 3.19. The neutral functional evolution problem (3.23)–(3.24) is said to be controllable if for every initial function 2 B and yO 2 E, there is some control u 2 L2 .Œ0; n; E/ such that the mild solution y./ of (3.23)–(3.24) satisfies y.n/ D yO . We consider the hypotheses (3.3.1)–(3.1.3) given in Sect. 3.2.2 and the assumption .3:9:1/ of Sect. 3.4.2 and we will need to introduce the following one which is assumed hereafter:

3.7 Controllability on Semi-infinite Interval for Neutral Evolution Equations

85

.G4/ The function g is completely continuous and for any bounded set Q BT the set ft ! g.t; xt / W x 2 Qg is equi-continuous in C.RC ; E/. Consider the following space ˚ BC1 D y W R ! E W yjŒ0;T 2 C.Œ0; T; E/; y0 2 B ; where yjŒ0;T is the restriction of y to any real compact interval Œ0; T. For every n 2 N, we define in BC1 the semi-norms by: kykn WD sup f e

where

Ln .t/

Z

t

D 0

Ln .t/

jy.t/j W t 2 Œ0; n g

b C ln .t/, and ln is the function from ln .s/ ds, ln .t/ D Kn MŒL

.3:1:3/: Then BC1 is a Fréchet space with the family of semi-norms k kn2N . Let us fix 1 < 1.

> 0 and assume that M 0 L Kn C

Theorem 3.20. Suppose that hypotheses (3.1.1)–(3.1.3), (3.9.1) and the assumptions .G1/–.G3/ are satisfied and moreover there exists a constant M > 0 with M b c14;n C Kn M

bM QM Q 1n C 1 M 1 M 0 LKn

> 1; ŒM C

(3.25)

.M /k kL1

with .t/ D max.L; p.t// and c14;n D c14 .˚; yO ; n/ D " C

i bM QM Q 1 n C 1/ h Kn .M b C 1/ C MLn b M 0 L.M 1 M 0 LKn

b Kn M 1 M 0 LKn

bM QM Q 1 n C 1/ C M QM Q 1 n.MH b C M 0 LMn / M 0 L.M

b C Mn / 1 C C .Kn MH bM QM Q 1n CKn M

1 C M 0 LKn 1 M 0 LKn

M 0 LKn 1 M 0 LKn

!# kkB

jOyj

Then the neutral functional evolution problem (3.23)–(3.24) is controllable on R.

86

3 Partial Functional Evolution Equations with Infinite Delay

Proof. Consider the operator N14 W BC1 ! BC1 defined by: 8 .t/; if t 0I ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ U.t; 0/ Œ.0/ g.0; / C g.t; yt / ˆ < Z t .N14 y/.t/ D ˆ C U.t; s/A.s/g.s; ys /ds ˆ ˆ ˆ 0 ˆ ˆ Z Z t t ˆ ˆ ˆ : C U.t; s/Cuy .s/ds C U.t; s/f .s; ys /ds; if t 0: 0

0

Using assumption .3:9:1/; for arbitrary function y./, we define the control Z uy .t/ D W 1 yO U.n; 0/ ..0/ g.0; // g.n; yn / Z

n

0

0

n

U.n; s/A.s/g.s; ys /ds

U.n; s/f .s; ys /ds .t/:

Noting that

juy .t/j kW 1 k jOyj C kU.t; 0/kB.E/ j.0/j C kA1 .0/kjA.0/g.0; /j Z n CkA1 .n/kjA.n/g.n; yn /j C kU.n; /kB.E/ jA. /g. ; y /jd

0

Z

n

C 0

kU.n; /kB.E/ jf . ; y /jd

h i b Q 1 jOyj C MHkk b M B C MM 0 L.kkB C 1/ C M 0 L.kyn kB C 1/ Z n Z n Q b b Q .ky kB C 1/d C M1 M jf . ; y /jd : CM1 ML 0

0

Applying .3:1:2/, we get h i

Q 1 jOyj C M b C 1/ C MLn b b H C M 0 L kkB C M 0 L.M juy .t/j M Z n Z n Q 1 ML Q 1M b b Q 1 M 0 Lkyn kB C M ky kB d C M jf . ; y /j d

CM 0

0

h i

b C 1/ C MLn b Q 1 jOyj C M b H C M 0 L kkB C M 0 L.M M Z n Z n Q 1 ML Q 1M b b Q 1 M 0 Lkyn kB C M ky kB d C M p. / .ky kB /d : CM 0

0

3.7 Controllability on Semi-infinite Interval for Neutral Evolution Equations

87

Using this control the operator N14 has a fixed point y./. Then y./ is a mild solution of the neutral functional evolution system (3.23)–(3.24). For 2 B, we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 0I

: U.t; 0/ .0/;

if t 0:

Then x0 D . For each function z 2 BC1 , set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 3.19 if and only if z satisfies z0 D 0 and for t 0, we get Z z.t/ D g.t; zt C xt / U.t; 0/ g.0; / C Z

t

C 0

Z U.t; s/ C uzCx .s/ ds C

t 0

t

0

U.t; s/ A.s/ g.s; zs C xs /ds

U.t; s/ f .s; zs C xs / ds:

Let B0C1 D fz 2 BC1 W z0 D 0g : Define the operators F; G W B0C1 ! B0C1 by: Z F.z/.t/ D g.t; zt C xt / U.t; 0/ g.0; / C Z t C U.t; s/ C uzCx .s/ ds:

0

t

U.t; s/ A.s/ g.s; zs C xs / ds

0

and Z G.z/.t/ D 0

t

U.t; s/ f .s; zs C xs / ds:

Obviously the operator N14 has a fixed point is equivalent to the operator sum F C G has one, so it turns to prove that F C G has a fixed point. The proof will be given in several steps. We can show as in above sections that the operator F is continuous and compact and we have shown in Sect. 3.4.2 (Step 4) that the operator G is a contraction. For applying Avramescu nonlinear alternative, we must check .S2/ in Theorem 1.30: i.e., it remains to show that the following set o n z for some 0 < < 1 E D z 2 B0C1 W z D F.z/ C G is bounded.

88

3 Partial Functional Evolution Equations with Infinite Delay

Let z 2 E. Then, by (3.1.1)–(3.1.3), (3.9.1), .G1/, and .G2/, we have for each t 2 Œ0; n jz.t/j

kA1 .t/k jA.t/g.t; zt C xt /j C kU.t; 0/kB.E/ kA1 .0/k jA.0/g.0; /j Z

t

C

kU.t; s/kB.E/ jA.s/g.s; zs C xs /j ds

0

Z

t

C

kU.t; s/kB.E/ kCk juzCx .s/j ds

0

Z

C

ˇ z ˇ ˇ ˇ s kU.t; s/kB.E/ ˇf s; C xs ˇ ds

t 0

b 0 L .kkB C 1/ M 0 L .kzt C xt kB C 1/ C MM b CML

Z

t

0

.kzs C xs kB C 1/ ds

Z

bM Q CM

t

0

b H C M 0 L kkB C M 0 L.M b C 1/ C MLn b Q 1 jOyj C M M

b CM 0 Lkzn C xn kB C ML Z

b CM

n 0

b CM

Z

Z 0

n

kz C x kB d

p. /

.kz C x kB /d

p.s/

z s ds : C xs B

t

0

ds

Then jz.t/j

h i b C 1/ C MLn b bM QM Q 1 n C 1/ C M bM QM Q 1 n jOyj M 0 L.M .M h i b M 0 L.M bM QM Q 1 n C 1/ C M bM QM Q 1 nH kkB CM bM QM Q 1 M 0 Lnkzn C xn kB C M 0 Lkzt C xt kB C ML b CM Z t Z n b 2M QM Q 1 Ln kzs C xs kB ds C M kzs C xs kB ds 0

b 2M QM Q 1n CM b CM

Z

t

p.s/ 0

0

Z

n

p.s/ 0

.kzs C xs kB / ds

z s ds : C x s B

3.7 Controllability on Semi-infinite Interval for Neutral Evolution Equations

89

Noting that we have kzn C xn kB Kn jOyj C Mn kkB and using the inequality (3.4), then by the nondecreasing character of , we obtain jz.t/j

nh i

QM Q 1 n C 1/ C b QM Q 1 n 1 C Kn M 0 L jOyj M 0 L.b M C 1/ C b M Ln .b MM MM i h QM Q 1 n C 1/ C M QM Q 1n b Cb M M 0 L.b MM M H C M 0 LMn kkB CM 0 L .Kn jz.t/j C ˛n / Z t Z QM Q 1n .Kn jz.s/j C ˛n / ds C b Cb ML MM 0

Z QM Q 1n b CM b MM

n 0

.Kn jz.s/j C ˛n / ds

n 0

Z

p.s/ .Kn jz.s/j C ˛n / ds C

t 0

p.s/

Kn jz.s/j C ˛n ds

M 0 LKn jz.t/j C M 0 L˛n nh i

QM Q 1 n C 1/ C b QM Q 1 n 1 C Kn M 0 L jOyj C M 0 L.b M C 1/ C b M Ln .b MM MM i h QM Q 1 n C 1/ C M QM Q 1n b Cb M M 0 L.b MM M H C M 0 LMn kkB Cb ML

Z

t 0

QM Q 1n .Kn jz.s/j C ˛n / ds C b MM

QM Q 1n b b CM MM

Z

n 0

Z

n 0

.Kn jz.s/j C ˛n / ds Z

p.s/ .Kn jz.s/j C ˛n / ds C

t 0

p.s/

Kn jz.s/j C ˛n ds :

Then,

jz.t/j 1 M 0 LKn h i

b C 1/ C MLn b bM QM Q 1 n C 1/ C M bM QM Q 1 n 1 C Kn M 0 L jOyj .M M 0 L˛n C M 0 L.M h i b M 0 L.M b C M 0 LMn kkB bM QM Q 1 n C 1/ C M QM Q 1 n MH CM

Z t Z n bM QM Q 1n b .Kn jz.s/j C ˛n / ds C M .Kn jz.s/j C ˛n / ds C ML 0

Z b b Q Q C M M M M1 n

0

n 0

p.s/ .Kn jz.s/j C ˛n / ds C

Z

t

p.s/ 0

Kn jz.s/j C ˛n ds :

Set c14;n WD ˛n C

Kn

1 M 0 LKn ˚ M 0 L˛n h i

b C 1/ C MLn b bM QM Q 1 n C 1/ C M bM QM Q 1 n 1 C Kn M 0 L jOyj C M 0 L.M .M h i o b M 0 L.M b C M 0 LMn kkB : bM QM Q 1 n C 1/ C M QM Q 1 n MH CM

90

3 Partial Functional Evolution Equations with Infinite Delay

Thus Kn jz.t/j Kn C ˛n c14;n C 1 M 0 LKn Z t Z bM QM Q 1n b .Kn jz.s/j C ˛n / ds C M ML 0

Z bM QM Q 1n b M CM

n 0

n 0

.Kn jz.s/j C ˛n / ds

p.s/ .Kn jz.s/j C ˛n / ds C

By the nondecreasing character of

Z

t

p.s/ 0

Kn jz.s/j C ˛n ds :

, we get for < 1

b Kn jz.t/j Kn M C ˛n c14;n C 1 M 0 LKn

Z t Z n Kn jz.s/j Kn bM QM Q 1n C ˛n ds C M jz.s/j C ˛n ds L 0 0 Z t Z n Kn jz.s/j Kn jz.s/j bM QM Q 1n C M p.s/ p.s/ C ˛n ds C C ˛n ds : 0 0 We consider the function defined by .t/ WD sup

Kn jz.s/j C ˛n W 0 s t

Let t 2 Œ0; t be such that .t/ D inequality, we have for t 2 Œ0; n .t/ c14;n C

b Kn M

; 0 t < C1:

Kn jz.t /j C ˛n : If t 2 Œ0; n, by the previous Z t Z bM QM Q 1n L .s/ds C M

n

.s/ds

1 M 0 LKn 0 0

Z n Z t bM QM Q 1n C M p.s/ ..s//ds C p.s/ ..s//ds : 0

0

Then, we have b .t/ c14;n C Kn M

Z n bM QM Q 1n C 1 Z n M L .s/ds C p.s/ ..s//ds : 1 M 0 LKn 0 0

Set .t/ WD max.L; p.t// for t 2 Œ0; n; then bM QM Q 1n C 1 Z n M b .t/ c14;n C Kn M

.s/ Œ.s/ C 1 M 0 LKn 0

..s// ds:

3.7 Controllability on Semi-infinite Interval for Neutral Evolution Equations

91

Consequently, kzkn bQ Q b M M M1 n C 1 Œkzkn C c14;n C Kn M 1 M 0 LKn

1: .kzkn /k kL1

Then by the condition (3.25), there exists a constant M such that .t/ M . Since kzkn .t/, we have kzkn M : This shows that the set E is bounded, i.e., the statement .S2/ in Theorem 1.30 does not hold. Then the nonlinear alternative of Avramescu [32] implies that .S1/ holds: i.e., the operator F C G has a fixed-point z . Then y .t/ D z .t/ C x.t/, t 2 R is a fixed point of the operator N14 , which is a mild solution of the problem (3.23)–(3.24). Thus the evolution system (3.23)–(3.24) is controllable on R: t u

3.7.3 An Example To illustrate the previous results, we consider the following model 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <

Z 0 @ v.t; / T. /w.t; v.t C ; //d @t 1 @2 v D a.t; / 2 .t; / C d./u.t/ @ Z C

0

P. /r.t; v.t C ; //d ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ v.t; 0/ D v.t; C1/ D 0 ˆ ˆ ˆ ˆ ˆ ˆ : v.; / D v0 .; /

t0 0

(3.26)

t0 0; 0;

where a.t; / is a continuous function and is uniformly Hölder continuous in t ; T; P W .1; 0 ! R ; w; r W RC R ! R ; v0 W .1; 0 RC ! R and d W Œ0; ! E are continuous functions. u./ W RC ! E is a given control. Consider E D L2 .RC ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w.C1/ D 0 g

Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .3:1:1/ and .G1/ (see [112, 149]). For the phase space B, we choose the well-known space BUC.R ; E/: the space of uniformly bounded continuous functions endowed with the following norm k'k D sup j'./j for 0

' 2 B:

92

3 Partial Functional Evolution Equations with Infinite Delay

If we put for ' 2 BUC.R ; E/ and 0 y.t/./ D v.t; /; t 0; 0; . /./ D v0 .; /; 0; 0; Z g.t; '/./ D

0

1

T. /w.t; '. /.//d; 0; 0;

and Z f .t; '/./ D

0

1

P. /r.t; '. /.//d; 0; 0:

Finally let C 2 L.R; E/ be defined as Cu.t/./ D d./u.t/; t 0; 0; u 2 R; d./ 2 E: Then, problem (3.26) takes the abstract neutral functional evolution form (3.23)–(3.24). Furthermore, more appropriate conditions on T, w, P, and r ensure the controllability of mild solutions on .1; C1/ of the system (3.26) by Theorems 3.20 and 1.30.

3.8 Notes and Remarks The results of Chap. 3 are taken from [15, 36]. Other results may be found in [108, 141, 145].

Chapter 4

Perturbed Partial Functional Evolution Equations

4.1 Introduction Perturbed partial functional and neutral functional evolution equations with finite and infinite delay are studied in this chapter on the semi-infinite interval RC :

4.2 Perturbed Partial Functional Evolution Equations with Finite Delay 4.2.1 Introduction In this section, we give the existence of mild solutions for the following perturbed partial functional evolution equations with finite delay y0 .t/ D A.t/y.t/ C f .t; yt / C h.t; yt /; a.e. t 2 J D RC

(4.1)

y.t/ D '.t/; t 2 H;

(4.2)

where r > 0, f ; h W J C.H; E/ ! E and ' 2 C.H; E/ are given functions and fA.t/gt0 is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators fU.t; s/g.t;s/2JJ for 0 s t < C1. Here we are interested to give the existence of mild solutions for the partial functional perturbed evolution equations (4.1)–(4.2). This result is an extension of the problem (2.1) in [33] when the delay is finite.

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_4

93

94

4 Perturbed Partial Functional Evolution Equations

4.2.2 Existence of Mild Solutions Before stating and proving the main result, we give first the definition of mild solution of our perturbed evolution problem (4.1)–(4.2). Definition 4.1. We say that the continuous function y./ W R ! E is a mild solution of (4.1)–(4.2) if y.t/ D '.t/ for all t 2 H and y satisfies the following integral equation Z t y.t/ D U.t; 0/ '.0/ C U.t; s/ Œf .s; ys / C h.s; ys / ds; for each t 2 RC : 0

We introduce the following hypotheses which are assumed hereafter: O 1 such that: (4.1.1) U.t; s/ is compact for t s > 0 and there exists a constant M O for every .t; s/ 2 : kU.t; s/kB.E/ M 1 (4.1.2) There exists a function p 2 Lloc .J; RC / and a continuous nondecreasing function W RC ! .0; 1/ such that:

jf .t; u/j p.t/

.kuk/ for a.e. t 2 J and each u 2 C.H; E/:

(4.1.3) There exists a function 2 L1 .J; RC / where kkL1 <

1 such that: O M

jh.t; u/ h.t; v/j .t/ku vk for a.e. t 2 J and all u; v 2 C.H; E/: For every n 2 N, we define in C.Œr; C1/; E/ the semi-norms by: kykn WD sup f e

where Ln .t/ D

Z 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

O ln .s/ ds; ln .t/ D M.t/.

Then C.Œr; C1/; E/ is a Fréchet space with the family of semi-norms fk kn gn2N . In what follows we will choose > 1. Theorem 4.2. Suppose that hypotheses (4.1.1)–(4.1.3) are satisfied and moreover Z

C1

c15;n

ds O >M s C .s/

Z 0

n

max.p.s/; .s// ds; for each n > 0

with c15;n

O O D Mk'k CM

Z 0

n

jh.s; 0/jds;

then the problem (4.1)–(4.2) has a mild solution.

(4.3)

4.2 Perturbed Partial Functional Evolution Equations with Finite Delay

95

Proof. Transform the problem (4.1)–(4.2) into a fixed point problem. Consider the operator N15 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by: 8 '.t/; ˆ ˆ ˆ ˆ ˆ Z t < U.t; s/ f .s; ys / ds N15 .y/.t/ D U.t; 0/ '.0/ C ˆ 0 Z t ˆ ˆ ˆ ˆ :C U.t; s/ h.s; ys / ds; 0

if t 2 H;

if t 0:

Clearly, the fixed points of the operator N15 are mild solutions of the problem (4.1)–(4.2). Define the operators F; G W C.Œr; C1/; E/ ! C.Œr; C1/; E/ by

.Fy/.t/ D

8 ˆ < '.t/;

if t 0;

ˆ : U.t; 0/ '.0/ C

Z

t 0

U.t; s/ f .s; ys / ds;

if t 0:

and Z .Gy/.t/ D

0

t

U.t; s/ h.s; ys / ds:

Obviously the operator N15 has a fixed point is equivalent to F C G has one, so it turns to prove that F C G has a fixed point. The proof will be given in several steps. Let us first show that the operator F is continuous and compact. Step 1: F is continuous. Let .yk /k be a sequence in C.Œr; C1/; E/ such that yk ! y in C.Œr; C1/; E/. Then Z t kU.t; s/kB.E/ jf .s; yks / f .s; ys /j ds jF.yk /.t/ F.y/.t/j 0

O M

Z 0

t

jf .s; yks / f .s; ys /j ds ! 0 as k ! C1:

Step 2: F maps bounded sets of C.Œr; C1/; E/ into bounded sets. It is enough to show that for any d > 0, there exists a positive constant ` such that for each y 2 Bd D fy 2 C.Œr; C1/; E/ W kyk1 dg we have F.y/ 2 B` . Let y 2 Bd . By .4:1:1/, .4:1:2/ and the nondecreasing character of , we have for each t 2 J

96

4 Perturbed Partial Functional Evolution Equations

Z jF.y/.t/j kU.t; s/kB.E/ j'.0/j C Z

O O Mk'k CM

t

O O Mk'k CM

Z

.d/

kU.t; s/kB.E/ jf .s; ys /j ds

.kys k/ ds

p.s/ 0

0

t

t

p.s/ds: 0

O O .d/ kpkL1 WD `: Hence F.Bd / B` . Then we have ŒjF.y/k1 Mk'k CM Step 3: F maps bounded sets into equi-continuous sets of C.Œr; C1/; E/. We consider Bd as in Step 2 and we show that F.Bd / is equi-continuous. Let

1 ; 2 2 J with 2 > 1 and y 2 Bd . Then, by .4:1:1/, .4:1:2/ and the nondecreasing character of , we get jF.y/. 2 / F.y/. 1 /j jU. 2 ; 0/ U. 1 ; 0/j j'.0/j ˇ ˇZ 1 ˇ ˇ ˇ ŒU. 2 ; s/ U. 1 ; s/ f .s; ys / dsˇˇ Cˇ ˇZ ˇ C ˇˇ

0

2

1

ˇ ˇ U. 2 ; s/ jf .s; ys /j dsˇˇ

kU. 2 ; 0/ U. 1 ; 0/kB.E/ k'k Z 1 C kU. 2 ; s/ U. 1 ; s/kB.E/ p.s/ Z

0

2

C

1

kU. 2 ; s/kB.E/ p.s/

.kys k/ ds

.kys k/ds

kU. 2 ; 0/ U. 1 ; 0/kB.E/ k'k Z 1 C .d/ kU. 2 ; s/ U. 1 ; s/kB.E/ p.s/ ds 0

O CM

.d/

Z

2

p.s/ ds:

1

The right-hand of the above inequality tends to zero as 2 1 ! 0, since U.t; s/ is a strongly continuous operator and the compactness of U.t; s/ for t > s implies the continuity in the uniform operator topology (see [20, 168]). As a consequence of Steps 1–3 together with the Arzelá–Ascoli theorem it suffices to show that the operator F maps Bd into a precompact set in E.

4.2 Perturbed Partial Functional Evolution Equations with Finite Delay

97

Let t 2 J be fixed and let be a real number satisfying 0 < < t. For y 2 Bd we define Z t U.t; s/ f .s; ys / ds F .y/.t/ D U.t; 0/ '.0/ C 0

Z

D U.t; 0/ '.0/ C U.t; t /

t 0

U.t ; s/ f .s; ys /ds:

Since U.t; s/ is a compact operator, the set Z .t/ D fF .y/.t/ W y 2 Bd g is precompact in E for every , 0 < < t. Moreover by the nondecreasing character of , we get Z

t

jF.y/.t/ F .y/.t/j

kU.t; s/kB.E/ jf .s; ys /jds

t

O M

Z .d/

t

p.s/ds: t

Therefore the set Z.t/ D fF.y/.t/ W y 2 Bd g is totally bounded. Hence the set fF.y/.t/ W y 2 Bd g is relatively compact E. So we deduce from Steps 1, 2, and 3 that F is a compact operator. Step 4: We can show that the operator G is a contraction for all n 2 N as in the proof of Theorem 2.2). Step 5: For applying Theorem 1.30, we must check .S2/: i.e., it remains to show that the set n y o E D y 2 C.Œr; C1/; E/ W y D F.y/ C G for some 0 < < 1 is bounded. Let y 2 E. By (4.1.1)–(4.1.3), we have for each t 2 Œ0; n Z jy.t/j U.t; 0/'.0/ C Z

0

t

kU.t; s/kB.E/ jf .s; ys /jds

ˇ ˇ y ˇ ˇ s h.s; 0/ C h.s; 0/ˇ ds kU.t; s/kB.E/ ˇh s; 0 Z t O O CM Mk'k p.s/ .kys k/ ds C

O CM

t

Z 0

0

t

Z t y s O .s/ ds C M jh.s; 0/jds : 0

98

4 Perturbed Partial Functional Evolution Equations

gives with the fact that 0 < < 1

The nondecreasing character of

Z n jy.t/j O O jh.s; 0/jds Mk'k CM 0 Z t Z t y s O O CM .s/ ds C M p.s/ 0 0 Set c15;n

O O WD Mk'k CM

Z

n

0

y s ds

jh.s; 0/jds: Thus

jy.t/j O c15;n C M

Z 0

t

Z t y s O .s/ ds C M p.s/ 0

y s ds:

Consider the function defined by jy.s/j .t/ WD sup W 0 s t ; 0 t < C1: jy.t /j : By the previous inequality, we have Z t Z t O O .t/ c15;n C M .s/.s/ds C M p.s/ ..s//ds; for t 2 Œ0; n:

Let t 2 Œ0; t be such that .t/ D

0

0

Let us take the right-hand side of the above inequality as v.t/. Then, we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we have v.0/ D c15;n

and

O O v 0 .t/ D M.t/.t/ C Mp.t/ ..t// a.e. t 2 Œ0; n:

Using the nondecreasing character of

, we get

O O v 0 .t/ Mp.t/ .v.t// C M.t/v.t/ a.e. t 2 Œ0; n: This implies that for each t 2 Œ0; n and using (4.3), we get Z

v.t/ c15;n

ds O M s C .s/ O M Z

Z Z

t 0 n 0

C1

< c15;n

max.p.s/; .s//ds max.p.s/; .s//ds ds : s C .s/

4.2 Perturbed Partial Functional Evolution Equations with Finite Delay

99

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n : Since kykn .t/, we have kykn n : This shows that the set E is bounded. Then statement .S2/ in Theorem 1.30 does not hold. The nonlinear alternative of Avramescu implies that .S1/ holds, we deduce that the operator F C G has a fixed point y the fixed point of the operator N15 , which is a mild solution of the problem (4.1)–(4.2). t u

4.2.3 An Example As an application of Theorem 4.2, we present the following partial functional differential equation 8 @z @2 z ˆ ˆ .t; x/ D a.t; x/ .t; x/ ˆ ˆ @t @x2 ˆ ˆ ˆ ˆ CQ.t; z.t r; x// C P.t; z.t r; x// t 2 Œ0; C1/; x 2 Œ0; < ˆ ˆ z.t; 0/ D z.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ : z.t; x/ D ˚.t; x/

t 2 Œ0; C1/ t 2 H; x 2 Œ0; ; (4.4)

where a.t; x/ W Œ0; 1/Œ0; ! R is a continuous function and is uniformly Hölder continuous in t, Q; P W Œ0; C1/ R ! R and ˚ W H Œ0; ! R are continuous functions. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumption .4:1:1/ (see [112, 149]). For x 2 Œ0; , we set y.t/.x/ D z.t; x/;

t 2 RC ;

f .t; yt /.x/ D Q.t; z.t r; x//;

t 2 RC

h.t; yt /.x/ D P.t; z.t r; x//;

t 2 RC

and '.t/.x/ D ˚.t; x/;

r t 0:

100

4 Perturbed Partial Functional Evolution Equations

Thus, under the above definitions of f , h, ', and A./, the system (4.4) can be represented by the abstract evolution problem (4.1)–(4.2). Furthermore, more appropriate conditions on Q and P ensure the existence of mild solutions for (4.4) by Theorems 4.2 and 1.30.

4.3 Perturbed Neutral Functional Evolution Equations with Finite Delay 4.3.1 Introduction In this section, we consider the following perturbed neutral functional evolution equations with finite delay d Œy.t/ g.t; yt / D A.t/y.t/ C f .t; yt / C h.t; yt /; a.e. t 2 J D RC dt y.t/ D '.t/; t 2 H;

(4.5) (4.6)

where r > 0, A./, f , h, and ' are as in problem (4.1)–(4.2) and g W J C.H; E/ ! E is a given function. Here we are interested to give the existence of mild solutions for the perturbed neutral functional evolution equations (4.5)–(4.6). This result is an extension of the problem (4.1) for the neutral case.

4.3.2 Existence of Mild Solutions In this section, we give an existence result for the perturbed neutral functional evolution problem (4.5)–(4.6). Firstly we define the mild solution. Definition 4.3. We say that the continuous function y./ W R ! E is a mild solution of (4.5)–(4.6) if y.t/ D '.t/ for all t 2 H and y satisfies the following integral equation Z y.t/ D U.t; 0/Œ'.0/ g.0; '/ C g.t; yt / C Z t C U.t; s/Œf .s; ys / C h.s; ys / ds; 0

0

t

U.t; s/A.s/g.s; ys /ds

for each t 2 RC :

4.3 Perturbed Neutral Functional Evolution Equations with Finite Delay

101

We consider the hypotheses (4.1.1)–(4.1.3) and in what follows we will need the following additional assumptions: .G1/

There exists a constant M 0 > 0 such that: kA1 .t/kB.E/ M 0

.G2/

There exists a constant 0 < L <

1 M0

for all t 2 J:

such that:

jA.t/ g.t; '/j L .k'k C 1/ for all t 2 J and ' 2 H: .G3/

There exists a constant L > 0 such that: jA.s/ g.s; '/ A.s/ g.s; '/j L .js sj C k' 'k/

for all s; s 2 J and '; ' 2 H. For every n 2 N, we define in C.Œr; C1/; E/ the semi-norms by: kykn WD sup f e

where Ln .t/ D

Z 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

O C .t/. ln .s/ ds; ln .t/ D Kn MŒL

Then C.Œr; C1/; E/ is a Fréchet space with the family of semi-norms fk

1 < 1. kn gn2N . Let us fix > 0 and assume that M 0 L Kn C

Theorem 4.4. Suppose that hypotheses (4.1.1)–(4.1.3) and the assumptions .G1/– .G3/ are satisfied and moreover Z

C1

c16;n

Z n O M ds > max.L; .s/; p.s//ds; s C .s/ 1 M0L 0

for each n > 0

(4.7)

with c16;n WD

1 1 M0L

Z t O O O O M 0 L.M C 1/ C MLn C MM 0 Lk'k C M jh.s; 0/j ds : 0

Then the problem (4.5)–(4.6) has a mild solution. Proof. Transform the problem (4.5)–(4.6) into a fixed point problem. Consider the operator N16 W C.Œr; C1/; E/ ! C.Œr; C1/; E/ defined by:

102

4 Perturbed Partial Functional Evolution Equations

N16 .y/.t/ D

8 '.t/; ˆ ˆ ˆ ˆ ˆ ˆ ˆ < U.t; Z 0/ Œ'.0/ g.0; '/ C g.t; yt /

if t 0I

t

C U.t; s/A.s/g.s; ys /ds ˆ ˆ ˆ ˆ Z0 t ˆ ˆ ˆ :C U.t; s/Œf .s; ys / C h.s; ys /ds; 0

if t 0:

Clearly, the fixed points of the operator N16 are mild solutions of the problem (4.5)–(4.6). Define the operators F; G W C.Œr; C1/; E/ ! C.Œr; C1/; E/ by

F.y/.t/ D

8 ˆ < '.t/; ˆ : U.t; 0/ '.0/ C

if t 0; Z 0

t

U.t; s/ f .s; ys / ds;

if t 0:

and Z G.y/.t/ D g.t; yt / U.t; 0/g.0; '/ C Z

t

C 0

0

t

U.t; s/A.s/g.s; ys /ds

U.t; s/ h.s; ys / ds:

Obviously the operator N16 has a fixed point is equivalent to F C G has one, so it turns to prove that F C G has a fixed point. The proof will be given in several steps. We can show that the operator F is continuous and compact. We can prove also that the operator G is a contraction for all n 2 N as in the proof of Theorem 2.6. For applying Theorem 1.30, we must check .S2/: i.e., it remains to show that the set o n y for some 0 < < 1 E D y 2 C.Œr; C1/; E/ W y D F.y/ C G is bounded. Let y 2 E. Then, we have Z jy.t/j

t 0

kU.t; s/kB.E/ jf .s; ys /j ds

nˇ y ˇ ˇ t ˇ C ˇg t; ˇ C kU.t; 0/kB.E/ jg.0; '/j Z t ˇ y ˇ ˇ s ˇ kU.t; s/kB.E/ ˇA.s/g s; C ˇ ds 0 Z t ˇ ˇ y ˇ ˇ s C h.s; 0/ C h.s; 0/ˇ ds : kU.t; s/kB.E/ ˇh s; 0

4.3 Perturbed Neutral Functional Evolution Equations with Finite Delay

103

By the hypotheses (4.1.1)–(4.1.3), .G1/, and .G2/ we obtain Z

t

Z

t

ˇ y ˇ ˇ t ˇ O 1 .s/kjA.t/g.0; '/j f .s; ys / ds C kA1 .s/k ˇA.t/g t; ˇ C MkA 0 Z tˇ Z tˇ ˇ y ˇ ys ˇ ˇ ˇ s ˇ O O CM h.s; 0/ˇ ds ˇ ds C M ˇA.s/g s; ˇh s; 0 0 Z t O CM jh.s; 0/j ds

jy.t/j O M

0

y t O 0 L.k'k C 1/ p.s/ .kys k/ ds C M 0 L C 1 C MM 0 Z t Z t Z t y ys s O O O CML .s/ ds C M jh.s; 0/j ds C 1 ds C M 0 0 0 Z t y t O O C 1/ C MLn O O 0 Lk'k M p.s/ .kys k/ ds C M 0 L C M 0 L.M C MM 0 Z t Z t Z t y ys s O O O CM jh.s; 0/j ds C M L ds C M .s/ ds: 0 0 0

O M

gives with the fact that 0 < < 1

The nondecreasing character of

Z t y jy.t/j t O O O O M 0 L C M 0 L.M C 1/ C MLn C MM 0 Lk'k C M jh.s; 0/j ds 0 Z t Z t Z t y y s ys s O O O CM L ds C M .s/ ds C M p.s/ ds: 0 0 0 Consider the function defined by .t/ WD sup

jy.s/j W 0st

; 0 t < C1:

jy.t /j ; by the previous inequality, we have Z t

O O O O 1 M 0 L .t/ M 0 L.M C 1/ C MLn C MM 0 Lk'k C M jh.s; 0/j ds

Let t 2 Œ0; t be such that .t/ D

O CM O CM

Z Z

t 0

0 t

0

O L.s/ ds C M

Z

p.s/ ..s// ds:

0

t

.s/.s/ ds

104

4 Perturbed Partial Functional Evolution Equations

1

Set c16;n WD

O C 1/ C MLn O O 0 Lk'k C M O M 0 L.M C MM

1 M0L Thus, for each t 2 Œ0; n we get .t/ c16;n C

Z

O M 1 M0L

t

Z L.s/ ds C

0

0

t

Z .s/.s/ ds C

0

t

Z

t 0

jh.s; 0/j ds :

p.s/ ..s// ds :

Let us take the right-hand side of the above inequality as v.t/. Thus, we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we have v.0/ D c16;n and v 0 .t/ D

O M 1 M0L

ŒL.t/ C .t/.t/ C p.t/ ..t//

a.e. t 2 Œ0; n: Using the nondecreasing character of ; we get O M

v 0 .t/

1 M0L

ŒLv.t/ C .t/v.t/ C p.t/ .v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using the condition (4.7), we get Z

v.t/

c16;n

Z t O M ds; max.L; .s/; p.s//ds s C .s/ 1 M0L 0 Z n O M max.L; .s/; p.s//ds 1 M0L 0 Z C1 ds : < s C .s/ c16;n

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kykn .t/, we have kykn n : This shows that the set E is bounded. Then the statement .S2/ in Theorem 1.30 does not hold. A consequence of the nonlinear alternative of Avramescu that .S1/ holds, we deduce that the operator FCG has a fixed point y . Then y .t/ D y .t/Cx.t/, t 2 Œr; C1/ is a fixed point of the operator N16 , which is the mild solution of the problem (4.5)–(4.6). t u

4.3 Perturbed Neutral Functional Evolution Equations with Finite Delay

105

4.3.3 An Example Consider the following model

Z t Z 8 @ @2 z ˆ ˆ z.t; x/ b.s t; u; x/ z.s; u/ du ds D a.t; x/ .t; x/ ˆ ˆ @t @x2 ˆ r 0 ˆ ˆ @z ˆ ˆ ˆ CQ t; z.t r; x/; .t r; x/ ˆ ˆ @x ˆ ˆ < @z CP t; z.t r; x/; .t r; x/ ; t 2 Œ0; C1/; x 2 Œ0; @x ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ t 2 Œ0; C1/ ˆ z.t; 0/ D z.t; / D 0; ˆ ˆ ˆ ˆ ˆ : z.t; x/ D ˚.t; x/; t 2 H; x 2 Œ0; (4.8) where r > 0; a.t; x/ is a continuous function and is uniformly Hölder continuous in t, Q; P W Œ0; C1/ R R ! R and ˚ W H Œ0; ! R are continuous functions. Let y.t/.x/ D z.t; x/; t 2 Œ0; 1/; x 2 Œ0; ; Z t Z g.t; yt /.x/ D b.s t; u; x/z.s; u/duds; x 2 Œ0; ; r

0

@z f .t; yt /.x/ D Q t; z.; x/; .; x/ ; 2 H; x 2 Œ0; ; t 0; @x @z h.t; yt /.x/ D P t; z.; x/; .; x/ ; 2 H; x 2 Œ0; ; t 0 @x

and './.x/ D ˚.; x/; 2 H; x 2 Œ0; : Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g: Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .4:1:1/ and .G1/ (see [112, 149]). Here we assume that ' W H ! E is Lebesgue measurable and h.s/j'.s/j2 is Lebesgue integrable on H where h W H ! R is a positive integrable function. The norm is defined here by: Z k'k D j˚.0/j C

0

2

h.s/j'.s/j ds r

12

:

106

4 Perturbed Partial Functional Evolution Equations

The function b is measurable on Œ0; 1/ Œ0; Œ0; ; b.s; u; 0/ D b.s; u; / D 0; .s; u/ 2 Œ0; 1/ Œ0; ; Z

Z

0

t

Z

r

0

b2 .s; u; x/ dsdudx < 1 h.s/

and sup N .t/ < 1; where t2Œ0;1/

Z N .t/ D

0

Z

t

r

Z

0

2 @2 1 a.s; x/ 2 b.s; u; x/ dsdudx: h.s/ @x

Thus, under the above definitions of f , g, h, ', and A./, the system (4.8) can be represented by the abstract evolution problem (4.5)–(4.6). Furthermore, more appropriate conditions on Q and P ensure the existence of mild solutions for (4.8) by Theorems 4.4 and 1.30.

4.4 Perturbed Partial Functional Evolution Equations with Infinite Delay 4.4.1 Introduction The existence of mild solutions is studied here for the following perturbed partial functional evolution equations with infinite delay y0 .t/ D A.t/y.t/ C f .t; yt / C h.t; yt /; a.e. t 2 J

(4.9)

y0 D 2 B;

(4.10)

where f ; h W J B ! E and 2 B are given functions and fA.t/g0t
4.4.2 Existence of Mild Solutions Before stating and proving the main result, we give first the definition of mild solution of our perturbed evolution problem (4.9)–(4.10).

4.4 Perturbed Partial Functional Evolution Equations with Infinite Delay

107

Definition 4.5. We say that the continuous function y./ W R ! E is a mild solution of (4.9)–(4.10) if y.t/ D .t/ for all t 2 .1; 0 and y satisfies the following integral equation Z y.t/ D U.t; 0/ .0/ C

t 0

U.t; s/ Œf .s; ys / C h.s; ys / ds;

for each t 2 RC :

We introduce the following hypotheses which are assumed hereafter: O 1 such that: (4.1.1) U.t; s/ is compact for t s > 0 and there exists a constant M O for every .t; s/ 2 : kU.t; s/kB.E/ M 1 .J; RC / and a continuous nondecreasing (4.1.2) There exists a function p 2 Lloc function W RC ! .0; 1/ and such that:

jf .t; u/j p.t/

.kukB / for a.e. t 2 J and each u 2 B:

(4.1.3) There exists a function 2 L1 .J; RC / where kkL1 <

1 such that: O M

jh.t; u/ h.t; v/j .t/ku vkB for a.e. t 2 J and all u; v 2 B: Consider the following space ˚ BC1 D y W R ! E W yjŒ0;T 2 C.Œ0; T; E/; y0 2 B ; where yjŒ0;T is the restriction of y to any real compact interval Œ0; T. For every n 2 N, we define in BC1 the semi-norms by: kykn WD sup f e

Ln .t/

where

Z D 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

O ln .s/ ds and ln .t/ D Kn M.t/.

Then BC1 is a Fréchet space with the family of semi-norms k kn2N . In what follows let us fix > 1. Theorem 4.6. Suppose that hypotheses (4.1.1)–(4.1.3) are satisfied and moreover Z

C1

c17;n

ds O > Kn M s C .s/

Z 0

n

max.p.s/; .s// ds; for each n > 0

with O C Mn /kkB C Kn M O c17;n D .Kn MH then the problem (4.9)–(4.10) has a mild solution.

Z

n 0

jh.s; 0/jds;

(4.11)

108

4 Perturbed Partial Functional Evolution Equations

Proof. Consider the operator N17 W BC1 ! BC1 defined by: 8 .t/; ˆ ˆ ˆ ˆ ˆ Z t < U.t; s/ f .s; ys / ds N17 .y/.t/ D U.t; 0/ .0/ C ˆ 0 Z t ˆ ˆ ˆ ˆ :C U.t; s/ h.s; ys / ds; 0

if t 0;

if t 0:

Clearly, the fixed points of the operator N17 are mild solutions of the problem (4.9)– (4.10). For 2 B, we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 2 .1; 0I

: U.t; 0/ .0/;

if t 2 J:

Then x0 D . For each function z 2 BC1 , set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 4.5 if and only if z satisfies z0 D 0 and Z

t

z.t/ D 0

Z U.t; s/ f .s; zs C xs / ds C

t 0

U.t; s/ h.s; zs C xs / ds;

for t 2 J:

Let B0C1 D fz 2 BC1 W z0 D 0g : Define the operators F; G W B0C1 ! B0C1 by Z

t

F.z/.t/ D 0

U.t; s/ f .s; zs C xs / ds;

for t 2 J

U.t; s/ h.s; zs C xs / ds;

for t 2 J:

and Z

t

G.z/.t/ D 0

Obviously the operator N17 has a fixed point is equivalent to F C G has one, so it turns to prove that F C G has a fixed point. The proof will be given in several steps. We can show that the operator F is continuous and compact. We can prove also that the operator G is a contraction for all n 2 N as in the proof of Theorem 3.2). For applying Theorem 1.30, we must check .S2/: i.e., it remains to show that the set o n z for some 0 < < 1 E D z 2 B0C1 W z D F.z/ C G is bounded.

4.4 Perturbed Partial Functional Evolution Equations with Infinite Delay

109

Let z 2 E. By (4.1.1)–(4.1.3), we have for each t 2 Œ0; n Z jz.t/j

t 0

kU.t; s/kB.E/ jf .s; zs C xs /jds

Z

ˇ ˇ z ˇ ˇ s kU.t; s/kB.E/ ˇh s; C xs h.s; 0/ C h.s; 0/ˇ ds

t

C 0

O M

Z

t

p.s/ .kzs C xs kB / ds

0

O C M

Z

t

0

Z t z s O .s/ C xs ds C M jh.s; 0/jds: B 0

Using the inequality (3.4) and the nondecreasing character of 1 O jz.t/j M

Z

t 0

O CM

,we get

p.s/ .Kn jz.s/j C ˛n /ds

Z 0

t

.s/

The nondecreasing character of

Z t Kn O jz.s/j C ˛n ds C M jh.s; 0/jds: 0

gives with the fact that 0 < < 1

Z t Z t Kn Kn O O jz.t/j C ˛n ˛n C Kn M jz.s/j C ˛n ds jh.s; 0/jds C Kn M p.s/ 0 0 Z t Kn O CKn M jz.s/j C ˛n ds: .s/ 0 O Set c17;n WD Kn M

Z

t 0

jh.s; 0/jds C ˛n ; thus

Z t Kn Kn O jz.t/j C ˛n c17;n C Kn M jz.s/j C ˛n ds p.s/ 0 Z t Kn O CKn M jz.s/j C ˛n ds: .s/ 0 We consider the function defined by .t/ WD sup

Kn jz.s/j C ˛n W 0 s t

; 0 t < C1:

110

4 Perturbed Partial Functional Evolution Equations

Kn jz.t /j C ˛n ; by the previous inequality, we Let t 2 Œ0; t be such that .t/ D have Z t Z t O O .t/ c17;n C Kn M p.s/ ..s//ds C Kn M .s/.s/ds; for t 2 Œ0; n 0

0

Let us take the right-hand side of the above inequality as v.t/. Then, we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we have v.0/ D c17;n

O O v 0 .t/ D Kn Mp.t/ ..t// C Kn M.t/.t/ a.e. t 2 Œ0; n:

and

Using the nondecreasing character of

, we get

O O .v.t// C Kn M.t/v.t/ a.e. t 2 Œ0; n: v 0 .t/ Kn Mp.t/ This implies that for each t 2 Œ0; n and using (4.11), we get Z

v.t/

c17;n

ds O Kn M s C .s/ O Kn M Z

Z 0

Z

C1

< c17;n

t

n

0

max.p.s/; .s//ds max.p.s/; .s//ds

ds : s C .s/

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kzkn .t/, we have kzkn n : This shows that the set E is bounded. Then statement .S2/ in Theorem 1.30 does not hold. The nonlinear alternative of Avramescu implies that .S1/ holds, we deduce that the operator F C G has a fixed point z . Then y .t/ D z .t/ C x.t/, t 2 R is a fixed point of the operator N17 , which is the mild solution of the problem (4.9)–(4.10). t u

4.4 Perturbed Partial Functional Evolution Equations with Infinite Delay

111

4.4.3 An Example Consider the following model 8 @v @2 v ˆ ˆ ˆ .t; / D a.t; / 2 .t; / ˆ ˆ @t ˆ Z 0 @ ˆ ˆ ˆ ˆ ˆ C P. /r.t; v.t C ; //d ˆ ˆ ˆ ˆ Z1 0 < Q. /s.t; v.t C ; //d t 2 RC ; 2 Œ0; C ˆ 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ v.t; 0/ D v.t; / D 0 t 2 RC ˆ ˆ ˆ ˆ ˆ ˆ : v.; / D v0 .; / 1 < 0; 2 Œ0; ; (4.12) where a.t; / is a continuous function and is uniformly Hölder continuous in t; P; Q W .1; 0 ! RI r; s W .1; 0 R ! R and v0 W .1; 0 Œ0; ! R are continuous functions. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumption .4:1:1/ (see [112, 149]). For the phase space B, we choose the well-known space BUC.R ; E/: the space of uniformly bounded continuous functions endowed with the following norm k'k D sup j'./j for 0

' 2 B:

If we put for ' 2 BUC.R ; E/ and 2 Œ0; y.t/./ D v.t; /; t 2 RC ; 2 Œ0; ; . /./ D v0 .; /; 1 < 0; 2 Œ0; ; Z 0 P. /r.t; '. /.//d; 1 < 0; 2 Œ0; f .t; '/./ D 1

and Z h.t; '/./ D

0 1

Q. /s.t; '. /.//d; 1 < 0; 2 Œ0; :

112

4 Perturbed Partial Functional Evolution Equations

Then, the problem (4.12) takes the abstract partial perturbed evolution form (4.9)–(4.10). In order to show the existence of mild solutions of problem (4.12), we suppose the following assumptions: – The function s is Lipschitz continuous with respect to its second argument. Let lip.s/ denote the Lipschitz constant of s. – There exist p 2 L1 .J; RC / and a nondecreasing continuous function W Œ0; 1/ ! Œ0; 1/ such that jr.t; u/j p.t/ .juj/; for 2 J; and u 2 R: – P and Q are integrable on .1; 0: By the dominated convergence theorem, one can show that f is a continuous function from B to E. Moreover the mapping h is Lipschitz continuous in its second argument, in fact, we have Z jh.t; '1 / h.t; '2 /j lip.s/

0 1

jQ. /j d j'1 '2 j ; for '1 ; '2 2 B:

On the other hand, we have for ' 2 B and 2 Œ0; Z jf .t; '/./j Since the function

0

1

jp.t/P. /j .j.'. //./j/d:

is nondecreasing, it follows that Z

jf .t; '/j p.t/

0 1

jP. /j d .j'j/; for ' 2 B:

Proposition 4.7. Under the above assumptions, if we assume that condition (4.11) in Theorem 4.6 is true, ' 2 B, then the problem (4.12) has a mild solution which is defined in .1; C1/.

4.5 Notes and Remarks The results of Chap. 4 are taken from Adimy et al. [12], Baghli et al. [34], and Balachandran and Anandhi [42]. Other results may be found in [8, 9, 50, 158].

Chapter 5

Partial Functional Evolution Inclusions with Finite Delay

5.1 Introduction In this chapter, we provide sufficient conditions for the existence of mild solutions on the semi-infinite interval J D RC for some classes of first order partial functional and neutral functional differential evolution inclusions with finite delay by using the recent nonlinear alternative of Frigon [114, 115] for contractive multi-valued maps in Fréchet spaces [116], combined with the semigroup theory [16, 20, 168].

5.2 Partial Functional Evolution Inclusions 5.2.1 Introduction We establish here the existence of mild solutions for the partial functional evolution inclusion of the form y0 .t/ 2 A.t/y.t/ C F.t; yt /; a.e. t 2 J D RC

(5.1)

y.t/ D '.t/; t 2 H;

(5.2)

where F W J C.H; E/ ! P.E/ is a multi-valued map with nonempty compact values, P.E/ is the family of all subsets of E, ' 2 C.H; E/ is a given function, and fA.t/g0t
© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_5

113

114

5 Partial Functional Evolution Inclusions with Finite Delay

5.2.2 Existence of Mild Solutions Let us introduce the definition of the mild solution of our partial functional evolution inclusion system (5.1)–(5.2) before stating and proving our main result. Definition 5.1. We say that the continuous function y./ W Œr; C1/ ! E is a mild solution of the evolution system (5.1)–(5.2) if y.t/ D '.t/ for all t 2 H and the restriction of y./ to the interval J is continuous and there exists f ./ 2 L1 .J; E/: f .t/ 2 F.t; yt / a.e. in J such that y satisfies the following integral equation: Z y.t/ D U.t; 0/ '.0/ C

t 0

U.t; s/ f .s/ ds;

for each t 2 RC :

We will introduce the following hypotheses which are assumed afterwards b 1 such that: (5.1.1) There exists a constant M b for every .t; s/ 2 : kU.t; s/kB.E/ M 1 -Carathéodory with (5.1.2) The multi-function F W J C.H; E/ ! P.E/ is Lloc compact and convex values for each u 2 C.H; E/ and there exist a function 1 p 2 Lloc .J; RC / and a continuous nondecreasing function W J ! .0; 1/ and such that:

kF.t; u/kP.E/ p.t/

.kuk/ for a.e. t 2 J and each u 2 C.H; E/:

1 .J; RC / such that: (5.1.3) For all R > 0, there exists lR 2 Lloc

Hd .F.t; u/ F.t; v// lR .t/ ku vk for each t 2 J and for all u; v 2 C.H; E/ with kuk R and kvk R and d.0; F.t; 0// lR .t/

a.e. t 2 J:

For every n 2 N, we define in C.Œr; C1/; E/ the family of semi-norms by kykn WD sup f e

where Ln .t/ D

Z 0

t

Ln .t/

jy.t/j W t 2 Œ0; n g

b n .t/ and ln is the function from .5:1:3/: Then ln .s/ ds; ln .t/ D Ml

C.Œr; C1/; E/ is a Fréchet space with the family of semi-norms k kn2N : In what follows we will choose > 1:

5.2 Partial Functional Evolution Inclusions

115

Theorem 5.2. Suppose that hypotheses (4.1.1)–(4.1.3) are satisfied and moreover Z

C1 c19

Z

ds b >M .s/

0

n

p.s/ ds; for each n > 0

(5.3)

b k'k: Then the evolution inclusion problem (5.1)–(5.2) has a mild with c19 D M solution. Proof. Transform the problem (5.1)–(5.2) into a fixed point problem. Consider the multi-valued operator N19 W C.Œr; C1/; E/ ! P.C.Œr; C1/; E// defined by: 8 8 9 '.t/; if t 2 H; > ˆ ˆ ˆ ˆ > ˆ ˆ > < < = 0/ '.0/ N19 .y/ D h 2 C.Œr; C1/; E/ W h.t/ D U.t; Z t ˆ ˆ > ˆ ˆ > ˆ ˆ > :C : U.t; s/ f .s/ ds; if t 0: ; 0

where f 2 SF;y D fv 2 L1 .J; E/ W v.t/ 2 F.t; yt / for a.e. t 2 Jg: Clearly, the fixed points of the operator N19 are mild solutions of the problem (5.1)–(5.2). We remark also that, for each y 2 C.Œr; C1/; E/, the set SF;y is nonempty since, by .5:1:2/, F has a measurable selection (see [94], Theorem III.6). Let y be a possible fixed point of the operator N19 . Given n 2 N and t n, then y should be solution of the inclusion y 2 N19 .y/ for some 2 .0; 1/ and there exists f 2 SF;y , f .t/ 2 F.t; yt / such that, for each t 2 RC , we have Z jy.t/j kU.t; 0/kB.E/ j'.0/j C b k'k C M b M

Z

t

p.s/ 0

t 0

kU.t; s/kB.E/ jf .s/j ds

.kys k/ ds:

Consider the function defined by .t/ WD sup fky.s/j W 0 s t g; 0 t < C1: Let t 2 Œr; t be such that .t/ D jy.t /j: If t 2 Œ0; n, by the previous inequality, we have Z t b k'k C M b p.s/ ..s// ds; for t 2 Œ0; n: .t/ M 0

If t 2 H, then .t/ D k'k and the previous inequality holds. Let us take the right-hand side of the above inequality as v.t/. Then, we have .t/ v.t/ for all t 2 Œ0; n:

116

5 Partial Functional Evolution Inclusions with Finite Delay

From the definition of v, we have b k'k and v 0 .t/ D Mp.t/ b c19 WD v.0/ D M Using the nondecreasing character of b p.t/ v 0 .t/ M

..t// a.e. t 2 Œ0; n:

, we get .v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using the condition (5.3), we get Z

v.t/

c19

Z

ds b M .s/

p.s/ ds 0

Z

b M Z

t

n

p.s/ ds 0 C1

< c19

ds : .s/

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kyt k .t/, we have kykn maxfk'kI n g WD n : Set U D f y 2 C.Œr; C1/; E/ W supf jy.t/j W 0 t ng < n C 1 for all n 2 Ng: Clearly, U is an open subset of C.Œr; C1/; E/. We shall show that N19 W U ! P.C.Œr; C1/; E// is a contraction and an admissible operator. First, we prove that N19 is a contraction; Let y; y 2 C.Œr; C1/; E/ and h 2 N19 .y/. Then there exists f .t/ 2 F.t; yt / such that for each t 2 Œ0; n Z h.t/ D U.t; 0/ '.0/ C

t 0

U.t; s/ f .s/ ds:

From .5:1:3/ it follows that Hd .F.t; yt /; F.t; yt // ln .t/ kyt yt k: Hence, there is 2 F.t; yt / such that jf .t/ j ln .t/ kyt yt k; t 2 Œ0; n: Consider U W Œ0; n ! P.E/, given by U D f 2 E W jf .t/ j ln .t/ kyt yt kg:

5.2 Partial Functional Evolution Inclusions

117

Since the multi-valued operator V.t/ D U .t/ \ F.t; yt / is measurable (in [94], see Proposition III.4), there exists a function f .t/, which is a measurable selection for V. So, f .t/ 2 F.t; yt / and we obtain for each t 2 Œ0; n jf .t/ f .t/j ln .t/ kyt yt k: Let us define, for each t 2 Œ0; n Z h.t/ D U.t; 0/ '.0/ C

0

t

U.t; s/ f .s/ds:

Then we can show as in previous sections that we have kh hkn

1 ky ykn :

By an analogous relation, obtained by interchanging the roles of y and y; it follows that Hd .N19 .y/; N19 .y//

1 ky ykn :

So, for > 1, N19 is a contraction for all n 2 N. It remains to show that N19 is an admissible operator. Let y 2 C.Œr; C1/; E/. Consider N19 W C.Œr; n; E/ ! P.C.Œr; n; E/, given by 8 ˆ ˆ '.t/; ˆ ˆ ˆ < N19 .y/ D h 2 C.Œr; C1/; E/ W h.t/ D U.t; 0/ '.0/ ˆ ˆ Z t ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :C : U.t; s/ f .s/ ds; 8 ˆ ˆ ˆ ˆ ˆ <

0

if t 2 HI

9 > > > > > =

> > > > ; if t 2 Œ0; n; >

n where f 2 SF;y D fv 2 L1 .Œ0; n; E/ W v.t/ 2 F.t; yt / for a.e. t 2 Œ0; ng: From (5.1.1) to (5.1.3) and since F is a multi-valued map with compact values, we can prove that for every y 2 C.Œr; n; E/; N19 .y/ 2 Pcp .C.Œr; n; E// and there exists y 2 C.Œr; n; E/ such that y 2 N19 .y /. Let h 2 C.Œr; n; E/, y 2 U and > 0. Assume that y 2 N19 .y/; then we have

ky.t/ y .t/k ky.t/ h.t/k C ky .t/ h.t/k e

Ln .t/

ky N19 .y/kn C ky .t/ h.t/k:

Since h is arbitrary, we may assume that h 2 B.y ; / D fh 2 C.Œr; n; E/ W kh y kn g: Therefore, ky y kn ky N19 .y/kn C :

118

5 Partial Functional Evolution Inclusions with Finite Delay

If y is not in N19 .y/, then ky N19 .y/k ¤ 0. Since N19 .y/ is compact, there exists x 2 N19 .y/ such that ky N19 .y/k D ky xk. Then we have ky.t/ x.t/k ky.t/ h.t/k C kx.t/ h.t/k e

Ln .t/

ky N19 .y/kn C kx.t/ h.t/k:

Thus, ky xkn ky N19 .y/kn C : So, N19 is an admissible operator contraction. From the choice of U there is no y 2 @U such that y D N19 .y/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.31 does not hold. A consequence of the nonlinear alternative of Frigon that .S1/ holds, we deduce that the operator N19 has a fixed point y which is a mild solution of the evolution inclusion problem (5.1)–(5.2). t u

5.2.3 An Example Consider the following model 8 @v @2 v ˆ ˆ ˆ .t; / 2 a.t; / 2 .t; / ˆ ˆ @t @ ˆ Z 0 ˆ ˆ ˆ ˆ ˆ C P. /R.t; v.t C ; //d < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

r

2 Œ0; (5.4)

v.t; 0/ D v.t; / D 0 v.; / D v0 .; /

t 2 RC r 0; 2 Œ0; ;

where r > 0, a.t; / is a continuous function and is uniformly Hölder continuous in t; P W H ! R and v0 W H Œ0; ! R are continuous functions and R W RC R ! P.R/ is a multi-valued map with compact convex values. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumption .4:1:1/; see [112, 149]. For 2 Œ0; , we have y.t/./ D v.t; /; t 2 RC ; '. /./ D v0 .; /; r 0;

5.3 Neutral Functional Evolution Inclusions

119

and Z F.t; /./ D

0

r

P. /R.t; . /.//d; r 0:

Then, the problem (5.4) takes the abstract partial functional evolution inclusion form (5.1)–(5.2). In order to show the existence of mild solutions of problem (5.4), we assume the following assumptions: – There exist p 2 L1 .J; RC / and a nondecreasing continuous function .0; C1/ such that

W RC !

jR.t; /j p.t/ .jj/; for 2 J; and 2 R: – P is integrable on H: By the dominated convergence theorem, one can show that f 2 SF;y is a continuous function from C.Œr; C1/; E/ to E. On the other hand, we have for 2 R and 2 Œ0; Z jF.t; /./j Since the function

0

r

jp.t/P. /j .j.. //./j/d:

is nondecreasing, it follows that Z

kF.t; /kP.E/ p.t/

0 r

jP. /j d .jj/; for 2 R:

Proposition 5.3. Under the above assumptions, if we assume that condition (5.3) in Theorem 5.2 is true, then the problem (5.4) has a mild solution which is defined in Œr; C1/.

5.3 Neutral Functional Evolution Inclusions 5.3.1 Introduction We investigate in this section the neutral functional evolution inclusion of the form d Œy.t/ g.t; yt / 2 A.t/y.t/ C F.t; yt /; a.e. t 2 J D RC dt y.t/ D '.t/; t 2 H;

(5.5) (5.6)

120

5 Partial Functional Evolution Inclusions with Finite Delay

where F W J C.H; E/ ! P.E/ is a multi-valued map with nonempty compact values, P.E/ is the family of all subsets of E, g W J C.H; E/ ! E and ' 2 C.H; E/ is a given function. This result is an extension of the problem (5.1) for the neutral case and also the multi-valued generalization of the neutral problem (2.8) in [37].

5.3.2 Existence of Mild Solutions Definition 5.4. We say that the function y./ W Œr; C1/ ! E is a mild solution of the neutral functional evolution system (5.5)–(5.6) if y.t/ D '.t/ for all t 2 H and the restriction of y./ to the interval J is continuous and there exists f ./ 2 L1 .J; E/: f .t/ 2 F.t; yt / a.e. in J such that y satisfies the following integral equation Z t y.t/ D U.t; 0/Œ'.0/ g.0; '/ C g.t; yt / C U.t; s/A.s/g.s; ys /ds 0 Z t C U.t; s/f .s/ ds; for each t 2 RC : 0

We consider the hypotheses (5.1.1)–(5.1.3) and we will need the following assumptions: .G1/

There exists a constant M 0 > 0 such that: kA1 .t/kB.E/ M 0

.G2/

There exists a constant 0 < L <

1 M0

for all t 2 J:

such that:

jA.t/ g.t; '/j L .k'k C 1/ for all t 2 J and ' 2 C.H; E/: .G3/

There exists a constant L > 0 such that: jA.s/ g.s; '/ A.s/ g.s; '/j L .js sj C k' 'k/ for all s; s 2 J and '; ' 2 C.H; E/.

b C ln .t/ for the family of For every n 2 N, let us take here ln .t/ D MŒL g defined in Sect. 5.3. In what follows we fix > 0 such that seminorm fk k n n2N

1 < 1. M 0 L C

5.3 Neutral Functional Evolution Inclusions

121

Theorem 5.5. Suppose that hypotheses (5.1.1)–(5.1.3) and the assumptions .G1/– .G3/ are satisfied and moreover Z

C1

c20;n

Z n b M ds > max.L; p.s//ds; s C .s/ 1 M0L 0

for each n > 0

(5.7)

with c20;n WD

b C 1/ C MLn b b C M 0 L/k'k C M 0 L.M M.1 1 M0L

then the neutral functional evolution problem (5.5)–(5.6) has a mild solution. Proof. Transform the neutral functional evolution problem (5.5)–(5.6) into a fixed point problem. Consider the multi-valued operator N20 W C.Œr; C1/; E/ ! P.C.Œr; C1/; E// defined by: 8 8 ˆ ˆ '.t/; if t 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Œ'.0/ g.0; '/ C g.t; yt / < U.t; < Z 0/ t N20 .y/ D h 2 C.Œr; C1/; E/ W h.t/ D ˆ ˆ U.t; s/A.s/g.s; ys /ds C ˆ ˆ ˆ ˆ ˆ Z0 t ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ U.t; s/f .s/ds; if t 0; :C : 0

9 > > > > > > > > = > > > > > > > > ;

where f 2 SF;y D fv 2 L1 .J; E/ W v.t/ 2 F.t; yt / for a.e. t 2 Jg: Clearly, the fixed points of the operator N20 are mild solutions of the problem (5.5)–(5.6). We remark also that, for each y 2 C.Œr; C1/; E/, the set SF;y is nonempty since, by .5:1:2/, F has a measurable selection (see [94], Theorem III.6). Let y be a possible fixed point of the operator N20 . Given n 2 N and t n, then y should be solution of the inclusion y 2 N20 .y/ for some 2 .0; 1/ and there exists f 2 SF;y , f .t/ 2 F.t; yt / such that, for each t 2 RC , we have jy.t/j kU.t; 0/kB.E/ j'.0/j C kU.t; 0/kB.E/ kA1 .0/kkA.0/ g.0; '/k Z t CkA1 .t/kB.E/ kA.t/g.t; yt /k C kU.t; s/kB.E/ kA.s/g.s; ys /kds 0

Z

t

C

kU.t; s/kB.E/ jf .s/jds

0

b b 0 L.k'k C 1/ C M 0 L.kyt k C 1/ C M b Mk'k C MM b CM

Z 0

t

p.s/ .kys k/ds

Z

t 0

L.kys k C 1/ds

122

5 Partial Functional Evolution Inclusions with Finite Delay

b C M 0 L/k'k C M 0 L.M b C 1/ C MLn b M.1 Z t Z t b b CM 0 Lkyt k C M Lkys kds C M p.s/ .kys k/ds: 0

0

We consider the function defined by .t/ WD supf jy.s/j W 0 s t g; 0 t < C1: Let t 2 Œr; t be such that .t/ D jy.t /j: If t 2 Œ0; n, by the previous inequality we have for t 2 Œ0; n b C 1/ C MLn b b C M 0 L/k'k C M 0 L.M .t/ M.1 Z t Z t b b CM 0 L.t/ C M L.s/ds C M p.s/ ..s//ds: 0

0

Then b C M 0 L/k'k C M 0 L.M b C 1/ C MLn b .1 M 0 L/.t/ M.1 Z t Z t b b CM L.s/ds C M p.s/ ..s//ds: 0

Set c20;n WD

0

b C 1/ C MLn b b C M 0 L/k'k C M 0 L.M M.1 1 M0L .t/ c20;n C

b M 1 M0L

Z

t 0

, thus

ŒL.s/ C p.s/ ..s// ds:

If t 2 H, then .t/ D k'k and the previous inequality holds. Let us take the right-hand side of the above inequality as v.t/. Then we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we have v.0/ D c20;n and v 0 .t/ D

b M 1 M0L

Using the nondecreasing character of v 0 .t/

b M 1 M0L

ŒL.t/ C p.t/ ..t// a.e. t 2 Œ0; n: , we get

ŒLv.t/ C p.t/ .v.t// a.e. t 2 Œ0; n:

5.3 Neutral Functional Evolution Inclusions

123

This implies that for each t 2 Œ0; n and using the condition (5.7), we get Z

v.t/

c20;n

Z t b ds M max.L; p.s//ds s C .s/ 1 M0L 0 Z n b M max.L; p.s//ds 1 M0L 0 Z C1 ds : < s C .s/ c20;n

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kyt k .t/, we have kykn max fk'k; n g WD n : We can show that N20 is an admissible operator and we shall prove now that N20 W U ! P.C.Œr; C1/; E// is a contraction. Let y; y 2 C.Œr; C1/; E/ and h 2 N20 .y/. Then there exists f .t/ 2 F.t; yt / such that for each t 2 Œ0; n Z h.t/ D U.t; 0/Œ'.0/ g.0; '/ C g.t; yt / C Z

t

C 0

0

t

U.t; s/A.s/g.s; ys /ds

U.t; s/f .s/ds:

From .5:1:3/ it follows that Hd .F.t; yt /; F.t; yt // ln .t/ kyt yt k: Hence, there is 2 F.t; yt / such that jf .t/ j ln .t/ kyt yt k t 2 Œ0; n: Consider U W Œ0; n ! P.E/, given by U D f 2 E W jf .t/ j ln .t/ kyt yt kg: Since the multi-valued operator V.t/ D U .t/ \ F.t; yt / is measurable (in [94], see Proposition III.4), there exists a function f .t/, which is a measurable selection for V. So, f .t/ 2 F.t; yt /, and we obtain for each t 2 Œ0; n jf .t/ f .t/j ln .t/ kyt yt k:

124

5 Partial Functional Evolution Inclusions with Finite Delay

Let us define, for each t 2 Œ0; n Z h.t/ D U.t; 0/Œ'.0/ g.0; '/ C g.t; yt / C Z

t

C 0

t

0

U.t; s/A.s/g.s; ys /ds

U.t; s/f .s/ds:

Then we can show as in previous sections that we have for each t 2 Œ0; n and n2N

1 ky ykn : kh hkn M 0 L C

By an analogous relation, obtained by interchanging the roles of y and y; it follows that

1 ky ykn : Hd .N20 .y/; N20 .y// M 0 L C

1 < 1, the operator N20 is a contraction for all n 2 N and an So, for M 0 L C

admissible operator. From the choice of U there is no y 2 @U such that y D N20 .y/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.31 does not hold. By the nonlinear alternative due to Frigon we get that .S1/ holds, we deduce that the operator N20 has a fixed point y which is a mild solution of the neutral functional evolution inclusion problem (5.5)–(5.6). t u

5.3.3 An Example Consider the following model 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <

Z 0 @ v.t; / T. /u.t; v.t C ; //d @t r @2 v 2 a.t; / 2 .t; / @ Z C

0

P. /R.t; v.t C ; //d r ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ v.t; 0/ D v.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ : v.; / D v0 .; /

t 2 RC ;

2 Œ0;

t 2 RC r 0; 2 Œ0; ; (5.8)

5.3 Neutral Functional Evolution Inclusions

125

where r > 0, a.t; / is a continuous function and is uniformly Hölder continuous in t; T; P W H ! R; u W H R ! R and v0 W H Œ0; ! R are continuous functions and R W RC R ! P.R/ is a multi-valued map with compact convex values. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .5:1:1/ and .G1/; see [112, 149]. For 2 Œ0; , we have y.t/./ D v.t; /; t 2 RC ; '. /./ D v0 .; /; r 0; Z 0 T. /u.t; . /.//d; r 0; g.t; /./ D r

and Z F.t; /./ D

0

r

P. /R.t; . /.//d; r 0:

Then, the problem (5.8) takes the abstract neutral functional evolution inclusion form (5.5)–(5.6). In order to show the existence of mild solutions of problem (5.8), we suppose the following assumptions: – u is Lipschitz with respect to its second argument. Let lip.u/ denotes the Lipschitz constant of u. – There exist p 2 L1 .J; RC / and a nondecreasing continuous function W RC ! .0; C1/ such that jR.t; /j p.t/ .jj/; for 2 J; and 2 R: – T; P are integrable on H: By the dominated convergence theorem, one can show that f 2 SF;y is a continuous function from C.H; E/ to E. Moreover the mapping g is Lipschitz continuous in its second argument, in fact, we have Z jg.t; 1 / g.t; 2 /j M 0 L lip.u/

0 r

jT. /j d j1 2 j ; for 1 ; 2 2 R:

On the other hand, we have for 2 R and 2 Œ0; Z jF.t; /./j

0

r

jp.t/P. /j .j.. //./j/d:

126

Since the function

5 Partial Functional Evolution Inclusions with Finite Delay

is nondecreasing, it follows that Z kF.t; /kP.E/ p.t/

0 r

jP. /j d .jj/:

Proposition 5.6. Under the above assumptions, if we assume that condition (5.7) in Theorem 5.5 is true, then the problem (5.8) has a mild solution which is defined in Œr; C1/:

5.4 Notes and Remarks The results of Chap. 5 are taken from Arara et al. [27, 28]. Other results may be found in [1, 3, 51, 53, 74, 75].

Chapter 6

Partial Functional Evolution Inclusions with Infinite Delay

6.1 Introduction We are interested in this chapter by the study of the existence of mild solutions of two classes of partial functional and neutral functional evolution inclusions with infinite delay on the semi-infinite interval RC . It is known that in the modeling of the evolution of some physical, biological, and economic systems using functional and partial functional differential equations, the response of the systems depends not only on the current state of the system but also on the past history of the system. We assume that the histories yt belongs to some abstract phase space B. Sufficient conditions are provided to get existence results of mild solutions of the partial functional and neutral functional differential evolution problems by applying the recent nonlinear alternative of Frigon [114, 115] for contractive multi-valued maps in Fréchet spaces [116], combined with the semigroup theory [16, 20, 168].

6.2 Partial Functional Evolution Inclusions 6.2.1 Introduction In this chapter, we consider the partial functional evolution inclusions with infinite delay of the form y0 .t/ 2 A.t/y.t/ C F.t; yt /; a.e. t 2 J D RC

(6.1)

y0 D 2 B;

(6.2)

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_6

127

128

6 Partial Functional Evolution Inclusions with Infinite Delay

where F W J B ! P.E/ is a multi-valued map with nonempty compact values, P.E/ is the family of all subsets of E, 2 B are given functions, and fA.t/g0t
6.2.2 Existence of Mild Solutions Definition 6.1. We say that the function y./ W R ! E is a mild solution of the evolution system (6.1)–(6.2) if y.t/ D .t/ for all t 2 .1; 0 and the restriction of y./ to the interval J is continuous and there exists f ./ 2 L1 .J; E/: f .t/ 2 F.t; yt / a.e. in J such that y satisfies the following integral equation: Z y.t/ D U.t; 0/ .0/ C

t 0

U.t; s/ f .s/ ds;

for each t 2 RC :

We will need to introduce the following hypotheses which are assumed hereafter: b 1 such that: (6.1.1) There exists a constant M b for every .t; s/ 2 : kU.t; s/kB.E/ M 1 -Carathéodory with compact (6.1.2) The multi-function F W J B ! P.E/ is Lloc 1 .J; RC / and convex values for each u 2 B and there exist a function p 2 Lloc and a continuous nondecreasing function W J ! .0; 1/ and such that:

kF.t; u/kP.E/ p.t/

.kukB / for a.e. t 2 J and each u 2 B:

1 .J; RC / such that: (6.1.3) For all R > 0, there exists lR 2 Lloc

Hd .F.t; u/ F.t; v// lR .t/ ku vkB for each t 2 J and for all u; v 2 B with kukB R and kvkB R and d.0; F.t; 0// lR .t/

a.e. t 2 J:

Consider the following space ˚ BC1 D y W R ! E W yjŒ0;T 2 C.Œ0; T; E/; y0 2 B ; where yjŒ0;T is the restriction of y to any real compact interval Œ0; T.

6.2 Partial Functional Evolution Inclusions

129

For every n 2 N, we define in BC1 the family of semi-norms by: kykn WD sup f e

where Ln .t/ D

Z

t

0

Ln .t/

jy.t/j W t 2 Œ0; n g

b n .t/ and ln is the function from .6:1:3/: ln .s/ ds; ln .t/ D Kn Ml

Then BC1 is a Fréchet space with the family of semi-norms kkn2N . In what follows we will choose > 1: Theorem 6.2. Suppose that hypotheses (6.1.1)–(6.1.3) are satisfied and moreover Z

C1

c21;n

ds b > Kn M .s/

Z

n 0

p.s/ ds; for each n > 0

(6.3)

b C Mn /kkB . Then evolution problem (6.1)–(6.2) has a mild with c21;n D .Kn MH solution. Proof. Consider the multi-valued operator N21 W BC1 ! P.BC1 / defined by: 8 8 9 if t 0; > ˆ ˆ < < .t/; = Z t N21 .y/ D h 2 BC1 W h.t/ D ˆ ˆ : : U.t; 0/ .0/ C ; U.t; s/ f .s/ ds; if t 0: > 0

where f 2 SF;y D fv 2 L1 .J; E/ W v.t/ 2 F.t; yt / for a.e. t 2 Jg: Clearly, the fixed points of the operator N21 are mild solutions of the problem (6.1)–(6.2). We remark also that, for each y 2 BC1 , the set SF;y is nonempty since, by .6:1:2/; F has a measurable selection (see [94], Theorem III.6). For 2 B; we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 2 .1; 0I

: U.t; 0/ .0/;

if t 2 J:

Then x0 D . For each function z 2 BC1 , set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 6.1 if and only if z satisfies z0 D 0 and Z z.t/ D 0

t

U.t; s/ f .s/ ds;

for t 2 J:

where f .t/ 2 F.t; zt C xt / a:e: t 2 J. Let B0C1 D fz 2 BC1 W z0 D 0g :

130

6 Partial Functional Evolution Inclusions with Infinite Delay

Define in B0C1 , the multi-valued operator F W B0C1 ! P.B0C1 / by: F.z/ D h 2

B0C1

Z

t

W h.t/ D 0

U.t; s/ f .s/ ds;

t2J ;

where f 2 SF;z D fv 2 L1 .J; E/ W v.t/ 2 F.t; zt C xt / for a.e. t 2 Jg: Obviously the operator inclusion N21 has a fixed point is equivalent to the operator inclusion F has one, so it turns to prove that F has a fixed point. Let z 2 B0C1 be a possible fixed point of the operator F. Given n 2 N, then z should be solution of the inclusion z 2 F.z/ for some 2 .0; 1/ and there exists f 2 SF;z , f .t/ 2 F.t; zt C xt / such that, for each t 2 Œ0; n, we have Z

t

jz.t/j

kU.t; s/kB.E/ jf .s/j ds

0

Z

b M

t

p.s/ 0

.kzs C xs kB / ds:

b C Mn /kkB D ˛n , then using the inequality (3.4) and the Set c21;n WD .Kn MH nondecreasing character of , we get b jz.t/j M

Z

t

p.s/ 0

.Kn jz.s/j C ˛n / ds:

Then b Kn jz.t/j C ˛n Kn M

Z

t 0

p.s/ .Kn jz.s/j C ˛n /ds C c21;n :

We consider the function defined by .t/ WD sup f Kn jz.s/j C ˛n W 0 s t g; 0 t < C1: Let t 2 Œ0; t be such that .t/ D Kn jz.t /j C ˛n : By the previous inequality, we have Z t b .t/ Kn M p.s/ ..s// ds C c21;n ; for t 2 Œ0; n: 0

Let us take the right-hand side of the above inequality as v.t/. Then, we have .t/ v.t/ for all t 2 Œ0; n: From the definition of v, we have b v.0/ D c21;n and v 0 .t/ D Kn Mp.t/

..t// a.e. t 2 Œ0; n:

6.2 Partial Functional Evolution Inclusions

131

Using the nondecreasing character of

, we get

b p.t/ v 0 .t/ Kn M

.v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using the condition (6.3), we get Z

v.t/ c21;n

ds b Kn M .s/ b Kn M Z

Z

p.s/ ds Z

C1

< c21;n

t 0 n

p.s/ ds 0

ds : .s/

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kzkn .t/, we have kzkn n : Set U D f z 2 B0C1 W supf jz.t/j W 0 t ng < n C 1 for all n 2 Ng: Clearly, U is an open subset of B0C1 . We shall show that F W U ! P.B0C1 / is a contraction and an admissible operator. First, we prove that F is a contraction; Let z; z 2 B0C1 and h 2 F.z/. Then there exists f .t/ 2 F.t; zt C xt / such that for each t 2 Œ0; n Z

t

h.t/ D 0

U.t; s/ f .s/ ds:

From .5:1:3/ it follows that Hd .F.t; zt C xt /; F.t; zt C xt // ln .t/ kzt zt kB : Hence, there is 2 F.t; zt C xt / such that jf .t/ j ln .t/ kzt zt kB t 2 Œ0; n: Consider U W Œ0; n ! P.E/, given by U D f 2 E W jf .t/ j ln .t/ kzt zt kB g:

132

6 Partial Functional Evolution Inclusions with Infinite Delay

Since the multi-valued operator V.t/ D U .t/\F.t; zt Cxt / is measurable (in [94], see Proposition III.4), there exists a function f .t/, which is a measurable selection for V. So, f .t/ 2 F.t; zt C xt / and using .A1 /, we obtain for each t 2 Œ0; n jf .t/ f .t/j ln .t/ kzt zt kB ln .t/ ŒK.t/ jz.t/ z.t/j C M.t/ kz0 z0 kB ln .t/ Kn jz.t/ z.t/j Let us define, for each t 2 Œ0; n Z

t

h.t/ D 0

U.t; s/ f .s/ ds:

Then we can show as in previous sections that we have kh hkn

1 kz zkn :

By an analogous relation, obtained by interchanging the roles of z and z; it follows that Hd .F.z/; F.z//

1 kz zkB :

So, for > 1, F is a contraction for all n 2 N. It remains to show that F is an admissible operator. Let z 2 B0C1 . Set, for every n 2 N, the space ˚ B0n WD y W .1; n ! E W yjŒ0;n 2 C.Œ0; n; E/; y0 2 B ; and let us consider the multi-valued operator F W B0n ! Pcl .B0n / defined by: F.z/ D h 2

B0n

Z

t

W h.t/ D 0

U.t; s/ f .s/ ds;

t 2 Œ0; n :

n where f 2 SF;y D fv 2 L1 .Œ0; n; E/ W v.t/ 2 F.t; yt / for a.e. t 2 Œ0; ng: From (6.1.1) to (6.1.3) and since F is a multi-valued map with compact values, we can prove that for every z 2 B0n , F.z/ 2 Pcl .B0n / and there exists z 2 B0n such that z 2 F.z /. Let h 2 B0n , y 2 U and > 0. Assume that z 2 F.z/, then we have

jz.t/ z .t/j jz.t/ h.t/j C jz .t/ h.t/j e

Ln .t/

kz F.z/kn C kz hk:

6.2 Partial Functional Evolution Inclusions

133

Since h is arbitrary, we may suppose that h 2 B.z ; / D fh 2 B0n W kh z kn g. Therefore, kz z kn kz F.z/kn C : If z is not in F.z/, then kz F.z/k ¤ 0. Since F.z/ is compact, there exists x 2 F.z/ such that kz F.z/k D kz xk. Then we have jz.t/ z .t/j jz.t/ h.t/j C jx.t/ h.t/j e

Ln .t/

kz F.z/kn C jx.t/ h.t/j:

Thus, kz xkn kz F.z/kn C : So, F is an admissible operator contraction. From the choice of U there is no z 2 @U such that z D F.z/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.31 does not hold. A consequence of the nonlinear alternative due to Frigon we get that .S1/ holds, we deduce that the operator F has a fixed point z . Then y .t/ D z .t/ C x.t/, t 2 R is a fixed point of the operator N21 , which is a mild solution of the evolution inclusion problem (6.1)–(6.2). t u

6.2.3 An Example Consider the following model 8 @v @2 v ˆ ˆ ˆ .t; / 2 a.t; / 2 .t; / ˆ ˆ @t @ ˆ Z 0 ˆ ˆ ˆ ˆ ˆ C P. /R.t; v.t C ; //d < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ :

1

v.t; 0/ D v.t; / D 0 v.; / D v0 .; /

2 Œ0; (6.4) t 2 RC 1 < 0; 2 Œ0; ;

where a.t; / is a continuous function and is uniformly Hölder continuous in tI P W .1; 0 ! R and v0 W .1; 0 Œ0; ! R are continuous functions and R W RC R ! P.R/ is a multi-valued map with compact convex values. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g

134

6 Partial Functional Evolution Inclusions with Infinite Delay

Then A.t/ generates an evolution system U.t; s/ satisfying assumption .6:1:1/ (see [112, 149]). For the phase space B, we choose the well-known space BUC.R ; E/: the space of uniformly bounded continuous functions endowed with the following norm k'k D sup j'./j for 0

' 2 B:

If we put for ' 2 BUC.R ; E/ and 2 Œ0; y.t/./ D v.t; /; t 2 RC ; 2 Œ0; ; . /./ D v0 .; /; 1 < 0; 2 Œ0; ; and Z F.t; '/./ D

0

1

P. /R.t; '. /.//d; 1 < 0; 2 Œ0; :

Then, the problem (6.4) takes the abstract partial functional evolution inclusion form (6.1)–(6.2). In order to show the existence of mild solutions of problem (6.4), we suppose the following assumptions: – There exist p 2 L1 .J; RC / and a nondecreasing continuous function .0; C1/ such that

W RC !

jR.t; u/j p.t/ .juj/; for 2 J; and u 2 R: – P is integrable on .1; 0: By the dominated convergence theorem, one can show that f 2 SF;y is a continuous function from B to .E/. On the other hand, we have for ' 2 B and 2 Œ0; Z jF.t; '/./j Since the function

0

1

jp.t/P. /j .j.'. //./j/d:

is nondecreasing, it follows that Z

kF.t; '/kP.E/ p.t/

0 1

jP. /j d .j'j/; for ' 2 B:

Proposition 6.3. Under the above assumptions, if we assume that condition (6.3) in Theorem 6.2 is true, ' 2 B, then the problem (6.4) has a mild solution which is defined in .1; C1/.

6.3 Neutral Functional Evolution Inclusions

135

6.3 Neutral Functional Evolution Inclusions 6.3.1 Introduction A generalization of existence result of mild solutions to the neutral case is developed in Sect. 6.3 where we look for the neutral functional evolution inclusions with infinite delay of the form d Œy.t/ g.t; yt / 2 A.t/y.t/ C F.t; yt /; a.e. t 2 J dt

(6.5)

y0 D 2 B;

(6.6)

where F W J B ! P.E/ is a multi-valued map with nonempty compact values, g W J B ! E and 2 B are given functions.

6.3.2 Existence of Mild Solutions Definition 6.4. We say that the function y./ W R ! E is a mild solution of the neutral functional evolution system (6.5)–(6.6) if y.t/ D .t/ for all t 2 .1; 0 and the restriction of y./ to the interval J is continuous and there exists f ./ 2 L1 .J; E/: f .t/ 2 F.t; yt / a.e. in J such that y satisfies the following integral equation Z t U.t; s/A.s/g.s; ys /ds y.t/ D U.t; 0/Œ.0/ g.0; / C g.t; yt / C Z t 0 U.t; s/f .s/ ds; for each t 2 RC : C 0

We consider the hypotheses (6.1.1)–(6.1.3) and in what follows we will need the following additional assumptions: .G1/

There exists a constant M 0 > 0 such that: kA1 .t/kB.E/ M 0

.G2/

There exists a constant 0 < L <

1 M 0 Kn

for all t 2 J: such that:

jA.t/ g.t; /j L .kkB C 1/ for all t 2 J and 2 B: .G3/

There exists a constant L > 0 such that: jA.s/ g.s; / A.s/ g.s; /j L .js sj C k kB / for all s; s 2 J and ; 2 B.

136

6 Partial Functional Evolution Inclusions with Infinite Delay

Consider the following space ˚ BC1 D y W R ! E W yjŒ0;T 2 C.Œ0; T; E/; y0 2 B ; where yjŒ0;T is the restriction of y to any real compact interval Œ0; T. b C ln .t/ for the family For every n 2 N, let us take here ln .t/ D Kn MŒL of seminorm fk k g : In what follows we fix

> 0 and assume that n n2N

1 < 1. M 0 L Kn C

Theorem 6.5. Suppose that hypotheses (6.1.1)–(6.1.3) and the assumptions .G1/– .G3/ are satisfied and moreover Z

C1

c22;n

Z n b Kn M ds > max.L; p.s//ds; s C .s/ 1 M 0 LKn 0

for each n > 0;

(6.7)

with " c22;n D

b M 0 LKn M 1 M 0 LKn

C

Kn 1 M 0 LKn

b C Mn / 1 C C .Kn MH

M 0 LKn 1 M 0 LKn

!# kkB

h i b C 1/ C MLn b M 0 L.M ;

then the neutral functional evolution problem (6.5)–(6.6) has a mild solution. Proof. Consider the multi-valued operator N22 W BC1 ! P.BC1 / defined by 8 8 9 .t/; if t 0; > ˆ ˆ ˆ ˆ > ˆ ˆ > ˆ ˆ > ˆ ˆ > ˆ ˆ > ˆ ˆ > 0/ Œ.0/ g.0; / C g.t; y / < < U.t; = t Z t N22 .y/ D h 2 BC1 W h.t/ D C U.t; s/A.s/g.s; ys /ds ˆ ˆ > ˆ ˆ > ˆ ˆ > ˆ ˆ > Z0 t ˆ ˆ > ˆ ˆ > ˆ ˆ : :C ; U.t; s/f .s/ds; if t 0; > 0

where f 2 SF;y D fv 2 L1 .J; E/ W v.t/ 2 F.t; yt / for a.e. t 2 Jg: Clearly, the fixed points of the operator N22 are mild solutions of the problem (6.5)–(6.6). We remark also that, for each y 2 BC1 , the set SF;y is nonempty since, by .6:1:2/, F has a measurable selection (see [94], Theorem III.6). For 2 B, we will define the function x.:/ W R ! E by x.t/ D

8 < .t/;

if t 2 .1; 0I

: U.t; 0/ .0/;

if t 2 J:

6.3 Neutral Functional Evolution Inclusions

137

Then x0 D . For each function z 2 BC1 , set y.t/ D z.t/ C x.t/: It is obvious that y satisfies Definition 6.4 if and only if z satisfies z0 D 0 and Z z.t/ D g.t; zt C xt / U.t; 0/g.0; / C Z

t

C 0

0

t

U.t; s/A.s/g.s; zs C xs /ds

U.t; s/f .s/ds:

where f .t/ 2 F.t; zt C xt / a:e: t 2 J. Let B0C1 D fz 2 BC1 W z0 D 0g : Define in B0C1 , the multi-valued operator F W B0C1 ! P.B0C1 / by:

F.z/ D h 2 B0C1 W h.t/ D g.t; zt C xt / U.t; 0/g.0; / Z

t

C 0

Z U.t; s/A.s/g.s; zs C xs /ds C

t 0

U.t; s/f .s/ds; t 2 J

where f 2 SF;z D fv 2 L1 .J; E/ W v.t/ 2 F.t; zt C xt / for a.e. t 2 Jg: Obviously the operator inclusion N22 has a fixed point is equivalent to the operator inclusion F has one, so it turns to prove that F has a fixed point. Let z 2 B0C1 be a possible fixed point of the operator F. Given n 2 N, then z should be solution of the inclusion z 2 F.z/ for some 2 .0; 1/ and there exists f 2 SF;z , f .t/ 2 F.t; zt C xt / such that, for each t 2 Œ0; n, we have jz.t/j kA1 .t/kB.E/ kA.t/g.t; zt C xt /k C kU.t; 0/kB.E/ kA1 .0/kkA.0/ g.0; /k Z t Z t C kU.t; s/kB.E/ kA.s/g.s; zs C xs /kds C kU.t; s/kB.E/ jf .s/jds 0

0

b 0 L.kkB C 1/ M 0 L.kzt C xt kB C 1/ C MM Z t Z t b b CM L.kzs C xs kB C 1/ds C M p.s/ .kzs C xs kB /ds 0

0

b C 1/ C MLn b C MM b 0 LkkB M 0 Lkzt C xt kB C M 0 L.M Z t Z t b b CM Lkzs C xs kB ds C M p.s/ .kzs C xs kB /ds: 0

0

138

6 Partial Functional Evolution Inclusions with Infinite Delay

Using the inequality (3.4) and the nondecreasing character of

, we obtain

b C 1/ C MLn b C MM b 0 LkkB jz.t/j M 0 L.Kn jz.t/j C ˛n / C M 0 L.M Z t Z t b b CM L.Kn jz.s/j C ˛n /ds C M p.s/ .Kn jz.s/j C ˛n /ds 0

0

b C 1/ C MLn b C M 0 L˛n C MM b 0 LkkB M 0 LKn jz.t/j C M 0 L.M Z t

Z t b CM L.Kn jz.s/j C ˛n /ds C p.s/ .Kn jz.s/j C ˛n /ds : 0

0

Then b C 1/M 0 L C MLn b C M 0 L˛n C MM b 0 LkkB .1 M 0 LKn /jz.t/j .M Z t

Z t b CM L.Kn jz.s/j C ˛n /ds C p.s/ .Kn jz.s/j C ˛n /ds : 0

0

Set c22;n WD ˛n C

Kn 1 M 0 LKn

h i b C 1/M 0 L C MLn b C M 0 L˛n C MM b 0 LkkB : .M

Thus Kn jz.t/j C ˛n c22;n b Kn M

C

Z

t

L.Kn jz.s/j C ˛n /ds 1 M 0 LKn 0

Z t C p.s/ .Kn jz.s/j C ˛n /ds : 0

We consider the function defined by .t/ WD sup f Kn jz.s/j C ˛n W 0 s t g; 0 t < C1: Let t 2 Œ0; t be such that .t/ D Kn jz.t /j C ˛n . By the previous inequality, we have .t/ c22;n C

Z

b Kn M 1 M 0 LKn

0

t

Z L.s/ds C 0

t

p.s/ ..s//ds

for t 2 Œ0; n:

Let us take the right-hand side of the above inequality as v.t/. Then, we have .t/ v.t/ for all t 2 Œ0; n:

6.3 Neutral Functional Evolution Inclusions

139

From the definition of v, we have v.0/ D c22;n and v 0 .t/ D

b Kn M 1 M 0 LKn

ŒL.t/ C p.t/ ..t// a.e. t 2 Œ0; n:

Using the nondecreasing character of b Kn M

v 0 .t/

1 M 0 LKn

, we get

ŒLv.t/ C p.t/ .v.t// a.e. t 2 Œ0; n:

This implies that for each t 2 Œ0; n and using the condition (6.7), we get Z

v.t/

c22;n

Z t b Kn M ds max.L; p.s//ds s C .s/ 1 M 0 LKn 0 Z n b Kn M max.L; p.s//ds 1 M 0 LKn 0 Z C1 ds : < c22;n s C .s/

Thus, for every t 2 Œ0; n, there exists a constant n such that v.t/ n and hence .t/ n . Since kzkn .t/, we have kzkn n : We can show that F is an admissible operator and we shall prove now that FQ W U ! P.B0C1 / is a contraction. Let z; z 2 B0C1 and h 2 F.z/. Then there exists f .t/ 2 F.t; zt C xt / such that for each t 2 Œ0; n, we have Z h.t/ D g.t; zt C xt / U.t; 0/g.0; / C Z

t

C 0

t 0

U.t; s/A.s/g.s; zs C xs /ds

U.t; s/f .s/ds:

From .5:1:3/ it follows that Hd .F.t; zt C xt /; F.t; zt C xt // ln .t/ kzt zt kB : Hence, there is 2 F.t; zt C xt / such that jf .t/ j ln .t/ kzt zt kB t 2 Œ0; n: Consider U W Œ0; n ! P.E/, given by U D f 2 E W jf .t/ j ln .t/ kzt zt kB g:

140

6 Partial Functional Evolution Inclusions with Infinite Delay

Since the multi-valued operator V.t/ D U .t/\F.t; zt Cxt / is measurable (in [94], see Proposition III.4), there exists a function f .t/, which is a measurable selection for V. So, f .t/ 2 F.t; zt C xt / and using .A1 /, we obtain for each t 2 Œ0; n jf .t/ f .t/j ln .t/ kzt zt kB ln .t/ ŒK.t/ jz.t/ z.t/j C M.t/ kz0 z0 kB ln .t/ Kn jz.t/ z.t/j: Let us define, for each t 2 Œ0; n Z h.t/ D g.t; zt C xt / U.t; 0/g.0; / C Z

t

C 0

t 0

U.t; s/A.s/g.s; zs C xs /ds

U.t; s/f .s/ds:

Then we can show as in previous sections that we have for each t 2 Œ0; n and n2N

1 kz zkn : kh hkn M 0 L Kn C

By an analogous relation, obtained by interchanging the roles of z and z; it follows that

1 kz zkB : Hd .F.z/; F.z// M 0 L Kn C

1 < 1, the operator F is a contraction for all n 2 N and an So, for M 0 L Kn C

admissible operator. From the choice of U there is no z 2 @U such that z D F.z/ for some 2 .0; 1/. Then the statement .S2/ in Theorem 1.31 does not hold. By the nonlinear alternative due to Frigon we get that .S1/ holds, we deduce that the operator F has a fixed point z . Then y .t/ D z .t/ C x.t/, t 2 R is a fixed point of the operator N22 , which is a mild solution of the neutral functional evolution inclusion problem (6.5)–(6.6). t u

6.3 Neutral Functional Evolution Inclusions

141

6.3.3 An Example Consider the following model 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <

Z 0 @ v.t; / T. /u.t; v.t C ; //d @t 1 2 @v 2 a.t; / 2 .t; / @ Z C

0

P. /R.t; v.t C ; //d 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ v.t; 0/ D v.t; / D 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ : v.; / D v0 .; /

t 2 RC ;

2 Œ0;

t 2 RC 1 < 0; 2 Œ0; ; (6.8)

where a.t; / is a continuous function and is uniformly Hölder continuous in t; T; P W .1; 0 ! R; u W .1; 0 R ! R and v0 W .1; 0 Œ0; ! R are continuous functions and R W RC R ! P.R/ is a multi-valued map with compact convex values. Consider E D L2 .Œ0; ; R/ and define A.t/ by A.t/w D a.t; /w00 with domain D.A/ D f w 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0 g Then A.t/ generates an evolution system U.t; s/ satisfying assumptions .4:1:1/ and .G1/; see [112, 149]. For the phase space B, we choose the well-known space BUC.R ; E/: the space of uniformly bounded continuous functions endowed with the following norm k'k D sup j'./j for 0

' 2 B:

If we put for ' 2 BUC.R ; E/ and 2 Œ0; y.t/./ D v.t; /; t 2 RC ; 2 Œ0; ; . /./ D v0 .; /; 1 < 0; 2 Œ0; ; Z 0 T. /u.t; '. /.//d; 1 < 0; 2 Œ0; ; g.t; '/./ D 1

and Z F.t; '/./ D

0

1

P. /R.t; '. /.//d; 1 < 0; 2 Œ0; :

142

6 Partial Functional Evolution Inclusions with Infinite Delay

Then, the problem (6.8) takes the abstract neutral functional evolution inclusion form (6.5)–(6.6). In order to show the existence of mild solutions of problem (6.8), we suppose the following assumptions: – u is Lipschitz with respect to its second argument. Let lip.u/ denotes the Lipschitz constant of u. – There exist p 2 L1 .J; RC / and a nondecreasing continuous function W RC ! .0; C1/ such that jR.t; x/j p.t/ .jxj/; for 2 J; and x 2 R: – T; P are integrable on .1; 0: By the dominated convergence theorem, one can show that f 2 SF;y is a continuous function from B to .E/. Moreover the mapping g is Lipschitz continuous in its second argument, in fact, we have Z jg.t; '1 / g.t; '2 /j M 0 L lip.u/

0 1

jT. /j d j'1 '2 j ; for '1 ; '2 2 B:

On the other hand, we have for ' 2 B and 2 Œ0; Z jF.t; '/./j Since the function

0

1

jp.t/P. /j .j.'. //./j/d:

is nondecreasing, it follows that Z

kF.t; '/kP.E/ p.t/

0 1

jP. /j d .j'j/; for ' 2 B:

Proposition 6.6. Under the above assumptions, if we assume that condition (6.7) in Theorem 6.5 is true, ' 2 B, then the problem (6.8) has a mild solution which is defined in .1; C1/.

6.4 Notes and Remarks The results of Chap. 6 are taken from [11, 12, 35, 45]. Other results may be found in [10, 11, 54, 142, 145, 182].

Chapter 7

Densely Defined Functional Differential Inclusions with Finite Delay

7.1 Introduction In this chapter, we are concerned by the existence of mild and extremal solutions of some first order classes of impulsive semi-linear functional differential inclusions with local and nonlocal conditions when the delay is finite in a separable Banach space .E; j j/: In the literature devoted to equations with finite delay, the phase space is much of time the space of all continuous functions on H, endowed with the uniform norm topology. We mention, for instance, the books of Ahmed [16], Engel and Nagel [106], Kamenskii et al. [144], Pazy [168], and Wu [184].

7.2 Existence of Mild Solutions with Local Conditions 7.2.1 Introduction In this section, we consider the following class of semi-linear impulsive differential inclusions: y0 .t/ Ay.t/ 2 F.t; yt /; t 2 J WD Œ0; b; t ¤ tk

(7.1)

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(7.2)

y.t/ D .t/; t 2 H;

(7.3)

where F W J D ! 2E is a closed, bounded, and convex valued multi-valued map,

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_7

143

144

7 Densely Defined Functional Differential Inclusions with Finite Delay

D D f W H ! E; continuous everywhere except for a finite number of points at which .s / and .sC / exist and .s / D .s/g; 2 D, A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 0, E a real separable Banach space endowed with the norm j:j; 0 D t0 < t1 < < tm < tmC1 D b; Ik 2 C.E; E/; .k D 1; 2; : : :; m/:

7.2.2 Main Result We assume that F is compact and convex valued multi-valued map. In order to define the mild solution to the problem (7.1)–(7.3), we shall consider the following space n PC D y W Œ0; b ! E W

yk 2 CŒJk ; E; k D 0; : : :; m

such that o y.tk /; y.tkC / exist with y.tk / D y.tk /; k D 1; : : :; m ;

which is a Banach space with the norm kykPC WD maxfkyk k1 W k D 0; : : :; mg; where yk is the restriction of y to Jk D Œtk ; tkC1 ; k D 0; : : :; m: Set ˝ D fy W Œr; b ! E W y 2 D \ PCg: Definition 7.1. A function y 2 ˝ is said to be a mild solution of system (7.1)– (7.3) if y.t/ D .t/ for all t 2 H, the restriction of y./ to the interval Œ0; b is continuous and there exists v./ 2 L1 .Jk ; E/; Ik 2 Ik .y.tk // such that v.t/ 2 F.t; yt / a.e t 2 Œ0; b, and such that y satisfies the integral equation, Z y.t/ D T.t/.0/ C 0

t

T.t s/v.s/ds C

X

T.t tk /Ik ; t 2 J:

0
We will need the following hypotheses which are assumed hereafter (7.1.1) A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup fT.t/g, t 2 J which is compact for t > 0 in the Banach space E, and there exists a constant M 1, such that Let kT.t/kB.E/ MI t 0 (7.1.2) There exist constants ck 0, k D 1; : : : ; m such that Hd .Ik .y/ Ik .x// ck jy xj for each x; y 2 E: (7.1.3) F is L1 -Carathéodory with compact convex values.

7.2 Existence of Mild Solutions with Local Conditions

145

(7.1.4) There exist a function k 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/k k.t/ .kxkD / for a.e. t 2 J and each x 2 D; with Z

1

ds > C1 kpkL1 ; .s/

C0

(7.4)

where m X

MŒkkD C C0 D 1

kIk .0/k

kD1 m X

(7.5)

ck

kD1

C1 D 1

M m X

:

(7.6)

ck

kD1

Theorem 7.2. Assume that (7.1.1)–(7.1.4) hold. If M

m X

ck < 1

(7.7)

kD1

then the IVP (7.1)–(7.3) has at least one mild solution on Œr; b: Proof. Transform the problem (7.1)–(7.3) into a fixed point problem. Consider the multi-valued operator: N W ˝ ! ˝ defined by 8 8 .t/; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z t ˆ ˆ < < T.t/.0/ C T.t s/v.s/ds N.y/ D h 2 ˝ W h.t/ D 0 ˆ ˆ ˆ ˆ X ˆ ˆ ˆ ˆ ˆ ˆ T.t tk /Ik ; v 2 SF;y ; Ik 2 Ik .y.tk // ˆ ˆ :C : 0
9 if t 2 H, > > > > > > = > > > > if t 2 J: > > ;

It is clear that the fixed points of N are mild solutions of the IVP (7.1)–(7.3). Consider these multi-valued operators: A; B W ˝ ! ˝

146

7 Densely Defined Functional Differential Inclusions with Finite Delay

defined by 8 8 ˆ ˆ < 0; < X A.y/ WD h 2 ˝ W h.t/ D T.t tk /Ik ; Ik 2 Ik .y.tk // ˆ ˆ : : 0
9 if t 2 H; = > if t 2 J; > ;

and 8 8 .t/; ˆ ˆ ˆ ˆ ˆ ˆ < < B.y/ WD h 2 ˝ W h.t/ D T.t/.0/ Z t ˆ ˆ ˆ ˆ ˆ ˆ : C : T.t s/v.s/ds; v 2 SF;y

9 if t 2 H; > > > = > > > if t 2 J: ;

0

The problem of finding mild solutions of (7.1)–(7.3) is then reduced to finding mild solutions of the operator inclusion y 2 A.y/ C B.y/: The proof will be given in several steps. A is a contraction. Let y1 ; y2 2 ˝, then from (7.1.2) we have 0 1 X X T.t tk /Ik .y1 .tk //; T.t tk /Ik .y2 .tk //A Hd .A.y1 /; A.y2 // D Hd @

Step 1:

0
M

kDm X

0
ck jŒy1 .tk / y2 .tk /j

kD0

M

kDm X

ck ky1 y2 k˝ :

kD0

From (7.7) it follows that A is a contraction. Step 2:

B has compact, convex values, and is completely continuous.

Claim 1: B has compact values. The operator B is equivalent to the composition L ı SF of two operators on L1 .J; E/, where L W L1 .J; E/ ! ˝ is the continuous operator defined by Z L.v.t// D T.t/.0/ C

t

T.t s/v.s/ds

0

Then, it suffices to show that L ı SF has compact values on ˝. Let y 2 ˝ arbitrary, vn a sequence in SF;y , then by definition of SF , vn .t/ belongs to F.t; yt /; a:e:t 2 J. Since F.t; yt / is compact, we may pass to a subsequence. suppose that vn ! v in L1 .J; E/, where v.t/ 2 F.t; yt /; a:e:t 2 J.

7.2 Existence of Mild Solutions with Local Conditions

147

From the continuity of L, it follows that Lvn .t/ ! Lv.t/ point wise on J and n ! 1: In order to show that the convergence is uniform, we first show that fLvn g is an equi-continuous sequence. Let 1 ; 2 2 J, then we have: Z 1 ˇ ˇ ˇ ˇL.vn . 1 // L.vn . 2 //ˇ D ˇT. 1 /.0/ T. 2 /.0/ C T. 1 s/vn .s/ds 0

Z

2

0

ˇ T. 2 s/vn .s/dsˇ

ˇ ˇ

ˇ T. 1 / T. 2 / .0/ˇ Z 1 ˇ

ˇ ˇ T. 1 s/ T. 2 s/ ˇjvn .s/jds C Z

0

2

C

1

jT. 2 s/jjvn .s/jds:

As 1 ! 2 , the right-hand side of the above inequality tends to zero. Since T.t/ is a strongly continuous operator and the compactness of T.t/I t > 0; implies the continuity in uniform topology. Hence fLvn g is equi-continuous, and an application of Arzéla–Ascoli theorem implies that there exists a subsequence which is uniformly convergent. Then we have Lvnj ! Lv 2 .L ı SF /.y/ as j 7! 1, and so .L ı SF /.y/ is compact . Therefore B has compact values. Claim 2: B.y/ is convex for each y 2 ˝: Let h1 ; h2 2 B.y/, then there exists v1 ; v2 2 SF;y such that, for each t 2 J we have

hi .t/ D

8 ˆ < .t/;

if t 2 H,

ˆ : T.t/.0/ C

Z 0

t

T.t s/vi .s/ds

9 > =

; if t 2 J;i D 1; 2: >

Let 0 ı 1: Then, for each t 2 J, we have 8 .t/; ˆ ˆ ˆ < .ıh1 C .1 ı/h2 /.t/ D T.t/.0/ ˆ ˆ ˆ Rt : C 0 T.t s/Œıv1 .s/ C .1 ı/v2 .s/ds Since F.t; yt / has convex values, one has ıh1 C .1 ı/h2 2 B.y/:

9 t 2 H, > > > = t 2 J;

> > > ;

148

7 Densely Defined Functional Differential Inclusions with Finite Delay

Claim 3: B maps bounded sets into bounded sets in ˝ Let B D fy 2 ˝I kyk1 qg; q 2 RC a bounded set in ˝. For each h 2 B.y/, for some y 2 B, there exists v 2 SF;y such that Z

t

h.t/ D T.t/.0/ C

T.t s/v.s/ds:

0

Thus Z

t

jh.t/j Mj.0/j C M 0

'q .s/ds

Mj.0/j C Mk'q kL1 ; this implies that: khk1 Mj.0/j C Mk'q kL1 : Hence B.B/ is bounded. Claim 4: B maps bounded sets into equi-continuous sets. Let B is a bounded set as in Claim 3 and h 2 B.y/ for some y 2 B: Then, there exists v 2 SF;y such that Z t T.t s/v.s/ds; t 2 J h.t/ D T.t/.0/ C 0

Let 1 ; 2 2 Jnft1 ; t2 ; : : :tm g; 1 < 2 . Thus if > 0, we have jh. 2 / h. 1 /j jŒT. 2 / T. 1 /.0/j Z 1 C kT. 2 s/ T. 1 s/kjv.s/jds Z

0

1

C Z

1

2

C

1

kT. 2 s/ T. 1 s/kjv.s/jds

kT. 2 s/kjv.s/jds

jŒT. 2 / T. 1 /.0/j Z 1 C kT. 2 s/ T. 1 s/k'q .s/ds Z

0

1

C

1

Z

CM

kT. 2 s/ T. 1 s/k'q .s/ds

2

1

'q .s/ds:

7.2 Existence of Mild Solutions with Local Conditions

149

As 1 ! 2 and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0 implies the continuity in the uniform operator topology. This proves the equi-continuity for the case where t ¤ ti ; i D 1; : : :; m C 1. It remains to examine the equi-continuity at t D ti . First we prove the equi-continuity at t D ti , we have for some y 2 B, there exists v 2 SF;y such that Z t T.t s/v.s/ds; t 2 J h.t/ D T.t/.0/ C 0

Fix ı1 > 0 such that ftk ; k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < < ı1 , we have jh.ti / h.ti /j jŒT.ti / T.ti /.0/j Z ti C kT.ti s/ T.ti s/kjv.s/jds 0 Z ti M'q .s/ds C ti

Which tends to zero as ! 0. Define hO 0 .t/ D h.t/; and hO i .t/ D

t 2 Œ0; t1

8 < h.t/;

9 if t 2 .ti ; tiC1 =

: h.tC /;

if t D ti :

i

;

Next, we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk ; k ¤ ig \ Œti ı2 ; ti C ı2 D ;. Then Z ti O i / D T.ti /.0/ C h.t T.ti s/v.s/ds: 0

For 0 < < ı2 ; we have O i /j jŒT.ti C / T.ti /.0/j O i C / h.t jh.t Z ti C kT.ti C s/ T.ti s/kjv.s/jds Z

0

ti C

C ti

M'q .s/ds:

150

7 Densely Defined Functional Differential Inclusions with Finite Delay

The right-hand side tends to zero as ! 0: The equi-continuity for the cases 1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval H As a consequence of Claims 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t . For y 2 B, we define

Z

h .t / D T.t /.0/ C T./

t 0

T.t s /v.s/ds;

where v 2 SF;y . Since T.t / is a compact operator, the set H .t / D fh .t / W h 2 B.y/g is precompact in E for every ; 0 < < t : Moreover, for every h 2 B.y/ we have ˇ jh.t / h .t /j D ˇ

Z

Z

t 0

t t

Z

M

T.t s/v.s/ds T./

0

Z Dj

t

ˇ T.t s /v.s/dsˇ

T.t s/v.s/dsj

t

t

'q .s/ds:

Therefore, there are precompact sets arbitrarily close to the set H.t / D fh.t / W h 2 B.y/g: Hence the set H.t / D fh.t / W h 2 B.B/g is precompact in E. Hence the operator B is completely continuous. Claim 5: B has closed graph. Let yn ! y , hn 2 B.yn /, and hn ! h . We shall show that h 2 B.y /: hn 2 B.yn / means that there exists vn 2 SF;yn such that Z hn .t/ D T.t/.0/ C

t

T.t s/vn .s/ds; t 2 J:

0

We must prove that there exists v 2 SF;y such that Z h .t/ D T.t/.0/ C

t 0

T.t s/v .s/ds:

Consider the linear and continuous operator K W L1 .J; E/ ! D defined by Z .Kv/.t/ D

0

t

T.t s/v.s/ds:

7.2 Existence of Mild Solutions with Local Conditions

151

We have j.hn .t/T.t/.0// .h .t/ T.t/.0//j D jhn .t/ h .t/j khn h k1 ! 0;

as n 7! 1:

From Lemma 1.11 it follows that K ı SF is a closed graph operator and from the definition of K one has hn .t/ T.t/.0/ 2 K ı SF;yn : As yn ! y and hn ! h , there is a v 2 SF;y such that Z h .t/ T.t/.0/ D

t 0

T.t s/v .s/ds:

Hence the multi-valued operator B is upper semi-continuous. Step 3:

A priori bounds on solutions. Now, it remains to show that the set E D fy 2 ˝j y 2 Ay C By; 0 1g

is unbounded. Let y 2 E be any element. Then there exist v 2 SF;y and Ik 2 Ik .y.tk // such that Z y.t/ D T.t/.0/ C

t 0

T.t s/v.s/ds C

X

T.t tk /Ik :

0
Then for each t 2 J Z

t

jy.t/j Mj.0/j C M

jv.s/jds C M

0

Z MkkD C M

0

Z MkkD C M

CM

m X kD0

0

jIk j

kD0

p.s/ .kys k/ds C M

m X

jIk j

kD0 t

0

Z MkkD C M

t

m X

p.s/ .kys k/ds C M

m X

ck jy.tk /j C M

kD0 t

p.s/ .kys k/ds C M

ck jy.tk /j:

m X kD0

kDm X kD0

jIk .0/j

jIk .0/j

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7 Densely Defined Functional Differential Inclusions with Finite Delay

Set X mjIk .0/j : C D M kkD C kD0

Then, we have: jy.t/j C C M

m X

ck jy.tk /j

Z

t

CM 0

kD0

p.s/ .kys k/ds:

(7.8)

Consider the function .t/ defined by .t/ D supfjy.s/j W r s tg; 0 t b Then, we have, for all t 2 J,kys k .t/. Let t 2 J such that .t/ D jy.t /j, then by the previous inequality we have, for t 2 J; .t/ C C M

m X

Z ck j.t/j C M

kD0

t 0

p.s/ .ksk/ds:

(7.9)

p.s/ ..s//ds:

(7.10)

Thus 1M

m X

!

Z

ck .t/ C C M

kD0

t 0

It follows that Z .t/ C0 C C1

t 0

p.s/ ..s//ds:

Then, we have .t/ v.t/

for all t 2 J;

v.0/ D C0 : Differentiating both sides of the above equality, we obtain v 0 .t/ D C1 p.t/ ..t//; a:e: t 2 J; and using the nondecreasing character of the function

, we obtain

v 0 .t/ C1 p.t/ .v.t//; a:e: t 2 J;

(7.11)

7.2 Existence of Mild Solutions with Local Conditions

153

that is v 0 .t/ C1 p.t/; a:e: t 2 J: .v.t//

(7.12)

Integrating both sides of the previous inequality from 0 to t we get Z

t 0

v 0 .s/ ds C1 .v.s//

Z

t

p.s/ds: 0

By a change of variables we get Z

v.t/

v.0/

du C1 kpkL1 .u/

Z

1

c0

du : .u/

Hence there exists a constant K such that .t/ v.t/ K; for all t 2 J: Now from the definition of it follows that kyk˝ D sup jy.t/j .b/ K t2Œr;b

for all y 2 E: This shows that the set E is bounded. As a consequence of Theorem 1.32, we deduce that A C B has a fixed point y on Œr; b which is a mild solution of our problem. t u

7.2.3 Existence of Extremal Mild Solutions In this subsection we prove the existence of maximal and minimal mild solutions of problem (7.1)–(7.3) under suitable monotonicity conditions on the multi-valued functions involved in it. Our proof is based upon the Theorem 1.37 due to Dhage. Let us introduce the concept of lower and upper mild solutions for problem (7.1)–(7.3). Definition 7.3. We say that a continuous function v W Œr; b ! E is a lower mild solution of problem (7.1)–(7.3) if there exist functions v 2 L1 .J; E/ such that v.t/ 2 F.t; yt /; a.e. on J; y.t/ D .t/; t 2 H; and Z y.t/ T.t/.0/ C 0

t

T.t s/v.s/ds C

X

T.t tk /Ik ..tk //; t 2 J; t ¤ tk

0
and v.tkC / v.tk Ik .v.tk //; t D tk ; k D 1; : : :m: Similarly an upper mild solution w of (7.1)–(7.3) is defined by reversing the order.

154

7 Densely Defined Functional Differential Inclusions with Finite Delay

Definition 7.4. A solution xM of IVP (7.1)–(7.3) is said to be maximal if for any other solution x of IVP (7.1)–(7.3) on J, we have that x.t/ xM .t/ for each t 2 J. Similarly a minimal solution of IVP (7.1)–(7.3) is defined by reversing the order of the inequalities. We consider the following assumptions in the sequel. (7.4.1) The multi-valued function F.t; y/ and is strictly monotone increasing in y for almost each t 2 J. (7.4.2) The IVP (7.1)–(7.3) has a lower mild solution v and an upper mild solution w with v w. (7.4.3) T.t/ is preserving the order, that is T.t/v 0 whenever v 0. (7.4.4) The multi-valued functions Ik ; k D 1; : : :m are continuous and nondecreasing. Theorem 7.5. Assume that assumptions (7.1.11)–(7.1.4) and (7.4.1)–(7.4.4) hold. Then IVP (7.1)–(7.3) has minimal and maximal solutions on Œr; b. Proof. It can be shown as in the proof of Theorem 7.2 that A is completely continuous and B is a contraction on Œv; w. We shall show that A and B are isotone increasing on Œv; w. Let y; y 2 Œv; w be such that y y; y 6D y: Then by (7.4.4), we have for each t 2 J X A.y/ D fh 2 ˝ W h.t/ D T.t tk /Ik ; Ik 2 Ik .y.tk //g 0
P fh 2 ˝ W h.t/ D

X

T.t tk /Ik ; Ik 2 Ik .y.tk //g

0
D A.y/: Similarly, by .7:4:1/; .7:4:3/ Z B.y/ D fh 2 ˝ W h.t/ D T.t/.0/ C

0

Z P fh 2 ˝ W h.t/ D T.t/.0/ C

t

0

t

T.t s/v.s/ds; v 2 SF;y g T.t s/v.s/ds; v 2 SF;y g

D B.y/: Therefore A and B are isotone increasing on Œv; w. Finally, let x 2 Œv; w be any element. By (7.4.2), (7.4.3) we deduce that v A.v/ C B.v/ A.x/ C B.x/ A.w/ C B.w/ w; which shows that A.x/ C B.x/ 2 Œv; w for all x 2 Œv; w. Thus, A and B satisfy all conditions of Theorem 7.5, hence IVP (7.1)–(7.3) has maximal and minimal solutions on J: t u

7.3 Existence of Mild Solutions with Nonlocal Conditions

155

7.3 Existence of Mild Solutions with Nonlocal Conditions In this section we consider the following class of semi-linear Impulsive differential inclusions: y0 .t/ Ay.t/ 2 F.t; yt /; t 2 J WD Œ0; b; t ¤ tk

(7.13)

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(7.14)

y.t/ C ht .y/ D .t/; t 2 Œr; 0 ;

(7.15)

where A F and Ik are as in the previous section, and ht W C.H; E/ ! E is a given function. The nonlocal Cauchy Problem was introduced by Byszewski in [89], and the importance of nonlocal conditions in different fields has been discussed in [89, 90].

7.3.1 Main Result Let us start by the definition of the mild solution of the problem (7.13)–(7.15) Definition 7.6. A function y 2 ˝ is said to be a mild solution of problem (7.13)– (7.15) if y.t/ D .t/ ht .y/ ; t 2 Œr; 0; and the restriction of y./ to the interval Œ0; b is continuous and there exist v./ 2 L1 .Jk ; E/ and Ik 2 Ik .y.tk // such that v.t/ 2 F.t; yt / a.e t 2 Œ0; b, and y satisfies the integral equation, y.t/ D T.t/ ..0/ h0 .y// C

Rt 0

T.t s/v.s/ds C

X

T.t tk /Ik

0
Let us introduce the following assumptions. (7.6.1) The function h is continuous with respect to t, and there exists a constant ˛ > 0 such that kht .u/k ˛;

u 2 C.H; E/

and for each k > 0 the set f.0/ h0 .y/; y 2 C.H; E/; kyk kg is precompact in E. Theorem 7.7. Assume that hypotheses (7.1.1)–(7.1.4) and (7.6.1) hold. Then the IVP (7.13)–(7.15) has at least one mild solution on Œr; b:

156

7 Densely Defined Functional Differential Inclusions with Finite Delay

Proof. Consider the two multi-valued operators A1 B1 : ˝ ! P.˝/ 8 8 .t/ ht .y/; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < T.t/ ..0/ h0 .y// < Z t B1 .y/ WD f 2 ˝ W f .t/ D ˆ ˆ ˆ ˆ C T.t s/v.s/ds; v 2 SF;y ˆ ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ : : 8 ˆ <

8 ˆ < 0; X A1 .y/ WD f 2 ˝ W f .t/ D T.t tk /Ik ; Ik 2 Ik .y.tk //; ˆ ˆ : : 0
9 if t 2 H; > > > > > > =

if t 2 J;

> > > > > > ;

9 if t 2 H; > = if t 2 J; > ;

Then the problem of finding the solution of problem (8.8)–(8.11) is reduced to finding the solution of the operator inclusion y 2 A1 .y/ C B1 .y/: By parallel steps of Theorem 7.2 we can show that the operators A1 and B1 satisfy all conditions of Theorem 1.32. t u

7.4 Application to the Control Theory This section is devoted to an application of the argument used in previous sections to the controllability of a semi-linear functional differential inclusions. More precisely we will consider the following IVP: y0 .t/ Ay.t/ 2 F.t; yt / C Bu.t/; t 2 J WD Œ0; b; t ¤ tk

(7.16)

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(7.17)

y.t/ D .t/; t 2 H;

(7.18)

where A and F are as in the previous section, the control function u./ is given in L2 .J; U/, a Banach space of admissible control functions, with U as a Banach space. Finally B is a bounded linear operator from U to E. In the case of single-valued functions Ik , the problem (7.16–7.18) has been recently studied in the monographs by Ahmed [16], and Benchohra et al. [81], and in the papers [18, 19].

7.4.1 Main Result Before stating and proving our result we give the meaning of mild solution of our problem (7.16)–(7.18).

7.4 Application to the Control Theory

157

Definition 7.8. A function y 2 ˝ is said to be a mild solution of system (7.16)– (7.18) if y.t/ D .t/ for all t 2 H, the restriction of y./ to the interval Œ0; b is continuous and there exists v./ 2 L1 .Jk ; E/ and Ik 2 Ik .y.tk //, such that v.t/ 2 F.t; yt / a.e Œ0; b, and such that y satisfies the integral equation, Z t Z t y.t/ D T.t/.0/ C T.t s/v.s/ds C T.t s/Buy .s/ds 0 0 X C T.t tk /Ik ; t 2 J: 0
Theorem 7.9. Assume that hypotheses (7.1.1)–(7.1.3) hold. Moreover we suppose that: (C1)

the linear operator W W L2 .J; U/ ! E; defined by Z b T.b s/Bu.s/ds; Wu D 0

has a bounded inverse operator W 1 which takes values in L2 .J; U/nKerW, and there exist positive constants M; M 1 such that kBk M and kW 1 k M 1 . (C2) F has closed, bounded and convex values, and there exists a function l 2 L1 .J; RC / such that

Hd F.t; y/; F.t; x/ l.t/ky xkD ; for a.e. t 2 J; x; y 2 D (C3) There exist a function k 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/kP k.t/ .kxkD / for a.e. t 2 J and each x 2 D; with Z

1 C0

ds D 1; s C .s/

where C0 D

C ; m X 1 ck kD0

with C D MkkD C MMM 1 b jy1 j C MkkD X jIk .0/j: C M 2 MM 1 b C M 0
(7.19)

158

7 Densely Defined Functional Differential Inclusions with Finite Delay

If m m X X MMM 1 b`kL1 C M 2 MM 1 b ck C M ck < 1; kD0

kD0

then the IVP (7.16)–(7.18) is controllable on Œr; b: Proof. Using hypothesis (C1) for each arbitrary function y./ define the control 2 3 Z b X uy .t/ D W 1 4y1 T.b/.0/ T.b s/v.s/ds T.b tk /Ik 5 .t/; 0

0
where v 2 SF;y and Ik 2 Ik .y.tk //. We shall show that the operator N W ˝ ! P.˝/ defined by 8 8 .t/; if t 2 H, ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z t ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ T.t/.0/ C T.t s/v.s/ds ˆ ˆ < < 0 Z N.y/ D f 2 ˝ W f .t/ D t ˆ ˆ ˆ ˆ C T.t s/.Buy /.s/ds ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ X ˆ ˆ ˆ ˆ ˆ ˆ T.t tk /Ik ; Ik 2 Ik .y.tk //; v 2 SF;y if t 2 J ˆ ˆ :C : 0
9 > > > > > > > > > > > = > > > > > > > > > > > ;

has a fixed point. This fixed point is then the mild solution of the IVP (7.16)–(7.18). Consider the multi-valued operators: A; B W ˝ ! P.˝/ defined by 8 8 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
9 if t 2 H; > > > > > > = > > > > if t 2 J; > > ;

and 8 8 9 .t/; if t 2 H; > ˆ ˆ ˆ ˆ > ˆ ˆ > ˆ ˆ > < < = Z t B.y/ WD f 2 ˝ W f .t/ D T.t/.0/ C T.t s/v.s/ds; v 2 S F;y ˆ ˆ > ˆ ˆ > 0 ˆ ˆ > ˆ ˆ > : : ; if t 2 J.

7.4 Application to the Control Theory

159

It is clear that N D A C B: Similarly, as in Theorem 7.2, we can prove that A is a contraction operator, and B is a completely continuous operator with compact convex values. Now, we prove that the set: E D fy 2 ˝j y 2 Ay C By; 0 1g is unbounded. Let y 2 E be any element. Then there exist v 2 SF;y and Ik 2 Ik .y.tk // such that Z y.t/ D T.t/.0/ C X

C

t 0

Z T.t s/v.s/ds C

t 0

T.t s/Buy .s/ds

T.t tk /Ik :

0
This implies by (C1)–(C3) that Z jy.t/j MkkD C M CM 2 MM 1 b

CM

kDm X

k.s/ .kys k/ds C MMM 1 b jy1 j C MkkD

t

0

Z

t 0

k.s/ .kys k/ds C M 2 MM 1

Z

t

X

0 0
jIk jds

jIk j:

kD0

Thus jy.t/j C C M 2 MM 1

CM

kDm X

Z

t

X

0 0
ck jy.tk /jds

ck jy.tk /j C M 2 MM 1 b C M

kD0

(7.20) Z

t 0

k.s/ .kys k/ds:

Consider the function .t/ defined by .t/ D supfjy.s/j W r s tg; 0 t b

160

7 Densely Defined Functional Differential Inclusions with Finite Delay

Then, we have, for all t 2 J,kys k .t/: Let t 2 J such that .t/ D jy.t /j; then by (7.20) we have, for t 2 J; Z

.t/ C C M 2 MM 1

CM

kDm X

X

t

0 0
ck .s/ds

(7.21)

ck .t/ C M 2 MM 1 b C M

Z

kD0

t 0

p.s/ ..s//ds:

From (7.21) we obtain m X

1M ck .t/ C C M 2 MM 1

Z

kD0

X

t

0 0
C M 2 MM 1 b C M

Z

t 0

ck .s/ds

(7.22)

p.s/ ..s//ds:

Let C0 D

C ; m X 1M ck

M 2 MM 1 ; m X 1M ck

C1 D

kD0

C2 D

M 2 MM 1 b C M (7.23) m X 1M ck

kD0

kD0

It follows from (7.22) and (7.23) that .t/ C0 C C1 CC2

C0

Z

t

C

t

X

0 0
ck .s/ds

p.s/ ..s//ds

0

Z

Z

t

0

O M.s/ .s/ C p.s/ ..s// ds;

where X

O M.s/ D max C1 ck ; C2 k.s/ .s/ 0
Let v.t/ D C0 C

Z 0

t

O M.s/ .s/ C

..s// ds:

(7.24)

7.4 Application to the Control Theory

161

Then, we have .t/ v.t/ for all t 2 J Differentiating both sides of (7.24), we obtain O C v 0 .t/ D M.t/Œ.t/

..t//; a:e: t 2 J

and v.0/ D C0 : Using the nondecreasing character of the function O C v 0 .t/ M.t/Œv.t/

, we obtain

.v.t//; a:e: t 2 J;

that is v 0 .t/ O M.t/; a:e: t 2 J: v.t/ C .v.t//

(7.25)

Integrating from 0 to t both sides of (7.25) we get Z

v 0 .s/ ds v.s/ C .v.s//

t

0

Z

t

O M.s/ds:

0

By a change of variables we get Z

v.t/

v.0/

du O L1 1: kMk u C .u/

From (7.19) there exists a constant K such that .t/ v.t/ K

for all t 2 J:

Now from the definition of it follows that kyk˝ D sup jy.t/j .b/ K

for all y 2 E:

t2Œr;b

This shows that the set E is bounded. As a consequence of Theorem 1.32 A C B has a fixed point which is a mild solution of problem (7.16)–(7.18). Thus, the problem (7.16)–(7.18) is controllable on the interval Œr; b: t u

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7 Densely Defined Functional Differential Inclusions with Finite Delay

7.4.2 Example As an application of our results we consider the following impulsive partial functional differential equation of the form @2 @ z.t; x/ D 2 z.t; x/ @t @x CQ.t; z.t r; x/ C Bu.t/; x 2 Œ0; ; t 2 Œ0; bnft1 ; t2 ; : : : ; tm g: (7.26) z.tkC ; x/ z.tk ; x/ 2 bk jz.tk ; x/jB.0; 1/; x 2 Œ0; ; k D 1; : : : ; m

(7.27)

z.t; 0/ D z.t; / D 0; t 2 J WD Œ0; b

(7.28)

z.t; x/ D .t; x/; t 2 H; x 2 Œ0; ;

(7.29)

where bk > 0; k D 1; : : : ; m; 2 D D f N W H Œ0; ! RI N is continuous everywhere except for a countable number of points at which N .s /; N .sC / exist with N .s / D N .s/g; 0 D t0 < t1 < t2 < < tm < tmC1 D b; z.tkC / D lim z.tk C h; x/; z.tk / D lim z.tk C h; x/, where Q W J R ! P.R/; .h;x/!.0 ;x/

.h;x/!.0C ;x/

is a multi-valued map with compact values. Here B.0; 1/ denotes the closure of the unit ball. Let y.t/ D z.t; :/I t 2 J; Ik W R ! PR such that Ik .y.tk // D bk jz.tk ; :/jB.0; 1/; k D 1; : : : ; m and F.t; yt /.x/ D Q.t; z.t r; x//; t 2 Œ0; b; x 2 Œ0; : Take E D L2 Œ0; , and define the linear operator A W D.A/ E ! E by Aw D w00 with domain D.A/ D fw 2 E; w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0g: Then Aw D

1 X nD1

n2 .w; wn /wn ; w 2 D.A/

7.5 Notes and Remarks

163

q 2 where .:; :/ is the inner product in L2 Œ0; and wn .s/ D sin ns: n D 1; 2P: is the orthogonal set eigenvectors in A. It is well known (see [168]) that A is the infinitesimal generator of an analytic semigroup T.t/; t 2 .0; b in E given by T.t/w D

1 X

exp.n2 t/.w; wn /wn ; w 2 E:

nD1

Since the analytic semigroup T.t/; t 2 .0; b is compact, there exists a constant M 1 such that kT.t/kB.E/ M: Assume that B W U ! Y; U Œ0; 1/ is a bounded linear operator and the operator W defined by Z

b

Wu D

T.b s/Bu.s/ds

0

has a bounded invertible operator W 1 which takes values in L2 .Œ0; b; U/nkerW. Also assume that there exists an integrable function W Œ0; b ! RC such that jQ.t; w.t r//j .t/˝.jwj/ where ˝ W Œ0; 1/ ! .0; 1/ is continuous and nondecreasing with Z

1 1

ds D1 s C ˝.s/

Assume that there exists Ql 2 L1 .Œ0; b; RC / such that N r; x/// Qljw wjI N t 2 Œ0; b; w; w_nR N Hd .Q.t; w.t r; x//; Q.t; w.t We can show that problem (7.16)–(7.18) is an abstract formulation of problem (7.26)–(7.29). Since all the conditions of Theorem 7.7 are satisfied, the problem (7.26)–(7.27) has a solution z on Œr; b Œ0; :

7.5 Notes and Remarks The results of Chap. 7 are taken from Abada et al. [4, 6]. Other results may be found in [53, 54].

Chapter 8

Non-densely Defined Functional Differential Inclusions with Finite Delay

8.1 Introduction In this chapter, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some non-densely defined impulsive semi-linear functional differential inclusions in separable Banach spaces with local and nonlocal conditions. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these inclusions with both multi-valued and single valued jump and the topological structure of the solutions set are considered too.

8.2 Integral Solutions of Non-densely Defined Functional Differential Inclusions with Local Conditions We will consider the following first order impulsive semi-linear differential inclusions of the form: y0 .t/ Ay.t/ 2 F.t; yt /; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m

(8.1)

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(8.2)

y.t/ D .t/; t 2 Œr; 0 ;

(8.3)

where F W J D ! P.E/, D, Ik W E ! P.E/ are as in the previous chapter and A W D.A/ E ! E is a non-densely defined closed linear operator on E.

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_8

165

166

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

In order to define a integral solution of problems (8.1)–(8.3) and (8.14)–(8.16), we shall consider the space n PC D y W Œ0; b ! D.A/ W yk 2 C.Jk ; D.A//; k D 0; : : : ; m such that o y.tk /; y.tkC / exist with y.tk / D y.tk /; k D 1; : : : ; m which is a Banach space with the norm kykPC D maxfkyk k1 ; k D 1; : : : ; mg where yk is the restriction of y to Jk D Œtk ; tkC1 ; k D 0; : : : ; m: Set ˝ D fy W Œr; b ! D.A/ W y 2 D \ PCg: Then ˝ is a Banach space with norm kyk˝ D max.kykD ; kykPC /:

8.2.1 Main Results We assume that the multi-valued F has compact and convex values. Let us first define the concept of integral solution of (8.1)–(8.2). Definition 8.1. We say that y W Œr; b ! E is an integral solution of (8.1)– (8.3) if (i) yZ 2 ˝: t (ii) y.s/ds 2 D.A/ for t 2 J,

0

(iii) y.t/ D .t/ for all t 2 H there exist v 2 L1 .J; E/ and Ik 2 Ik y.tk / such that v.t/ 2 F.t; yt / a:e t 2 J and

y.t/ D S0 .t/.0/ C

d dt

Z

t

S.t s/v.s; /ds C

0

X

S0 .t tk / Ik t 2 J:

(8.4)

0
We notice also that if y satisfies (8.4), then y.t/ D S0 .t/.0/ C lim

Z

!1 0

t

S0 .t s/B v.s/ds C

X 0
S0 .t tk / Ik ; t 2 J:

8.2 Integral Solutions of Non-densely Defined Functional Differential. . .

167

In what follows we will assume (without lost of generality) that w > 0. Let us introduce the following hypotheses: (8.1.1) A satisfies Hille–Yosida condition; (8.1.2) There exist constants ck > 0; k D 1; : : : ; m such that for each y; x 2 D.A/ Hd .Ik .y/; Ik .x// ck jy xj (8.1.3) The multi-valued map F is L1 -Carathéodory, with compact convex values. (8.1.4) The operator S0 .t/ is compact in D.A/ wherever t > 0I (8.1.5) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/k p.t/ .kxkD /;

a:e: t 2 J;

for all x 2 D

and Z C1

b

0

e!t p.t/dt <

Z

1 C0

du ; .u/

(8.5)

;

(8.6)

;

(8.7)

where Me!b m X

C1 D 1

Me!b

!tk

e

ck

kD1

C0 D 1

Me!b

C m X

!tk

e

ck

kD1

and C D Me!b kk C

m X

! e!tk ck jIk .0/j :

(8.8)

kD1

Theorem 8.2. Assume that (8.1.1)–(8.1.5) hold and .0/ 2 D.A/. If Me!b

m X

e!tk ck < 1;

kD1

then the problem (8.1)–(8.3) has at least one integral solution on Œr; b :

(8.9)

168

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

Proof. Consider the multi-valued operator N W ˝ ! P.˝/ defined by 8 8 9 .t/; if t 2 H; > ˆ ˆ ˆ ˆ > ˆ ˆ > ˆ ˆ > ˆ ˆ > ˆ ˆ > 0 ˆ ˆ > ˆ ˆ > S .t/.0/ < < = Z t d N.y/ D h 2 ˝ W h.t/ D C S.t s/v.s/ds ˆ ˆ > ˆ ˆ > dt 0 ˆ ˆ > ˆ ˆ > X ˆ ˆ > ˆ ˆ > 0 ˆ ˆ > C S .t t / I ; I 2 I .y.t //I v 2 S if t 2 J: ˆ ˆ > k k k k F;y k : : ; 0
Obviously the fixed points of the operator N are integral solutions of the IVP (8.1)–(8.3). Consider the multi-valued operators A; B W ˝ ! P.˝/ defined by 8 8 ˆ ˆ < < 0; X A.y/ WD h 2 ˝ W h.t/ D S0 .t tk /Ik ; ˆ ˆ : :

9 if t 2 HI > = Ik 2 Ik .y.tk //;

0
if t 2 J; > ;

and 8 8 ˆ ˆ < .t/; < Z t B.y/ WD h 2 ˝ W h.t/ D ˆ ˆ : S0 .t/.0/ C : S.t s/v.s/ds; 0

9 if t 2 H; > = v 2 SF;y

; if t 2 J: >

It is clear that N DACB The problem of finding integral solutions of (8.1)–(8.3) is reduced to finding integral solutions of the operator inclusion y 2 A.y/CB.y/. We shall show that the operators A and B satisfy all conditions of the Theorem 1.32. The proof will be given in several steps. Step 1: A is a contraction. Let y1 ; y2 2 ˝, then by (8.1.2) we have 0 1 X X

S0 .t tk /Ik .y1 .tk //; S0 .t tk /Ik .y2 .tk //A Hd A.y1 /; A.y2 / D Hd @ 0
Me!t

m X

e!tk ck jy1 .tk / y2 .tk /j

kD1 !b

Me

m X

0
e!tk ck ky1 y2 kD :

kD1

Hence by (8.9), A is a contraction.

8.2 Integral Solutions of Non-densely Defined Functional Differential. . .

169

Step 2: B has compact, convex values, and it is completely continuous. This will be given in several claims. Claim 1: B has compact values. The operator B is equivalent to the composition LıSF of two operators on L1 .J; E/; where L W L1 .J; E/ ! ˝ is the continuous operator defined by d L.v.t// D S .t/.0/ C dt 0

Z

t

S.t s/v.s/ds; t 2 J:

0

Then, it suffices to show that L ı SF has compact values on ˝. Let y 2 ˝ arbitrary, vn a sequence in SF;y , that is vn .t/ 2 F.t; yt /; a:e: t 2 J. Since F.t; yt / is compact, we may pass to a subsequence if necessary to get that vn ! v weakly in Lw1 .J; E/ and v.t/ 2 F.t; yt /, a.e. t 2 J. An application of Mazur’s Lemma implies that vn converges strongly to v in L1 .J; E/. From the continuity of L, it follows that Lvn .t/ ! Lv.t/ pointwise on J as n ! 1. In order to show that the convergence is uniform, we first show that fLvn g is an equi-continuous sequence. Let 1 ; 2 2 J, then we have: ˇ ˇ ˇ ˇL.vn . 1 // L.vn . 2 //ˇ D ˇS0 . 1 /.0/ S0 . 2 /.0/ Z d 1 C S. 1 s/vn .s/ds dt 0 Z ˇ d 2 S. 2 s/vn .s/dsˇ dt 0 ˇ 0 ˇ

ˇ S . 1 / S0 . 2 / .0/ˇ Z 1 ˇ ˇ 0 ˇ C lim S . 1 s/ S0 . 2 s/ B vn .s/jdsˇ !1 0 Z 2 ˇ

ˇ lim

!1 1

ˇ S0 . 2 s/B vn .s/jdsˇ:

As 1 ! 2 , the right-hand side of the above inequality tends to zero. Since S0 .t/ is a strongly continuous operator and the compactness of S0 .t/; t > 0, implies the continuity in uniform topology (see [16], Lemma 3.4.1, p. 104, [168]). Hence fLvn g is equi-continuous, and an application of Arzelá– Ascoli theorem implies that there exists a subsequence which is uniformly convergent. Then we have Lvnj ! Lv 2 .L ı SF /.y/ as j 7! 1, and so .LıSF /.y/ is compact. Therefore B is a compact valued multi-valued operator on ˝. Claim 2: B.y/ is convex for each y 2 ˝: Let h1 ; h2 2 B.y/, then there exist v1 ; v2 2 SF;y such that, for each t 2 J we have

170

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

hi .t/ D

8 ˆ < .t/; d ˆ : S0 .t/.0/ C dt

Z 0

t

S.t s/vi .s/ds;

if t 2 H,

9 > =

if t 2 J;

; i D 1; 2: >

Let 0 ı 1: Then, for each t 2 J, we have 8 .t/; ˆ ˆ ˆ < .ıh1 C.1ı/h2 /.t/ D S0 .t/.0/ Z t ˆ ˆ ˆ :C d S.t s/Œıv1 .s/ C .1 ı/v2 .s/ds dt 0

9 if t 2 H, > > > = > > > if t 2 J: ;

Since F.t; yt / has convex values, one has ıh1 C .1 ı/h2 2 B.y/: Claim 3: B maps bounded sets into bounded sets in ˝ Let Bq D fy 2 ˝I kyk˝ qg; q > 0 be a bounded set in ˝. For each h 2 B.y/, there exists v 2 SF;y such that d h.t/ D S .t/.0/ C dt 0

Z

t

S.t s/v.s/ds:

0

Then for each t 2 J !b

!t

Z

t

jh.t/j Me j.0/j C Me

0

!b

!b

Z

e!s 'q .s/ds

b

Me j.0/j C Me

0

e!s 'q .s/ds;

this further implies that khk1 Me!b j.0/j C Me!b Then, for all h 2 B.y/ B.Bq / D

S y2Bq

Z

b 0

e!s 'q .s/ds:

B.y/: Hence B.Bq / is bounded.

Claim 4: B maps bounded sets into equi-continuous sets. Let Bq be, as above, a bounded set and h 2 B.y/ for some y 2 Bq : Then, there exists v 2 SF;y such that h.t/ D S0 .t/.0/ C lim

Z

!1 0

t

S0 .t s/B v.s/ds;

t 2 J:

8.2 Integral Solutions of Non-densely Defined Functional Differential. . .

171

Let 1 ; 2 2 Jnft1 ; t2 ; : : :; tm g; 1 < 2 . Thus if > 0, we have jh. 2 / h. 1 /j jŒS0 . 2 / S0 . 1 /.0/j ˇ ˇ Z 1 ˇ ˇ 0 0 ˇ C ˇ lim kS . 2 s/ S . 1 s/B v.s/dsˇˇ !1 0

ˇ ˇ Z 1 ˇ ˇ 0 C ˇˇ lim S . 2 s/ S0 . 1 s/ B v.s/dsˇˇ !1 1 ˇ ˇ Z 2 ˇ ˇ C ˇˇ lim S0 . 2 s/B v.s/dsˇˇ : !1 1

As 1 ! 2 and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since S0 .t/ is a strongly continuous operator and the compactness of S0 .t/ for t > 0 implies the continuity in the uniform operator topology (see [168]). This proves the equi-continuity for the case where t ¤ ti ; i D 1; : : :; m C 1. It remains to examine the equi-continuity at t D ti . First we prove the equicontinuity at t D ti , we have for some y 2 Bq , there exists v 2 SF;y such that Z t 0 S0 .t s/B v.s/ds; t 2 J: h.t/ D S .t/.0/ C lim !1 0

Fix ı1 > 0 such that ftk ; k ¤ ig \ Œti ı1 ; ti C ı1 D ;. Let 0 < < ı1 : First we prove equi-continuity at t D ti . Fix ı1 > 0 such that ftk W k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < < ı1 we have

jh.ti / h.ti /j j S0 .ti / S0 .ti / .0/j Z ti

C lim j S0 .ti s/ S0 .ti s/ B v.s/jds !1 0 !b

CMe

Z

.q/

ti

e!s p.s/ dsI

ti

which tends to zero as ! 0. Define hO 0 .t/ D h.t/;

t 2 Œ0; t1

and hO i .t/ D

h.t/; if t 2 .ti ; tiC1 h.tiC /; if t D ti :

172

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

Next we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk W k ¤ ig \ Œti ı2 ; ti C ı2 D ;. For 0 < < ı2 we have

O i /j j S0 .ti C / S0 .ti / .0/j O i C / h.t jh.t Z ti

C lim j S0 .ti C s/ S0 .ti s/ B v.s/jds !1 0

CMe!b .q/

Z

ti C

e!s p.s/ ds:

ti

The right-hand side tends to zero as ! 0. The equi-continuity for the cases

1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval Œr; 0. As consequence of Claims 3 and 4 together with Arzelá– Ascoli theorem it suffices to show that B maps Bq into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t . For y 2 Bq we define Z t 0 0 h .t / D S .t/.0/ C S ./ lim S0 .t s /B v.s/ds: !1 0

where v 2 SF;y . Since ˇ ˇ Z Z t ˇ ˇ 0 ˇ lim ˇ Me!b .q/ S .t s /B v.s/ ds ˇ ˇ !1 0

t

e!s p.s/ds:

0

and S0 .t/ is a compact operator for t > 0, the set H .t / D fh .t / W h 2 B.y/g is precompact in E for every ; 0 < < t : Moreover, for every h 2 B.y/ we have Z t !b .q/ e!s p.s/ds: jh.t / h .t /j Me t

Therefore, there are precompact sets arbitrarily close to the set H .t / D fh.t / W h 2 B.y/g: Hence the set H.t / D fh.t / W h 2 B.Bq /g is precompact in E. Hence the operator B W ˝ ! P.˝/ is completely continuous. Claim 5: B has closed graph. Let fyn g be a sequence such that yn ! y in ˝; hn 2 B.yn /, and hn ! h . We shall show that h 2 B.y /. hn 2 B.yn / means that there exists vn 2 SF;yn such that hn .t/ D S0 .t/.0/ C lim

Z

!1 0

t

S0 .t s/B vn .s/ds; t 2 J:

8.2 Integral Solutions of Non-densely Defined Functional Differential. . .

173

We must prove that there exists v 2 SF;y such that Z

h .t/ D S0 .t/.0/ C lim

t

!1 0

S0 .t s/B v .s/ds; t 2 J:

Consider the linear and continuous operator K W L1 .J; E/ ! C.J; E/ defined by Z .Kv/.t/ D lim

t

S0 .t s/B v.s/ds; t 2 J:

!1 0

Then we have j.hn .t/ S0 .t/.0// .h .t/ S0 .t/.0//j D jhn .t/ h .t/j khn h k1 ! 0;

as n 7! 1:

From Lemma 1.11 it follows that K ı SF is a closed graph operator and from the definition of K one has hn .t/ S0 .t/.0/ 2 K ı SF;yn : As yn ! y and hn ! h , there is a v 2 SF;y such that Z

0

h .t/ S .t/.0/ D lim

!1 0

t

S0 .t s/B v .s/ds; t 2 J:

Hence the multi-valued operator B is upper semi-continuous. Step 3: A priori bounds. Now it remains to show that the set E D fy 2 ˝ W y 2 ˛A.y/ C ˛B

for some 0 < ˛ < 1g

is bounded. Let y 2 E; then there exist v 2 SF;y and Ik 2 Ik .y.tk // such that y.t/ D ˛S0 .t/.0/ C ˛ lim

Z

!1 0

t

S0 .t s/B v.s/ds C ˛

X

S0 .t tk / Ik

0
for some 0 < ˛ < 1: Thus, by (8.1.2), (8.1.5) for each t 2 J; we have jy.t/j Me!t j.0/j C Me!t

Z

t 0

CMe!t

m X kD1

e!tk jIk j

e!s p.s/ .kys k/ds

174

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

!b

!b

Z

t

Me kk C Me

0

CMe!b

m X

e!s p.s/ .kys k/ds

e!tk ck jy.tk /j C Me!t

kD1

C C Me!b

Z

m X

e!tk ck jIk .0/j

kD1 t

0

CMe!b

m X

e!s p.s/ .kys k/ds

e!tk ck jy.tk /j:

kD1

Now we consider the function defined by .t/ D supfjy.s/j W r s tg; 0 t b: Then kys k .t/ for all t 2 J and there is a point t 2 Œr; t such that .t/ D jy.t /j. If t 2 Œ0; b, by the previous inequality we have for t 2 Œ0; b (note t t) !b

.t/ C C Me

Z

t

0

e!s p.s/ ..s//ds C Me!b

m X

e!tk ck .t/:

kD1

Then !b

1 Me

m X

! !tk

e

!b

ck .t/ C C Me

kD1

Z

t

0

e!s p.s/ ..s//ds:

Thus by (8.6) and (8.7) we have Z .t/ C0 C C1

t 0

e!s p.s/ ..s//ds:

Let us take the right-hand side of (8.10) as v.t/: Then we have .t/ v.t/ for all t 2 J; with v.0/ D C0 ; and v 0 .t/ D C1 e!t p.t/ ..t//; a:e: t 2 J:

(8.10)

8.2 Integral Solutions of Non-densely Defined Functional Differential. . .

Using the increasing character of

175

we get

v 0 .t/ C1 e!t p.t/ .v.t//; a:e: t 2 J: Integrating from 0 to t we get Z t Z t .v.s//0 ds C1 e!t p.s//ds: .v.s// 0 0 By a change of variable we get Z t Z v.t/ du e!t p.s//ds C1 .u/ v.0/ 0 Z b C1 e!t p.s//ds: 0

Hence by (8.5) there exist a constant N such that .t/ v.t/ N

for all t 2 J:

Now from the definition of it follows that kyk˝ max.kkD ; N/;

for all y 2 E:

This shows that the set E is bounded. As a consequence of Theorem 1.32 we deduce that A C B has a fixed point y defined on the interval Œr; b which is the integral solution of problem (8.1)–(8.3). t u We now present another existence result for the problem (8.1)–(8.3) where a Lipschitz condition on the multi-valued F with respect to its second variable is assumed instead of a Wintner growth condition used in Theorem 8.2. Theorem 8.3. Assume that (8.1.1)–(8.1.4), ˚.0/ 2 D.A/ hold and the condition (8.3.1) There exists a function l 2 L1 .J; RC / such that: Hd .F.t; u/; F.t; uN // l.t/ku uN kD a.e. t 2 J; and for all u; uN 2 D; and Hd .0; F.t; 0// l.t/ for a.e. t 2 J; Z

b

where 0

e!s l.s/ds < 1,

Me!b kk C C0

D

m X

!tk

e

Z ck jIk .0/j C

kD1

1

Me!b

m X kD1

!tk

e

ck

b

! !s

e 0

l.s/ds (8.11)

176

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

and Me!b m X

C1 D 1

Me!b

: !tk

e

(8.12)

ck

kD1

If Me!b

m X

e!tk ck < 1;

(8.13)

kD1

then the problem (8.1)–(8.3) has at least one integral solution on Œr; b : Proof. Let A and B the operators defined in Theorem 8.2. It can be shown, as in the proof of Theorem 8.2 that B is completely continuous and upper semi-continuous and A is a contraction. Now we prove that E D fy 2 ˝ W y 2 ˛A.y/ C ˛B.y/; for some 0 < ˛ < 1g is bounded. Let y 2 E; then there exist v 2 SF;y and Ik 2 Ik .y.tk // such that for each t 2 J y.t/ D ˛S0 .t/.0/ C ˛

d dt

Z

t 0

S.t s/v.s/ds C ˛

X

S0 .t tk / Ik ;

0
for some 0 < ˛ < 1: Thus, by (8.1.2), (8.3.1), for each t 2 J; we have !t

Z

!t

jy.t/j Me j.0/j C Me

t

e!s jv.s/jds C Me!t

0

Z

Me!t j.0/j C Me!t !t

t

CMe

!s

e

e!s l.s/kys kds !t

l.s/ds C Me

0

CMe!t

e!tk jIk j

kD1 t

0

Z

m X

m X

e!tk ck jy.tk /j

kD1

m X

e!tk ck jIk .0/j

kD1 !b

Me

Z

t

kk C

e 0

CMe!b

Z

t 0

!s

l.s/ds C

m X

! !tk

e

ck jIk .0/ j

kD1

e!s l.s/kys kds C Me!b

m X kD1

e!tk ck y tk :

8.2 Integral Solutions of Non-densely Defined Functional Differential. . .

177

Now we consider the function defined by .t/ D supfjy.s/j W r s tg; 0 t b: Then kys k .t/ for all t 2 J and there is a point t 2 Œr; t such that .t/ D jy.t /j. If t 2 Œ0; b, by the previous inequality we have for t 2 Œ0; b (note t t) !b

.t/ Me

Z

t

kk C

!s

e

l.s/ds C

0

CMe!b

Z

t

!

m X

!tk

e

ck jIk .0/ j

kD1

e!s l.s/.s/ds C Me!b

0

m X

e!tk ck .t/:

kD1

Then .t/ C0 C C1

Z

t

e!s l.s/.s/ds:

0

By Gronwall inequality ([131]) we get for each t 2 J .t/

C0 exp

C1

Z

t

!s

e 0

l.s/ds :

Hence Z kk1 C0 exp C1

b

0

e!s l.s/ds WD M :

Thus kyk˝ max.kkD ; M /: This shows that the set E is bounded. As a consequence of Theorem 1.32 we deduce that A C B has a fixed point which is a integral solution of problem (8.1)–(8.3). The following result concerns the compactness property of the solutions set of problem (8.1)–(8.3). t u Theorem 8.4. Under assumptions (8.1.1)–(8.1.4), and (8.4.1) There exists p 2 C.J; RC / such that kF.t; u/k p.t/ for each t 2 J; and each u 2 D: the solution set of (8.1)–(8.3) in not empty and compact in ˝.

178

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

Proof. Let S D fy 2 ˝ W y is solution of (8.1)–(8.3)g: From Theorem 8.2, S 6D ;. Now, we prove that S is compact. Let .yn /n2N 2 S; then there exist vn 2 SF;yn and Ikn 2 Ik .yn .tk // such that Z X d t 0 S.t s/vn .s/ds C S0 .t tk / Ikn : yn .t/ D S .t/.0/ C dt 0 0
From .8:1:2/; .8:4:1/; we can prove that there exists an M1 > 0 such that kyn k1 M1 ; for every n 1: As in Claim 4 in Theorem 8.2, we can easily show using (7.1.2), (8.4.1) that the set fyn W n 1g is equi-continuous in ˝; hence by Arzelá–Ascoli Theorem we can conclude that, there exists a subsequence (denoted again by fyn g) of fyn g such that yn converges to y in ˝: We shall show that there exist v.:/ 2 F.:; y: / and Ik 2 Ik .y.tk // such that Z X d t y.t/ D S0 .t/.0/ C S.t s/v.s/ds C S0 .t tk / Ik : dt 0 0
Since F.t; :/ is upper semi-continuous, then for every " > 0, there exists n0 ./ 0 such that for every n n0 ; we have vn .t/ 2 F.t; ynt / F.t; yt / C "B.0; 1/; a.e. t 2 J: Since F.:; :/ has compact values, there exists subsequence vnm .:/ such that vnm .:/ ! v.:/ as m ! 1 and v.t/ 2 F.t; yt /; a.e. t 2 J: It is clear that jvnm .t/j p.t/; a.e. t 2 J: By Lebesgue’s dominated convergence theorem, we conclude that v 2 L1 .J; E/ which implies that v 2 SF;y : Also, since Ik has closed graph we get Ik 2 Ik .y.tk //. Thus Z X d t S.t s/v.s/ds C S0 .t tk / Ik : y.t/ D S0 .t/.0/ C dt 0 0
Then S 2 Pcp .˝/:

t u

8.3 Extremal Integral Solutions with Local Conditions

179

8.3 Extremal Integral Solutions with Local Conditions In this section we shall prove the existence of maximal and minimal integral solutions of problem (8.1)–(8.3) under suitable monotonicity conditions on the functions involved in it. Let us give the definition of the extremal integral solutions of the problem (8.1)–(8.3) Definition 8.5. We say that a continuous function u W Œr; b ! E is a lower integral solution of problem (8.1)–(8.3) if there exist v 2 L1 .J; E/ and Ik 2 Ik .u.tk // such that v.t/ 2 F.t; ut / a.e. on J; y.t/ D .t/; t 2 H; and u.t/ S0 .t/.0/ C

d dt

Z

t

S.t s/v.s/ds C

0

X

S0 .t tk / Ik ; t 2 J; t ¤ tk

0
and u.tkC / u.tk / Ik ; k D 1; : : : ; m. Similarly an upper integral solution w of problem (8.1)–(8.3) is defined by reversing the order. Definition 8.6. A solution xM of problem (8.1)–(8.3) is said to be maximal if for any other solution x of problem (8.1)–(8.3) on J, we have that x.t/ xM .t/ for each t 2 J. Similarly a minimal solution of problem (8.1)–(8.3) is defined by reversing the order of the inequalities. Definition 8.7. A multi-valued function F.t; x/ is called strictly monotone increasing in x almost everywhere for t 2 J, if F.t; x/ F.t; y/ a.e. t 2 J for all x; y 2 D with x < y. Similarly F.t; x/ is called strictly monotone decreasing in x almost everywhere for t 2 J, if F.t; x/ F.t; y/ a.e. t 2 J for all x; y 2 D with x < y. Let us the following assumptions. (8.7.1) The multi-valued function F.t; y/ is strictly monotone increasing in y for almost each t 2 J. (8.7.2) S0 .t/ is preserving the order, that is S0 .t/v 0 whenever v 0. (8.7.3) The multi-valued functions Ik , k D 1; : : :; m are strictly monotone increasing. (8.7.4) The problem (8.1)–(8.3) has a lower integral solution v and an upper integral solution w with v w. Theorem 8.8. Assume that assumptions (8.1.1)–(8.1.4) and (8.7.1)–(8.7.4) hold. Then problem (8.1)–(8.3) has a minimal and a maximal integral solutions on Œr; b. Proof. It can be shown, as in the proof of Theorem 8.3, that B is completely continuous and upper semi-continuous and A is a contraction on Œv; w. We shall show that A and B are isotone increasing on Œv; w. Let y; y 2 Œv; w be such that y y; y 6D y: Then by (8.7.2), (8.7.3), we have for each t 2 J

180

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

8 9 < = X A.y/ D h 2 ˝ W h.t/ D S0 .t tk /Ik ; Ik 2 Ik .y.tk //; : ; 0
9 8 = < X S0 .t tk /Ik ; Ik 2 Ik .Ny.tk //; h 2 ˝ W h.t/ D ; : 0
D A.Ny/: Similarly, by (8.7.1) and (8.7.2) and

Z d t B.y/ W D h 2 ˝ W h.t/ D S .t/.0/ C S.t s/v.s/ds; dt 0 Z d t h 2 ˝ W h.t/ D S0 .t/.0/ C S.t s/v.s/ds; dt 0 0

v 2 SF;y v 2 SF;Ny

D B.Ny/: Therefore A and B are isotone increasing on Œv; w. Finally, let x 2 Œv; w be any element. By (8.7.4) we deduce that v A.v/ C B.v/ A.x/ C B.x/ A.w/ C B.w/ w; which shows that A.x/ C B.x/ 2 Œv; w for all x 2 Œv; w. Thus, A and B satisfy all conditions of Theorem 1.32, hence problem (8.1)–(8.3) has a maximal and a minimal integral solutions on Œr; b. This completes the proof. t u

8.4 Integral Solutions with Nonlocal Conditions In this section we prove existence results for problem of the form y0 .t/ Ay.t/ 2 F.t; yt /; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m

(8.14)

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(8.15)

y.t/ C ht .y/ D .t/; t 2 Œr; 0 ;

(8.16)

where ht W ˝ ! D.A/ is a given function, A, F, and Ik are as above.

8.4 Integral Solutions with Nonlocal Conditions

181

8.4.1 Main Result Definition 8.9. A function y 2 ˝ is said to be an integral solution of problem (8.14)–(8.16) if y.t/ D .t/ ht .y/ ; t 2 Œr; 0; and there exist v.:/ 2 L1 .J; E/ and Ik 2 Ik .y.tk // such that v.t/ 2 F.t; yt / a.e. t 2 J, and y satisfies the integral equation, d y.t/ D S .t/ ..0/ h0 .y// C dt 0

Z

t

X

S.t s/v.s/ds C

0

S0 .t tk /Ik :

0
Theorem 8.10. Assume that hypotheses (8.1.1)–(8.1.4) hold and moreover (A1) The function h is continuous with respect to t, and there exists a constant ˛ > 0 such that jht .u/j ˛; u 2 ˝ and for each k > 0 the set f.0/ h0 .y/; y 2 ˝; kyk˝ kg is precompact in E; (A2) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/k p.t/ .kxkD /;

a:e: t 2 J;

for all x 2 D

with Z

1 Q0 C

du > CQ 1 .u/

Z

b

e!s p.s/ds;

(8.17)

0

where Me!b ŒkkD C ˛ C CQ 0 D

m X

e!tk ck jIk .0/j

kD1

1 Me!b

m X

;

(8.18)

e!tk ck

kD1

and CQ 1 D 1

Me!b m X

Me!b

kD1

: !tk

e

ck

(8.19)

182

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

Moreover, we suppose that Me!b

m X

e!tk ck < 1:

(8.20)

kD1

Then the problem (8.14)–(8.16) has at least one integral solution on Œr; b: Proof. Consider the multi-valued operators A1 ; B1 W ˝ ! P.˝/: 8 8 ˆ ˆ .t/ ht .y/; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < 0 < B1 .y/ WD f 2 ˝ W f .t/ D S .t/ ..0/ h0 .y// ˆ ˆ Z ˆ ˆ ˆ ˆ d t ˆ ˆ ˆ ˆ :C : S.t s/v.s/ds; dt 0

9 if t 2 H; > > > > > = v 2 SF;y

> > > > ; if t 2 J; >

and 8 8 ˆ ˆ < 0; < X A1 .y/ WD f 2 ˝ W f .t/ D S0 .t tk /Ik ; Ik 2 Ik .y.tk // ˆ ˆ : : 0
9 if t 2 H; > = if t 2 J: > ;

Then the problem of finding the solution of problem (8.14)–(8.16) is reduced to finding the solution of the operator inclusion y 2 A1 .y/ C B1 .y/. As in the previous section, it can be shown that the operators A1 and B1 satisfy all conditions of Theorem 1.32. t u

8.5 Application to the Control Theory In this section we treat the controllability of impulsive functional differential inclusions using the argument of the previous sections. More precisely we will consider the following problem: y0 .t/ Ay.t/ 2 F.t; yt / C Bu.t/; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m

(8.21)

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(8.22)

y.t/ D .t/; t 2 Œr; 0 ;

(8.23)

where A, F, and Ik are as above, the control function u ./ is given in L2 .J; U/ a Banach space of admissible control functions with U as a Banach. Finally B is a bounded linear operator from U to D.A/:

8.5 Application to the Control Theory

183

Definition 8.11. A function y 2 ˝ is said to be an integral solution of problem (8.21)–(8.23) if y.t/ D .t/; t 2 Œr; 0; and there exist v.:/ 2 L1 .J; E/ and Ik 2 Ik .y.tk // such that v.t/ 2 F.t; yt / a.e. t 2 J, and y satisfies the impulsive integral equation, Z d t y.t/ D S0 .t/.0/ C S.t s/v.s/ds dt 0 Z X d t C S.t s/Bu.s/ds C S0 .t tk /Ik : dt 0 0
Definition 8.12. The system (8.21)–(8.23) is said to be controllable on the interval Œr; b if for every initial function 2 D and every y1 2 D.A/, there exists a control u 2 L2 .J; U/, such that the integral solution y.t/ of system (8.21)–(8.23) satisfies y.b/ D y1 .

8.5.1 Main Result Let us the following assumptions (B1)

The linear operator W W L2 .J; U/ ! D.A/, defined by Z d b S.b s/Bu.s/ds; Wu D dt 0

has a bounded inverse operator W 1 which takes values in L2 .J; U/nKerW, and there exist positive constants M; M1 ; such that kBk M and kW 1 k M1 : (B2) F has compact and convex values, and there exists a function l 2 L1 .J; RC / such that Hd .F.t; x/; F.t; y// l.t/kx ykD for a.e. t 2 J; and for all x; y 2 D; with Hd .0; F.t; 0// l.t/; a:e: t 2 J: (B3) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/k p.t/ .kxkD /; Z with 0

b

a:e: t 2 J;

for all x 2 D

e!s p.s/ds < 1, Z

1

C0

ds D1 s C .s/

(8.24)

184

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

where C m X

C0 D 1

Me!b

; !tk

e

(8.25)

ck

kD1

C D Me!b .1 C MMM1 e!b b/kkD C Me!b MM 1 bjy1 j C.Me!b C MMM1 e2!b b/

m X

e!tk ck jIk .0/j:

kD1

Theorem 8.13. Assume that hypotheses (8.1.1)–(8.1.4) hold. Moreover we suppose that Z b m X e!s l.s/ds C Me!b .1 C Me!b MM1 b/ e!tk ck < 1: (8.26) M 2 e2!b MM1 b 0

kD1

Then the problem (8.21)–(8.23) is controllable on Œr; b. Remark 8.14. The construction of operator W 1 and its properties are discussed in [170]. Proof. Using hypothesis (B1) for an arbitrary function y .:/ we define the control Z uy .t/ D W 1 y1 S0 .b/.0/ lim

m X

!1 0

#

b

S0 .b s/B v.s/ds

S0 .b tk /Ik .t/;

kD1

where v 2 SF;y and Ik 2 Ik .y.tk //. Consider the multi-valued operators defined from ˝ to P.˝/ by: 8 8 9 0; if t 2 H; > ˆ ˆ ˆ ˆ > ˆ ˆ > ˆ ˆ > ˆ ˆ > Z ˆ ˆ > t < d < = S.t s/.Bu /.s/ds y A.y/ WD f 2 ˝ W f .t/ D dt 0 ˆ ˆ > ˆ ˆ > X ˆ ˆ > ˆ ˆ > 0 ˆ ˆ > S .t t /I ; if t 2 J; C k k ˆ ˆ > : : ; 0
and 8 8 if t 2 H; ˆ ˆ < < .t/; Z B.y/ WD f 2 ˝ W f .t/ D d t ˆ ˆ : : S0 .t/.0/ C S.t s/v.s/ds if t 2 J: dt 0

9 > = > ;

8.5 Application to the Control Theory

185

As in Theorem 8.2, we can prove that the operators A is a contraction operator, and B is completely continuous and upper semi-continuous with compact convex values. Now, we prove that the set E D fy 2 ˝j y 2 ˛Ay C ˛By; 0 ˛ 1g is bounded. Let y 2 E: be any element, then there exists v 2 SF;y and such that Z d t 0 S.t s/v.s/ds y.t/ D ˛S .t/.0/ C ˛ dt 0 Z X d t C˛ S.t s/Buy .s/ds C ˛ S0 .t tk /Ik : dt 0 0
This implies by (B1)–(B3) that, for each t 2 J; we have " jy.t/j Me!b .1 C MMM 1 be!b /kkD CMM 1 bjy1 j C .Me!b C M 2 MM 1 be!b / CŒMe!b C M 2 MM 1 be2!b CM 2 MM 1 be2!b CMe!b

m X

Z

t

# e!tk ck jIk .0/j

kD1

Z

t 0

e!s p.s/ .kys k/ds

X

e!s

0

m X

e!tk ck jy.tk //jds

0
e!tk ck jy.tk /j:

kD1

Consider the function defined by .t/ D supfjy.s/j W r s tg; 0 t b: Then kys k .t/ for all t 2 J and there is a point t 2 Œr; t such that .t/ D jy.t /j. If t 2 Œ0; b, by the previous inequality we have for t 2 Œ0; b (note t t) Z t !s .t/ C C Me!b C M 2 MM 1 be2!b e p.s/ ..s//ds 0

CMe!b

m X

e!tk ck .t/

kD1 2

2!b

CM MM 1 be

Z

t 0

e!s

X 0
e!tk ck .s/ds:

186

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

Then " !b

1 Me

m X

# !tk

e

ck .t/ C C M 2 MM 1 be2!b

Z

t

0

kD1

X

e!s

C Me!b C M 2 MM 1 be2!b

Z

e!tk ck .s/ds

0
e!s p.s/ ..s//ds:

Thus we have Z

.t/ C0 C C1 CC2

Z

C0 C

t

e!s

0

X

e!tk ck .s/ds

0
e!s p.s/ ..s//ds

0

Z

t

t

O M.s/Œ.s/ C

0

where O M.s/ D max.C1 e!s

X

..s//ds;

e!tk ck ; C2 e!s p.s//:

0
Set v.t/ D C0 C

Z

t

O M.s/Œ.s/ C

0

..s//ds:

Then we have .t/ v.t/ for all t 2 J: Differentiating the both sides of (8.27) we get O v 0 .t/ D M.t/Œ.t/ C

..t//; a:e: t 2 J;

and v.0/ D C0 : Using the nondecreasing character of

we obtain

O v 0 .t/ M.t/Œv.t/ C

.v.t//; a:e: t 2 J;

that is v 0 .t/ O M.t/: v.t/ C .v.t//

(8.27)

8.5 Application to the Control Theory

187

Integrating from 0 to t both sides of this inequality, we get Z

v 0 .s/ ds v.s/ C .v.s//

t

0

Z

t

O M.s/ds:

0

By a change of variables we get Z

v.t/

v.0/

du O L1 < 1: kMk u C .u/

Consequently, by (8.24), there exists a constant d such that .t/ v.t/ d; t 2 J and hence from the definition of it follows that kyk˝ max.kkD ; d/: This shows that the set E is bounded. As a consequence of Theorem 1.32 we deduce that A C B has a fixed point which is a integral solution of problem (8.21)–(8.23). Thus the system (8.21)–(8.23) is controllable on Œr; b. t u

8.5.2 An Example As an application of our results we consider the following impulsive partial functional differential equation of the form @2 @ z.t; x/ 2 2 z.t; x/ @t @x C ŒQ1 .t; z.t r; x//; Q2 .t; z.t r; x//; x 2 Œ0; ; t 2 Œ0; bnft1 ; t2 ; : : : ; tm g: (8.28) z.tkC ; x/ z.tk ; x/ 2 bk jz.tk ; x/jB.0; 1/; x 2 Œ0; ; k D 1; : : : ; m (8.29) z.t; 0/ D z.t; / D 0; t 2 J WD Œ0; b

(8.30)

z.t; x/ D .t; x/; t 2 H; x 2 Œ0; ;

(8.31)

where bk > 0; k D 1; : : : ; m; 2 D D f N W H Œ0; ! RI N is continuous everywhere except for a countable number of points at which N .s /; N .sC / exist with N .s / D N .s/g; 0 D t0 < t1 < t2 < < tm < tmC1 D b; z.tkC / D lim z.tk C h; x/; z.tk / D lim z.tk C h; x/, where Q1 ; Q2 W J R ! .h;x/!.0C ;x/

.h;x/!.0 ;x/

R; are given functions, and B.0; 1/ the closed unit ball. We assume that for each t 2 J; Q1 .t; / is lower semi-continuous (i.e, the set fy 2 R W Q1 .t; y/ > g is open for each 2 R), and assume that for each t 2 J; Q2 .t; / is upper semi-continuous (i.e., the set fy 2 R W Q2 .t; y/ < g is open for each 2 R).

188

8 Non-densely Defined Functional Differential Inclusions with Finite Delay

Let y.t/.x/ D z.t; x/; t 2 J; x 2 Œ0; ; Ik .y.tk //.x/

D bk z.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m

F.t; /.x/ D ŒQ1 .t; .; x//; Q2 .t; .; x//; 2 H; x 2 Œ0; ; and ./.x/ D .; x/; 2 H; x 2 Œ0; : It is clear that F is compact and convex valued, and it is upper semi-continuous W Œ0; 1/ ! .0; 1/ (see [101]). Assume that there are p 2 C.J; RC / and continuous and nondecreasing such that max.jQ1 .t; y/j; jQ2 .t; y/j/ p.t/ .jyj/;

t 2 J; and y 2 R;

and Z 1

1

ds D C1: .s/

Consider E D C.Œ0; /, the Banach space of continuous function on Œ0; with values in R. Define the linear operator A on E by Az D

@2 z; @x2

on D.A/ D fz 2 C.Œ0; / W z.0/ D z./ D 0;

@2 z 2 C.Œ0; /g: @x2

Now, we have D.A/ D C0 .Œ0; / D fv 2 C.Œ0; / W v.0/ D v./ D 0g ¤ C.Œ0; /: It is well known from [100] that A is sectorial, .0; C1/ .A/ and for > 0 kR.; A/kB.E/

1 :

It follows that A generates an integrated semigroup .S.t//t0 and that kS0 .t/kB.E/ et for t 2 J for some constant > 0 and A satisfied the Hille–Yosida condition. Assume that there exist functions e l1 ; e l2 2 L1 .J; RC / such that l1 .t/jw wj; t 2 J; w; w 2 R; jQ1 .t; w/ Q1 .t; w/j e

8.6 Notes and Remarks

189

and l2 .t/jw wj; t 2 J; w; w 2 R: jQ2 .t; w/ Q2 .t; w/j e We can show that problem (8.21)–(8.23) is an abstract formulation of problem (8.28)–(8.31). Since all the conditions of Theorem 8.2 are satisfied, the problem (8.28)–(8.31) has a solution z on Œr; b Œ0; :

8.6 Notes and Remarks The results of Chap. 8 are taken from Abada et al. [2, 4]. Other results may be found in [51, 54, 74, 109, 110].

Chapter 9

Impulsive Semi-linear Functional Differential Equations

9.1 Introduction In this chapter, we shall prove the existence of mild solutions of first order impulsive functional equations in a separable Banach space. Our approach will be based for the existence of mild solutions, on a fixed point theorem of Burton and Kirk [88] for the sum of a contraction map and a completely continuous map.

9.2 Semi-linear Differential Evolution Equations with Impulses and Delay 9.2.1 Introduction In this section, we shall establish sufficient conditions for the existence of mild and extremal mild solutions of first order impulsive functional equations in a separable Banach space .E: j:j/ of the form: y0 .t/ Ay.t/ D f .t; yt /; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m

(9.1)

yjtDtk D Ik .y.tk //; k D 1; : : : ; m

(9.2)

y.t/ D .t/; t 2 Œr; 0 ;

(9.3)

where f W J D ! E is a given function, D D f W Œr; 0 ! E; everywhere except for a finite number of points s at which .s / ; .s / D .s/g, 2 D; 0 < r < 1; 0 D t0 < t1 < < tm

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_9

isCcontinuous s exist and < tmC1 D b;

191

192

9 Impulsive Semi-linear Functional Differential Equations

Ik 2 C .E; E/ ; k D 1; 2; : : : ; m; A W D.A/ E ! E is the infinitesimal generator of a C0 -semigroup T.t/; t 0, and E a real separable Banach space with norm j:j : In the case where the impulses are absent (i.e., Ik D 0; k D 1; : : : ; m) and F is a single or multi-valued map and A is a densely defined linear operator generating a C0 -semigroup of bounded linear operators the problem (9.1)–(9.3) has been investigated on compact intervals in, for instance, the monographs by Ahmed [16], Hu and Papageorgiou [143], Kamenskii et al. [144], and Wu [184], and the papers of Benchohra and Ntouyas [58, 60, 63]. Next, we study the impulsive functional differential equations with nonlocal initial conditions of the form y0 .t/ Ay.t/ D f .t; yt /; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m

(9.4)

yjtDtk D Ik .y.tk //; k D 1; : : : ; m

(9.5)

y.t/ C ht .y/ D .t/; t 2 Œr; 0 ;

(9.6)

where ht W PC.Œr; b; E/ ! E is a given function. The nonlocal condition can be applied in physics with better effect than the classical initial condition y .0/ D y0 . For example, ht .y/ may be given by ht .y/ D

p X

ci y.ti C t/; t 2 Œr; 0

iD1

where ci ; i D 1; : : : ; p; are given constants and 0 < t1 < < tp b:

9.2.2 Existence of Mild Solutions Definition 9.1. A function y 2 PC .Œr; b ; E/ is said to be a mild solution of problem (11.15)–(11.17) if y.t/ D .t/; t 2 Œr; 0; and y is a solution of impulsive integral equation Z t X T.t s/f .t; ys /ds C T.t tk /Ik .y.tk //; t 2 J: y.t/ D T.t/.0/ C 0

0
Let us introduce the following hypotheses: (9.1.1) A W D.A/ E ! E is the infinitesimal generator of a C0 -semigroup fT.t/g, t 2 J which is compact for t > 0 in the Banach space E. Let M D supfkT.t/kB.E/ W t 2 Jg;

9.2 Semi-linear Differential Evolution Equations with Impulses and Delay

(9.1.2) There exist constants dk > 0; k D 1; : : : ; m with M

m X

193

dk < 1 such that

kD1

for each y; x 2 E jIk .y/ Ik .x/j dk jy xj

(9.1.3) The function f W J D ! E is Carathéodory; (9.1.4) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that jf .t; x/j p.t/ .kxkD /;

a:e: t 2 J;

for all x 2 D;

with Z

1

ds > D1 kpkL1 ; .s/

Do

where M.kk C

m X

jIk .0/j/

kD1 m X

D0 D

1M

;

D1 D

dk

M : m X 1M dk

kD1

kD1

Theorem 9.2. Assume that (9.1.1)–(9.1.4) hold. Then the problem (9.1)–(9.3) has at least one mild solution on Œr; b: Proof. Consider the two operators: A; B W PC .Œr; b ; E/ ! PC .Œr; b ; E/ : defined by 8 ˆ < 0; X

A.y/ .t/ WD T .t tk / Ik y tk ; ˆ :

if t 2 H; if t 2 J;

0
and

B.y/ .t/ WD

8 ˆ < .t/; ˆ : T.t/.0/ C

if t 2 H; Z

t 0

T.t s/f .s; ys / ds;

if t 2 J:

194

9 Impulsive Semi-linear Functional Differential Equations

Then, the problem of finding the solution of problem (9.1)–(9.3) is reduced to finding the solution of the operator equation A .y/ .t/ CB .y/ .t/ D y .t/ ; t 2 Œr; b. We shall show that the operators A and B satisfy all the conditions of Theorem 1.32 For better readability, we break the proof into a sequence of steps. Step 1: B is continuous. Let fyn g be a sequence such that yn ! y in PC.Œr; b; E/. Then for t 2 J ˇZ t ˇ ˇ ˇ ˇ jB.yn /.t/ B.y/.t/j D ˇ T.t s/Œf .s; yns / f .s; ys /dsˇˇ 0

Z

b

M 0

jf .s; yns / f .s; ys /j ds:

Since f .s; / is continuous for a.e. s 2 J, we have by the Lebesgue dominated convergence theorem jB.yn /.t/ B.y/.t/j ! 0 as n ! 1: Thus B is continuous. Step 2: B maps bounded sets into bounded sets in PC.Œr; b; E/. It is enough to show that for any q > 0 there exists a positive constant l such that for each y 2 Bq D fy 2 PC.Œr; b; E/ W kyk qg we have kB .y/k l: So choose y 2 Bq ; then we have for each t 2 J; ˇ ˇ Z t ˇ ˇ ˇ jB.y/.t/j D ˇT.t/.0/ C T.t s/f .s; ys /dsˇˇ 0

Mj.0/j C M .q/

Z

b

p.s/ ds: 0

Then we have kB.y/k Mkk C M .q/kpkL1 WD l: Step 3: B maps bounded sets into equi-continuous sets of PC.Œr; b; E/: We consider Bq as in step 2 and let 1 ; 2 2 Jn ft1 ; : : : ; tm g ; 1 < 2 . Thus if > 0 and 1 < 2 we have jB.y/. 2 / B.y/. 1 /j jT. 2 /.0/ T. 1 /.0/j Z 1 C .q/ kT. 2 s/ T. 1 s/kB.E/ p.s/ds Z C .q/ Z C .q/

0

1

1

2

1

kT. 2 s/ T. 1 s/kB.E/ p.s/ds

kT. 2 s/kB.E/ p.s/ds:

9.2 Semi-linear Differential Evolution Equations with Impulses and Delay

195

As 1 ! 2 and become sufficiently small, the right-hand side of the above inequality tends to zero, since T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0 implies the continuity in the uniform operator topology [16]. This proves the equi-continuity for the case where t ¤ ti ; k D 1; 2; : : : ; m C 1. It remains to examine the equi-continuity at t D ti . First we prove equi-continuity at t D ti . Fix ı1 > 0 such that ftk W k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < h < ı1 we have jB.y/.ti h/ B.y/.ti /j j .T.ti h/ T.ti // .0/j Z ti h C j .T.ti h s/ T.ti s// f .s; ys /jds 0

Z

C .q/M

ti

p.s/dsI ti h

which tends to zero as h ! 0. Define BO0 .y/.t/ D B.y/.t/; t 2 Œ0; t1 ; and BOi .y/.t/ D

B.y/.t/; if t 2 .ti ; tiC1 B.y/.tiC /; if t D ti :

Next we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk W k ¤ ig \ Œti ı2 ; ti C ı2 D ;. For 0 < h < ı2 we have O O jB.y/.t i C h/ B.y/.ti /j j .T.ti C h/ T.ti // .0/j Z ti C j .T.ti C h s/ T.ti s// f .s; ys /jds 0

Z

ti Ch

C .q/M

p.s/ds: ti

The right-hand side tends to zero as h ! 0. The equi-continuity for the cases

1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval H: As a consequence of Steps 1–3 together with Arzelá–Ascoli theorem, it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For y 2 Bq we define Z B .y/.t/ D T.t/.0/ C T./

t 0

T.t s /f .s; ys /ds:

196

9 Impulsive Semi-linear Functional Differential Equations

Since T.t/ is a compact operator, the set Y .t/ D fB .y/.t/ W y 2 Bq g is precompact in E for every ; 0 < < t: Moreover, for every y 2 Bq we have Z jB.y/.t/ B .y/.t/j

t

.q/

kT.t s/kB.E/ p.s/ds

t

Z

.q/M

t

p.s/ds: t

Therefore, there are precompact sets arbitrarily close to the set Y .t/ D fB .y/.t/ W y 2 Bq g: Hence the set Y.t/ D fB.y/.t/ W y 2 Bq g is precompact in E. Hence the operator B W PC .Œr; b ; E/ ! PC .Œr; b ; E/ is completely continuous. Step 4: A is a contraction Let x; y 2 PC.Œr; b; E/. Then for t 2 J ˇ ˇ ˇX ˇ ˇ

ˇ

ˇ ˇ jA.y/.t/ A.x/.t/j D ˇ T .t tk / Ik y tk Ik x tk ˇ ˇ0

ˇ ˇIk y t Ik x t ˇ M k k 0
M

m X

ˇ ˇ dk ˇy tk x tk ˇ

kD1

M

m X

dk ky xk :

kD1

Then kA.y/ A.x/k M

m X

dk ky xk ;

kD1

which is a contraction, since M

m P

dk < 1.

kD1

Step 5: A priori bounds. Now it remains to show that the set o n y for some 0 < < 1 E D y 2 PC.Œr; b; E/ W y D B.y/ C A y is bounded. Let y 2 E; then y D B.y/ C A for some 0 < < 1: Thus, for each t 2 J;

9.2 Semi-linear Differential Evolution Equations with Impulses and Delay

Zt y.t/ D T.t/.0/ C

T.t s/f .s; ys /ds C

X

197

T .t tk / Ik

0
0

y t : k

This implies by (9.1.2) and (9.1.4) that, for each t 2 J; we have Zt jy.t/j Mj.0/j C M

p.s/ .kys k/ds C M 0

m ˇ X y ˇˇ ˇ t ˇ ˇIk k kD1

Zt Mkk C M

p.s/ .kys k/ds 0

m ˇ X

m ˇ y X ˇ ˇ CM tk Ik .0/ˇ C M jIk .0/j ˇIk kD1 kD1

M kk C

m X

!

Zt

jIk .0/j C M

kD1

p.s/ .kys k/ds 0

ˇ y ˇ ˇ ˇ CM dk ˇ tk ˇ kD1 m X

M kk C

m X

3 2 t Z m X ˇ ˇ jIk .0/j C M 4 p.s/ .kys k/ds C dk ˇy tk ˇ5 : !

kD1

kD1

0

Now we consider the function defined by .t/ D supfjy.s/j W r s tg; 0 t b: Then kys k .t/ for all t 2 J and there is a point t 2 Œr; t such that .t/ D jy.t /j. If t 2 Œ0; b, by the previous inequality we have for t 2 Œ0; b (note t t) ! Zt m m X X .t/ M kk C jIk .0/j C M p.s/ ..s//ds C M dk .t/: kD1

kD1

0

Then 1M

m X kD1

Thus we have

! dk .t/ M kk C

m X kD1

!

Zt

jIk .0/j C M

p.s/ ..s//ds: 0

198

9 Impulsive Semi-linear Functional Differential Equations

Zt .t/ D0 C D1

p.s/ ..s//ds: 0

Let us take the right-hand side of the above inequality as v.t/. Then we have .t/ v.t/ for all t 2 J; v.0/ D D0 ; and v 0 .t/ D D1 p.t/ ..t//; a:e: t 2 J: Using the nondecreasing character of

we get

v 0 .t/ D1 p.t/ .v.t//; a:e: t 2 J: That is v 0 .t/ D1 p .t/ ; a:e: t 2 J: .v.t// Integrating from 0 to t we get Z t Z t v 0 .s/ ds D1 p .s/ ds: .v.s// 0 0 By a change of variable we get Z

v.t/

v.0/

du D1 .u/

Z

b 0

Z

1

p .s/ ds D D1 kpkL1 < D0

du : .u/

Hence there exists a constant N such that .t/ v.t/ N

for all t 2 J:

Now from the definition of it follows that kyk D sup jy.t/j .b/ N;

for all y 2 E:

t2Œr;b

This shows that the set E is bounded. As a consequence of Theorem 1.32 we deduce that A C B has a fixed point which is a mild solution of problem (11.15)–(11.17). t u

9.2 Semi-linear Differential Evolution Equations with Impulses and Delay

199

9.2.3 Existence of Extremal Mild Solutions In this section we shall prove the existence of maximal and minimal solutions of problem (9.1)–(9.3) under suitable monotonicity conditions on the functions involved in it. We need the following definitions in the sequel. Definition 9.3. We say that a function v 2 PC.Œr; b; E/ is a lower mild solution of problem (9.1)–(9.3) if v.t/ D .t/; t 2 H; and Z t X v.t/ T.t/.0/ C T.t s/f .s; vs / ds C T.t tk /Ik .v.tk //; t 2 J; t ¤ tk 0

0
and v.tkC / v.tk / Ik .v.tk //; t D tk ; k D 1; : : : ; m. Similarly an upper mild solution w of problem (9.1)–(9.3) is defined by reversing the order. Definition 9.4. A solution xM of problem (9.1)–(9.3) is said to be maximal if for any other solution x of problem (9.1)–(9.3) on J, we have that x.t/ xM .t/ for each t 2 J. Similarly a minimal solution of problem (9.1)–(9.3) is defined by reversing the order of the inequalities. Definition 9.5. A function f .t; x/ is called strictly monotone increasing in x almost everywhere for t 2 J, if .t; x/ f .t; y/ a.e. t 2 J for all x; y 2 D with x < y. Similarly f .t; x/ is called strictly monotone decreasing in x almost everywhere for t 2 J, if f .t; x/ f .t; y/ a.e. t 2 J for all x; y 2 D with x < y. We consider the following assumptions in the sequel. (9.10.1) The function f .t; y/ is strictly monotone increasing in y for almost each t 2 J. (9.10.2) T.t/ is preserving the order, that is T.t/v 0 whenever v 0. (9.10.3) The function Ik , k D 1; : : : ; m are continuous and nondecreasing. (9.10.4) The problem (9.1)–(9.3) has a lower mild solution v and an upper mild solution w with v w. Theorem 9.6. Assume that assumptions (9.1.1)–(9.1.4) and (9.10.1)–(9.10.4) hold. Then problem (9.1)–(9.3) has minimal and maximal solutions on Œr; b. Proof. It can be shown, as in the proof of Theorem 9.2, that B is completely continuous and A is a contraction on Œv; w. We shall show that A and B are isotone increasing on Œv; w. Let y; y 2 Œa; b be such that y y; y 6D y: Then by (9.10.1), (9.10.2), we have for each t 2 J Z t B.y/ .t/ D T.t/.0/ C T.t s/f .s; ys / ds Z

0

t

T.t/.0/ C 0

D B.y/ .t/ :

T.t s/f .s; ys / ds

200

9 Impulsive Semi-linear Functional Differential Equations

and by (9.10.3), we have for each t 2 J X

A.y/ .t/ D T .t tk / Ik y tk 0

X

T .t tk / Ik y tk

0
D A.y/ .t/ : Therefore A and B are isotone increasing on Œv; w. Finally, let x 2 Œv; w be any element. By (9.10.4) we deduce that v A.v/ C B.v/ A.x/ C B.x/ A.w/ C B.w/ w; which shows that A.x/ C B.x/ 2 Œv; w for all x 2 Œv; w. Thus, A and B satisfy all conditions of Theorem 1.32, hence problem (9.1)–(9.3) has maximal and minimal solutions on Œr; b: t u

9.2.4 Impulsive Differential Equations with Nonlocal Conditions In this section we shall prove the existence results for problem (9.4)–(9.6). Nonlocal conditions were initiated by Byszewski [89] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. Definition 9.7. A function y 2 PC .Œr; b ; E/ is said to be a mild solution of problem (9.4)–(9.6) if y.t/ D .t/ ht .y/ ; t 2 Œr; 0; and Z t T.t s/f .s; ys / ds y.t/ D T.t/ ..0/ h0 .y// C C

X

0

T .t tk / Ik y tk ; t 2 J:

0
Theorem 9.8. Assume that hypotheses (9.1.1)–(9.1.3) hold and moreover (A1) The function h is continuous with respect to t, and there exists a constant ˛ > 0 such that jht .u/j ˛;

u 2 PC.Œr; b; E/

and for each k > 0 the set f.0/ h0 .y/; y 2 PC.Œr; b; E/; kyk kg is precompact in E

9.2 Semi-linear Differential Evolution Equations with Impulses and Delay

201

(A2) There exists a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that jf .t; x/j p.t/ .kxkD /; with

Z

1

Q0 D

a:e: t 2 J;

for all x 2 D

ds > D1 kpkL1 ; .s/

and MŒkkD C ˛ C Q0 D D

m X

jIk .0/j

kD1

1M

m X

:

dk

kD1

Then the problem 9.4)–(9.6) has at least one mild solution on Œr; b: Proof. Consider the two operators: B1 W PC .Œr; b ; E/ ! PC .Œr; b ; E/ defined by 8 if t 2 H; ˆ < .t/ ht .y/; Z t B1 .y/.t/ D ˆ : T.t/ ..0/ h0 .y// C if t 2 J; T.t s/f .s; ys / ds; 0

and 8 ˆ < 0; X A1 .y/.t/ D T.t tk /Ik .y.tk //; ˆ :

if t 2 H; if t 2 J:

0
Then the problem of finding the solution of problem (9.4)–(9.6) is reduced to finding the solution of the operator equation A1 .y/ .t/ CB2 .y/ .t/ D y .t/ ; t 2 Œr; b. As in Sect. 9.3, we can show that the operators A1 and B1 satisfy all conditions of Theorem 1.32. t u

9.2.5 An Example As an application of our results we consider the following impulsive partial functional differential equation of the form

202

9 Impulsive Semi-linear Functional Differential Equations

@2 @ z.t; x/ D 2 z.t; x/ @t @x CQ.t; z.t r; x//; x 2 Œ0; ; t 2 Œ0; bnft1 ; t2 ; : : : ; tm g:

(9.7)

z.tkC ; x/ z.tk ; x/ D bk z.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m

(9.8)

z.t; 0/ D z.t; / D 0; t 2 Œ0; b z.t; x/ D .t; x/; t 2 H; x 2 Œ0; ;

(9.9) (9.10)

where bk > 0; k D 1; : : : ; m; 2 D D f W H Œ0; ! RI is continuous everywhere except for a countable number of points at which .s /; .sC / exist with .s / D .s/g; 0 D t0 < t1 < t2 < < tm < tmC1 D b; z.tkC / D lim z.tk C h; x/; z.tk / D lim z.tk C h; x/ and Q W Œ0; b R ! R is a .h;x/!.0 ;x/

.h;x/!.0C ;x/

given function. Let y.t/.x/ D z.t; x/; t 2 J; x 2 Œ0; ; Ik .y.tk //.x/ D bk z.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m F.t; /.x/ D Q.t; .; x//; 2 H; x 2 Œ0; ; ./.x/ D .; x/; 2 H; x 2 Œ0; : Take E D L2 Œ0; and define A W D.A/ E ! E by Aw D w00 with domain D.A/ D fw 2 E; w; w0 are absolutely continuous, w00 2 E; w.0/ D w./ D 0g: Then Aw D

1 X

n2 .w; wn /wn ; w 2 D.A/

nD1

q 2 sin ns; n D 1; 2; : : : is where ( , ) is the inner product in L2 and wn .s/ D the orthogonal set of eigenvectors in A: It is well known (see [168]) that A is the infinitesimal generator of an analytic semigroup T.t/; t 2 Œ0; b in E and is given by T.t/w D

1 X

exp.n2 t/.w; wn /wn ; w 2 E:

nD1

Since the analytic semigroup T.t/ is compact, there exists a constant M 1 such that kT.t/kB.E/ M: Also assume that there exists an integrable function W Œ0; b ! RC such that jQ.t; w.t r; x//j .t/˝.jwj/

9.3 Impulsive Semi-linear Functional Differential Equations. . .

203

where ˝ W Œ0; 1/ ! .0; 1/ is continuous and nondecreasing with Z 1 ds D C1: s C ˝.s/ 1 Assume that there exists a function Ql 2 L1 .Œ0; b; RC / such that jQ.t; w/ Q.t; w/j Ql.t/jw wj; t 2 Œ0; b; w; w 2 R: We can show that problem (11.15)–(11.17) is an abstract formulation of problem (9.7)–(9.10). Since all the conditions of Theorem 9.2 are satisfied, the problem (9.7)–(9.10) has a solution z on Œr; b Œ0; :

9.3 Impulsive Semi-linear Functional Differential Equations with Non-densely Defined Operators 9.3.1 Introduction In this section, we shall be concerned with the existence of integral solutions and extremal integral solutions defined on a compact real interval for first order impulsive semi-linear functional equations in a separable Banach space. We will consider the following first order impulsive semi-linear differential equations of the form: y0 .t/ Ay.t/ D f .t; yt /; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m

(9.11)

yjtDtk D Ik .y.tk //; k D 1; : : : ; m

(9.12)

y.t/ D .t/; t 2 Œr; 0 ;

(9.13)

where f W J D ! E is a given function, D D f W Œr; 0 ! E; is continuous

everywhere except for a finite number of points s at which .s / ; sC exist and .s / D .s/g, 2 D, .0 < r < 1/, 0 D t0 < t1 < < tm < tmC1 D b, Ik W E ! E .k D 1; 2; : : : ; m/, A W D.A/ E ! E is a non-densely defined closed linear operator on E, and E a real separable Banach space with norm j:j: We shall prove the existence of extremal integral solutions of the problem (11.15)–(11.17), and our approach here is based on the concept of upper and lower solutions combined with a fixed point theorem on ordered Banach spaces established recently by Dhage [102]. Next, we study the impulsive functional differential equations with nonlocal initial conditions of the form y0 .t/ Ay.t/ D f .t; yt /; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m yjtDtk D

Ik .y.tk //;

(9.14)

k D 1; : : : ; m

(9.15)

y.t/ C ht .y/ D .t/; t 2 Œr; 0 ;

(9.16)

204

9 Impulsive Semi-linear Functional Differential Equations

where ht W PC.Œr; b; D.A// ! D.A/ is a given function. The nonlocal condition can be applied in physics with better effect than the classical initial condition y .0/ D y0 . For example, ht .y/ may be given by ht .y/ D

p X

ci y.ti C t/; t 2 Œr; 0

(9.17)

iD1

where ci ; i D 1; : : : ; p; are given constants and 0 < t1 < < tp p:

9.3.2 Examples of Operators with Non-dense Domain In this section we shall present examples of linear operators with non-dense domain satisfying the Hille–Yosida estimate. More details can be found in the paper by Da Prato and Sinestrari [100]. Example 9.9. Let E D C.Œ0; 1; R/ and the operator A W D.A/ ! E defined by Ay D y0 , where D.A/ D fy 2 C1 ..0; 1/; R/ W y.0/ D 0g: Then D.A/ D fy 2 C..0; 1/; R/ W y.0/ D 0g 6D E: Example 9.10. Let E D C.Œ0; 1; R/ and the operator A W D.A/ ! E defined by Ay D y00 , where D.A/ D fy 2 C2 ..0; 1/; R/ W y.0/ D y.1/ D 0g: Then D.A/ D fy 2 C..0; 1/; R/ W y.0/ D y.1/ D 0g 6D E: Example 9.11. Let us set for some ˛ 2 .0; 1/ E D C0˛ .Œ0; 1; R/ D fy W Œ0; 1 ! R W y.0/ D 0 and

jy.t/ y.s/j < 1g jt sj˛ 0t<s1 sup

and the operator A W D.A/ ! E defined by Ay D y0 , where D.A/ D fy 2 C1C˛ ..0; 1/; R/ W y.0/ D y0 .0/ D 0g: Then D.A/ D h˛0 .0; 1/; R/ D fy W Œ0; 1 ! R W lim

sup

ı!0 0<jtsjı

jy.t/ y.s/j D 0g 6D E: jt sj˛

9.3 Impulsive Semi-linear Functional Differential Equations. . .

205

Here C1C˛ .Œ0; 1; R/ D fy W Œ0; 1 ! R W y0 2 C˛ .Œ0; 1; R/g: The elements of h˛ ..0; 1/; R/ are called little Holder functions and it can be proved that the closure of C1 ..0; 1/; R/ in C˛ ..0; 1/; R/ is h˛ ..0; 1/; R/ (see [175], Theorem 5.3). Example 9.12. Let ˝ Rn be a bounded open set with regular boundary and define E D C.˝; R/ and the operator A W D.A/ ! E defined by Ay D y, where D.A/ D fy 2 C.˝; R/ W y D 0 on I y 2 C.˝; R/g: Here is the Laplacian in the sense of distributions on ˝. In this case we have D.A/ D fy 2 C.˝; R/ W y D 0 on g 6D E:

9.3.3 Existence of Integral Solutions Definition 9.13. We say that y W Œr; T ! E is an integral solution of (9.11)– (9.13) if Z t Z t X

y.s/ds C f .s; ys /ds C Ik y tk ; t 2 J: (i) y.t/ D .0/ C A Z (ii) 0

0

t

0

0
y.s/ds 2 D.A/ for t 2 J, and y.t/ D .t/; t 2 H:

From the definition it follows that y.t/ 2 D.A/, for each t 0; in particular .0/ 2 D.A/: Moreover, y satisfies the following variation of constants formula: y.t/ D S0 .t/.0/ C

d dt

Z

t 0

S.t s/f .s; ys /ds C

X

t 0: S0 .t tk / Ik y tk

0
(9.18) We notice also that if y satisfies (9.18), then y.t/ D S0 .t/.0/ C lim C

X

Z

!1 0

t

S0 .t s/B f .s; ys /ds

S0 .t tk / Ik y tk ; t 0:

0
Let us introduce the following hypotheses: (9.17.1) A satisfies Hille–Yosida condition;

206

9 Impulsive Semi-linear Functional Differential Equations

(9.17.2) There exist constants dk > 0; k D 1; : : : ; m such that for each y; x 2 D.A/ jIk .y/ Ik .x/j dk jy xj (9.17.3) The function f W J D ! E is Carathéodory; (9.17.4) The operator S0 .t/ is compact in D.A/ wherever t > 0I (9.17.5) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that jf .t; x/j p.t/ .kxkD /; Z

b

with 0

a:e: t 2 J;

for all x 2 D

e!s p.s/ds < 1, Z

1 c0

where

du > c1 .u/

Z

b

e!s p .s/ ds:

0

(9.19)

m P e!b M kk C jIk .0/j c0 D

1

kD1 m P

Me!b

;

(9.20)

dk

kD1

and c1 D

Me!b : m P 1 Me!b dk

(9.21)

kD1

Theorem 9.14. Assume that (9.17.1)–(9.17.5) hold. If Me!b

m X

dk < 1;

(9.22)

kD1

then the problem (9.11)–(9.13) has at least one integral solution on Œr; b. Proof. Consider the two operators: A; B W PC Œr; b ; D.A/ ! PC Œr; b ; D.A/ defined by 8 ˆ < 0; X

A.y/ .t/ WD S0 .t tk / Ik y tk ; ˆ : 0
if t 2 H; if t 2 J;

9.3 Impulsive Semi-linear Functional Differential Equations. . .

207

and 8 .t/; ˆ ˆ ˆ < B.y/ .t/ WD S0 .t/.0/ Z t ˆ ˆ ˆ :C d S.t s/f .s; ys / ds; dt 0

if t 2 H;

if t 2 J:

The problem of finding the solution of problem (9.11)–(9.13) is reduced to finding the solution of the operator equation A .y/ .t/ CB .y/ .t/ D y .t/ ; t 2 Œr; b. We shall show that the operators A and B satisfy all the conditions of Theorem 1.32. For better readability, we break the proof into a sequence of steps. Step 1: B is continuous. Let fyn g be a sequence such that yn ! y in PC.Œr; b; D.A//. Then for ! > 0 (if ! < 0 one has e!t < 1) ˇ Z t ˇ ˇd ˇ jB.yn /.t/ B.y/.t/j D ˇˇ S.t s/Œf .s; yns / f .s; ys /dsˇˇ dt 0 Z b Me!b e!s jf .s; yns / f .s; ys /j ds: 0

Since f .s; / is continuous for a.e. s 2 J, we have by the Lebesgue dominated convergence theorem jB.yn /.t/ B.y/.t/j ! 0 as n ! 1: Thus B is continuous. Step 2: B maps bounded sets into bounded sets in PC.Œr; b; D.A//: It is enough to show that for any q > 0 there exists a positive constant l such that for each y 2 Bq D fy 2 PC.Œr; b; D.A// W kyk qg we have kB .y/k l: So choose y 2 Bq ; then we have for each t 2 J ˇ ˇ Z ˇ 0 ˇ d t ˇ jB.y/.t/j D ˇS .t/.0/ C S.t s/f .s; ys /dsˇˇ dt 0 Z b !b !b Me j.0/j C Me .q/ e!s p.s/ ds: 0

Then we have jB.y/.t/j Me!b kk C Me!b .q/

Z

b

e!s p.s/ ds WD l:

0

Step 3: B maps bounded sets into equi-continuous sets of PC.Œr; b; D.A//:

208

9 Impulsive Semi-linear Functional Differential Equations

We consider Bq as in Step 2 and let 1 ; 2 2 Jn ft1 ; : : : ; tm g ; 1 < 2 .Thus if > 0 and 1 < 2 we have jB.y/. 2 / B.y/. 1 /j jS0 . 2 /.0/ S0 . 1 /.0/j ˇ ˇ Z 1 ˇ ˇ 0 0 ˇ C ˇ lim ŒS . 2 s/ S . 1 s/B f .s; ys /dsˇˇ !1 0

ˇ ˇ Z 1 ˇ ˇ C ˇˇ lim ŒS0 . 2 s/ S0 . 1 s/B f .s; ys / dsˇˇ !1 1 ˇ ˇ Z 2 ˇ ˇ 0 ˇ C ˇ lim S . 2 s/B f .s; ys / dsˇˇ : !1 1

As 1 ! 2 and become sufficiently small, the right-hand side of the above inequality tends to zero, since S0 .t/ is a strongly continuous operator and the compactness of S0 .t/ for t > 0 implies the continuity in the uniform operator topology. This proves the equi-continuity for the case where t ¤ ti ; k D 1; 2; : : : ; m C 1. It remains to examine the equi-continuity at t D ti . First we prove equi-continuity at t D ti . Fix ı1 > 0 such that ftk W k ¤ ig \ Œti ı1 ; ti C ı1 D ;: For 0 < h < ı1 we have

jB.y/.ti h/ B.y/.ti /j j S0 .ti h/ S0 .ti / .0/j Z ti h

C lim k S0 .ti h s/S0 .ti s/ B f .s; ys /kds !1 0 !b

CMe

Z

.q/

ti

e!s p.s/ dsI

ti h

which tends to zero as h ! 0. Define BO0 .y/.t/ D B.y/.t/; t 2 Œ0; t1 and

BOi .y/.t/ D

B.y/.t/; if t 2 .ti ; tiC1 B.y/.tiC /; if t D ti :

Next we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk W k ¤ ig \ Œti ı2 ; ti C ı2 D ;. For 0 < h < ı2 we have 0

0 O O jB.y/.t i C h/ B.y/.ti /j j S .ti C h/ S .ti / .0/j Z ti

C lim k S0 .ti C h s/ S0 .ti s/ B f .s; ys /kds !1 0 !b

CMe

Z

ti Ch

.q/ ti

e!s p.s/ ds:

9.3 Impulsive Semi-linear Functional Differential Equations. . .

209

The right-hand side tends to zero as h ! 0. The equi-continuity for the cases

1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval Œr; 0. As a consequence of steps 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps Bq into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For y 2 Bq we define B .y/.t/ D S0 .t/.0/ C S0 ./ lim

Z

!1 0

t

S0 .t s /B f .s; ys /ds:

Note

Z lim

!1 0

t

S0 .t s /B f .s; ys / ds W y 2 Bq

is a bounded set since ˇ ˇ Z t Z ˇ ˇ 0 !b ˇ lim ˇ S .t s /B f .s; ys / dsˇ Me .q/ ˇ !1 0

t

e!s p.s/ds:

0

Since S0 .t/ is a compact operator, the set Y .t/ D fB .y/.t/ W y 2 Bq g is precompact in E for every ; 0 < < t: Moreover, for every y 2 Bq we have Z t !b jB.y/.t/ B .y/.t/j Me .q/ e!s p.s/ds: t

Therefore, there are precompact sets arbitrarily close to the set Y .t/ D fB .y/.t/ W y 2 Bq g: Hence the set Y.t/ D fB.y/.t/W y 2 Bq g is precompact in E. Hence the

operator B W PC Œr; b ; D.A/ ! PC Œr; b ; D.A/ is completely continuous. Step 4: A is a contraction. Let x; y 2 PC.Œr; b; D.A//. Then for t 2 J ˇ ˇ ˇX ˇ ˇ ˇ

jA.y/.t/ A.x/.t/j D ˇˇ S0 .t tk / Ik y tk Ik x tk ˇˇ ˇ0

ˇ ˇI k y t I k x t ˇ Me!b k k 0
Me!b

m X

ˇ ˇ dk ˇy tk x tk ˇ

kD1

Me!b

m X kD1

dk ky xk :

210

9 Impulsive Semi-linear Functional Differential Equations

Then !b

kA.y/ A.x/k Me

m X

dk ky xk ;

kD1

which is a contraction from (9.22). Step 5: A priori bounds. Now it remains to show that the set o y n for some 0 < < 1 E D y 2 PC.Œr; b; D.A// W y D B.y/ C A is bounded. y for some 0 < < 1: Thus, for each Let y 2 E: Then y D B.y/ C A t 2 J; d y.t/ D S .t/.0/ C dt 0

Zt S.t s/f .s; ys /ds C

X

S0 .t tk / Ik

0
0

y t : k

This implies by (9.17.2), (9.17.5) that, for each t 2 J; we have !t

!t

Zt

jy.t/j Me j.0/j C Me

e!s p.s/ .kys k/ds

0

m ˇ X y ˇˇ ˇ t CMe!t ˇ ˇIk k kD1

Me!t kk C Me!t

Zt

e!s p.s/ .kys k/ds

0

m ˇ ˇ X y ˇ ˇ tk Ik .0/ˇ CMe!t ˇIk kD1

CMe!t

m X

jIk .0/j

kD1 !t

Me

kk C

m X

! jIk .0/j

kD1 !t

Zt

CMe

e!s p.s/ .kys k/ds

0

ˇ y ˇ ˇ ˇ dk ˇ tk ˇ kD1 3 2 t Z m X ˇ

ˇ ce!t C Me!t 4 e!s p.s/ .kys k/ds C dk ˇy tk ˇ5 ; !t

CMe

m X

0

kD1

9.3 Impulsive Semi-linear Functional Differential Equations. . .

where c D M kk C

m X

211

! jIk .0/j :

(9.23)

kD1

Now we consider the function defined by .t/ D supfjy.s/j W r s tg; 0 t b: Then kys k .t/ for all t 2 J and there is a point t 2 Œr; t such that .t/ D jy.t /j. If t 2 Œ0; b, by the previous inequality and (9.23) we have for t 2 Œ0; b (note t t). .t/ ce!b C Me!b

Zt

e!s p.s/ ..s//ds C Me!b

m X

dk .t/:

kD1

0

Then !b

1 Me

m X

! !b

dk .t/ ce

!b

Zt

C Me

kD1

e!s p.s/ ..s//ds:

0

Thus from (9.20) and (9.21) we have Zt .t/ c0 C c1

e!s p.s/ ..s//ds:

0

Let us take the right-hand side of (9.24) as v.t/. Then we have .t/ v.t/ for all t 2 J; v.0/ D c0 ; and v 0 .t/ D c1 e!t p.t/ ..t//; a:e: t 2 J: Using the nondecreasing character of

we get

v 0 .t/ c1 e!t p.t/ .v.t//; a:e: t 2 J: That is v 0 .t/ c1 e!t p.t/; a:e: t 2 J: .v.t// Integrating from 0 to t we get Z 0

t

v 0 .s/ ds c1 .v.s//

Z

t 0

e!s p .s/ ds:

(9.24)

212

9 Impulsive Semi-linear Functional Differential Equations

By a change of variable and (9.19) we get Z

v.t/

v.0/

du c1 .u/

Z

b

!s

e 0

Z

1

p .s/ ds < c0

du : .u/

Hence there exists a constant N such that .t/ v.t/ N

for all t 2 J:

Now from the definition of it follows that kyk D sup jy.t/j max.kk; N/ for all y 2 E: t2Œr;b

This shows that the set E is bounded. As a consequence of Theorem 1.32 we deduce that A C B has a fixed point which is a integral solution of problem (9.11)–(9.13). t u Now we give a result where f Lipschitz with respect to y. Theorem 9.15. Assume that (9.17.1)–(9.17.4) hold and the condition (9.19.1) There exists a function l 2 L1 .J; RC / such that: jf .t; x/ f .t; y/j l.t/kx ykD a.e. t 2 J; and for all x; y 2 D; Z

b

with 0

e!s l.s/ds < 1, Z m P Me!b kk C jIk .0/j C c0 D

kD1

1

m P

Me!b

b

0

e!s jf .s; 0/jds

(9.25)

dk

kD1

and c1 D

Me!b : m P 1 Me!b dk kD1

If Me!b

m X

dk < 1;

kD1

then the problem (9.11)–(9.13) has at least one integral solution on Œr; b.

(9.26)

9.3 Impulsive Semi-linear Functional Differential Equations. . .

213

Proof. Let A and B the operator defined in Theorem 9.14. It can be shown that B is completely continuous and A is a contraction. Now we prove that o y n for some 0 < < 1 E D y 2 PC.Œr; b; D.A// W y D B.y/ C A is bounded. y for some 0 < < 1: Thus, for each Let y 2 E: Then y D B.y/ C A t 2 J; d y.t/ D S .t/.0/ C dt 0

Zt S.t s/f .s; ys /ds C

X

S0 .t tk / Ik

0
0

y t : k

This implies by (9.17.2) and (9.19.1) that, for each t 2 J; we have !t

!t

Zt

jy.t/j Me j.0/j C Me

e!s jf .s; ys / f .s; 0/jds

0

CMe!t

Zt

e!s jf .s; 0/jds C Me!t

0 !t

!t

Zt

Me kk C Me

!s

e

m ˇ X y ˇˇ ˇ t ˇ ˇIk k kD1

!t

Zt

l.s/kys kds C Me

0

e!s jf .s; 0/jds

0

m ˇ X

m ˇ y X ˇ ˇ CMe!t tk Ik .0/ˇ C Me!t jIk .0/j ˇIk kD1 kD1 1 0 Zt m X Me!t @kk C e!s jf .s; 0/jds C jIk .0/ jA 0 !t

Zt

CMe

e!s l.s/kys kds C Me!t

0

kD1 m X

dk y tk :

kD1

Now we consider the function defined by .t/ D supfjy.s/j W r s tg; 0 t b:

214

9 Impulsive Semi-linear Functional Differential Equations

Then kys k .t/ for all t 2 J and there is a point t 2 Œr; t such that .t/ D jy.t /j. If t 2 Œ0; b, by the previous inequality we have for t 2 Œ0; b (note t t) 0 .t/ Me!t @kk C

Zt

e!s jf .s; 0/jds C

Zt

CMe

1 jIk .0/ jA

kD1

0 !t

m X

e!s l.s/.s/ds C Me!t

m X

dk .t/:

kD1

0

Then !b

1 Me

m X

0

!

dk .t/ Me @kk C !t

kD1

Zt

e!s jf .s; 0/jds

0

C

m X

! jIk .0/ j

kD1 !b

Zt

CMe

e!s l.s/.s/ds:

0

Thus by (9.25) and (9.26) we have .t/

c0

C

c1

Zt

e!s l.s/.s/ds:

0

By Gronwall inequality [131] we get for each t 2 J .t/

c0 exp

Z t !s c1 e l.s/ds : 0

Thus Z kyk c0 exp c1

b 0

e!s l.s/ds WD M :

This shows that the set E is bounded. As a consequence of Theorem 1.32 we deduce that A C B has a fixed point which is a integral solution of problem (9.11)–(9.13). t u

9.3 Impulsive Semi-linear Functional Differential Equations. . .

215

9.3.4 Existence of Extremal Integral Solutions In this section we shall prove the existence of maximal and minimal integral solutions of problem (9.11)–(9.13) under suitable monotonicity conditions on the functions involved in it. We need the following definitions in the sequel. Definition 9.16. We say that a continuous function v W Œr; b ! E is a lower integral solution of problem (9.11)–(9.13) if v.t/ D .t/; t 2 H; and v.t/ S0 .t/.0/ C A C

X

Z

t 0

Z v.s/ds C

t 0

f .s; vs /ds

S0 .t tk / Ik v tk ; t 2 J; t ¤ tk

0
and v.tkC / v.tk / Ik .v.tk //; t D tk ; k D 1; : : : ; m. Similarly an upper integral solution w of problem (11.15)–(11.17) is defined by reversing the order. Definition 9.17. A solution xM of problem (9.11)–(9.13) is said to be maximal if for any other solution x of problem (9.11)–(9.13) on J, we have that x.t/ xM .t/ for each t 2 J. Similarly a minimal solution of problem (9.11)–(9.13) is defined by reversing the order of the inequalities. Definition 9.18. A function f .t; x/ is called strictly monotone increasing in x almost everywhere for t 2 J, if f .t; x/ f .t; y/ a.e. t 2 J for all x; y 2 D with x < y. Similarly f .t; x/ is called strictly monotone decreasing in x almost everywhere for t 2 J, if f .t; x/ f .t; y/ a.e. t 2 J for all x; y 2 D with x < y: We consider the following assumptions in the sequel. (9.22.1) The function f .t; y/ is strictly monotone increasing in y for almost each t 2 J. (9.22.2) S0 .t/ is preserving the order, that is S0 .t/v 0 whenever v 0. (9.22.3) The functions Ik , k D 1; : : : ; m are continuous and nondecreasing. (9.22.4) The problem (11.15)–(11.17) has a lower integral solution v and an upper integral solution w with v w. Theorem 9.19. Assume that assumptions (9.17.1)–(9.17.5) and (9.12.1)–(9.12.4) hold. Then problem (9.11)–(9.13) has a minimal and a maximal integral solutions on Œr; b. Proof. It can be shown, as in the proof of Theorem 9.14, that B is completely continuous and A is a contraction on Œv; w. We shall show that A and B are isotone increasing on Œv; w. Let y; y 2 Œa; b be such that y y; y 6D y: Then by (9.22.1), (9.22.2), we have for each t 2 J

216

9 Impulsive Semi-linear Functional Differential Equations

d B.y/ .t/ D S .t/.0/ C dt 0

S0 .t/.0/ C

d dt

Z Z

t

S.t s/f .s; ys / ds

0 t

S.t s/f .s; ys / ds

0

D B.y/ .t/ : and by (9.22.3), we have for each t 2 J A.y/ .t/ D

X 0

X

S0 .t tk / Ik y tk

S0 .t tk / Ik y tk

0
D A.y/ .t/ : Therefore A and B are isotone increasing on Œv; w. Finally, let x 2 Œv; w be any element. By (9.22.4) we deduce that v A.v/ C B.v/ A.x/ C B.x/ A.w/ C B.w/ w; which shows that A.x/ C B.x/ 2 Œv; w for all x 2 Œv; w. Thus, problem (9.11)– (9.13) has a maximal and a minimal integral solutions on Œr; b: t u

9.3.5 Impulsive Differential Equations with Nonlocal Conditions In this section we shall prove existence results for problem (9.14)–(9.16). Definition 9.20. A function y 2 PC Œr; b ; D.A/ is said to be a integral solution of problem (9.14)–(9.16) if y.t/ D .t/ ht .y/ ; t 2 H; and y.t/ D S0 .t/ ..0/ h0 .y// C C

X

Z 0

t

T.t s/f .s; ys / ds

T .t tk / Ik y tk ; t 2 J:

0
Theorem 9.21. Assume that hypotheses (9.17.1)–(9.17.4) and the following hypotheses hold: (A1) The function h is continuous with respect to t, and there exists a constant ˛ > 0 such that

9.3 Impulsive Semi-linear Functional Differential Equations. . .

jht .u/j ˛;

217

u 2 PC.Œr; b; D.A//

and for each k > 0 the set f.0/ h0 .y/; y 2 PC.Œr; b; D.A//; kyk kg is precompact in E (A2) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that jf .t; x/j p.t/ .kxkD /;

a:e: t 2 J;

for all x 2 D

with Z

1 cQ0

du > cQ 1 .u/

Z

b

e!s p.s/ds;

0

where Me!b ŒkkD C ˛ C

m X

jIk .0/j

kD1

cQ 0 D 1

Me!b

m X

;

dk

kD1

and cQ 1 D

Me!b : m X !b 1 Me dk kD1

Moreover, we suppose that Me!b

m X

dk < 1;

kD1

then the problem (9.14)–(9.16) has at least one integral solution on Œr; b: Proof. Consider the two operators: B1 W PC Œr; b ; D.A/ ! PC Œr; b ; D.A/

B1 .y/.t/ WD

8 ˆ < .t/ ht .y/; d ˆ : S0 .t/ ..0/ h0 .y// C dt

if t 2 H; Z

t 0

S.t s/f .s; ys / ds

if t 2 J;

218

9 Impulsive Semi-linear Functional Differential Equations

8 ˆ < 0; X A1 .y/.t/ D S0 .t tk /Ik .y.tk //; ˆ :

if t 2 H; if t 2 J:

0
Then the problem of finding the solution of problem (9.14)–(9.16) is reduced to finding the solution of the operator equation A1 .y/ .t/ CB2 .y/ .t/ D y .t/ ; t 2 Œr; b. As in Sect. 9.3, we can show that the operators A1 and B1 satisfy all conditions of Theorem 1.32. t u

9.3.6 Applications to Control Theory This section is devoted to an application of the argument used in the previous sections to the controllability of impulsive functional differential equations. More precisely we will consider the following problem: y0 .t/ Ay.t/ D f .t; yt / C Bu.t/; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m

(9.27)

yjtDtk D Ik .y.tk //; k D 1; : : : ; m

(9.28)

y.t/ D .t/; t 2 Œr; 0 ;

(9.29)

where A, f , and Ik are as in Sect. 9.3, the control function u ./ is given in L2 .J; U/ a Banach space of admissible control functions with U as a Banach. Finally B is a bounded linear operator from U to D.A/. Definition 9.22. A function y 2 PC Œr; b ; D.A/ is said to be a integral solution of problem (9.27)–(9.29) if y.t/ D .t/; t 2 Œr; 0; and y is a solution of impulsive integral equation Z Z d t d t y.t/ D S .t/.0/ C S.t s/f .s; ys / ds C S.t s/Bu .s/ ds dt 0 dt 0 X

S0 .t tk / Ik y tk ; t 2 J: C 0

0
Definition 9.23. The system (9.27)–(9.29) is said to be controllable on the interval Œr; b if for every initial function 2 D and every y1 2 D.A/, there exists a control u 2 L2 .J; U/, such that the mild solution y .t/ of system (9.27)–(9.29) satisfies y .b/ D y1 . Our main result in this section is the following.

9.3 Impulsive Semi-linear Functional Differential Equations. . .

219

Theorem 9.24. Assume that hypotheses (9.17.1)–(9.17.4) hold. Moreover we suppose that (B1) The linear operator W W L2 .J; U/ ! D.A/ defined by Zb T .b s/ Bu .s/ ds;

Wu D 0

has a bounded inverse operator W 1 which takes values in L2 .J; U/ nKerW, and there exist positive constants M; M1 ; such that kBk M and W 1 M1 : (B2) There exists a function l 2 L1 .J; RC / such that jf .t; x/ f .t; y/j l.t/kx ykD for a.e. t 2 J; and for all x; y 2 D; with 2 2!b

M e

Zb MM1 b

e!s l.s/ds C Me!b .1 C Me!b MM1 b/

m X

dk < 1

kD0

0

(B3) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that jf .t; x/j p.t/ .kxkD /; Z

b

with 0

a:e: t 2 J;

for all x 2 D

e!s p.s/ds < 1, Z1 c3

ds > kmk O L1 s C .s/

(9.30)

where c3 D

c2 1

Me!b .1

C MMM1

be!b /

;

m P

(9.31)

dk

kD1

c2 D M.1 C MMM1 e!b b/kk CMMM1 bjy1 j C M.1 C MMM1 e!b b/

m X

jIk .0/j;

(9.32)

kD1

m O .s/ D maxf!; c4 p.s/g;

(9.33)

220

9 Impulsive Semi-linear Functional Differential Equations

and c4 D

M C M 2 MM1 e!b b 1 Me!b .1 C MMM1 be!b /

m P

:

(9.34)

dk

kD1

Then the problem (9.27)–(9.29) is controllable on Œr; b: Remark 9.25. The construction of operator W 1 and its properties are discussed in [93]. Proof. Using hypothesis (B1) for an arbitrary function y .:/ we define the control Z uy .t/ D W 1 y1 S0 .b/ .0/ lim

!1 0

X

b

S0 .b s/B f .s; ys /ds

3

S0 .b tk / Ik y tk 5 .t/ :

0
Consider the two operators: A; B W PC Œr; b ; D.A/ ! PC Œr; b ; D.A/ defined by 8 0; ˆ ˆ ˆ ˆ ˆ < d Z t S.t s/Bu.s/ds A.y/ .t/ WD dt X ˆ 0 ˆ

ˆ ˆ S0 .t tk / Ik y tk ; ˆ :C

if t 2 H;

if t 2 J;

0
and 8 .t/; ˆ ˆ ˆ < B.y/ .t/ WD S0 .t/.0/ Z t ˆ ˆ ˆ :C d S.t s/f .s; ys / ds; dt 0

if t 2 H;

if t 2 J:

We can prove that A is a contraction operator and B is completely continuous. Now, we prove that o y n for some 0 < < 1 E D y 2 PC.Œr; b; D.A// W y D B.y/ C A is bounded.

9.3 Impulsive Semi-linear Functional Differential Equations. . .

221

y for some 0 < < 1: Thus, for each Let y 2 E: Then y D B.y/ C A t 2 J; d y.t/ D S .t/.0/ C dt 0

C C

d dt

Zt S.t s/f .s; ys /ds 0

Zt S.t s/Buy .s/ ds 0

X

S0 .t tk / Ik

0
y t : k

This implies by (B1)–(B3) that, for each t 2 J; we have !t

!t

Zt

jy.t/j Me j.0/j C Me

e!s p.s/ .kys k/ds

0

CMe!t

!t

Me

Zt 0

m ˇ X ˇ ˇ y ˇˇ ˇ t e!s ˇBuy .s/ˇ ds C Me!t ˇ ˇIk k kD0

.1 C MMM1 be!b /kk C MM1 bjy1 j

!t

Zt

CMe

e!s p.s/ .kys k/ds

0 2

!t !b

Zt

CM M1 Mbe e

e!s p.s/ .kys k/ds

0

CM 2 M1 Mbe!t e!b CMe!t

m X

y jIk . .tk //j kD1

m ˇ X y ˇˇ ˇ t ˇ ˇIk k kD1

Me!t Œ.1 C MMM1 be!b /kk CMM1 bjy1 j C Me!t .1 C MMM1 be!b /

m X kD1

jIk .0/j

222

9 Impulsive Semi-linear Functional Differential Equations

!t

Zt

e!s p.s/ .kys k/ds

CMe

0 2 !t

!b

Zt

CM e MM1 be

e!s p.s/ .kys k/ds

0

CMe!t .1 C MMM1 be!b /

m X

jy.tk //j:

kD1

Set ˛ D M.1 C MMM1 e!b b/kk C MMM1 bjy1 j m X

CM.1 C MMM1 e!b b/

jIk .0/j:

kD1

Consider the function defined by .t/ D supfjy.s/j W r s tg; 0 t b: Then kys k .t/ for all t 2 J and there is a point t 2 Œr; t such that .t/ D jy.t /j: If t 2 Œ0; b; by the previous inequality we have for t 2 Œ0; b (note t t) !t

!t

Zt

e!s p.s/ ..s//ds

.t/ ˛e C Me

0 2 !t

!b

Zt

CM e MM1 be

e!s p.s/ ..s//ds

0

CMe!t .1 C MMM1 be!b /

m X

dk .t/:

kD1

Then Œ1 Me!b .1 C MMM1 be!b /

m X

dk .t/ ˛e!t

kD1

CMe!t .1 C MMM1 be!b /

Zt p.s/ ..s//ds: 0

9.3 Impulsive Semi-linear Functional Differential Equations. . .

223

Thus by (9.31), (9.32), (9.34) we have !t

e

Zt .t/ c3 C c4

p.s/ ..s//ds:

(9.35)

0

Let us take the right-hand side of (9.35) as v.t/. Then we have .t/ e!t v.t/ for all t 2 J; v.0/ D c3 ; and v 0 .t/ D c4 p.t/ ..t//; a:e: t 2 J: Using the nondecreasing character of

we get

v 0 .t/ c4 p.t/ .e!t v.t//; a:e: t 2 J: Then by (9.33) for a.e. t 2 J we have .e!t v.t//0 D !e!t v.t/ C v 0 .t/e!t !e!t v.t/ C c4 p.t/e!t .e!t v.t// !t m.t/Œe O v.t/ C

.e!t v.t//:

Thus (9.30) gives Z

e!t v.t/

v.0/

du u C .u/

Z

b 0

Z

1

m.s/ds O D kmk O L1 < c3

du : u C .u/

Consequently, by (B3), there exists a constant d such that e!t v.t/ d; t 2 J and hence kyk d: This shows that the set E is bounded. As a consequence of Theorem 9.14 we deduce that A C B has a fixed point which is a integral solution of problem (9.27)–(9.29). Thus the system (9.27)–(9.29) is controllable on Œr; b: u t

9.3.7 An Example As an application of our results we consider the following impulsive partial functional differential equation of the form

224

9 Impulsive Semi-linear Functional Differential Equations

@2 @ z.t; x/ D 2 z.t; x/ @t @x CQ.t; z.t r; x//CBu.t/; x 2 Œ0; ; t 2 Œ0; bnft1 ; t2 ; : : :; tm g: (9.36) z.tkC ; x/ z.tk ; x/ D bk z.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m

(9.37)

z.t; 0/ D z.t; / D 0; t 2 Œ0; b

(9.38)

z.t; x/ D .t; x/; t 2 H; x 2 Œ0; ;

(9.39)

where bk > 0; k D 1; : : : ; m; 2 D D f W H Œ0; ! RI is continuous everywhere except for a countable number of points at which .s /; .sC / exist with .s / D .s/g; 0 D t0 < t1 < t2 < < tm < tmC1 D b; z.tkC / D lim z.tk C h; x/; z.tk / D lim z.tk C h; x/ and Q W Œ0; b R ! R; is a .h;x/!.0 ;x/

.h;x/!.0C ;x/

given function. Let y.t/.x/ D z.t; x/; t 2 Œ0; b; x 2 Œ0; ; Ik .y.tk //.x/ D bk z.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m F.t; /.x/ D Q.t; .; x//; 2 H; x 2 Œ0; ; and ./.x/ D .; x/; 2 H; x 2 Œ0; : Consider E D C.˝/, the Banach space of continuous function on ˝ with values in R. Define the linear operator A on E by Az D

@2 z; @x2

in D.A/ D fz 2 C.˝/ W z D 0 on @˝;

@2 z 2 C.˝g @x2

Now, we have D.A/ D C0 .˝/ D fv 2 C.˝/ W v D 0 on @˝g ¤ C.˝/: It is well known from [100] that A is sectorial, .0; C1/ .A/ and for > 0 kR.; A/kB.E/

1 :

It follows that A generates an integrated semigroup .S.t//t0 and that kS0 .t/kB.E/ et for t 0 for some constant > 0 and A satisfied the Hille–Yosida condition. Assume that the operator B W U ! Y; U Œ0; 1/; is a bounded linear operator and the operator

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

Z

b

Wu D

225

T.b s/Bu.s/ds

0

has a bounded inverse operator W 1 which takes values in L2 .Œ0; b; U/nkerW: Also assume that there exists an integrable function W Œ0; b ! RC such that jQ.t; w.t r; x//j .t/˝.jwj/ where ˝ W Œ0; 1/ ! .0; 1/ is continuous and nondecreasing with Z 1

1

ds D C1: s C ˝.s/

Assume that there exists a function Ql 2 L1 .Œ0; b; RC / such that jQ.t; w.t r; x// Q.t; w.t r; x//j Ql.t/jw wj; t 2 Œ0; b; w; w 2 R: We can show that problem (8.21)–(8.23) is an abstract formulation of problem (9.36)–(9.39). Since all the conditions of Theorem 9.24 are satisfied, the problem (9.36)–(9.39) has a solution z on Œr; b Œ0; :

9.4 Impulsive Semi-linear Neutral Functional Differential Equations with Infinite Delay 9.4.1 Introduction In this section we shall be concerned with the existence of mild solutions as well as integral solutions defined on a compact real interval for first order impulsive semilinear functional equations in a separable Banach space. More precisely we consider the initial value problem d Œy.t/ g.t; yt / D AŒy.t/ g.t; yt / (9.40) dt Cf .t; yt /; a:e: t 2 J D Œ0; b ; t ¤ tk ; k D 1; : : : ; m yjtDtk D Ik .y.tk //; k D 1; : : : ; m

(9.41)

y.t/ D .t/; t 2 .1; 0;

(9.42)

where f ; g W JD ! E is a given function, D D f W Œ1; 0 ! E; is continuous

everywhere except for a finite number of points s at which .s / ; sC exist and .s / D .s/g, 2 D, .0 < r < 1/, 0 D t0 < t1 < < tm < tmC1 D b, Ik 2 C .E; E/ .k D 1; 2; : : : ; m/, A is a closed linear operator on E, and E a real separable Banach space with norm j:j.

226

9 Impulsive Semi-linear Functional Differential Equations

9.4.2 Existence of Mild Solutions This section is devoted to the case when the operator A generates a .C0 /-semigroup. Before starting and proving our main result, we will give the definition of the mild solution. Definition 9.26. We say that a function y W .1; b ! E is a mild solution of problem (9.40)–(9.42) if y0 D and the restriction of y./ to the interval Œ0; b is continuous; and Z

t

T.t s/f .s; ys /ds y.t/ D g.t; yt / C T.t/Œ.0/ g.0; / C X0 X

T.t tk /Ik y tk T.t tk /g.tk ; ytk /; t 2 J: C 0
(9.43)

0
Let us introduce the following hypotheses: (9.30.1) A is the infinitesimal generator of a .C0 /semigroup fT.t/gt2J , which is compact for t > 0 in the Banach space E. Let M D supfkT.t/kB.E/ W t 2 Jg: (9.30.2) There exist constants ˛1 ; ˛2 0 and lg 0 such that: (i) jg.t; u/ g.t; u/j lg ku ukD ; t 2 J; and u; u 2 D, and (ii) jg.t; u/j ˛1 kukD C ˛2 ; t 2 J; for a.e. t 2 J; and each u 2 D: (9.30.3) f W J D ! E is Carathéodory function; (9.30.4) There exist constants dk > 0; k D 1; : : : ; m such that for each y; x 2 E jIk .y/ Ik .x/j dk jy xj (9.30.5) There exists a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! Œ0; 1/ such that jf .t; u/j p.t/ .kukD /;

a:e: t 2 J;

for all u 2 D:

with Z

1

C1

ds > C2 kpkL1 ; .s/

where C1 D Kb

C1 C .MKb C Mb /kk; C

C2 D

MKb C

with C D 1 ˛1 Kb M

m X kD1

dk 2Mmlg Kb ;

(9.44)

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

227

and C1 D ˛1 .MKb C Mb /kk C .1 C M/˛2 C M

m X

dk kk

kD1

CM

m X

jIk .0/j C 2Mmlg .MKb C Mb /kk C M

kD1

m X

jg.tk ; 0/j

kD1

Theorem 9.27. Assume that (9.30.1)–(9.30.5) hold. Suppose that Kb max .lg ; ˛1 / C M

m X

dk C 2mMlg Kb < 1

(9.45)

kD1

Then the problem (9.40)–(9.42) has at least one mild solution. Proof. Consider the operator N W Db ! Db defined by: 8 .t/; t 2 .1; 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ T.t/Œ.0/ g.0; / C g.t; yt / ˆ ˆ < .Ny/.t/ D C R t T.t s/f .s; y /ds C X T.t t /I y t

s k k ˆ k 0 ˆ ˆ ˆ 0
For 2 D, we define the function: Q D .t/

8 < .t/I

t 2 .1; 0;

: T.t/.0/I t 2 J;

Then Q 2 Db . Set Q y.t/ D x.t/ C .t/: It is clear to see that y satisfies (9.43) if and only if x satisfies x0 D 0; and x.t/ D g.t; xt C Q t / T.t/g.0; / Z t X

C T.t s/f .s; xs C Q s /ds C T.t tk /Ik x tk C Q tk 0

X 0
0
T.t tk /g.tk ; xtk C Q tk /; t 2 J:

228

9 Impulsive Semi-linear Functional Differential Equations

Let D0b D fx 2 Db W x0 D 0 2 Dg : For any x 2 D0b ,we have kxkb D kx0 kD C sup fjx.s/j W 0 s bg D sup fjx.s/j W 0 s bg : Thus .D0b ; k kb / is a Banach space. Define the two operators A; B W D0b ! D0b by: Z B.x/.t/ D

0

t

T.t s/f .s; xs C Q s /ds; t 2 J

and A.x/.t/ D g.t; xt C Q t / T.t/g.0; / C

X

X

T.t tk /Ik x tk C Q tk

0
T.t tk /g.tk ; xtk C Q tk /; t 2 J:

0
Obviously the operator N has a fixed point is equivalent to A C B has one, so it turns to prove that A C B has a fixed point.We shall show that the operators A and B satisfy all the conditions of Theorem 1.32. The proof will be given in several steps. Step 1: B is continuous. Let fxn g be a sequence such that xn ! x in D0b . Then ˇZ t ˇ ˇ ˇ ˇ Q Q jB.xn /.t/ B.x/.t/j D ˇ T.t s/Œf .s; xns C s / f .s; xs C s /dsˇˇ 0 Z b ˇ ˇ ˇf .s; xn C Q s / f .s; xs C Q s /ˇ ds: M s 0

Since f .s; / is continuous for a.e. s 2 J, we have by the Lebesgue dominated convergence theorem jB.xn /.t/ B.x/.t/j ! 0 as n ! 1: Thus B is continuous. Step 2: B maps bounded sets into bounded sets in D0b . It is enough to show that for any q > 0 there exists a positive constant l such that for each x 2 Bq D fx 2 D0b W kxk qg we have kB .x/k l. Let x 2 Bq ; then

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

229

kxs C Q s kD kxs kD C kQ s kD Kb q C Kb Mj.0/j C Mb kkD D q : Then we have for each t 2 J Z

t

jB.x/.t/j D j 0

T.t s/f .s; xs C Q s /dsj

Z

t

M Z

0 t

M 0

p.s/ .kxs C Q s k/ds p.s/ .q /ds Z

M .q /

t

p.s/ds: 0

Taking the supremum over t we obtain kB.x/kb M .q /kpkL1 WD l: Step 3: B maps bounded sets into equi-continuous sets inD0b . We consider Bq as in step 2 and let 1 ; 2 2 Jn ft1 ; : : : ; tm g ; 1 < 2 .Thus if > 0 and 1 < 2 we have We consider Bq as in step 2 and let 1 ; 2 2 Jn ft1 ; : : : ; tm g ; 1 < 2 .Thus if > 0 and 1 < 2 we have Z jB.x/. 2 / B.x/. 1 /j

.q /

1 0

Z

C .q / Z C .q /

kT. 2 s/ T. 1 s/kB.E/ p.s/ds

1

1

2

1

kT. 2 s/ T. 1 s/kB.E/ p.s/ds

kT. 2 s/kB.E/ p.s/ds:

As 1 ! 2 and become sufficiently small, the right-hand side of the above inequality tends to zero, since T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0 implies the continuity in the uniform operator topology. This proves the equi-continuity for the case where t ¤ ti ; k D 1; 2; : : : ; m C 1. It remains to examine the equi-continuity at t D ti . First we prove equi-continuity at t D ti . Fix ı1 > 0 such that ftk W k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < h < ı1 we have

230

9 Impulsive Semi-linear Functional Differential Equations

Z jB.x/.ti h/ B.x/.ti /j

ti h

0

j .T.ti h s/ T.ti s// f .s; xs C Q s /jds Z

C .q /M

ti

p.s/dsI ti h

which tends to zero as h ! 0. Define BO0 .x/.t/ D B.x/.t/; t 2 Œ0; t1 and

BOi .x/.t/ D

B.x/.t/; if t 2 .ti ; tiC1 B.x/.tiC /; if t D ti :

Next we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk W k ¤ ig \ Œti ı2 ; ti C ı2 D ;. For 0 < h < ı2 we have O O jB.x/.t i C h/ B.x/.ti /j

Z

ti

0

j .T.ti C h s/ T.ti s// f .s; xs C Q s /jds Z

ti Ch

C .q /M

p.s/ds: ti

The right-hand side tends to zero as h ! 0. The equi-continuity for the cases

1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval Œr; 0. As a consequence of steps 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For y 2 Bq we define Z B .x/.t/ D T./

t 0

T.t s /f .s; xs C Q s /ds:

Since T.t/ is a compact operator, the set X .t/ D fB .x/.t/ W x 2 Bq g is precompact in E for every ; 0 < < t: Moreover, for every y 2 Bq we have Z jB.x/.t/ B .x/.t/j

.q /

t

kT.t s/kB.E/ p.s/ds

t

Z

.q /M

t

p.s/ds: t

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

231

Therefore, there are precompact sets arbitrarily close to the set X .t/ D fB .x/.t/ W x 2 Bq g: Hence the set X.t/ D fB.x/.t/ W x 2 Bq g is precompact in E. Hence the operator B is completely continuous. Step 4: A is a contraction Let x1 ; x2 2 D0b . Then for t 2 J jA.x1 /.t/ A.x2 /.t/j jg.t; x1t C Q t / g.t; x2t C Q t /j X ˇ

ˇ ˇIk x1 t Ik x2 t ˇ CM k k 0
X

CM

jg.tk ; x1tk C Q tk / g.tk ; x2tk C Q tk /j

0
lg kx1t x2t k C M

m X

ˇ ˇ dk ˇx1 tk x2 tk ˇ

kD1

C2Mlg

m X

jx1tk x2tk j

kD1

M

m X

!

dk C lg Kb C 2mMlg Kb kx1 x2 k :

kD1

Then kA.x1 / A.x2 /k .M

m X

dk C lg Kb C 2mMlg Kb / kx1 x2 k ;

kD1

which is a contraction, since Kb lg C M

m X

dk C 2mMlg Kb Kb max .lg ; ˛1 / C M

kD1

m X

dk C 2mMlg Kb < 1:

kD1

Step 5: A priori bounds. Now it remains to show that the set xE is bounded. Let x 2 E; then x D B.x/ C A for some 0 < < 1: Thus, for each t 2 J; Zt x.t/ D 0

C

T.t s/f .s; xs C Q s /ds T.t/g.0; / C g.t; X 0

X

0
T .t tk / Ik

x Q k/ tk C .t

T .t tk / g.tk ;

xtk C Q tk /:

xt C Q t /

232

9 Impulsive Semi-linear Functional Differential Equations

This implies by (9.30.2) and (9.30.4) that, for each t 2 J; we have Zt

p.s/ .kxs C Q s k/ds

jx.t/j M 0

Cjg.t;

CM

m ˇ X x Q ˇˇ xt ˇ t C .tk / ˇ C Q t /j C M ˇIk k kD1

m X

jg.tk ;

kD1

xtk xt C Q tk /j C Mg.tk ; k C Q tk /j C Mjg.0; /j

Zt p.s/ .Kb jxs j C .MKb C Mb /kk/ds C ˛1 k

M 0

CM

m X

jIk

x

kD1

C2M

m X kD1

lg j

xt C Q t k C ˛2

m X Q k / Ik .0/j C M .tk / C .t jIk .0/j kD1

m X xtk jg.tk ; 0/j C M˛2 C Q tk j C M kD1

Zt M

p.s/ .Kb jx.s/j C .MKb C Mb /kk/ds 0

ˇ ˇ ˇ x.t/ ˇ ˇ ˇ C .MKb C Mb /kk C˛1 Kb ˇ ˇ ˇ ˇ m m X X ˇ x.t/ ˇ ˇ C kk C M ˇ C˛2 C M dk ˇ jIk .0/j ˇ kD1 kD1 C2M

CM

ˇ ˇ ˇ x.t/ ˇ ˇ C .MKb C Mb /kk lg Kb ˇˇ ˇ kD1

m X

m X

jg.tk ; 0/j C M˛2

kD1

Zt M

p.s/ .Kb jx.s/j C .MKb C Mb /kk/ds 0

C˛1 .Kb jx.t/j C .MKb C Mb /kk/

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

C˛2 C M

m X

dk jx.t/j C M

kD1

m X

dk kk C M

kD1

m X

233

jIk .0/j C M˛2

kD1

C2Mmlg Kb jx.t/j C 2Mmlg .MKb C Mb /kk C M

m X

jg.tk ; 0/j:

kD1

Then 1 ˛1 Kb M

m X

! dk 2Mmlg Kb jx.t/j

kD1

Zt M

p.s/ .Kb jx.s/j C .MKb C Mb /kk/ds 0

C˛1 .MKb C Mb / kk C .1 C M/˛2 CM

m X

dk kk C M

kD1

m X

jIk .0/j

kD1

C2Mmlg .MKb C MB / kk C M

m X

jg.tk ; 0/j:

kD1

Thus Kb jx.t/j C .MKb C Mb / kk

C1

C

C2

Zt p.s/ .Kb jx.s/j 0

C.MKb C Mb /kk/ds We consider the function defined by .t/ D supfKb jx.s/j C .MKb C Mb /kk W 0 s tg; 0 t b: Let t 2 Œ0; t be such that .t/ D Kb jx.t /j C .MKb C Mb /kk, by the previous inequality we have for t 2 Œ0; b .t/

C1

C

C2

Zt p.s/ ..t//ds: 0

Let us take the right-hand side of (9.46) as v.t/. Then we have v.0/ D C1 ;

.t/ v.t/ for all t 2 J;

(9.46)

234

9 Impulsive Semi-linear Functional Differential Equations

and v 0 .t/ D C2 p.t/ ..t//; a:e: t 2 J: Using the nondecreasing character of

we get

v 0 .t/ p.t/ .v.t//; a:e: t 2 J: Thus Z

v.t/

C1

ds .s/

Z 0

b

Z p.s/ds D kpkL1 <

1

C1

ds : .s/

Consequently, by assumption .9:30:5/, there exists a constant N such that v.t/ N; t 2 J and hence there exists a constant such that kzkb : This shows that the set E is bounded. As a consequence of Theorem 1.32, we deduce Q that F C G has a fixed point z . Then z .t/ D x .t/ C .t/; t 2 .1; b is a fixed point of the operator N, which gives rise to a mild solution of the problem (9.40)–(9.42). t u

9.4.3 Existence of Integral Solutions In the previous section we considered the same problem, when the operator was non-densely defined. However, as indicated in [100], we sometimes need to deal with non-densely defined operators. For example, when we look at a one@2 dimensional heat equation with Dirichlet conditions on Œ0; 1 and consider A D 2 @x in C.Œ0; 1; R/ in order to measure the solutions in the sup-norm, then the domain, D.A/ D f 2 C2 .Œ0; 1; R/ W .0/ D .1/ D 0g; is not dense in C.Œ0; 1; R/ with the sup-norm. Before starting and proving this one, we give the definition of its integral solution. Definition 9.28. We say that y W .1; b ! E is an integral solution of (9.40)– (9.42) if Z t (i) Œy.s/ g.s; ys /ds 2 D.A/ for t 2 J, 0

(ii) y.t/ D .t/; t 2 .1; 0:

Z t Z t f .s; ys /ds C (iii) y.t/ D .0/ g.0; / C g.t; yt / C A Œy.s/ g.s; ys /ds C 0 0 X X Ik .y.tk // g.tk ; ytk /; t 2 J: 0
0
9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

235

From the definition it follows that y.t/ g.t; yt / 2 D.A/; 8 t 0; in particular .0/ g.0; / 2 D.A/. Moreover, y satisfies the following variation of constants formula: Z d t S.t s/f .s; ys /ds y.t/ D g.t; yt / C S0 .t/..0/ g.0; // C dt 0 X X (9.47) S0 .t tk /Ik .y.tk // S0 .t tk /g.tk ; ytk /; t 0: C 0
0
Let B D R.; A/ WD .I A/1 : Then [146] for all x 2 D.A/; B x ! x as ! 1: Also from the Hille–Yosida condition (with n D 1) it easy to see that lim jB xj Mjxj; since

!1

jB j D j.I A/1 j

M : !

Thus lim jB j M. Also if y is given by (9.47), then !1

Z

t

y.t/ D g.t; yt / C S .t/..0/ g.0; // C lim S0 .t s/B f .s; ys /ds !1 0 X X S0 .t tk /Ik .y.tk // S0 .t tk /g.tk ; ytk /; t 2 J: C 0

0
(9.48)

0
The key tool in our approach is the following form of the fixed point theorem of Dhage [102]. Let Db the set of all functions that belong in Db and have values in D.A/. Let us introduce the following hypotheses: (C1) A satisfies Hille–Yosida condition; (C2) The operator S0 .t/ is compact in D.A/ whenever t > 0I (C3) There exists a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! Œ0; 1/ such that jf .t; u/j p.t/ .kukD /; for a.e. t 2 J; and each u 2 D: with Z

1 C1

du > C2 .u/

Z

b

e!t p.t/dt

0

where C1 D Kb

C1 C .MKb C Mb /kk; C

C2 D

Me!b Kb C

(9.49)

236

9 Impulsive Semi-linear Functional Differential Equations

with C D 1 ˛1 Kb Me!b

m X

e!tk dk 2Me!b lg Kb

kD1

m X

e!tk

(9.50)

kD1

and C1 D ˛1 .MKb C Mb /kk C ˛2 C Me!b

m X

e!tk dk kk

kD1

CMe!b

m X

e!tk jIk .0/j C Me!b ˛2

kD1

C2Me!b lg .MKb C Mb /

m X

e!tk kk C Me!b

kD1

m X

e!tk jg.tk ; 0/j:

kD1

Theorem 9.29. Assume that (9.30.2)–(9.30.4) and (C1)–(C3) hold. If Kb max .lg ; ˛1 / C Me!b

m X

e!tk dk C 2Mlg Kb e!b

kD1

m X

e!tk < 1;

(9.51)

kD1

then the problem (9.40)–(9.42) has at least one integral solution on .1; b: Proof. Transform the problem (9.40)–(9.42) into a fixed point problem. Consider the operator N W Db ! Db defined by: 8 ˆ .t/; t 2 .1; 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ S0 .t/Œ.0/ g.0; / C g.t; yt / ˆ ˆ < Z X

d t .Ny/.t/ D ˆ C S.t s/f .s; ys /ds C S0 .t tk /Ik y tk ˆ ˆ dt ˆ 0 ˆ 0
For 2 D, we define the function: Q D .t/

8 < .t/;

t 2 .1; 0;

: S0 .t/.0/; t 2 J;

Then Q 2 Db . Set Q y.t/ D x.t/ C .t/:

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

237

It is clear that x satisfies x0 D 0 and x.t/ D g.t; xt C Q t / S0 .t/g.0; / Z X

d t C S.t s/f .s; xs C Q s /ds C S0 .t tk /Ik x tk C Q tk dt 0 0
k

X

S0 .t tk /g.tk ; xtk C Q tk /; t 2 J:

0
Let D0b D fx 2 Db W x0 D 0g : For any x 2 D0b we have kxkb D kx0 kD C sup fjz.s/j W 0 s bg D sup fjx.s/j W 0 s bg : Thus .D0b ; k kb / is a Banach space. Define the two operators A; B W D0b ! D0b by: d B.x/.t/ D dt

Z 0

t

S.t s/f .s; xs C Q s /ds; t 2 J

and A.z/.t/ D g.t; xt C Q t / S0 .t/g.0; / C

X

X

S0 .t tk /Ik x tk C Q tk

0
S0 .t tk /g.tk ; xtk C Q tk /; t 2 J:

0
Obviously the operator N has a fixed point is equivalent to A C B has one, so it turns to prove that A C B has a fixed point. We shall show that the operators A and B satisfy all the conditions of Theorem 1.32. The proof will be given in several steps. Step 1: B is continuous. Let fxn g be a sequence such that xn ! x in D0b . Then ˇ Z t ˇ ˇd ˇ ˇ Q Q jB.xn /.t/ B.x/.t/j D ˇ S.t s/Œf .s; xns C s / f .s; xs C s /dsˇˇ dt 0 Z b ˇ ˇ Me!b e!s ˇf .s; xns C Q s / f .s; xs C Q s /ˇ ds: 0

Since f .s; / is continuous for a.e. s 2 J, we have by the Lebesgue dominated convergence theorem jB.xn /.t/ B.x/.t/j ! 0 as n ! 1: Thus B is continuous.

238

9 Impulsive Semi-linear Functional Differential Equations

Step 2: B maps bounded sets into bounded sets in D0b . It is enough to show that for any q > 0 there exists a positive constant l such that for each x 2 Bq D fy 2 D0b W kxk qg we have kB .x/k l. Let x 2 Bq ; then kxs C Q s kB kxs kD C kQ s kD Kb q C Kb Mj.0/j C Mb kkD D q : Then we have for each t 2 J jB.x/.t/j D j

Z

d dt

t

S.t s/f .s; xs C Q s /dsj

0 !b

Z

b

Me

0 !b

e!s p.s/ .kxs C Q s k/ds Z

.q /

Me

b

e!s p.s/ds WD l:

0

Step 3: B maps bounded sets into equi-continuous sets in D0b . We consider Bq as in step 2 and let 1 ; 2 2 Jn ft1 ; : : : ; tm g ; 1 < 2 .Thus if > 0 and 1 < 2 we have We consider Bq as in step 2 and let 1 ; 2 2 Jn ft1 ; : : : ; tm g ; 1 < 2 . Thus if > 0 and 1 < 2 we have ˇ ˇ Z 1 ˇ ˇ 0 0 ˇ Q ŒS . 2 s/ S . 1 s/B f .s; xs C s /dsˇˇ jB.x/. 2 / B.x/. 1 /j ˇ lim !1 0 ˇ ˇ Z 1 ˇ ˇ C ˇˇ lim ŒS0 . 2 s/ S0 . 1 s/B f .s; xs C Q s / dsˇˇ !1 1

ˇ ˇ Z 2 ˇ ˇ C ˇˇ lim S0 . 2 s/B f .s; xs C Q s / dsˇˇ !1 1 Z 1 .q / kS0 . 2 s/ S0 . 1 s/kB.E/ p.s/ds 0

Z

C .q / Z C .q /

1

1

2

1

kS0 . 2 s/ S0 . 1 s/kB.E/ p.s/ds

kS0 . 2 s/kB.E/ p.s/ds:

As 1 ! 2 and become sufficiently small, the right-hand side of the above inequality tends to zero, since S0 .t/ is a strongly continuous operator and the compactness of S0 .t/ for t > 0 implies the continuity in the uniform operator

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

239

topology. This proves the equi-continuity for the case where t ¤ ti ; k D 1; 2; : : : ; m C 1: It remains to examine the equi-continuity at t D ti : First we prove equi-continuity at t D ti . Fix ı1 > 0 such that ftk W k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < h < ı1 we have Z

ti h

jB.x/.ti h/ B.x/.ti /j lim

!1 0

j S0 .ti h s/S0 .ti s/ B f .s; xs CQ s /jds

CMe!b .q /

Z

ti

e!s p.s/dsI

ti h

which tends to zero as h ! 0. Define BO0 .x/.t/ D B.x/.t/; t 2 Œ0; t1 and BOi .x/.t/ D

B.x/.t/; if t 2 .ti ; tiC1 B.x/.tiC /; if t D ti :

Next we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk W k ¤ ig \ Œti ı2 ; ti C ı2 D ;. For 0 < h < ı2 we have O O jB.x/.t i C h/B.x/.ti /j lim

Z

ti

!1 0

k S0 .ti C h s/S0 .ti s/ B f .s; xs C Q s /kds Z

CMe!b .q/

ti Ch

e!s p.s/ds:

ti

The right-hand side tends to zero as h ! 0: The equi-continuity for the cases

1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval Œr; 0 : As a consequence of steps 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For y 2 Bq we define B .x/.t/ D S0 ./ lim

Z

!1 0

t

S0 .t s /B f .s; xs C Q s /ds:

Since S0 .t/ is a compact operator, the set X .t/ D fB .x/.t/ W x 2 Bq g

240

9 Impulsive Semi-linear Functional Differential Equations

is precompact in E for every ; 0 < < t: Moreover, for every y 2 Bq we have Z t !b jB.x/.t/ B .x/.t/j Me .q / e!s p.s/ds: t

Therefore, there are precompact sets arbitrarily close to the set X .t/ D fB .x/.t/ W x 2 Bq g: Hence the set X.t/ D fB.x/.t/ W x 2 Bq g is precompact in E. Hence the operator B is completely continuous. Step 4: A is a contraction. Let x1 ; x2 2 D0b . Then for t 2 J jA.x1 /.t/ A.x2 /.t/j jg.t; x1t C Q t / g.t; x2t C Q t /j X ˇ

ˇ CMe!b e!tk ˇIk x1 tk Ik x2 tk ˇ 0
CMe!b

X

e!tk jg.tk ; x1tk C Q tk /

0
g.tk ; x2tk C Q tk /j lg kx1t x2t k C Me!b

m X

ˇ ˇ e!tk dk ˇx1 tk x2 tk ˇ

kD1

C2Me!b lg

m X

e!tk jx1tk x2tk j

kD1

.Me!b

m X

e!tk .dk C 2lg Kb / C lg Kb / kx1 x2 kB :

kD1

Then kA.x1 / A.x2 /k M

m X

! dk C lg Kb C 2mMlg Kb kx1 x2 k ;

kD1

which is a contraction, since Kb lg C M

m X

dk C 2mMlg Kb Kb max .lg ; ˛1 / C M

kD1

m X

dk C 2mMlg Kb < 1:

kD1

Step 5: A priori bounds. Now it remains to show that the set o n x for some 0 < < 1 E D x 2 D0b W x D B.x/ C A is bounded.

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

Let x 2 E; then x D B.x/ C A Zt

for some 0 < < 1: Thus, for each t 2 J;

T.t s/f .s; xs C Q s /ds T.t/g.0; / C g.t;

x.t/ D 0

X

C

T .t tk / Ik

0
X

x

241

xt C Q t /

x Q k/ tk C .t

T .t tk / g.tk ;

0
xtk C Q tk /:

This implies by (9.30.2) and (9.30.4) that, for each t 2 J; we have Zt

p.s/ .kxs C Q s k/ds

jx.t/j M 0

Cjg.t;

CM

m ˇ X xt x Q ˇˇ ˇ C Q t /j C M t C .tk / ˇ ˇIk k kD1

m X

jg.tk ;

kD1

xtk xt C Q tk /j C Mg.tk ; k C Q tk /j C Mjg.0; /j

Zt M

p.s/ .Kb jxs j C .MKb C Mb /kk/ds 0

xt C Q t k C ˛2 m m x X X Q k / Ik .0/j C M .tk / C .t CM jIk jIk .0/j kD1 kD1 C˛1 k

C2M

m X kD1

lg j

m X xtk C Q tk j C M jg.tk ; 0/j C M˛2 kD1

Zt M

p.s/ .Kb jx.s/j C .MKb C Mb /kk/ds 0

ˇ ˇ ˇ x.t/ ˇ ˇ ˇ C .MKb C Mb /kk C˛1 Kb ˇ ˇ ˇ ˇ m m X X ˇ x.t/ ˇ ˇ C kk C M ˇ dk ˇ jIk .0/j C˛2 C M ˇ kD1 kD1

242

9 Impulsive Semi-linear Functional Differential Equations

C2M

CM

ˇ ˇ ˇ x.t/ ˇ ˇ C .MKb C Mb /kk lg Kb ˇˇ ˇ kD1

m X

m X

jg.tk ; 0/j C M˛2

kD1

Zt M

p.s/ .Kb jx.s/j C .MKb C Mb /kk/ds 0

C˛1 .Kb jx.t/j C .MKb C Mb /kk/ C˛2 C M

m X

dk jx.t/j C M

kD1

m X

dk kk C M

kD1

m X

jIk .0/j C M˛2

kD1

C2Mmlg Kb jx.t/j C 2Mmlg .MKb C Mb /kk C M

m X

jg.tk ; 0/j

kD1

Then 1 ˛1 Kb M

m X

! dk 2Mmlg Kb jx.t/j

kD1

Zt M

p.s/ .Kb jx.s/j C .MKb C Mb /kk/ds 0

C˛1 .MKb C Mb / kk C .1 C M/˛2 CM

m X

dk kk C M

kD1

m X

jIk .0/j

kD1

C2Mmlg .MKb C MB / kk C M

m X

jg.tk ; 0/j:

kD1

Thus by (9.49) we have Kb jx.t/j C .MKb C Mb / kk C1 CC2

Zt p.s/ .Kb jx.s/j C .MKb C Mb /kk/ds 0

We consider the function defined by .t/ D supfKb jx.s/j C .MKb C Mb /kk W 0 s tg; 0 t b:

9.4 Impulsive Semi-linear Neutral Functional Differential Equations...

243

Let t 2 Œ0; t be such that .t/ D Kb jx.t /j C .MKb C Mb /kk, by the previous inequality we have for t 2 Œ0; b .t/

C1

C

C2

Zt p.s/ ..t//ds

(9.52)

0

Let us take the right-hand side as v.t/. Then we have v.0/ D C1 ;

.t/ v.t/ for all t 2 J;

and v 0 .t/ D C2 p.t/ ..t//; a:e: t 2 J: Using the nondecreasing character of

we get

v 0 .t/ p.t/ .v.t//; a:e: t 2 J; Thus Z

v.t/

C1

ds .s/

Z 0

b

Z p.s/ds D kpkL1 <

1

C1

ds : .s/

Consequently, by assumption .9:30:5/, there exists a constant N such that v.t/ N; t 2 J and hence there exists a constant such that kzkb : This shows that the set E is bounded. As a consequence of Theorem 1.32, we deduce that F C G has Q a fixed point z . Then z .t/ D x .t/ C .t/; t 2 .1; b is a fixed point of the operator N, which gives rise to a mild solution of the problem (9.40)–(9.42). t u

9.4.4 An Example In this section we apply some of the results established in this section. We begin by mentioning an example of phase space.

The Phase Space Let h.:/ W .1; r ! R be a positive Lebesgue integrable function and D WD PCr L2 .h; E/; r 0; be the space formed of all classes of functions ' W .1; 0 ! E such that 'jH 2 PC.H; E/, '.:/ is Lebesgue-measurable on .1; r and hj'jp is Lebesgue integrable on .1; r. The semi-norm in k:kD is defined by

244

9 Impulsive Semi-linear Functional Differential Equations

Z k'kD D sup k'./k C 2H

r 1

h. /k'./kp d

1=p (9.53)

Assume that h.:/ satisfies conditions (g-6) and (g-7) in the terminology of [142]. Proceeding as in the proof of ([142], Theorem 1.3.8) it follows that D is a phase space which verifies the axioms (A1)–(A2) and (A3). Moreover, when r D 0 this space coincides with C0 L2 .h; E/ and the parameters H D 1I M.t/ D .t/1=2 and 1=2 R 0 , for t 0 (see [142]). Let E D L2 .Œ0; / and let A be K.t/ D 1 C t h./d the operator given by Af D f 00 with domain ˚ D.A/ WD f 2 L2 .Œ0; / W f 00 2 L2 .Œ0; /; f .0/ D f ./ D 0

(9.54)

It is well known that A is the infinitesimal generator of a C0 -semigroup on E, which will be denoted by .T.t//t0 . Moreover, A has discrete spectrum, the eigenvalues are n2 ; n 2 N; with corresponding normalized eigenvectors zn ./ WD . 2 /1=2 sin.n/ and the following properties hold: (a) fzn W n 2 Ng is an orthonormal basis of E. P1 n P1 2t 2 (b) For f 2 E; T.t/f D e hf ; zn i and Af D nD1 nD1 n hf ; zn izn when f 2 D.A/:

A First Order Neutral Equation We study the first order neutral differential equation with unbounded delay

Z t Z d u.t; / C b.t s; ; /u.s; /dds dt 1 0

Z t Z @2 b.t s; ; /u.s; /dds D 2 u.t; / C @ 1 0 Z t C F.t; t s; ; u.s; //ds; t 2 Œ0; a; 2 Œ0;

(9.55)

1

u.t; 0/ D u.t; / D 0; t 2 Œ0; a;

(9.56)

u. ; / D '. ; /; 0; 0 ; R ti u.ti /./ D 1 ai .ti s/u.s; /ds;

(9.57)

where ' 2 C0 L2 .h; E/; 0 < t1 < < tn < a and

(9.58)

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential. . .

245

(a) The function b.s; ; /; @b.s;;/ are measurable, b.s; ; / D b.s; ; 0/ D 0 and @ ( Z lg WD max

Z

0

Z

0

1

0

1 @i b.s; ; / /ddsd . h.s/ @ i

)

1=2

W i D 0; 1 < 1 (9.59)

(b) The function F W R4 !* R is continuous and there are continuous functions W R3 ! R and W R2 ! R such that jF.t; s; ; x/j .t; s; / C .t; s/jxj;

.t; s; ; x/ 2 R3 I

(c) The functions ak 2 C.Œ0; 1/; R/ and dk WD . i D 1; : : : ; m:

R0

.ak /2 1=2 1 h.s/ ds/

(9.60) < 1 for all

Assuming that conditions .a/ .c/ are verified, our problem can be modeled as the abstract impulsive problem (9.40)–(9.42) by defining g.t; /./ WD

R0

R 1 0

f .t; /./ WD

R0

Ik . /./ WD

1

R0

b.s; ; / .s; /dds;

(9.61)

F.t; s; ; .s; //ds;

(9.62)

ak .s/ .s; /ds:

(9.63)

1

Moreover, f .t; :/; Ik ; i D 1; : : : ; m; are bounded linear operators, and kg.t; /k ˛2 C ˛1 k kB , where Z ˛1 WD

Z

0

0

1

Z

0

1 jb.s; ; /j2 ddsd h.s/

1=2 ; ˛2 WD 0:

Hence, the problem has a mild solution in .1; b:

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential Equations with State-Dependent Delay 9.5.1 Introduction In this section, we shall be concerned with existence of integral solutions defined on a compact real interval for first order impulsive semi-linear functional equations with state-dependent delay in a separable Banach space of the form:

246

9 Impulsive Semi-linear Functional Differential Equations

y0 .t/ D Ay.t/ C f .t; y .t;yt / /;

t 2 I D Œ0; b;

y.t/ D t 2 .1; 0;

y.ti / D Ik .ytk /;

(9.64) (9.65)

k D 1; 2; : : : ; m;

(9.66)

where f W J D ! E is a given function, D D f W .1; 0 ! E; is continuous

sC exist and everywhere except for a finite number of points s at which .s / ; .s / D .s/g, 2 D, .0 < r < 1/, 0 D t0 < t1 < < tm < tmC1 D b, Ik W D ! E .k D 1; 2; : : : ; m/, W I D ! .1; b, A W D.A/ E ! E is a non-densely defined closed linear operator on E, and E a real separable Banach space with norm j:j.

9.5.2 Existence of Integral Solutions Definition 9.30. We say that y W .1; T ! E is an integral solution of (9.64)– (9.66) if Z t Z t X y.s/ds C f .s; y .s;ys / /ds C Ik .ytk / ; t 2 J: (i) y.t/ D .0/ C A Z (ii) 0

0

t

0

0
y.s/ds 2 D.A/ for t 2 J, and y.t/ D .t/; t 2 .1; 0:

From the definition it follows that y.t/ 2 D.A/, for each t 0; in particular .0/ 2 D.A/. Moreover, y satisfies the following variation of constants formula: y.t/ D S0 .t/.0/ C

d dt

Z

t 0

S.t s/f .s; y .s;ys / /ds C

X

S0 .t tk / Ik .ytk / t 0:

0
(9.67) We notice also that if y satisfies (9.67), then Z

0

y.t/ D S .t/.0/ C lim C

X

!1 0

t

S0 .t s/B f .s; y .s;ys / /ds

S0 .t tk / Ik .ytk / ; t 0:

0
Our main result in this section is based upon the fixed point theorem due to Burton and Kirk [88]. We always assume that W I D ! .1; b is continuous. Additionally, we introduce the following hypotheses: (H') The function t ! 't is continuous from R. / D f .s; '/ W .s; '/ 2 J D; .s; '/ 0g into D and there exists a continuous and bounded function L W R. / ! .0; 1/ such that kt kD L .t/kkD for every t 2 R. /.

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential. . .

247

(9.34.1) A satisfies Hille–Yosida condition; (9.34.2) There exist constants dk > 0; k D 1; : : : ; m such that for each y; x 2 D kIk .y/ Ik .x/ k dk ky xkD (9.34.3) The function f W J D ! E is Carathéodory; (9.34.4) The operator S0 .t/ is compact in D.A/ wherever t > 0I (9.34.5) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that jf .t; x/j p.t/ .kxkD /; Z

b

with 0

a:e: t 2 J;

for all x 2 D

e!s p.s/ds < 1, Z

1

du > c2 .u/

c1

Z 0

b

e!s p .s/ ds:

(9.68)

where c1 D

ce!b Kb 1 Me!b Kb

m P

C Mb C L C MKb kk;

(9.69)

dk

kD1

and cD

m X

jIk .0/j C dk Mb C L C MKb kkD :

(9.70)

kD1

c2 D

MKb e!b : m P 1 Me!b Kb dk

(9.71)

kD1

The next result is a consequence of the phase space axioms. Lemma 9.31 ([139], Lemma 2.1). If y W .1; b ! E is a function such that y0 D and yjJ 2 PC.J W D.A//, then kys kD .Ma C L /kkD C Ka supfky. /kI 2 Œ0; maxf0; sgg;

s 2 R. / [ J;

where L D supt2R. / L .t/, Ma D supt2J M.t/ and Ka D supt2J K.t/.

248

9 Impulsive Semi-linear Functional Differential Equations

Theorem 9.32. Assume that (H') and (9.34.1)–(9.34.5) hold. If Me!b Kb

m X

dk < 1;

(9.72)

kD1

then the problem (9.64)–(9.66) has at least one integral solution on .1; b: Proof. Consider the operator N W PC .1; b; D.A/ ! PC .1; b; D.A/ defined by: 8 t 2 .1; 0; ˆ ˆ .t/; ˆ ˆ ˆ Z ˆ <

d t 0 S.t s/f s; y .s;ys / ds .Ny/.t/ D S .t/.0/ C dt 0 ˆ ˆ X ˆ ˆ 0 ˆ S .t t /I t 2 J: C k k .ytk /; ˆ : 0
Q W .1; b ! E be the function defined by Let .:/ Q D .t/

8 < .t/;

t 2 .1; 0;

: S0 .t/.0/; t 2 J:

Then Q 0 D . For each x 2 Bb with x.0/ D 0, we denote by x the function defined by 8 < 0; t 2 .1; 0; x.t/ D : x.t/; t 2 J; Q C x.t/; 0 t b, which implies yt D xt C Q t , We can decompose it as y.t/ D .t/ for every 0 t b and the function x.:/ satisfies Z d t x.t/ D S.t s/f s; x .s;xs CQs / C Q .s;xs CQs / ds dt 0 X

C S0 .t tk /Ik xtk C Q tk t 2 J: 0
Let Bb0 D fx 2 Bb W x0 D 0 2 Dg: For any x 2 Bb0 we have kxkb D kx0 kD C supfjx.s/j W 0 s bg D supfjx.s/j W 0 s bg:

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential. . .

249

Thus .Bb0 ; kkb / is a Banach space. We define the two operators A; B W Bb0 ! Bb0 by: d B.x/.t/ D dt

Z

t 0

S.t s/f s; x .s;xs CQs / C Q .s;xs CQs / ds; t 2 J

and A.x/.t/ D

X

S0 .t tk /Ik xtk C Q tk ; t 2 J:

0
Obviously the operator N has a fixed point is equivalent to A C B has one, so it turns to prove that A C B has a fixed point. We shall show that the operators A and B satisfies all the conditions of Theorem 1.32. For better readability, we break the proof into a sequence of steps. Step 1: B is continuous. Let fxn g be a sequence such that xn ! x in Bb0 . Then for ! > 0 (if ! < 0 one has e!t < 1). n At first, we study the convergence of the sequences .x .s;x n /n2 N; s 2 J: If s 2 J s is such that .s; xs / > 0 for every n > N. In the case, for n > N we see that n n kx .s;x n x .s;xs / kD kx .s;xn / x .s;xsn / kD C kx .s;xsn / x .s;xs / kD s/ s

Kb kxn xkD C kx .s;xsn / x .s;xs / kD : n Which prove that x .s;x n ! x .s;xs / in D as n ! 1 for every s 2 J such that s/ .s; xs / > 0: Similarly, if .s; xs / < 0 and n 2 N is such that .s; xsn / < 0 for every n > N, we get n kx .s;x n x .s;xs / kD D k .s;xsn / .s;xs / kD D 0 s/ n Which also shows that x .s;x n ! x .s;xs / in D as n ! 1 for every s 2 J such that s/ n .s; xs / < 0. Combining the previous arguments, we can prove that x .s;x n ! s/ for every s 2 J such that .s; xs / D 0: Finalely,

ˇ Z t h ˇd n jB.xn /.t/ B.x/.t/j D ˇˇ S.t s/ f s; x .s;x C Q .s;xsn CQs / n C Q / s s dt 0 i ˇ ˇ f s; x .s;xs CQs / C Q .s;xs CQs / dsˇ Z t ˇ ˇ Me!b e!s ˇf s; xn n C Q n Q 0

.s;xs CQs /

ˇ ˇ f s; x .s;xs CQs / C Q .s;xs CQs / ˇ ds

.s;xs Cs /

250

9 Impulsive Semi-linear Functional Differential Equations

!b

Z

Me

0

t

ˇ ˇ n Q n Q e!s ˇf s; x .s;x C n C .s;x C / Q/ s s s

s

ˇ ˇ f s; x .s;xsn CQs / C Q .s;xsn CQs / ˇ ds Z t ˇ ˇ !b CMe e!s ˇf s; x .s;xsn CQs / C Q .s;xsn CQs / 0

ˇ ˇ f s; x .s;xs CQs / C Q .s;xs CQs / ˇ ds n We infer that f .s; x .s;x n / ! f .s; x .s;xs / / as n ! 1, for every s 2 J. Now, s/ a standard application of the Lebesgue dominated convergence theorem proves that

kB.xn /.t/ B.x/.t/kb ! 0 as n ! 1: Thus B is continuous. Step 2: B maps bounded sets into bounded sets in Bb0 . It is enough to show that for any q > 0 there exists a positive constant l such that for each x 2 Bq D fx 2 Bb0 W kxkb qg we have kB .y/kb l: So choose x 2 Bq ; then kx .t;xt CQt / C Q .t;xt CQt / kD Kb q C .Mb C L /kkD C Kb Mj.0/j D q Then we have for each t 2 J ˇ Z t ˇˇ ˇd ˇ Q S.t s/f s; x .s;xs CQs / C .s;xs CQs / dsˇˇ jB.x/.t/j D ˇ dt 0 Z b Me!b e!s p.s/ kx .t;xt CQt / C Q .t;xt CQt / kD : 0

Then we have kB.x/.t/kb Me!b .q /

Z

b

e!s p.s/ds WD l:

0

Step 3: B maps bounded sets into equi-continuous sets of Bb0 : We consider Bq as in Step 2 and let 1 ; 2 2 Jn ft1 ; : : : ; tm g ; 1 < 2 .Thus if > 0 and 1 < 2 we have jB.x/. 2 / B.x/. 1 /j ˇ Z 1 ˇˇ ˇ ˇˇ lim ŒS0 . 2 s/ S0 . 1 s/B f s; x .s;xs CQs / C Q .s;xs CQs / dsˇˇ !1 0

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential. . .

251

ˇ Z 1 ˇˇ ˇ 0 0 ˇ Q C ˇ lim ŒS . 2 s/ S . 1 s/B f s; x .s;xs CQs / C .s;xs CQs / dsˇˇ !1 1 ˇ Z 2 ˇˇ ˇ 0 ˇ Q C ˇ lim S . 2 s/B f s; x .s;xs CQs / C .s;xs CQs / dsˇˇ !1 1

Z

.q /

1 0

Z

C .q / !b

CMe

jS0 . 2 s/ S0 . 1 s/jp.s/ds

1

1

.q /

jS0 . 2 s/ S0 . 1 s/jp.s/ds Z

2

e!s p.s/ds:

1

As 1 ! 2 and become sufficiently small, the right-hand side of the above inequality tends to zero, since S0 .t/ is a strongly continuous operator and the compactness of S0 .t/ for t > 0 implies the continuity in the uniform operator topology. This proves the equi-continuity for the case where t ¤ ti ; k D 1; 2; : : : ; m C 1. It remains to examine the equi-continuity at t D ti . First we prove equi-continuity at t D ti . Fix ı1 > 0 such that ftk W k ¤ ig \ Œti ı1 ; ti C ı1 D ;: For 0 < h < ı1 we have jB.x/.ti h/ B.x/.ti /j Z ti h

lim k S0 .ti h s/ S0 .ti s/ !1 0

B f s; x .s;xs CQs / C Q .s;xs CQs / kds Z ti CMe!b .q / e!s p.s/ dsI ti h

which tends to zero as h ! 0. Define BO0 .x/.t/ D B.x/.t/; t 2 Œ0; t1 and BOi .x/.t/ D

B.x/.t/; if t 2 .ti ; tiC1 B.x/.tiC /; if t D ti :

252

9 Impulsive Semi-linear Functional Differential Equations

Next we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk W k ¤ ig \ Œti ı2 ; ti C ı2 D ;. For 0 < h < ı2 we have O O jB.x/.t i C h/ B.x/.ti /j Z ti

lim k S0 .ti C h s/ S0 .ti s/ !1 0

B f s; x .s;xs CQs / C Q .s;xs CQs / kds CMe!b .q /

Z

ti Ch

e!s p.s/ dsI

ti

The right-hand side tends to zero as h ! 0. The equi-continuity for the cases

1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval Œr; 0. As a consequence of steps 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps Bq into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For x 2 Bq we define B .x/.t/ D S0 ./ lim

Z

t

!1 0

S0 .t s /B f .s; x .s;xs CQs / C Q .s;xs CQs / /ds:

Note

Z lim

!1 0

t

S0 .t s /B f .s; x .s;xs CQs / C Q .s;xs CQs / / ds W y 2 Bq

is a bounded set since ˇ ˇ Z t ˇ ˇ 0 ˇ lim Q S .t s /B f .s; x .s;xs CQs / C .s;xs CQs / / dsˇˇ ˇ !1 0

Me!b .q /

Z

t

e!s p.s/ds:

0

Since S0 .t/ is a compact operator, the set X .t/ D fB .x/.t/ W x 2 Bq g is precompact in E for every ; 0 < < t: Moreover, for every y 2 Bq we have jB.x/.t/ B .x/.t/j Me!b .q /

Z

t t

e!s p.s/ds:

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential. . .

253

Therefore, there are precompact sets arbitrarily close to the set X .t/ D fB .x/.t/ W x 2 Bq g: Hence the set X.t/ D fB.x/.t/ W x 2 Bq g is precompact in E. Hence the operator B W Bb0 ! Bb0 is completely continuous. Step 4: A is a contraction Let x1 ; x2 2 Bb0 . Then for t 2 J ˇ ˇ ˇ ˇX ˇ ˇ

0 1 2 ˇ Q Q jA.x1 /.t/ A.x2 /.t/j D ˇ S .t tk / Ik .xtk C tk / Ik .xtk C tk / ˇˇ ˇ ˇ0
Me

m X

dk kxt1k xt2k kD

kD1

Me!b Kb

m X

dk kx1 x2 kD :

kD1

Then kA.x1 / A.x2 /kb Me!b Kb

m X

dk kx1 x2 kD ;

kD1

which is a contraction from (9.72). Step 5: A priori bounds. Now it remains to show that the set n o x E D x 2 Bb0 W x D B.x/ C A for some 0 < < 1 is bounded. x for some 0 < < 1: Thus, for each Let x 2 E: Then x D B.x/ C A t 2 J; d x.t/ D dt C

Zt

S.t s/f .s; x .s;xs CQs / C Q .s;xs CQs / /ds

0

X

0
S0 .t tk / Ik

x

tk

C Q tk :

254

9 Impulsive Semi-linear Functional Differential Equations

This implies by (9.34.2), (9.34.5) that, for each t 2 J; we have !t

Zt

jx.t/j Me

e!s p.s/ .kx .s;xs CQs / C Q .s;xs CQs / kD /ds

0

CMe!t

!t

Zt

Me

m ˇ ˇ X xtk ˇ ˇ C Q tk ˇ ˇIk kD1

Kb jx.s/j C .Mb C L C MKb /kkD ds

e!s p.s/

0

CMe!t CMe!t

m ˇ ˇ X xtk ˇ ˇ C Q tk Ik .0/ˇ ˇIk kD1 m X

jIk .0/j

kD1 !t

Zt

Me

Kb jx.s/j C .Mb C L C MKb /kkD ds

e!s p.s/

0

CMe!t

m X kD1

jIk .0/j C Me!t 2

ce!t C Me!t 4

CKb

dk Kb jx.s/j C .Mb C L C MKb /kkD

kD1

Zt 0

m X

m X

e!s p.s/

Kb jx.s/j C .Mb C L C MKb /kkD ds

#

dk jx.t/j ;

kD1

where cDM

m X

jIk .0/j C dk .Mb C L C MKb /kkD

!

:

kD1

Therefore, !b

1 Me Kb

m X

! !t

!t

Zt

dk jx.t/j ce C Me

kD1

0

e!s p.s/ .Kb jx.s/j

C.Mb C L C MKb /kkD ds:

(9.73)

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential. . .

255

Thus from (9.69) and (9.71) we have Zt

.Mb C L C MKb /kkD C Kb jx.s/j c1 C c2

e!s p.s/ .Kb jx.t/j

0

C.Mb C L C MKb /kkD ds: We consider the function defined by .t/ D supfKb jx.s/j C .Mb C L C MKb /kkD W 0 s tg; 0 t b: t 2 Œ0; t be such that .t/ D Kb jx.t /j C .Mb C L C MKb /kkD , by the previous inequality we have for t 2 Œ0; b Zt .t/ c1 C c2

e!s p.s/ ..s//ds:

0

Let us take the right-hand side of (9.74) as v.t/. Then we have .t/ v.t/ for all t 2 J; v.0/ D c1 ; and v 0 .t/ D c2 e!t p.t/ ..t//; a:e: t 2 J: Using the nondecreasing character of

we get

v 0 .t/ c2 e!t p.t/ .v.t//; a:e: t 2 J: That is v 0 .t/ c2 e!t p .t/ ; a:e: t 2 J: .v.t// Integrating from 0 to t we get Z

t 0

v 0 .s/ ds c2 .v.s//

Z

t

e!s p.s/ds:

0

By a change of variable and (9.68) we get Z

v.t/

v.0/

du c2 .u/

Z 0

b

e!s p .s/ ds <

Z

1 c1

du : .u/

(9.74)

256

9 Impulsive Semi-linear Functional Differential Equations

Hence there exists a constant N such that .t/ v.t/ N

for all t 2 J:

Now from the definition of it follows that kxkb N for all x 2 E: This shows that the set E is bounded. As a consequence of Theorem 1.32 we deduce that A C B has a fixed point which is a integral solution of problem (9.64)–(9.66). t u

Phase Spaces Let g W .1; 0 ! Œ1; 1/ be a continuous, nondecreasing function with g.0/ D 1, which satisfies the conditions (g-1), (g-2) of [142]. This means that the function G.t/ D

g.t C / g. / 1<t sup

is locally bounded for t 0 and that lim g. / D 1: !1

We said that W Œ1; 0 ! E is normalized piecewise continuous, if is left continuous and the restriction of to any interval H is piecewise continuous. Next we modify slightly the definition of the spaces Cg ; Cg0 of [142]. We denote by PC g .E/ the space formed by the normalized piecewise continuous functions such that is bounded on .1; 0 and by PC 0g the subspace of PCg .E/ formed by g the functions such that lim

!1

./ D 0: g. /

It is easy to see that D D PC g .E/ and D D PC 0g .E/ endowed with the norm kkD D

k./k 2.1;0 g. / sup

are phase spaces. Moreover, in these cases K.s/ D 1 for s 0. Let 1 p < 1; 0 r < 1, and g./ is a Borel nonnegative measurable function on .1; r/ which satisfies the conditions (g-5)–(g-6) in the terminology of [142]. This means that g./ is locally integrable on .1; r/ and there exists a nonnegative and locally bounded function G on .1; 0 such that g. C / G./g. / for all 0 and 2 .1; r/nN , where N .1; r/ is a set with

9.5 Non-densely Defined Impulsive Semi-linear Functional Differential. . .

257

Lebesgue measure 0. Let D WD PCr Lp .g; E/; r 0; p > 1, be the space formed of all classes of functions W .1; 0 ! E such that jH 2 PC.H; E/; ./ is Lebesgue measurable on .1; r and gjjp is Lebesgue integrable on .1; r: The seminorm in k kD is defined by Z kkD WD sup k./k C

r 1

2H

g. /k./kp d

1p

:

Proceeding as in the proof of ([142], Theorem 1.3.8), it follows that D is a phase space which satisfies Axioms (A) and (B). Moreover, for r D 0 and p D 2 this space 1 coincides (see [142]) with C0 L2 .g; E/; H D 1; M.t/ D G.t/ 2 and Z K.t/ D 1 C

12

0

g.s/ds t

; for t 0:

A First Order Partial Functional Differential Equations To apply our abstract results, we consider the partial functional differential equations with state dependent delay of the form @ @ v.t; / D v.t; / C m.t/a.v.t .v.t; 0//; //; 2 Œ0; ; t 2 Œ0; b; @t @ (9.75) v.t; 0/ D v.t; / D 0; t 2 Œ0; b; (9.76) v.; / D v0 .; /; 2 Œ0; ; 2 .1; 0; Z v.ti /./ D

ti

1

i .ti s/v.s; /ds

(9.77) (9.78)

where v0 W .1; 0 Œ0; ! R is an appropriate function, i 2 CŒ0; 1/; R/; 0 < t1 < t2 < < tn < b: The functions m W Œ0; b ! R; a W R J ! R; W R ! RC are continuous and we assume the existence of positive constants b1 ; b2 such that jb.t/j b1 jtj C b2 for every t 2 R. Let A be the operator defined on E D C.Œ0; ; R/ by D.A/ D fg 2 C1 .Œ0; ; R/ W g.0/ D 0gI Ag D g0 : Then D.A/ D C0 .Œ0; ; R/ D fg 2 C.Œ0; ; R/ W g.0/ D 0g:

258

9 Impulsive Semi-linear Functional Differential Equations

It is well known from [100] that A is sectorial, .0; C1/ .A/ and for > 0 kR.; A/kB.E/

1 :

It follows that A generates an integrated semigroup .S.t//t0 and that kS0 .t/kB.E/ et for t 0 for some constant > 0 and A satisfied the Hille–Yosida condition. Set > 0. For the phase space, we choose D to be defined by D D C D f 2 C..1; 0; E/ W lim e ./ exists in Eg !1

with norm kk D

sup 2.1;0

e j./j; 2 C :

By making the following change of variables y.t/./ D v.t; /; t 0; 2 .0; ; . /./ D v0 .; /; 0; 2 Œ0; 1; F.t; '/./ D m.t/b.'.0; //; 2 Œ0; ; 2 C .t; '/ D t .'.0; 0// R0 Ik .ytk / D 1 k .s/v.s; /ds; the problem (9.75)–(9.78) takes the abstract form (9.64)–(9.66). Moreover, a simple estimate shows that kf .t; '/k m.t/Œb1 k'kD C b2 1=2 for all .t; '/ 2 I D with Z

1 1

ds D .s/

Z

1 1

ds D C1: b1 s C b2 1=2

and Z dk D

0 1

.k .s//2 ds es

1=2 <1

9.6 Notes and Remarks

259

Theorem 9.33. Let ' 2 D be such that H' is valid and t ! 't is continuous on R. /, then there exists a integral solution of (9.75)–(9.78) whenever Z 1C

0

s

e ds 1

1=2 ! X m

dk < 1:

kD1

9.6 Notes and Remarks The results of Chap. 9 are taken from Abada et al. [1, 3, 4]. Other results may be found in [124, 151, 152, 159].

Chapter 10

Impulsive Functional Differential Inclusions with Unbounded Delay

10.1 Introduction In this chapter, we shall establish sufficient conditions for the existence of mild, extremal mild, integral, and extremal integral solutions for some impulsive semilinear neutral functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators.

10.2 Densely Defined Impulsive Functional Differential Inclusions 10.2.1 Introduction We shall be concerned with existence of mild solutions, integral, and extremal integral solutions defined on a compact real interval for first order impulsive semi-linear neutral functional inclusions in a separable Banach space. We will consider the following first order impulsive semi-linear neutral functional differential inclusions of the form: d Œy.t/ g.t; yt / AŒy.t/ g.t; yt / 2 F.t; yt /; dt a:e: t 2 J D Œ0; b; t ¤ tk ; k D 1; : : : ; m

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_10

(10.1)

261

262

10 Impulsive Functional Differential Inclusions with Unbounded Delay

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(10.2)

y.t/ D .t/; t 2 .1; 0;

(10.3)

where F W J D ! 2E is a closed, bounded, and convex valued multi-valued map, g W J D ! E is a given function, 2 D where D is the phase space that will be specified later Ik 2 C .E; E/ ; .k D 1; 2; : : : ; m/ ; A W D.A/ E ! E is a densely defined closed linear operator on E, and E a real separable Banach space with norm j:j: Consider the space n PC D y W .1; b ! E; y.tk /; y.tkC /; exist with y.tk / D y.tk /; o y.t/ D .t/; t 0; yk 2 C.Jk ; E/ ; where yk is the restriction of y to Jk D .tk ; tkC1 ; k D 0; : : : ; m: Let k kPC be the norm in PC defined by kykPC D supfjy.s/j W 0 s bg; y 2 PC: We will assume that D satisfies the following axioms: (A) If y W .1; b ! E; b > 0 and y.tk /; y.tkC /, exist with y.tk / D y.tk /; k D 1; : : : ; m and y0 2 D; then for every t in Œ0; b/nft1 ; : : : ; tm g the following conditions hold: (i) yt is in DI and yt is continuous on Œ0; bnft1 ; : : : ; tm g (ii) kyt kD K.t/ supfjy.s/j W 0 s tg C M.t/ky0 kD ; (iii) jy.t/j Hkyt kD where H 0 is a constant, K W Œ0; 1/ ! Œ0; 1/ is continuous, M W Œ0; 1/ ! Œ0; 1/ is locally bounded and H; K; M are independent of y./. (A-1) For the function y./ in .A/; yt is a D-valued continuous function on Œ0; b/nft1 ; : : : ; tm g: (A-2) The space D is complete. Set Db D fy W .1; b ! Ej y 2 PC \ Dg; and let k kb be the seminorm in Db defined by kykb WD ky0 kD C supfjy.t/j W 0 s bg; y 2 Db : Denote Kb D supfK.t/ W t 2 Jg and Mb D supfM.t/ W t 2 Jg:

10.2 Densely Defined Impulsive Functional Differential Inclusions

263

10.2.2 Mild Solutions In order to define the mild solutions of the problem (10.1)–(10.3) we assume that F is compact and convex valued multi-valued map. Now, we can define a meaning of the mild solution of problem (10.1)–(10.3). Definition 10.1. A function y 2 Db is said to be a mild solution of system (10.1)– (10.3) if y.t/ D .t/ for all t 2 .1; 0, the restriction of y./ to the interval Œ0; b is continuous, and there exist v./ 2 L1 .Jk ; E/ and Ik 2 Ik .y.tk //, such that v.t/ 2 F.t; yt / a.e t 2 Œ0; b, and y satisfies the integral equation, Z t y.t/ D T.t/ ..0/ g.0; .0/// C g.t; yt / C T.t s/v.s/ds 0 X T.t tk /Ik ; t 2 J: C 0
We introduce the following hypotheses: (10.1.1) A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup fT.t/g, t 2 J which is compact for t > 0 in the Banach space E, and there exist constant M, such that: kT.t/kB.E/ MI t 2 J (10.1.2) There exist constants ck 0, k D 1; : : : ; m such that Hd .Ik .y/; Ik .x//j ck jy xj for each x; y 2 E: (10.1.3) F is L1 -Carathéodory with compact convex values. (10.1.4) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/k D supfjvj=v 2 F.t; x/g p.t/ .kxkD / for a.e. t 2 J and each x 2 D;

with Z

1

C0

ds > C1 kpkL1 ; .s/

where " C0 D M˛1 C M 2

m X

# ck kkD

kD0

CM

m X kD0

jIk .0/j C ˛2 .1 C M/

264

10 Impulsive Functional Differential Inclusions with Unbounded Delay

C1 D

C0 m X 1M ck Kb ˛1 kD0

.MKb C Mb /M

m X

ck

kD0

C .1 M

m X

./kkD

ck Kb ˛1 /

kD0

C2 D

M m X 1M ck Kb ˛1 kD0

(10.1.5) The function g.t; :/ is continuous on J and there exists a constant lg > 0 such that jg.t; u/ g.t; v/j lg ku vk for each u; v 2 D: (10.1.6) There exist constants ˛1 and ˛2 such that kg.t; u/k ˛1 kukD C ˛2 for each .t; u/ 2 Œ0; b D: Theorem 10.2. Assume that (10.1.1)–(10.1.6) hold. If lg C M ˛1 Kb C

m X

.1; b:

m X

ck < 1; and

1

ck < 1, then the IVP (10.1)–(10.3) has at least one mild solution on

1

Proof. Consider the multi-valued operator: N W D ! P.D/ defined by 8 8 .t/; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z t ˆ ˆ < < T.t s/v.s/ds N.y/ D h 2 D W h.t/ D T.t/ ..0/ g.0; .0/// C g.t; yt / C 0 ˆ ˆ ˆ ˆ ˆ ˆ X ˆ ˆ ˆ ˆ ˆ ˆ T.t tk /Ik ; v 2 SF;y ; Ik 2 Ik .y.tk // : : C 0
9 if t 0; > > > > > > = > > > > > if t 2 J; > ;

Has a fixed point . This fixed point is then the mild solution of the IVP (10.1)–(10.3). For 2 D define the function x./ W .1; b ! E such that: x.t/ D

8 < .t/;

if t 0

: T.t/.0/;

if t 2 J

10.2 Densely Defined Impulsive Functional Differential Inclusions

265

Then x./ is an element of Db , and x0 D .0/. Set y.t/ D z.t/ C x.t/: Obviously if y satisfies the integral equation Z y.t/ D T.t/ ..0/ g.0; .0/// C g.t; yt / C X T.t tk /Ik ; t 2 J; C

t

T.t s/v.s/ds

0

0
then z satisfies z0 D 0 and z.t/ D g.t; zt C xt / T.t/g.0; .0// Z t X C T.t s/v.s/ds C T.t tk /Ik ; t 2 J: 0

0
where v.t/ 2 F.t; zt C xt / a.e. t 2 Œ0; b and Ik 2 Ik .z.tk C x.tk /. Let D0b D fz 2 Db W z0 D 0g : For any z 2 D0b , we have kzkb D kz0 kD C sup fjz.s/j W 0 s bg D sup fjz.s/j W 0 s bg : Thus .D0b ; k kb / is a Banach space. Let the operator P W D0b ! P.D0b / defined by 8 8 0 if t 2 .1; 0I ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < < P.z/ D h 2 D0b jh.t/ D g.t; zt C xt / T.t/g.0; .0// X Rt ˆ ˆ ˆ ˆ ˆ ˆ T.t tk /Ik ; if t 2 J: C 0 T.t s/v.s/ds C ˆ ˆ : : 0
9 > > > > = > > > > ;

The operator N has a fixed point is equivalent to P has one, so it turns to prove that P has a fixed point. Consider these multi-valued operators: A; B W D0b ! P.D0b /

266

10 Impulsive Functional Differential Inclusions with Unbounded Delay

defined by 9 if t 0; > > > > =

8 8 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < < zt C xt / T.t/g.0; .0// A.z/ WD h 2 D0b W h.t/ D g.t;X ˆ ˆ ˆ ˆ ˆ ˆ T.t tk /Ik ; Ik 2 Ik .z.tk / C x.tk // C ˆ ˆ : :

> > if t 2 J; > > ;

0
and 8 8 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <
9 if t 0; > > > > = > > > > ;

if t 2 J;

where v 2 SF;z D fv 2 L1 .Œ0; b; E/ W v.t/ 2 F.t; zt C xt / for a.e. t 2 Œ0; bg: It is clear that PDACB Then the problem of finding mild solutions of (10.1)–(10.3) is then reduced to finding mild solutions of the operator inclusion z 2 A.z/ C B.z/. We shall show that the operators A and B satisfy all conditions of the Theorem 1.32. The proof will be given in several steps. Step 1: A is a contraction Let z1 ; z2 2 D0b , then from .10:1:1/

Hd A.z1 /; A.z2 / kg.t; z1t C xt / g.t; z2t C xt /k CHd

X

T.t tk /Ik .z1 .tk / C x.tk /;

0
X

!

T.t

tk /Ik .z2 .tk /

C

x.tk /

0
lg kz1 z2 k C M

m X

jz1 .tk / z2 .tk /j

1

lg C M

m X 1

ck kz1 z2 k;

10.2 Densely Defined Impulsive Functional Differential Inclusions

267

which is a contraction since lg C M

m X

ck < 1:

1

Step 2: B has compact, convex values, and it is completely continuous. This will be given in several claims. Claim 1: B has compact values. The operator B is equivalent to the composition L ı SF of two operators on L1 .J; E/ L W L1 .J; E/ ! D0b is the continuous operator defined by Z L.v.t// D

t

T.t s/v.s/ds;

t 2 J:

0

Then, it suffices to show that L ı SF has compact values on D0b . Let z 2 D0b arbitrary, vn a sequence in SF;z , then by definition of SF , vn .t/ belongs to F.t; zt /; a:e:t 2 J. Since F.t; zt / is compact, we may pass to a subsequence. Suppose that vn ! v in L1 .J; E/, where v.t/ 2 F.t; zt /; a:e:t 2 J. From the continuity of L, it follows that Lvn .t/ ! Lv.t/ pointwise on J as n ! 1. In order to show that the convergence is uniform, we first show that fLvn g is an equi-continuous sequence. Let 1 ; 2 2 J, then we have: Z ˇ ˇ ˇ 1 ˇL.vn . 1 // L.vn . 2 //ˇ D ˇ T. 1 s/vn .s/ds 0

Z

Z 0

C

2

0

1 ˇ

ˇ T. 2 s/vn .s/dsˇ

ˇ ˇ T. 1 s/ T. 2 s/ ˇjvn .s/jds

Z

2

1

jT. 2 s/jjvn .s/jds

As 1 ! 2 , the right-hand side of the above inequality tends to zero. Since T.t/ is a strongly continuous operator and the compactness of T.t/,t > 0, implies the continuity in uniform topology. Hence fLvn g is equi-continuous, and an application of Arzéla-Ascoli theorem implies that there exists a subsequence which is uniformly convergent. Then we have Lvnj ! Lv 2 .L ı SF /.z/ as j 7! 1, and so .L ı SF /.z/ is compact . Therefore B is a compact valued multivalued operator on D0b .

268

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Claim 2: B.z/ is convex for each z 2 D0b : Let h1 ; h2 2 B.z/, then there exists v1 ; v2 2 SF;z such that, for each t 2 J we have 8 9 0; if t 2 .1; 0, > ˆ ˆ > ˆ > ˆ > ˆ > ˆ > ˆ 0 > : ; if t 2 J;i D 1; 2: Let 0 ı 1: Then, for each t 2 J, we have

.ıh1 C .1 ı/h2 /.t/ D

8 0; ˆ ˆ ˆ ˆ
0

9 if t 2 .1; 0, > > > > = T.t s/Œıv1 .s/ C .1 ı/v2 .s/ds if t 2 J;

> > > > ;

Since F.t; zt / has convex values, one has ıh1 C .1 ı/h2 2 B.z/: Claim 3: B maps bounded sets into bounded sets in D0b Let B D fz 2 D0b I kzk1 qg; q 2 RC a bounded set in D0b . We know that for each h 2 B.z/, for some z 2 B, there exists v 2 SF;z such that Z t h.t/ D T.t s/v.s/ds: 0

v 2 SF;z D fv 2 L1 .Œ0; b; E/ W v.t/ 2 F.t; zt C xt / From (10.1.4) we have kzs C xs kD kzs kD C kxs kD Kb q C Kb Mj.0/j C Mb kkD D q : Then Z jh.t/j M .q /

t

p.s/ds 0

M .q /kpkL1 D l; This further implies that: khk1 l Then, for all h 2 B.z/ B.B/ D

S z2B

B.z/: Hence B.B/ is bounded.

10.2 Densely Defined Impulsive Functional Differential Inclusions

269

Claim 4: B maps bounded sets into equi-continuous sets. Let B be, as above, a bounded set and h 2 B.z/ for some z 2 B: Then, there exists v 2 SF;z such that Z

t

h.t/ D

T.t s/v.s/ds;

t2J

0

Let 1 ; 2 2 Jnft1 ; t2 ; : : : ; tm g; 1 < 2 . Thus if > 0, we have Z jh. 2 / h. 1 /j

1 0

Z

C Z C

1

kT. 2 s/ T. 1 s/kjv.s/jds

1

2

1

kT. 2 s/ T. 1 s/kjv.s/jds

kT. 2 s/kjv.s/jds Z

.q /

1

0

Z

C .q / Z C .q /

kT. 2 s/ T. 1 s/kB.E/ p.s/ds

1

1

2

1

kT. 2 s/ T. 1 s/kB.E/ p.s/ds

kT. 2 s/kB.E/ p.s/ds:

As 1 ! 2 and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0 implies the continuity in the uniform operator topology. This proves the equi-continuity for the case where t ¤ ti ; i D 1; : : : ; m C 1. It remains to examine the equi-continuity at t D ti . First we prove the equi-continuity at t D ti , we have for some z 2 B, there exists v 2 SF;z such that Z t T.t s/v.s/ds; t 2 J h.t/ D 0

Fix ı1 > 0 such that ftk ; k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < < ı1 , we have Z ti jh.ti / h.ti /j kT.ti s/ T.ti s/kjv.s/jds 0

C .q /M which tends to zero as ! 0.

Z

ti

p.s/ds; ti

270

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Define hO 0 .t/ D h.t/; and hO i .t/ D

t 2 Œ0; t1

8 < h.t/;

9 if t 2 .ti ; tiC1 =

: h.tC /;

if

i

t D ti

;

Next, we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk ; k ¤ ig \ Œti ı2 ; ti C ı2 D ;. Then Z ti Oh.ti / D T.ti s/v.s/ds; 0

For 0 < < ı2 , we have O i C / h.t O i /j jh.t

Z

ti

0

kT.ti C s/ T.ti s/kjv.s/jds Z

ti C

C .q /M

p.s/ds ti

The right-hand side tends to zero as ! 0. The equi-continuity for the cases 1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval .1; 0 As a consequence of Claims 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For z 2 B, we define Z t T.t s /v.s/ds; h .t/ D T./ 0

where v 2 SF;z . Since T.t/ is a compact operator, the set H .t/ D fh .t/ W h 2 B.z/g is precompact in E for every ; 0 < < t: Moreover, for every h 2 B.z/ we have Z Z t ˇ t ˇ ˇ jh.t/ h .t/j D T.t s/v.s/ds T./ T.t s /v.s/dsˇ 0

Z Dj

0

t

T.t s/v.s/dsj

t

M .q /

Z

t

p.s/ds t

10.2 Densely Defined Impulsive Functional Differential Inclusions

271

Therefore, there are precompact sets arbitrarily close to the set H.t/ D fh.t/ W h 2 B.z/g: Hence the set H.t/ D fh.t/ W h 2 B.B/g is precompact in E. Hence the operator B is totally bounded. Claim 5: B has closed graph. Let zn ! z , hn 2 B.zn /, and hn ! h . We shall show that h 2 B.z /: hn 2 B.zn / means that there exists vn 2 SF;zn such that Z t hn .t/ D T.t s/vn .s/ds; t 2 J: 0

We must prove that there exists v 2 SF;z such that Z t T.t s/v .s/ds: h .t/ D 0

Consider the linear and continuous operator K W L1 .J; E/ ! D0b defined by Z t .Kv/.t/ D T.t s/v.s/ds: 0

We have j.hn .t/ .h .t// khn h k1 ! 0;

as n 7! 1:

From Lemma 1.11 it follows that K ı SF is a closed graph operator and from the definition of K one has hn .t/ 2 K ı SF;zn : As zn ! z and hn ! h , there is a v 2 SF;z such that Z t T.t s/v .s/ds: h .t/ D 0

Hence the multi-valued operator B is upper semi-continuous. Step 3: A priori bounds on solutions. Now, it remains to show that the set E D fz 2 D0b j z 2 Az C Bz; 0 1g is unbounded. Let z 2 E be any element. Then there exist v 2 SF;z and Ik 2 Ik .z.tk // such that z.t/ D g.t; zt C xt / T.t/g.0; .0// Z t X C T.t s/v.s/ds C T.t tk /Ik : 0

0
272

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Then jz.t/j ˛1 kzt C xt k C ˛2 C M.˛1 kkD C ˛2 / Z t m kDm X X CM p.s/ .kzs C xs k/ds C M ck jz.tk / C x.tk /j C M jIk .0/j 0

kD0

kD0

˛1 Kb jz.t/j C .MKb C Mb /kkD C ˛2 Z t

CM.˛1 kkD C ˛2 / C M p.s/ Kb jz.s/j C .MKb C Mb /kkD /ds 0

CM

m X

ck jz.t/j C M

kD0

CM

kDm X

m X

ck jx.t/j

kD0

jIk .0/j

kD0

˛1 Kb jz.t/j C .MKb C Mb /kkD C ˛2 C M ˛1 kkD C ˛2 Z t m X CM p.s/ .Kb jz.s/j C .MKb C Mb /kkD //ds C M ck jz.tj/ 0

CM 2

kD0

m X kD0

ck kkD C M

kDm X

jIk .0/j:

kD0

Then, we have:

jz.t/j C0 C ˛1 .Kb jz.t/j C .MKb C Mb /kkD / Z t m X

CM p.s/ Kb jz.s/j C .MKb C Mb /kkD /ds C M ck jz.tj/ 0

kD0

Since M

m X

ck < 1;

kD0

then Z Kb jz.t/j C .MKb C Mb /kkD C1 C C2

t

p.s/ 0

Kb jz.s/j

C.MKb C Mb /kkD ds

10.2 Densely Defined Impulsive Functional Differential Inclusions

273

Consider the function .t/ defined by .t/ D supfKb jz.s/j C .MKb C Mb /kkD W 0 s tg; 0 t b Then, we have, for all t 2 J,kKb jz.t/j C .MKb C Mb /kkD k .t/. Let t 2 J such that .t/ D Kb jz.t /j C .MKb C Mb /kkD , then by the previous inequality we have, for t 2 J; Z .t/ C1 C C2

t 0

p.s/ .ksk/ds:

Let us note the right-hand side of the above inequality by v.t/, i.e., Z v.t/ D C1 C C2

0

t

p.s/ ..s//ds:

Then, we have .t/ v.t/ for all t 2 J v.0/ D C1 Differentiating both sides of the above equality, we obtain v 0 .t/ D C2 p.t/ ..t//; a:e: t 2 J and using the nondecreasing character of the function

, we obtain

v 0 .t/ C2 p.t/ .v.t//; a:e: t 2 J; that is v 0 .t/ C2 p.t/; a:e: t 2 J: .v.t// Integrating from 0 to t we get Z 0

t

v 0 .s/ ds C2 .v.s//

Z

t

p.s/ds: 0

By a change of variables we get Z

v.t/

v.0/

du C2 kpkL1 .u/

Z

1

c1

du : .u/

274

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Hence there exists a constant K such that .t/ v.t/ K

for all t 2 J:

Now from the definition of it follows that kKb jz.t/j C .MKb C Mb /kkD k .b/ K

for all z 2 E;

which means that E is bounded. As a consequence of Theorem 1.32, A.z/ C B.z/ has a fixed point z on the interval .1; b, so y D z C x is a fixed point of the operator N which is the mild solution of problem (10.1)–(10.3). t u

10.2.3 Extremal Mild Solutions In this section we shall prove the existence of maximal and minimal solutions of problem (10.1)–(10.3) under suitable monotonicity conditions on the multi-valued functions involved in it. We need the following definitions in the sequel. Definition 10.3. We say that a continuous function vQ 2 Db is a lower mild solution of problem (10.1)–(10.3) if v.t/ Q D .t/; t 2 .1; 0; and there exist v./ 2 L1 .Jk ; E/ and Ik 2 Ik .v.t Q k //, such that v.t/ 2 F.t; vQ t / a.e t 2 Œ0; b, and vQ satisfies, Z v.t/ Q T.t/ ..0/ g.0; .0/// C g.t; vQ t / C X T.t tk /Ik ; t 2 J; t ¤ tk : C

t

T.t s/v.s/ds

0

0
and v.t Q kC / v.t Q k Ik where Ik 2 Ik .v.t Q k //; t D tk ; k D 1; : : : ; m Similarly an upper mild solution wQ of IVP (10.1)–(10.3) is defined by reversing the order. Definition 10.4. A solution xM of IVP (10.1)–(10.3) is said to be maximal if for any other solution x of IVP (10.1)–(10.3) on J, we have that x.t/ xM .t/ for each t 2 J. Similarly a minimal solution of IVP (10.1)–(10.3) is defined by reversing the order of the inequalities. We consider the following assumptions in the sequel. (10.4.1) The multi-valued function F.t; y/ is strictly monotone increasing in y for almost each t 2 J. (10.4.2) The IVP (10.1)–(10.3) has a lower mild solution vQ and an upper mild solution wQ with vQ w. Q (10.4.3) T.t/ is preserving the order, that is T.t/v 0 whenever v 0. (10.4.4) The functions Ik ; k D 1; : : : ; m are continuous and nondecreasing.

10.2 Densely Defined Impulsive Functional Differential Inclusions

275

Theorem 10.5. Assume that assumptions (10.1.1)–(10.1.6) and (10.4.1)–(10.4.4) hold. Then IVP (10.1)–(10.3) has minimal and maximal solutions on Db . Proof. We can write vQ and w Q as v.t/ Q D v .t/ C x.t/ w.t/ Q D w .t/ C x.t/ where v 2 D0b and w 2 D0b and x.t/ is defined in the above section. Then vQ is lower solution to IVP (10.1)–(10.3) if v satisfies Z t T.t s/v.s/ds v .t/ T.t/g.0; .0// C g.t; vt C x.t// C 0 X T.t tk /Ik ; t 2 J; t ¤ tk : C 0
and v .tkC / v .tk Ik such that Ik Ik .v .tk //; t D tk ; k D 1; : : : ; m respectively (w) Q is upper solution to IVP (10.1)–(10.3) if w satisfies the reversed inequality. It can be shown, as in the proof of Theorem 10.2, that A is completely continuous and B is a contraction on Œv ; w . We shall show that A and B are isotone increasing on Œv ; w . Let z; z 2 Œv ; w be such that z z; z 6D z: Then by (10.4.4), we have for each t 2 J ( h 2 D0b W h.t/ D T.t/g.0; .0// C g.t; zt C xt /

A.z/ D

C

X

) T.t tk /Ik ; 2

Ik z.tk /

0
(

h 2 D0b W h.t/ D T.t/g.0; .0// C g.t; zt C xt /

C

X

) T.t tk /Ik ; Ik 2

Ik .z.tk //

0
D A.z/: Similarly, by .10:4:1/; .10:4:3/ ( h 2 D0b W h.t/ D

B.z/ D (

h2

D B.z/:

D0b

Z

t 0

Z

t

W h.t/ D 0

) T.t s/v.s/ds; v 2 SF;z ) T.t s/v.s/ds; f 2 SF;z

276

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Therefore A and B are isotone increasing on Œv ; w . Finally, let y 2 Œv ; w be any element. By (10.4.2), (10.4.3) we deduce that v A.v / C B.v / A.y/ C B.y/ A.w / C B.w / w ; which shows that A.y/ C B.y/ 2 Œv ; w for all y 2 Œv ; w . Thus, A and B satisfy all the conditions of Theorem 1.37, hence IVP (10.1)–(10.3) has maximal and minimal solutions on J: t u

10.2.4 Example As an application of our results we consider the following impulsive partial functional differential equation of the form

Z 0 @ v.t; / K1 . /g1 .t C ; /d @t 1

Z 0 @2 K1 . /g1 .t C ; /d D 2 v.t; / @ 1 Z 0 K2 . / ŒQ1 .t; v.t C ; /; Q2 .t; v.t C ; /d C 1

for 2 Œ0; ; t 2 Œ0; bnft1 ; t2 ; : : : ; tm g: v.tkC ; / z.tk ; / 2 bk jz.tk ; /jB.0; 1/; 2 Œ0; ; k D 1; : : : ; m Z v.t; 0/ Z v.t; /

0 1

0

(10.4) (10.5)

K1 . /g1 .t C ; 0/d D 0; t 2 J WD Œ0; b

(10.6)

K1 . /g1 .tC; /d D .t; x/; t 2 J WD Œ0; b; 2 Œ0; ;

(10.7)

1

v.; ; / D v0 .; / fot 1 < 0 and 2 Œ0; ;

(10.8)

where bk > 0; k D 1; : : : ; m; K1 W .1; 0 ! R; K2 W .1; 0 ! R and g1 W J R ! R and v0 W .1; 0XŒ0; ! R are continuous functions, 0 D t0 < t1 < t2 < < tm < tmC1 D b; v.tkC / D lim v.tk C h; x/; v.tk / D .h;x/!.0C ;x/

lim v.tk C h; x/, where Q1 ; Q2 W J R ! R; are given functions, and B.0; 1/

.h;x/!.0 ;x/

the closed unit ball. We assume that for each t 2 J; Q1 .t; / is lower semi-continuous (i.e., the set fy 2 R W Q1 .t; y/ > g is open for each 2 R), and assume that for each t 2 J; Q2 .t; / is upper semi-continuous (i.e., the set fy 2 R W Q2 .t; y/ < g is open for each 2 R).

10.2 Densely Defined Impulsive Functional Differential Inclusions

277

Let y.t/./ D v.t; /; t 2 J; 2 Œ0; ; Ik .y.tk //./

D bk v.tk ; /; 2 Œ0; ; k D 1; : : : ; m R0 F.t; /.x/ D 1 K2 . / Q1 .t; v.t C ; /; Q2 .t; v.t C ; /d ; 2 .1; 0; 2 Œ0; ; R0 h..t; / D 1 K1 . /g1 .t C ; /d; and . /./ D .; /; 2 .1; 0; 2 Œ0; : E D L2 Œ0; and define A W D.A/ E ! E by Aw D w00 with domain D.A/ D fw 2 E; w; w0

are absolutely continuous, w00 2 E; w.0/ D w./ D 0g:

Then Aw D

1 X

n2 .w; wn /wn ; w 2 D.A/

nD1

q 2 sin ns; n D 1; 2; : : : is where .; / is the inner product in L2 and wn .s/ D the orthogonal set of eigenvectors in A: It is well known (see [168]) that A is the infinitesimal generator of an analytic semigroup T.t/; t 2 .0; b in E and is given by T.t/w D

1 X

exp.n2 t/.w; wn /wn ; w 2 E:

nD1

Since the analytic semigroup T.t/; t 2 .0; b is compact, there exists a constant M 1 such that kT.t/kB.E/ M: It is clear that F is compact and convex valued, and it is upper semi-continuous (see [101]). Assume that there are p 2 C.J; RC / and W Œ0; 1/ ! .0; 1/ continuous and nondecreasing such that max.jQ1 .t; y/j; jQ2 .t; y/j/ p.t/ .jyj/;

t 2 J; and y 2 R:

278

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Assume that there exist functions lQ1 ; lQ2 2 L1 .J; RC / such that jQ1 .t; w/ Q1 .t; w/j lQ1 .t/jw wj; t 2 J; w; w 2 R; and jQ2 .t; w/ Q2 .t; w/j lQ2 .t/jw wj; t 2 J; w; w 2 R: We can show that problem (10.1)–(10.3) is an abstract formulation of problem (10.4)–(10.8). Since all the conditions of Theorem 10.2 are satisfied, the problem (10.4)–(10.8) has a solution z on .1; b Œ0; :

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions In this section, we use the extrapolation method combined with a fixed point theorem for the sum of completely continuous and contraction operators, to establish sufficient conditions for the existence of mild solutions and extremal mild solutions for some classes of non-densely defined impulsive semi-linear neutral functional differential inclusions in separable Banach spaces with infinite delay. More precisely, we will consider the following first order impulsive semi-linear neutral functional differential inclusions of the form: d Œy.t/ g.t; yt / AŒy.t/ g.t; yt / 2 F.t; yt /; dt a:e: t 2 J D Œ0; b; t ¤ tk ; k D 1; : : : ; m

(10.9)

yjtDtk 2 Ik .y.tk //; k D 1; : : : ; m

(10.10)

y.t/ D .t/; t 2 .1; 0;

(10.11)

where F W JD ! 2E is a compact and convex valued multi-valued map,g W JD ! E is a given function, 2 D where D is the phase space that will be specified later Ik 2 C .E; E/ ; .k D 1; 2; : : : ; m/ are bounded valued multi-valued maps, P.E/ is the collection of all E-subsets, A W D.A/ E ! E is a non-densely defined closed linear operator on E, and E a real separable Banach space with norm j:j:

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions

279

10.3.1 Mild Solutions We shall consider the space Db D fy W .1; b ! Ej y 2 PC \ Dg; and let k kb be the seminorm in Db defined by kykb WD ky0 kD C supfjy.t/j W 0 s bg; y 2 Db : Assume that F is compact and convex valued multi-valued map. Let us start by defining what we mean by a solution of problem (10.9)–(10.11). Definition 10.6. A function y 2 Db is said to be a mild solution of system (10.9)– (10.11) if y.t/ D .t/ for all t 2 .1; 0, the restriction of y./ to the interval Œ0; b is continuous, and there exist v./ 2 L1 .Jk ; E/ and Ik 2 Ik .y.tk //, such that v.t/ 2 F.t; yt / a.e t 2 Œ0; b, and y satisfies the integral equation, Z y.t/ D T0 .t/ ..0/ g.0; .0/// C g.t; yt / C C

X

t

0

T1 .t s/v.s/ds

T1 .t tk /Ik ; t 2 J:

(10.12)

0
Before beginning our result, we shall introduce the following hypotheses: (10.6.1) There exists a constant M, such that: kT1 .t/kB.E/ MI t 2 J (10.6.2) There exist constants ck 0, k D 1; : : : ; m such that Hd .Ik .y/; Ik .x//j ck jy xj for each x; y 2 E: (10.6.3) F is L1 -Carathéodory with compact convex values. (10.6.4) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/k D supfjvj=v 2 F.t; x/g p.t/ .kxkD / for a.e. t 2 J and each x 2 D;

with 1 ˛1 Kb M lim sup u!1

m X

! ck u

kD1

C0 C MkpkL1 .Kb u C .MKb C Mb /kkD /

> 1;

(10.13)

280

10 Impulsive Functional Differential Inclusions with Unbounded Delay

where C0 D ˛1 .MKb C Mb /kkD C ˛2 C M.˛1 kkD C ˛2 / CM 2

m X

ck kkD C M

kD0

m X

jIk .0/j:

kD0

(10.6.5) The function g.t; :/ is continuous on J and there exists a constant lg > 0 such that jg.t; u/ g.t; v/j lg ku vk for each u; v 2 D: (10.6.6) There exist constants ˛1 and ˛2 such that kg.t; u/k ˛1 kukD C ˛2 for each .t; u/ 2 Œ0; b D: Theorem 10.7. Assume that (10.6.1)–(10.6.6), 2 D and .0/ 2 X0 ; g.0; .0// 2 m X X0 hold. If lg C M ck < 1; then the IVP (10.9)–(10.11) has at least one mild 1

solution on .1; b:

Proof. Transform the problem (10.9)–(10.11) into a fixed point problem. Consider the multi-valued operator: N W D ! P.D/ defined by 8 8 ˆ ˆ .t/; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z t ˆ ˆ < < T1 .t s/v.s/ds N.y/ D h 2 D W h.t/ D T0 .t/ ..0/ g.0; .0/// C g.t; yt / C 0 ˆ ˆ ˆ ˆ ˆ ˆ X ˆ ˆ ˆ ˆ ˆ ˆ T1 .t tk /Ik ; v 2 SF;y ; Ik 2 Ik .y.tk // ˆ ˆC : : 0
9 if t 0; > > > > > > > = > > > > > if t 2 J: > > ;

Now we shall show that the operator N has a fixed point . This fixed point is then the mild solution of the IVP (10.9)–(10.11). For 2 D define the function x./ W .1; b ! E such that: x.t/ D

8 < .t/;

if t 0

: T .t/.0/; 0

if t 2 J:

Then x is an element of Db ,and x0 D .0/. Set y.t/ D z.t/ C x.t/:

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions

281

Obviously if y satisfies the integral equation Z y.t/ D T0 .t/ ..0/ g.0; .0/// C g.t; yt / C X T1 .t tk /Ik ; t 2 J: C

t 0

T1 .t s/v.s/ds

0
then z satisfies z0 D 0 and z.t/ D g.t; zt C xt / T0 .t/g.0; .0// Z t X C T1 .t s/v.s/ds C T1 .t tk /Ik ; t 2 J: 0

0
where v.t/ 2 F.t; zt C xt / a.e t 2 Œ0; b. and Ik 2 Ik .z.tk C x.tk / Let D0b D fz 2 Db W z0 D 0g : For any z 2 D0b ; we have kzkb D kz0 kD C sup fjz.s/j W 0 s bg D sup fjz.s/j W 0 s bg : Thus .D0b ; k kb / is a Banach space. Let the operator P W D0b ! P.D0b / defined by 8 8 0 if t 2 .1; 0I ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < < P.z/ D h 2 D0b jh.t/ D g.t; zt C xt / T0 .t/g.0; .0// X Rt ˆ ˆ ˆ ˆ ˆ ˆ T1 .t tk /Ik ; if t 2 J: C 0 T1 .t s/v.s/ds C ˆ ˆ : : 0
9 > > > > = > > > > ;

The operator N has a fixed point is equivalent to P has one, so it turns to prove that P has a fixed point. Consider these multi-valued operators: A; B W D0b ! P.D0b / defined by 8 8 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < < zt C xt / T0 .t/g.0; .0// A.z/ WD h 2 D0b W h.t/ D g.t;X ˆ ˆ ˆ ˆ ˆ ˆ T1 .t tk /Ik ; Ik 2 Ik .z.tk / C x.tk // C ˆ ˆ : : 0
9 if t 0; > > > > = > > if t 2 J; > > ;

282

10 Impulsive Functional Differential Inclusions with Unbounded Delay

and 8 8 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ <
9 if t 0; > > > > = if t 2 J;

> > > > ;

where v 2 SF;z D fv 2 L1 .Œ0; b; E/ W v.t/ 2 F.t; zt C xt / for a.e. t 2 Œ0; bg: It’s clear that PDACB Then the problem of finding mild solutions of (10.9)–(10.11) is then reduced to finding mild solutions of the operator inclusion z 2 A.z/ C B.z/. We shall show that the operators A and B satisfy all conditions of the Theorem 1.32. The proof will be given in several steps. Step 1: A is a contraction Let z1 ; z2 2 D0b , then from .10:6:2/ and 10:6:5 Hd .A.z1 /; A.z2 // kg.t; z1t C xt / g.t; z2t C xt /k X

CHd

T1 .t tk /Ik .z1 .tk / C x.tk /;

0
X

T1 .t

! tk /Ik .z2 .tk /

C

x.tk /

0
lg kz1 z2 k C M lg C M

m X

m X

!

jz1 .tk / z2 .tk /j

1

ck kz1 z2 k;

1

which is a contraction since lg C M

m X

ck < 1:

1

Step 2 B has compact, convex values, and it is completely continuous. This will be given in several claims.

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions

283

Claim 1: B has compact values. The operator B is equivalent to the composition L ı SF of two operators on L1 .J; E/ L W L1 .J; E/ ! D0b is the continuous operator defined by Z L.v.t// D

t 0

T1 .t s/v.s/ds;

t2J

Then, it suffices to show that L ı SF has compact values on D0b . Let z 2 D0b arbitrary, vn a sequence in SF;z , then by definition of SF , vn .t/ belongs to F.t; zt /; a:e:t 2 J. Since F.t; zt / is compact, we may pass to a subsequence. Suppose that vn ! v in L1 .J; E/, where v.t/ 2 F.t; zt /; a:e:t 2 J. From the continuity of L, it follows that Lvn .t/ ! Lv.t/ point wise on J as n ! 1. In order to show that the convergence is uniform, we first show that fLvn g is an equi-continuous sequence. Let 1 ; 2 2 J, then we have: Z ˇ ˇ ˇ 1 ˇL.vn . 1 // L.vn . 2 //ˇ D ˇ T1 . 1 s/vn .s/ds 0

Z

2

0

1 ˇ

Z 0

C

ˇ T1 . 2 s/vn .s/dsˇ

ˇ ˇ T1 . 1 s/ T1 . 2 s/ ˇjvn .s/jds

Z

2

1

jT1 . 2 s/jjvn .s/jds:

As 1 ! 2 , the right-hand side of the above inequality tends to zero. Since T1 .t/ is a strongly continuous operator and the compactness of T1 .t/,t > 0, implies the continuity in uniform topology. Hence fLvn g is equi-continuous, and an application of Arzéla-Ascoli theorem implies that there exist a subsequence which is uniformly convergent. Then we have Lvnj ! Lv 2 .L ı SF /.z/ as j 7! 1, and so .L ı SF /.z/ is compact. Therefore B is a compact valued multivalued operator on D0b . Claim 2: B.z/ is convex for each z 2 D0b : Let h1 ; h2 2 B.z/, then there exists v1 ; v2 2 SF;z such that, for each t 2 J we have

hi .t/ D

8 0; ˆ ˆ ˆ ˆ
0

9 if t 2 .1; 0, > > > > = T1 .t s/vi .s/ds if t 2 J;i D 1; 2:

> > > > ;

284

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Let 0 ı 1: Then, for each t 2 J, we have

.ıh1 C .1 ı/h2 /.t/ D

8 0; ˆ ˆ ˆ ˆ
0

9 if t 2 .1; 0, > > > > = T1 .t s/Œıv1 .s/ C .1 ı/v2 .s/ds if t 2 J;

> > > > ;

Since F.t; zt / has convex values, one has ıh1 C .1 ı/h2 2 B.z/: Claim 3: B maps bounded sets into bounded sets in D0b Let B D fz 2 D0b I kzk1 qg; q 2 RC a bounded set in D0b . We know that for each h 2 B.z/, for some z 2 B, there exists v 2 SF;z such that Z

t

h.t/ D 0

T1 .t s/v.s/ds:

v 2 SF;z D fv 2 L1 .Œ0; b; E/ W v.t/ 2 F.t; zt C xt / From (10.6.4) we have kzs C xs kD kzs kD C kxs kD Kb q C Kb Mj.0/j C Mb kkD D q : Then Z jh.t/j M .q /

t

p.s/ds 0

M .q /kpkL1 D l; This further implies that: khk1 l Then, for all h 2 B.z/ B.B/ D

S z2B

B.z/: Hence B.B/ is bounded.

Claim 4: B maps bounded sets into equi-continuous sets. Let B be, as above, a bounded set and h 2 B.z/ for some z 2 B: Then, there exists v 2 SF;z such that Z h.t/ D 0

t

T.t s/v.s/ds;

t2J

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions

285

Let 1 ; 2 2 Jnft1 ; t2 ; : : : tm g; 1 < 2 . Thus if > 0, we have Z jh. 2 / h. 1 /j

1

0

Z

1

C Z

1

2

C

1

kT1 . 2 s/ T1 . 1 s/kjv.s/jds kT1 . 2 s/ T1 . 1 s/kjv.s/jds

kT1 . 2 s/kjv.s/jds Z

.q /

1 0

Z

C .q / Z C .q /

kT1 . 2 s/ T1 . 1 s/kB.E/ p.s/ds

1

1

2

1

kT1 . 2 s/ T1 . 1 s/kB.E/ p.s/ds

kT1 . 2 s/kB.E/ p.s/ds:

As 1 ! 2 and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since T1 .t/ is a strongly continuous operator and the compactness of T1 .t/ for t > 0 implies the continuity in the uniform operator topology. This proves the equi-continuity for the case where t ¤ ti ; i D 1; : : : ; m C 1. It remains to examine the equi-continuity at t D ti . First we prove the equi-continuity at t D ti , we have for some z 2 B, there exists v 2 SF;z such that Z t T1 .t s/v.s/ds; t 2 J h.t/ D 0

Fix ı1 > 0 such that ftk ; k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < < ı1 , we have Z ti jh.ti / h.ti /j kT1 .ti s/ T1 .ti s/kjv.s/jds 0

Z

C .q /M

ti

p.s/ds ti

which tends to zero as ! 0. Define hO 0 .t/ D h.t/;

t 2 Œ0; t1

and hO i .t/ D

8 < h.t/;

9 if t 2 .ti ; tiC1 =

: h.tC /;

if

i

t D ti

;

286

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Next, we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk ; k ¤ ig \ Œti ı2 ; ti C ı2 D ;. Then Z ti O i/ D h.t T1 .ti s/v.s/ds; 0

For 0 < < ı2 , we have O i C / h.t O i /j jh.t

Z

ti

kT1 .ti C s/ T1 .ti s/kjv.s/jds

0

Z

C .q /M

ti C

p.s/ds ti

The right-hand side tends to zero as ! 0. The equi-continuity for the cases 1 < 2 0 and 1 0 2 follows from the uniform continuity of on the interval .1; 0 As a consequence of Claims 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E. Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For z 2 B, we define Z t h .t/ D T1 ./ T1 .t s /v.s/ds; 0

where v 2 SF;z . Since T1 .t/ is a compact operator, the set H .t/ D fh .t/ W h 2 B.z/g is precompact in E for every ; 0 < < t: Moreover, for every h 2 B.z/ we have Z Z t ˇ t ˇ jh.t/ h .t/j D ˇ T1 .t s/v.s/ds T1 ./ T1 .t s /v.s/dsˇ 0 0 Z t Dj T1 .t s/v.s/dsj t Z t M .q / p.s/ds: t

Therefore, there are precompact sets arbitrarily close to the set H.t/ D fh.t/ W h 2 B.z/g: Hence the set H.t/ D fh.t/ W h 2 B.B/g is precompact in E. Hence the operator B is totally bounded. Claim 5: B has closed graph. Let zn ! z , hn 2 B.zn /, and hn ! h . We shall show that h 2 B.z /: hn 2 B.zn / means that there exists vn 2 SF;zn such that Z hn .t/ D

0

t

T1 .t s/vn .s/ds; t 2 J:

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions

287

We must prove that there exists v 2 SF;z such that Z t h .t/ D T1 .t s/v .s/ds: 0

Consider the linear and continuous operator K W L1 .J; E/ ! D0b defined by Z t T1 .t s/v.s/ds: .Kv/.t/ D 0

We have j.hn .t/ .h .t// khn h k1 ! 0;

as n 7! 1:

From Lemma 1.11 it follows that K ı SF is a closed graph operator and from the definition of K one has hn .t/ 2 K ı SF;zn : As zn ! z and hn ! h , there is a v 2 SF;z such that Z t h .t/ D T1 .t s/v .s/ds: 0

Hence the multi-valued operator B is upper semi-continuous. Step 3: A priori bounds on solutions. Now, it remains to show that the set E D fz 2 D0b j z 2 Az C Bz; 0 1g is unbounded. Let z 2 E be any element. Then there exist v 2 SF;z and Ik 2 Ik .z.tk // such that z.t/ D g.t; zt C xt / T0 .t/g.0; .0// Z t X C T1 .t s/v.s/ds C T1 .t tk /Ik : 0

0
Then

jz.t/j ˛1 kzt C xt k C ˛2 C M ˛1 kkD C ˛2 Z t m X p.s/ .kzs C xs k/ds C M ck jz.tk / C x.tk /j CM 0

CM

kDm X kD0

kD0

kIk .0/k

288

10 Impulsive Functional Differential Inclusions with Unbounded Delay

˛1 Kb jz.t/j C .MKb C Mb /kkD C ˛2 C M ˛1 kkD C ˛2 Z t

CM p.s/ Kb jz.s/j C .MKb C Mb /kkD /ds 0

CM

m X

ck jz.t/j C M

kD0

m X

ck jx.t/j C M

kD0

kDm X

kIk .0/k

kD0

˛1 Kb jz.t/j C .MKb C Mb /kkD C ˛2 C M ˛1 kkD C ˛2 Z t m X

CM p.s/ Kb jz.s/j C .MKb C Mb /kkD ds C M ck jz.tj/ 0

CM

m X

kD0

ck jx.t/j C M

kD0

kDm X

kIk .0/k:

kD0

Then, we have: jz.t/j ˛1 ..Kb jz.t/j C .MKb C Mb /kkD // Z t m X

p.s/ Kb jz.s/j C .MKb C Mb /kkD ds C M ck jz.tj/ CM 0

CM 2

m X

kD0

ck kkD C M

kD0

kDm X

kIk .0/k:

kD0

Thus m X

ck kzkD0 ˛1 C .MKb C Mb /kkD C ˛2 C M.˛1 kkD C ˛2 / 1 ˛1 Kb M b

kD0

CMkpkL1 CM 2

m X

Kb kforzkD0 C .MKb C Mb /kkD b

ck kkD C M

kD0

D C0 C MkpkL1

kDm X kD0

kIk .0/k

Kb kzkD0 C .MKb C Mb /kkD b

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions

289

By the previous inequality we have, 1 ˛1 Kb M

m X

! ck kzkD0 b

kD0

C0 C MkpkL1

1 Kb kzkD0 C .MKb C Mb /kkD

(10.14)

b

From (10.13) it follows that there exists a constant R > 0 such that for each z 2 E with kzkD0 > R the propriety is not satisfied. Hence kzkD0 R for each z 2 E which b b means that E is bounded. As a consequence of Theorem 1.32, A C B has a fixed point z on the interval .1; b, so y D z C x is a fixed point of the operator N, which is the mild solution of problem (10.9)–(10.11). t u

10.3.2 Extremal Mild Solutions In this section we shall prove the existence of maximal and minimal solutions of problem (10.9)–(10.11) under suitable monotonicity conditions on the multi-valued functions involved in it. We need the following definitions in the sequel. Definition 10.8. We say that a continuous function vQ 2 Db is a lower mild solution of problem (10.9)–(10.11) if v.t/ Q D .t/; t 2 .1; 0; and there exist v./ 2 L1 .Jk ; E/ and Ik 2 Ik .v.t Q k //, such that v.t/ 2 F.t; vQ t / a.e t 2 Œ0; b, and vQ satisfies, Z v.t/ Q T0 .t/ ..0/ g.0; .0/// C g.t; vQ t / C X T1 .t tk /Ik ; t 2 J; t ¤ tk ; C

0

t

T1 .t s/v.s/ds

0
and v.t Q kC / v.t Q k Ik where Ik 2 Ik .v.t Q k //; t D tk ; k D 1; : : : ; m Similarly an upper mild solution wQ of IVP (10.9)–(10.11) is defined by reversing the order. Definition 10.9. A solution xM of IVP (10.9)–(10.11) is said to be maximal if for any other solution x of IVP (10.9)–(10.11) on J, we have that x.t/ xM .t/ for each t 2 J. Similarly a minimal solution of IVP (10.9)–(10.11) is defined by reversing the order of the inequalities. We consider the following assumptions in the sequel. (10.9.1) The multi-valued function F.t; y/ is strictly monotone increasing in y for almost each t 2 J. (10.9.2) The IVP (10.9)–(10.11) has a lower mild solution vQ and an upper mild solution wQ with vQ w. Q (10.9.3) T1 .t/ is preserving the order, that is T1 .t/v 0 whenever v 0. (10.9.4) The functions Ik ; k D 1; : : : ; m are continuous and nondecreasing.

290

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Theorem 10.10. Assume that assumptions (10.6.1)–(10.6.6) and (10.9.1)–(10.9.4) hold. If 2 D, .0/ 2 X0 and g.0; .0// 2 X0 , then IVP (10.9)–(10.11) has minimal and maximal solutions on Db . Proof. We can write vQ and w Q as v.t/ Q D v .t/ C x.t/ w.t/ Q D w .t/ C x.t/; where v 2 D0b and w 2 D0b and x.t/ is defined in the above section. Then vQ is a lower solution to IVP (10.9)–(10.11) if v satisfies v .t/ T0 .t/g.0; .0// C g.t; vt C x.t// C X T1 .t tk /Ik ; t 2 J; t ¤ tk ; C

Z

t 0

T1 .t s/v.s/ds

0
and v .tkC / v .tk Ik such that Ik 2 Ik .v .tk //; t D tk ; k D 1; : : : ; m. Respectively w Q is upper solution to IVP (10.9)–(10.11) if w satisfies the reversed inequality. It can be shown, as in the proof of Theorem 10.2, that A is completely continuous and B is a contraction on Œv ; w . We shall show that A and B are isotone increasing on Œv ; w . Let z; z 2 Œv ; w be such that z z; z 6D z: Then by (10.9.4), we have for each t 2 J ( h 2 D0b W h.t/ D T0 .t/g.0; .0// C g.t; zt C xt /

A.z/ D

C

) T1 .t tk /Ik ; 2

Ik z.tk /

0
(

X

h 2 D0b W h.t/ D T0 .t/g.0; .0// C g.t; zt C xt / C

X 0
D A.z/:

) T1 .t tk /Ik ; Ik 2 Ik .z.tk //

10.3 Non-densely Defined Impulsive Neutral Functional Differential Inclusions

291

Similarly, by .10:9:1/; .10:9:3/ ( B.z/ D

h2

D0b

Z W h.t/ D 0

(

h2

D0b

t

Z

t

W h.t/ D 0

) T1 .t s/v.s/ds; v 2 SF;z ) T1 .t s/v.s/ds; f 2 SF;z

D B.z/: Therefore A and B are isotone increasing on Œv ; w . Finally, let y 2 Œv ; w be any element. By (10.9.2), (10.9.3) we deduce that v A.v / C B.v / A.y/ C B.y/ A.w / C B.w / w ; which shows that A.y/ C B.y/ 2 Œv ; w for all y 2 Œv ; w . Thus, A and B satisfy all the conditions of Theorem 10.7, hence IVP (10.9)–(10.11) has maximal and minimal solutions on J: t u

10.3.3 Example To apply our previous results, we consider the following impulsive partial neutral functional differential equation

Z 0 @ v.t; / K1 .; v.t C /; /d @t 1

Z 0 2 @ K1 .; v.t C /; /d D 2 v.t; / @t 1 Z 0 K2 . /ŒQ1 .t; .; //; Q2 .t; .; //dI C 1

t 2 J D Œ0; b; t ¤ tk ; k D 1; : : : ; m; 0 1; N 1/; 2 Œ0; 1; k D 1; : : : ; m v.tkC ; / v.tk ; / 2 bk jv.tk ; /jB.0; Z v.t; 0/

1

Z v.t; 1/

0

0

1

(10.15) (10.16)

K1 .; v.t C /; 0/d D 0; t 2 J;

(10.17)

K1 .; v.t C /; 1/d D 0; t 2 J;

(10.18)

v.; / D v0 .; / 1 < 0; 0 1;

(10.19)

292

10 Impulsive Functional Differential Inclusions with Unbounded Delay

where bk > 0; k D 1; : : : ; m; K1 W .1; 0 R ! R; K2 W .1; 0 ! R; Q1 ; Q2 W J R ! R and v0 W .1; 0 Œ0; 1 ! R are continuous functions, N 1/ the v.tk / D lim v.tk C h; /; v.tkC / D lim v.tk C h; / and B.0; .h;/!.0 ;/

.h;/!.0C ;/

closed unit ball. We assume that for each t 2 J; Q1 .t; / is lower semi-continuous (i.e., the set fy 2 R W Q1 .t; y/ > g is open for each 2 R), and assume that for each t 2 J; Q2 .t; / is upper semi-continuous (i.e., the set fy 2 R W Q2 .t; y/ < g is open for each 2 R). We choose E D C.Œ0; 1; R/ endowed with the uniform topology and consider the operator A W D.A/ E ! E defined by: D.A/ D fy 2 C2 .Œ0; 1; R/ W y.0/ D y.1/ D 0g

Ay D y00 :

It is well known (see [100]) that the operator A satisfies the Hille–Yosida condition with .0; C1/ .A/; k.I A/1 k 1 for > 0, and X0 D D.A/ D fy 2 E W y.0/ D y.1/ D 0g ¤ E: So the extrapolation method can be applied. We define: N 1/; 2 Œ0; 1; k D 1; : : : ; m Ik .y.tk //./ D bk jv.tk ; /jB.0; Z 0 K2 . /ŒQ1 .t; .; //; Q2 .t; .; //d; t 2 J; 2 Œ0; 1; F.t; /./ D 1

g.t; /./ D

R0

1

K1 .; . /.//d; t 2 J; 2 Œ0; 1;

y.t/./ D v.t; /; t 2 J; 2 Œ0; 1; . /./ D v0 .; /; 0; 2 Œ0; 1: Then problem (10.1)–(10.3) is an abstract formulation of the problem (10.15)– (10.19) with F compact and convex values, and it is upper semi continuous (see [101]). Assume that there are p 2 C.J; RC / and W Œ0; 1/ ! .0; 1/ continuous and nondecreasing such that max.jQ1 .t; y/j; jQ2 .t; y/j/ p.t/ .jyj/;

t 2 J; and y 2 R:

Under suitable conditions, the problem (10.15)–(10.19) has by Theorem 10.7 a solution on .1; b Œ0; 1:

10.4 Controllability of Impulsive Semi-linear Differential Inclusions in. . .

293

10.4 Controllability of Impulsive Semi-linear Differential Inclusions in Fréchet Spaces In this section, we use the extrapolation method combined with a recent nonlinear alternative of Leray-Schauder type for multi-valued admissible contractions in Fréchet spaces to study the existence of the mild solution for a class of non-densely defined first order semi-linear impulsive functional differential inclusions with finite delay in the semi-infinite interval J WD Œ0; 1/; and with single valued jump. More precisely we consider the first order semi-linear impulsive functional differential inclusions of the form: y0 .t/ Ay.t/ 2 F.t; yt / C Bu.t/; a:e: t 2 Jnft1 ; t2 ; : : :g

(10.20)

yjtDtk D Ik .y.tk //; k D 1; : : : ;

(10.21)

y.t/ D .t/; t 2 H;

(10.22)

where J WD Œ0; 1/, F W J D ! P.E/ is a multi-valued map with compact values, .E; j j/, D, P.E/,B, u./, , yt are as in the above section and A W D.A/ E ! E is a non-densely defined closed linear operator on E Let A0 the dense part of A, and let .T0 .t//t0 the strongly continuous semigroup generated by A0 defined on X0 D D.A/and let .T1 .t//t0 the extrapolated semigroup of .T0 .t//t0 whose generator is .A1 ; D.A1 //:

10.4.1 Main Result We shall consider the space n PC D y W Œr; 1/ ! E W tk

y.t/ is continuous everywhere except for some o at wich y.tk /; y.tkC / exist with y.tk / D y.tk /; k D 1; : : :

Set ˝ D fy W Œr; 1/ ! E W y 2 PC \ Dg Definition 10.11. We say that the function y 2 ˝ is a mild solution of system (10.20)–(10.22) if y.t/ D .t/ for all t 2 Œr; 0, the restriction of y./ to the interval 1 .Œ0; 1/; E/, such that v.t/ 2 F.t; yt / Œ0; 1/ is continuous and there exists v./ 2 Lloc a.e Œ0; 1/, and such that y satisfies the integral equation, Z

t

Z

t

y.t/ D T0 .t/.0/ C T1 .t s/v.s/ds C T1 .t s/Buy .s/ds 0 0 X C T1 .t tk /Ik .y.tk //; 0 t < 1: 0
(10.23)

294

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Definition 10.12. The system (10.20)–(10.22) is said to be infinite controllable on the interval Œr; 1/nftk g; k D 1; : : : if for every initial function 2 D and every y1 2 E; and for each n 2 N, there exists a control u 2 L2 .Œ0; tn ; U/; such that the mild solution y.t/ of (10.20)–(10.22) satisfies y.tn / D y1 : Let us introduce the following hypotheses: (10.12.1) The function F W J ˝ ! Pcp .E/ is an L1 -Carathéodory map. (10.12.2) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! Œ0; 1/ such that kF.t; x/k p.t/ .kxkD / for a.e. t 2 J and each x 2 D; with Z

1

1

ds D 1: s C .s/

(10.12.3) There exists M > 0 such that kT1 .t/kB.E/ M for each t > 0: 1 (10.12.4) For all R > 0 there exists lR 2 Lloc .Œ0; 1/; RC / such that

Hd .F.t; x/; F.t; x// lR .t/kx xkD for all x; x 2 D with kxk; kxk R; and d.0; F.t; 0// lR .t/ for a.e. t 2 J: (10.12.5) There exist constants ck 0, k D 1; : : : ; such that jIk .y/ Ik .x/j ck jx xj for each x; x 2 E: (10.12.6) For every n > 0, the linear operator W W L2 .Jn ; U/ ! E .Jn D Œ0; tn /; defined by Z Wu D 0

tn

T.tn s/Bu.s/ds;

has a bounded inverse operator W 1 which takes values in L2 .Jn ; U/n KerW, and there exist positive constants M; M 1 such that kBk M and kW 1 k M 1 : Theorem 10.13. Assume that hypotheses (10.12.1)–(10.12.6) hold. If M then the IVP (10.20)–(10.22) is infinite controllable on Œr; 1/:

1 X kD1

ck < 1,

10.4 Controllability of Impulsive Semi-linear Differential Inclusions in. . .

295

Proof. Using hypothesis (10.12.6) for each y./ define the control 2 uy .t/ D W

1

4y1 T.tn /.0/

X

Z

tn 0

T.tn s/v.s/ds

3 T.s tk /Ik .y.tk //5 .t/;

0
where v 2 SF;y D fv 2 L1 .J; E/ W v.t/ 2 F.t; yt / a:e t 2 Jg; We shall now show that when using this control, the operator N W ˝ ! P.˝/ defined by 8 8 .t/; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z t ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ T .t/.0/ C T1 .t s/v.s/ds 0 ˆ ˆ < < 0 Z t N.y/ D h 2 ˝ W h.t/ D ˆ ˆ ˆ ˆ C T1 .t s/.Buy /.s/ds ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ X ˆ ˆ ˆ ˆ ˆ ˆ T1 .t tk /Ik .y.tk //; ˆ ˆ :C : 0
9 if t 2 H, > > > > > > > > > > = > > > > > > > > if t 2 J; > > ;

has a fixed point. This fixed point is then the mild solution of the IVP (10.20)– (10.22) We define on ˝ a family of semi-norms, thus rendering ˝ into Fréchet space. Let be sufficiently large, then 8n 2 N we define in ˝ the semi-norm: ˚ kykn D sup e Ln .t/ jy.t/j W r t tn ; where Z Ln D

t r

Oln .s/ds;

with

Oln .t/ D

8 ˆ ˆ 0; < 2 ˆ 2 ˆ : ln .t/ M M M1 tn C M C M M M 1

if t 2 H, t X kD0

ck ;

if t 2 Œ0; tn ;

296

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Thus ˝ D

[

˝n where

n1

n ˝n D y W Œr; tn ! E W

y2D

\

PCn .J; E/

o

and

n PCn D y W Œ0; tn ! E W

y.t/ is continuous everywhere except for some o at wich y.tk /; y.tkC / exist with y.tk / D y.tk /; k D 1; : : : n 1

tk

Then ˝ is a Fréchet space with the family of the semi-norms fk:kn gn2IN . Now, using the Frigon alternative, we are able to prove that the operator N has a fixed point. Let y 2 N.y/ for some 2 Œ0:1, and for some v 2 SF;y . For each n 2 IN and t 2 Œ0; tn we have: Z t Z t y.t/ D T0 .t/.0/ C T1 .t s/v.s/ds C T1 .t s/Buy .s/ds 0 0 3 X C T1 .t tk /Ik .y.tk //5 0
then, we have Z

Z

t

jy.t/j kT0 kkkD C jT1 .t s/jkv.s/kds C 0 X C jT1 .t tk /jjIk .y.tk //j 0
Z

MkkD C M CM

n X

t 0

Z MkkD C M tn

0

CM

t 0

kBkkuy .s/kds

0

X 0
kD1

t

jT1 .t s/jjBuy .s/jds

2 Z tˇ ˇ 1 4 y1 T0 .tn /.0/ p.s/ .kys k/ds C MM ˇW

T1 .tn /v. /d

n X

0

0

jIk .y.tk //j

kD1

Z

Z p.s/ .kys k/ds C M

t

jIk .y.tk //j

3

ˇ ˇ T1 . tk /Ik .y.tk //5 .s/ˇds

10.4 Controllability of Impulsive Semi-linear Differential Inclusions in. . .

Z MkkD C M Z

tn

CM 0

CM

n X

0

p.s/ .kys k/ds C MM M1 X

t

0

4jy1 j C kT0 .tn /kk.0/k 3

jIk .y.tk //j 5 ds

0
jIk .y.tk //j Z

MkkD C M CM 2 M M1 tn n X

2

Z

p. / .ky k/d C M

kD1

CM

t

297

Z

t 0 t

0

p.s/ .kys k/ds C MM M1 tn jy1 j C MM M1 tn MkkD p. / .ky k/d C M 2 M M 1

Z

t

X

0 0
jIk .y.tk //jds

jIk .y.tk //j:

kD1

It follows that i h jy.t/j MM M1 tn jy1 j C M C MM M 1 tn kkD iZ t h C M C M 2 M M 1 tn p.s/ .kys k/ds CM 2 M M 1

Z

t

X

0 0
0

jIk .y.tk //jds C M

n X

jIk .y.tk //j

kD1

i h MM M 1 tn jy1 j C M C MM M 1 tn kkD iZ t h 2 C M C M M M 1 tn p.s/ .kys k/ds CM 2 M M 1 CM

Z

0

t

X

jIk .y.tk // Ik .0/j C jIk .0/j ds

0 0
n X

jIk .y.tk // Ik .0/j C jIk .0/j kD1

i h MM M 1 tn jy1 j C M C MM M 1 tn kkD iZ t h 2 C M C M M M 1 tn p.s/ .kys k/ds CM 2 M M 1

Z

0

t

X

0 0
jIk .y.tk // Ik .0/jds

298

10 Impulsive Functional Differential Inclusions with Unbounded Delay

CM

n X

jIk .y.tk // Ik .0/j C M 2 M M 1

Z

kD1

CM

n X

t

X

0 0
jIk .0/jds

kIk .0/k

kD1

i h MM M 1 tn jy1 j C M C MMtn M 1 kkD X C M C M 2 M M 1 tn kIk .0/k kD1n

h C M C M 2 Mtn M1 CM

n X

iZ 0

t

2

p.s/ .kys k/ds C M M M1

Z tX n 0 kD1

ck jy.tk /jds

ck jy.tk /j:

kD1

Set n i iX h h C D MM M1 tn jy1 j C M C MMtn M1 kkD C M C M 2 Mtn M1 kIk .0/k: kD1

Now, we consider the function defined by: n o .t/ D sup jy.s/j W r s t ; t tn Let t 2 Œr; tn such that .t/ D jy.t j It is clear that: if t 2 H, then .t/ D kkD if t 2 Œ0; tn , we have for each t 2 Œ0; tn h iZ t .t/ C C M C M 2 Mtn M1 p.s/ ..s //ds 2

Z

CM M M1

0

t

X

0 0
ck .s/ds C M

n X

ck .t/:

kD1

Then n h i h iZ t X 1M ck .t/ C C M C M 2 Mtn M1 p.s/ ..s//ds 0

kD1

CM 2 M M1

Z

t

X

0 0
ck .s/ds:

10.4 Controllability of Impulsive Semi-linear Differential Inclusions in. . .

Thus we have Z .t/ C1 C

t 0

O M.s/ .s/ C

..s// ;

where C1 D

C ; n X 1M ck kD1

and 2 3 h i X 1 O 4 M C M 2 M M 1 tn p.s/ C M 2 M M 1 M.s/ D ck 5 : n X 0
Let us take the right-hand side of the above inequality as v.t/, then we have: v.0/ D C1 ;

.t/ v.t/

8t 2 Œ0; tn

and 0

O C v .t/ D M.t/..t/ Using the nondecreasing character of

..t///:

, we get:

0

O C v .t/ M.t/.v.t/

.v.t///a:et 2 Œ0; tn :

This implies that for each t 2 Œ0; tn Z

v.t/

v.0/

ds s C .s/

Z

tn

O M.s/ds

Z

1 v.0/

0

ds : s C .s/

Thus from .10:12:2/ there exists a constant Mn such that v.t/ Mn ;

8t 2 Œ0; tn

From the definition of , we conclude that n sup jy.t/j;

o t 2 Œ0; tn Mn

299

300

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Set

n U0 D y 2 ˝n ;

o kykn Mn C 1

Clearly U0 is a closed subset of ˝n . We shall show that N W U0 ! P.U0 / is a contraction and an admissible operator. First, we prove that N is a contraction; that is, there exists < 1; such that Hd .N.y/; N.y // ky y kn ; for each y; y 2 U0 : Let y; y 2 U0 and h 2 N.y/. Then there exists v.t/ 2 F.t; yt / such that for each t 2 Œ0; tn Z t Z t h.t/ D T0 .t/.0/ C T1 .t s/v.s/ds C T1 .t s/Buy .s/ds 0 0 X C T1 .t tk /Ik .y.tk /: 0
From (10.12.4) it follows that Hd .F.t; yt /; F.t; yt // ln .t/kyt yt kD : Hence there exists v 2 F.t; yt / such that jv.t/ v .t/j ln.t/kyt yt kD ;

8t 2 Œ0; tn :

Let us define 8t 2 Œ0; tn Z

h .t/ D T0 .t/.0/ C C

X

t

0

Z

T1 .t s/v .s/ds C

0

t

T1 .t s/Buy .s/ds

T1 .t tk /Ik .y .tk //:

0
Then we have Z t ˇZ t ˇ T1 .t s/ŒB.uy uy /.s/ds jh.t/ h .t/j D ˇ T1 .t s/Œv.s/ v .s/ds C

0

C

X

0
0

T1 .t

tk /ŒIk .y.tk /

ˇ ˇ Ik .y .tk //ˇ

10.4 Controllability of Impulsive Semi-linear Differential Inclusions in. . .

Z

t

M 0

X

CM Z

ln .s/kys

ys kD ds

t

C MM 0

juy .s/ uy .s/jds

ck jy.tk / y .tk /j

0
M 0

ln .s/kys ys kD ds

Z tˇ hZ ˇ 1 CMM ˇW C

Z

X

0

T1 .t / v. / v . / d

tn

0

iˇ ˇ T1 .t /Ik .y.tk / y .tk // ˇds

0
X

CM Z

ck jy.tk / y .tk /j

0
M 0

ln .s/kys ys kD ds

CM 2 M M 1 tn CM M M 1 X

t

0

Z

2

C

Z

ln .s/kys ys kD ds

X

t

0 0
ck jy.tk / y .tk /jds

ck jy.tk / y .tk /jds

0
iZ t h 2 M C M M M 1 tn ln .s/kys ys kD ds CM 2 M M 1 C

X

0

Z

X

t

0 0
ck jy.tk / y .tk /jds

ck jy.tk / y .tk /jds

0
iZ t h M C M 2 M M 1 tn ln .s/e Ln .s/ ky y kn ds CM 2 M M 1 CMe Ln .t/

0

Z

t

e Ln .s/

0

X 0
X

ck ky y kn ds

0
ck ky y kn ds;

301

302

10 Impulsive Functional Differential Inclusions with Unbounded Delay

which gives: 2 jh.t/ h .t/j 43

Z

t 0

Oln .s/e Ln .s/ ds C Me Ln .t/

"

#

X

3 ck 5 ky y kn

0
n X 3 Ln .s/ t e j0 C Me Ln .t/ ck ky y kn

kD1 " # n X 3 Ln .t/ 3

Ln .t/ e C Me ck ky y kn :

kD1

As is sufficiently large, thus "

jh.t/ h .t/j

# n X 3 CM ck e Ln .t/ ky y kn :

kD1

Then, it follows "

Ln .t/

jh.t/ h .t/je

# n X 3 CM ck ky y kn :

kD1

Therefore, "

kh h kn

# n X 3 CM ck ky y kn :

kD1

By an analogous relation, obtained by interchanging the roles of y and y ; it follows that ! n X 3 Hd .N.y/; N.y // CM ck ky y kn :

kD1 So, N is a contraction. Now, N W ˝n ! Pcp .˝n / is given by 8 8 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z ˆ ˆ t ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < 0 T1 .t s/v.s/ds < Z t N.y/ D h 2 ˝n W h.t/ D ˆ ˆ ˆ ˆ C T1 .t s/.Buny /.s/ds ˆ ˆ ˆ ˆ ˆ ˆ 0 ˆ ˆ X ˆ ˆ ˆ ˆ ˆ ˆ T1 .t tk /Ik .y.tk //; ˆ ˆ :C : 0
if t 2 H,

9 > > > > > > > > > > =

> > > > > > > > if t 2 Œ0; tn ; > > ;

10.4 Controllability of Impulsive Semi-linear Differential Inclusions in. . .

303

n where v 2 SF;y D fu 2 L1 .Œ0; tn ; E/ W u 2 F.t; yt / a:e: t 2 Œ0; tn g: From (10.12.4)– (10.12.6) and since F is compact valued, we can prove that for every y 2 ˝n , N.y/ 2 Pcp .˝n /, and there exists y 2 ˝n such that y 2 N.y /: Let h 2 ˝n ; y 2 U0 and " > 0. Now, if yN 2 N.y /; then we have

ky yN kn ky hkn C kNy hkn : Since h is arbitrary we may suppose that h 2 B.Ny; "/ D fk 2 ˝n W kk yN kn "g: Therefore, ky yN kn ky N.y /kn C ": On the other hand, if yN 62 N.y /; then kNy N.y /k 6D 0: Since N.y / is compact, there exists x 2 N.y / such that kNy N.y /kn D kNy xkn : Then we have ky xkn ky hkn C kx hkn : Therefore, ky xkn ky N.y /kn C ": So, N is an admissible operator contraction. Finally, by Lemma 1.27, N has at least one fixed point, y; which is a mild solution to (10.20)–(10.22). t u

10.4.2 Example As an application of our above result, we consider the following impulsive partial functional inclusion, @z.t; x/ d4z.t; x/ 2 F.t r; x/ C Bu.t/; a:e: t 2 Jnft1 ; t2 ; : : :g; x 2 ˝ @t

(10.24)

bk z.tk ; x/ D z.tkC / z.tk /; k D 1; : : : ; x 2 @˝

(10.25)

z.t; x/ D 0; t 2 Œ0; 1/nft1 ; t2 ; : : :g; x 2 ˝

(10.26)

z.t; x/ D .t; x/; t 2 H; x 2 ˝

(10.27)

where d,r,bk are positive constants, ˝ is a bounded open in IRn with regular n X @2 , 2 D D f W H ˝ ! IRI is continuous boundary @˝, 4 D @xi2 iD1

everywhere except for a countable number of points at which .s /; .sC / exist with .s / D .s/, and j .; x/j < 1g, 0 D t0 < t1 < t2 < < tm < , z.tkC / D lim z.tk C h; x/ ,z.tk / D lim z.tk h; x/, F W Œ0; 1/ IRn ! .h;x/!.0C ;x/

.h;x/!.0 ;x/

P.IRn / is a multi-valued map with compact values.

304

10 Impulsive Functional Differential Inclusions with Unbounded Delay

Consider E D C.˝; IRn / the Banach space of continuous functions on ˝ with values in IRn , y.t/ D z.t; :/. Let A the operator defined in E by Ay D d4y; Ik W E ! D.A/ such that Ik .y.tk // D bk y.tk /, then the problem (10.24)–(10.27) can be written as y0 .t/ Ay.t/ 2 F.t; yt / C Bu.t/

a:e: t 2 Jnft1 ; t2 ; : : :g;

(10.28)

yjtDtk D Ik .y.tk //; k 2 f1; 2; : : :g

(10.29)

y.t/ D .t/ t 2 H

(10.30)

We have, D.A/ D fy W

y 2 E;

4y 2 E

and

yj@˝ D 0g;

and X0 D D.A/ D fy W

y 2 E;

yj@˝ D 0g ¤ E

So, we can apply the extrapolation method. It is well known from [100] that 4 satisfies the properties: i) .0; 1/ .4/ ii) kR.; 4/k 1 for some > 0 It follows that 4 satisfies .Hy/. Also from [106], the family T0 .t/f .s/ D .4/

n 2

Z e IRn

js j2 4t

f . /d

for t > 0; s 2 Rn , and f 2 X0 with T.0/ D I, define a strongly continuous semigroup on E, its generator A0 coincides with the closure of the Laplacian operator with domain X0 , and there exist constants N0 > 0,! > 0 such that kT0 k N0 e!t for t > 0. Thus under appropriate conditions on the function F and the operator B as those mentioned in hypotheses (10.12.1)–(10.12.6) the problem (10.24)–(10.27) has at least one mild solution.

10.5 Notes and Remarks The results of Chap. 10 are taken from Abada et al. [1, 3]. Other results may be found in [54, 74, 107].

Chapter 11

Functional Differential Inclusions with Multi-valued Jumps

11.1 Introduction In this chapter, we are concerned by the existence of mild solutions of functional differential inclusions with delay and multi-valued jumps in a Banach space.

11.2 Semi-linear Functional Differential Inclusions with State-Dependent Delay and Multi-valued Jump 11.2.1 Introduction In this section, we shall be concerned with the existence of integral solutions defined on a compact real interval for first order impulsive semi-linear functional inclusions with state-dependent delay in a separable Banach space of the form: y0 .t/ 2 Ay.t/ C F.t; y .t;yt / /; y.ti / 2 Ik .ytk /;

t 2 I D Œ0; b;

k D 1; 2; : : : ; m;

y.t/ D ; t 2 .1; 0;

(11.1) (11.2) (11.3)

where F W J D ! E is a given multi-valued function, D D f W .1; 0 ! E; is continuous everywhere except for a finite number of points s at which C .s / ; s exist and .s / D .s/g, 2 D,where D is the phase space that will be specified later .0 < r < 1/, 0 D t0 < t1 < < tm < tmC1 D b,

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_11

305

306

11 Functional Differential Inclusions with Multi-valued Jumps

Ik W D ! E .k D 1; 2; : : : ; m/, W I D ! .1; b, A W D.A/ E ! E is a non-densely defined closed linear operator on E, and E a real separable Banach space with norm j:j:

11.2.2 Existence of Integral Solutions In this section, we will employ an axiomatic definition for the phase space D which is similar to those introduced in [142]. Specifically, D will be a linear space of functions mapping .1; 0 into E endowed with a semi norm k:kD , and satisfies the following axioms introduced at first by Hale and Kato in [132]: (A1) There exist a positive constant H and functions K./, M./ W RC ! RC with K continuous and M locally bounded, such that for any b > 0, if y W .1; b ! E, y 2 D, and y./ is continuous on Œ0; b, then for every t 2 Œ0; b the following conditions hold: (i) yt is in DI (ii) jy.t/j Hkyt kD I (iii) kyt kD K.t/ supfjy.s/j W 0 s tg C M.t/ky0 kD ; and H; K and M are independent of y./: Denote Kb D supfK.t/ W t 2 Jg and Mb D supfM.t/ W t 2 Jg: (A) The space D is complete.

11.2.3 Main Results Before starting and proving our main theorem for the initial value problem (11.1)– (11.3), we give the definition of the integral solution. Definition 11.1. We say that y W .1; b ! E is an integral solution of (11.1)– (11.3) if y.t/ D .t/ for all t 2 .1; 0, the restriction of y./ to the interval Œ0; b is continuous, and there exist v./ 2 L1 .Jk ; E/ and Ik 2 Ik .y.tk //; such that v.t/ 2 F.t; y .t;yt / / a.e t 2 Œ0; b, and y satisfies the integral equation, Z t Z t X (i) y.t/ D .0/ C A y.s/ds C v.s/ds C S0 .t tk /Ik ; t 2 J: Z (ii) 0

0

t

y.s/ds 2 D.A/ for t 2 J,

0

0
11.2 Semi-linear Functional Differential Inclusions with State-Dependent. . .

307

From the definition it follows that y.t/ 2 D.A/, for each t 0; in particular .0/ 2 D.A/. Moreover, y satisfies the following variation of constants formula: d y.t/ D S .t/.0/ C dt 0

Z

t

S.t s/v.s/ds C

0

X

S0 .t tk / Ik t 0:

(11.4)

0
We notice also that if y satisfies (11.4), then Z

0

y.t/ D S .t/.0/ C lim

t

!1 0

S0 .t s/B v.s/ds C

X

S0 .t tk / Ik ; t 0:

0
we always assume that W I D ! .1; b is continuous. Additionally, we introduce following hypotheses: (H') The function t ! 't is continuous from R. / D f .s; '/ W .s; '/ 2 J D; .s; '/ 0g into D and there exists a continuous and bounded function L W R. / ! .0; 1/ such that kt kD L .t/kkD for every t 2 R. /. (11.1.1) A satisfies Hille–Yosida condition; (11.1.2) There exist constants ck 0, k D 1; : : : ; m such that Hd .Ik .y/; Ik .x//j ck jy xj for each x; y 2 D: (11.1.3) The valued multi-valued map F W J D ! E is convex and Carathéodory; (11.1.4) the operator S0 .t/ is compact in D.A/ wherever t > 0I (11.1.5) There exist a function p 2 L1 .J; RC / and a continuous nondecreasing function W Œ0; 1/ ! .0; 1/ such that kF.t; x/kP D supfjvj W v 2 F.t; x/g p.t/ .kxkD / for a.e. t 2 J and each x 2 D;

Z

b

with 0

e!s p.s/ds < 1, .Mb C L C MKb /kkD C Kb u

lim sup u!C1

c1 C c2

Rt

e!s p.s/

0

.Kb u C .Mb C

L

> 1;

C MKb /kkD / ds (11.5)

where c1 D

ce!b Kb 1 Me!b Kb

m P kD1

C Mb C L C MKb kkD ; ck

(11.6)

308

11 Functional Differential Inclusions with Multi-valued Jumps

and cD

m

P jIk .0/j C ck Mb C L C MKb kkD :

(11.7)

kD1

c2 D

MKb e!b : m P 1Me!b Kb ck

(11.8)

kD1

The next result is a consequence of the phase space axioms. Lemma 11.2 ([139], Lemma 2.1). If y W .1; b ! E is a function such that y0 D and yjJ 2 PC.J W D.A//, then kys kD .Ma C L /kkD C Ka supfky. /kI 2 Œ0; maxf0; sgg;

s 2 R. / [ J;

where L D supt2R. / L .t/, Ma D supt2J M.t/ and Ka D supt2J K.t/. Theorem 11.3. Assume that (H') and (11.1.1)–(11.1.5) hold. If Me!b Kb

m X

ck < 1;

(11.9)

kD1

then the problem (11.1)–(11.3) has at least one integral solution on .1; b: Proof. Consider the multi-valued operator: N W PC .1; b; D.A/ ! P.PC .1; b; D.A/ / defined by 8 8 .t/; if t 0; ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Z t ˆ ˆ ˆ ˆ ˆ ˆ < < S0 .t/.0/ C d S.t s/v.s/ds

dt 0 N.y/ D h 2 PC .1; b; D.A/ W h.t/ D X ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ S0 .t tk /Ik ; v 2 SF;y .s;ys / ; Ik 2 Ik .y.tk // C ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 0
For 2 D define the function Q W .1; b ! E such that: Q D t

8 < .t/;

if t 0

: S0 .t/.0/;

if t 2 J:

9 > > > > > > > > > = > > > > > > > > > ;

11.2 Semi-linear Functional Differential Inclusions with State-Dependent. . .

309

Then Q 0 D . For each x 2 Bb with x.0/ D 0, we denote by x the function defined by x.t/ D

8 < 0;

t 2 .1; 0;

: x.t/; t 2 J;

Q C x.t/; 0 t b, which implies yt D Q t C xt , We can decompose it as y.t/ D .t/ for every 0 t b and the function x.:/ satisfies x.t/ D

d dt C

Z

t

S.t s/v.s/ds

0

X

S0 .t tk /Ik t 2 J;

0
where: v.s/ 2 SF;x

Q

.s;xs CQs / C .s;xs CQs /

and Ik 2 Ik xtk C Q tk Let

Bb0 D fx 2 Bb W x0 D 0 2 Dg: For any x 2 Bb0 we have kxkb D kx0 kD C supfjx.s/j W 0 s bg D supfjx.s/j W 0 s bg: Thus .Bb0 ; k kb / is a Banach space. We define the two multi-valued operators A; B W Bb0 ! P.Bb0 / by: 8 ˆ ˆ <

8 ˆ 0; ˆ < 0 X A.x/ WD h 2 Bb W h.t/ D ˆ ˆ S0 .t tk /Ik ; ˆ ˆ : :

Ik 2 Ik xtk C Qtk ;

9 > if t 2 .1; 0; > = if t 2 J;

0
> > ;

and 8 8 ˆ ˆ 0; if ˆ ˆ < < 0 Z B.x/ WD h 2 Bb W h.t/ D d t ˆ ˆ ˆ ˆ : : S.t s/v.s/ds; v.s/ 2 SF;x if Q .s;xs CQs / C .s;xs CQs / dt 0

9 t 2 .1; 0I > > = t 2 J:

> > ;

Obviously to prove that the multi-valued operator N has a fixed point is reduced that the operator inclusion x 2 A.x/ C B.x/ has one, so it turns to show that the multi-valued operators A and B satisfy all conditions of Theorem 1.32. For better readability, we break the proof into a sequence of steps.

310

11 Functional Differential Inclusions with Multi-valued Jumps

Step 1: A is a contraction. Let x1 ; x2 2 Bb0 . Then for t 2 J 0 Hd .A.x1 /; A.x2 // D Hd @

X

S0 .t tk /Ik .xt1k C Q tk /;

0
X 0
Me!b

1

S0 .t tk / Ik .xt2k C Q tk /A X ˇ ˇ ˇIk .x1 / Ik .x2 /ˇ tk

tk

0tk t !b

Me

m X

ck kxt1k xt2k kD

kD1

Me!b Kb

m X

ck kx1 x2 kD :

kD1

Hence by (11.9) A is a contraction. Step 2: B has compact, convex values, and it is completely continuous. This will be given in several claims. Claim 1: B has compact values. The operator B is equivalent to the composition L ı SF on L1 .J; E/, where L W L1 .J; E/ ! Bb0 is the continuous operator defined by L.v/.t/ D

d dt

Z

t

S.t s/v.s/ds; t 2 J:

0

Then, it suffices to show that L ı SF has compact values on Bb0 . Let x 2 Bb0 arbitrary and vn a sequence such that vn .t/ 2 SF;x , a.e. Q .t;xt CQt / C .t;xt CQt / t 2 J. Since F.t; x .t;xt CQt / C Q .t;xt CQt / / is compact, we may pass to a subsequence. Suppose that vn ! v in Lw1 .J; E/ (the space endowed with the weak topology), where v.t/ 2 F.t; x .t;xt CQt / C Q .t;xt CQt / /, a.e. t 2 J. An application of Mazur’s theorem [185] implies that the sequence vn converges strongly to v and hence v.t/ 2 SF;x : From the continuity of L, it follows that Lvn .t/ ! Lv.t/ Q .t;xt CQt / C .t;xt CQt / pointwise on J as n ! 1. In order to show that the convergence is uniform, we first show that fLvn g is an equi-continuous sequence. Let 1 ; 2 2 J, then we have: ˇ Z 1 ˇd S. 1 s/vn .s/ds jL.vn . 1 // L.vn . 2 //j D ˇˇ dt 0 ˇ Z ˇ d 2 S. 2 s/vn .s/dsˇˇ dt 0

11.2 Semi-linear Functional Differential Inclusions with State-Dependent. . .

311

ˇ ˇ Z 1 ˇ ˇ 0 0 ˇ ˇ lim ŒS . 1 s/ S . 2 s/B vn .s//dsˇˇ !1 0 ˇ ˇ Z 2 ˇ ˇ C ˇˇ lim S0 . 2 s/B vn .s/dsˇˇ : !1 1

As 1 ! 2 , the right-hand side of the above inequality tends to zero. Since S0 .t/ is a strongly continuous operator and the compactness of S0 .t/; t > 0, implies the uniform continuity (see [16, 168]). Hence fLvn g is equi-continuous, and an application of Arzéla-Ascoli theorem implies that there exists a subsequence which is uniformly convergent. Then we have Lvnj ! Lv 2 .L ı SF /.x/ as j 7! 1, and so .L ı SF /.x/ is compact. Therefore B is a compact valued multi-valued operator on Bb0 . Claim 2: B.x/ is convex for each z 2 D0b : Let h1 ; h2 2 B.x/, then there exist v1 ; v2 2 SF;x ; such that, for each t 2 J we have Q Q C Q .t;xt Ct /

hi .t/ D

.t;xt Ct /

8 ˆ < 0; ˆ :

d dt

Z 0

9 if t 2 .1; 0, > = t

S.t s/vi .s/ds

if t 2 J;

> ;

; i D 1; 2:

Let 0 ı 1: Then, for each t 2 J, we have .ıh1 C .1 ı/h2 /.t/ D

8 ˆ ˆ 0; < ˆ ˆ :

d dt

9 if t 2 .1; 0, > > =

Z

t 0

S.t s/Œıv1 .s/ C .1 ı/v2 .s/ds

if t 2 J:

> > ;

Since F has convex values, one has ıh1 C .1 ı/h2 2 B.x/: Claim 3: B maps bounded sets into bounded sets in Bb0 Let Bq D fx 2 Bb0 W kxkb qg q 2 RC a bounded set in Bb0 . It is equivalent to show that there exists a positive constant l such that for each x 2 Bq we have kB .x/kb l: So choose x 2 Bq ; then for each h 2 B.x/, and each x 2 Bq , there exists v 2 SF;x . such that Q Q C Q .t;xt Ct /

d h.t/ D dt

.t;xt Ct /

Z

t

S.t s/v.s/ds:

0

From (A) we have kx .t;xt CQt / C Q .t;xt CQt / kD Kb q C .Mb C L /kkD C Kb Mj.0/j D q

312

11 Functional Differential Inclusions with Multi-valued Jumps

Then by (11.1.6) we have jh.t/j Me!b .q /

Z

t

e!s p.s/ds WD l:

0

This further implies that khkB0 l: b

Hence B.B/ is bounded. Claim 4: B maps bounded sets into equi-continuous sets. Let Bq be, as above, a bounded set and h 2 B.x/ for some x 2 B: Then, there exists v 2 SF;x . such that Q Q C Q .t;xt Ct /

.t;xt Ct /

h.t/ D

d dt

Z

t

S.t s/v.s/ds;

t 2 J:

0

Let 1 ; 2 2 Jnft1 ; t2 ; : : : ; tm g; 1 < 2 . Thus if > 0, we have ˇ ˇ Z 1 ˇ ˇ 0 0 ˇ ŒS . 2 s/ S . 1 s/B v.s/dsˇˇ jh. 2 / h. 1 /j ˇ lim !1 0 ˇ ˇ Z 1 ˇ ˇ 0 0 ˇ C ˇ lim ŒS . 2 s/ S . 1 s/B v.s/dsˇˇ !1 1

ˇ ˇ Z 2 ˇ ˇ C ˇˇ lim S0 . 2 s/B v.s/dsˇˇ !1 1 Z 1 .q / kS0 . 2 s/ S0 . 1 s/kB.E/ p.s/ds 0

Z

C .q /

1

1

CMe!b .q /

kS0 . 2 s/ S0 . 1 s/kB.E/ p.s/ds Z

2

e!s p.s/ds:

1

As 1 ! 2 and becomes sufficiently small, the right-hand side of the above inequality tends to zero, since S0 .t/ is a strongly continuous operator and the compactness of S0 t/ for t > 0 implies the uniform continuity. This proves the equicontinuity for the case where t ¤ ti ; i D 1; : : : ; m C 1. It remains to examine the equi-continuity at t D ti . First we prove the equi-continuity at t D ti , we have for some x 2 Bq , there exists v 2 SF;x ; such that Q Q C Q .t;xt Ct /

h.t/ D

d dt

Z 0

t

.t;xt Ct /

S.t s/v.s/ds;

t 2 J:

11.2 Semi-linear Functional Differential Inclusions with State-Dependent. . .

313

Fix ı1 > 0 such that ftk ; k ¤ ig \ Œti ı1 ; ti C ı1 D ;. For 0 < < ı1 , we have Z

ti

jh.ti / h/.ti /j lim

!1 0

k S0 .ti s/ S0 .ti s/ B v.s/kds

CMe!b .q /

Z

ti

e!s p.s/ dsI

ti

which tends to zero as ! 0: Define hO 0 .t/ D h.t/; and hO i .t/ D

t 2 Œ0; t1

8 < h.t/;

9 if t 2 .ti ; tiC1 =

: h.tC /;

if

i

t D ti :

;

Next, we prove equi-continuity at t D tiC . Fix ı2 > 0 such that ftk ; k ¤ ig \ Œti ı2 ; ti C ı2 D ;. Then Z ti Oh.ti / D T.ti s/v.s/ds: 0

For 0 < < ı2 , we have Z

O i C / h.t O i /j lim jh.t

k S0 .ti C s/ S0 .ti s/ B v.s/ds

ti

!1 0

Z

!b

CMe

ti C

.q /

e!s p.s/ dsI

ti

The right-hand side tends to zero as ! 0. The equi-continuity for the cases 1 <

2 0 and 1 0 2 follows from the uniform continuity of on the interval .1; 0 As a consequence of Claims 1–3 together with Arzelá–Ascoli theorem it suffices to show that B maps B into a precompact set in E: Let 0 < t < b be fixed and let be a real number satisfying 0 < < t. For x 2 Bq , we define 0

Z

h .t/ D S ./ lim

!1 0

where v 2 SF;x

Q

.t;xt CQt / C .t;xt CQt /

ˇ Z ˇ ˇ lim ˇ!1

0

t

t

S0 .t s /B v.s/ds;

. Since

ˇ Z ˇ S0 .t s /B v.s/dsˇˇ Me!b .q /

t 0

e!s p.s/ds:

314

11 Functional Differential Inclusions with Multi-valued Jumps

the set

Z

t

lim

!1 0

0

S .t s /B v.s/ds W v 2 SF;x

Q

.t;xt CQt / C .t;xt CQt /

:x 2 Bq

is bounded Since S0 .t/ is a compact operator for t > 0, the set H .t/ D fh .t/ W h 2 B.x/g is precompact in E for every ; 0 < < t: Moreover, for every h 2 B.x/ we have jh.t/ h .t/j Me!b .q /

Z

t

e!s p.s/ds:

t

Therefore, there are precompact sets arbitrarily close to the set H.t/ D fh.t/ W h 2 B.x/g: Hence the set H.t/ D fh.t/ W h 2 B.Bq /g is precompact in E. Hence the operator B is totally bounded. Step 3: A priori bounds. Now it remains to show that the set ˚ E D x 2 Bb0 W x 2 A.x/ C B.x/

for some 0 < < 1

is bounded.

and Ik 2 Ik xtk C Q tk such Let x 2 E: Then there exist v 2 SF;x Q .t;xt CQt / C .t;xt CQt / that for each t 2 J; d x.t/ D dt

Zt S.t s/v.s/ C

X

S0 .t tk / Ik :

0
0

This implies by (11.1.2), (11.1.5) that, for each t 2 J; we have !t

Zt

jx.t/j Me

e!s p.s/ .kx .s;xs CQs / C Q .s;xs CQs / kD /ds

0

CMe!t

m X ˇ

ˇ ˇIk xt C Q t ˇ k k kD1

Me!t

Zt 0

e!s p.s/

Kb jx.s/j C .Mb C L C MKb /kkD ds

11.2 Semi-linear Functional Differential Inclusions with State-Dependent. . .

CMe!t

315

m X ˇ ˇ

ˇIk xt C Q t Ik .0/ˇ k k kD1

CMe!t

m X

jIk .0/j

kD1

Zt

!t

Me

e!s p.s/

Kb jx.s/j C .Mb C L C MKb /kkD ds

0 !t

CMe

m X kD1

!t

jIk .0/j C Me 2

ce!t C Me!t 4

CKb

ck Kb jx.s/j C .Mb C L C MKb /kkD

kD1

Zt 0

m X

m X

e!s p.s/

Kb jx.s/j C .Mb C L C MKb /kkD ds

#

ck jx.t/j :

kD1

Hence from (11.6) to (11.8) we have Zt

.Mb C L C MKb /kkD C Kb jx.s/j c1 C c2

e!s p.s/ .Kb jx.t/j

0

C.Mb C L C MKb /kkD ds: Thus .Mb C L C MKb /kkD C Kb kxkB0 c1 C c2

Rt 0

b

e!s p.s/

.Kb jx.t/j C .Mb C

L

1:

(11.10)

C MKb /kkD / ds:

From (11.5) it follows that there exists a constant R > 0 such that for each x 2 E with kxkB0 > R the condition (11.10) is violated. Hence kxkB0 R for each x 2 E, which b b means that the set E is bounded. As a consequence of Theorem 1.32, A C B has a fixed point x on the interval .1; b, so y D x C Q is a fixed point of the operator N which is the mild solution of problem (11.1)–(11.3). t u

316

11 Functional Differential Inclusions with Multi-valued Jumps

11.2.4 Example To illustrate our previous result we consider the partial functional differential equations with state dependent delay of the form @ @ v.t; / D v.t; / C m.t/a.v.t .v.t; 0//; //; @t @ 2 Œ0; ; t 2 Œ0; b;

(11.11)

v.t; 0/ D v.t; / D 0; t 2 Œ0; b;

(11.12)

v.; / D v0 .; /; 2 Œ0; ; 2 .1; 0; R ti v.ti /./ D 1 i .ti s/v.s; /ds;

(11.13) (11.14)

where v0 W .1; 0 Œ0; ! R is an appropriate function, i 2 CŒ0; 1/; R/; 0 < t1 < t2 < < tn < b: the function m W Œ0; b ! R; a W R J ! R; W R ! RC are continuous and we assume the existence of positive constants b1 ; b2 such that jb.t/j b1 jtj C b2 for every t 2 R. Let A be the operator defined on E D C.Œ0; ; R/ by D.A/ D fg 2 C1 .Œ0; ; R/ W g.0/ D 0gI Ag D g0 : Then D.A/ D C0 .Œ0; ; R/ D fg 2 C.Œ0; ; R/ W g.0/ D 0g: It is well known from [100] that A is sectorial, .0; C1/ .A/ and for > 0 kR.; A/kB.E/

1 :

It follows that A generates an integrated semigroup .S.t//t0 and that kS0 .t/kB.E/ et for t 0 for some constant > 0 and A satisfied the Hille–Yosida condition. Set > 0. For the phase space, we choose D to be defined by D D C D f 2 C..1; 0; E/ W lim e ./ exists in Eg !1

with norm kk D

sup 2.1;0

e j./j; 2 C :

11.3 Impulsive Evolution Inclusions with Infinite Delay and Multi-valued Jumps

317

By making the following change of variables y.t/./ D v.t; /; t 0; 2 .0; ; . /./ D v0 .; /; 0; 2 Œ0; 1; F.t; '/./ D m.t/b.'.0; //; 2 Œ0; ; 2 C .t; '/ D t .'.0; 0// R0 Ik .ytk / D 1 k .s/v.s; /ds; the problem (11.11)–(11.14) takes the abstract form (11.1)–(11.3). Moreover, a simple estimates shows that kf .t; '/k m.t/Œb1 k'kD C b2 1=2 for all .t; '/ 2 I D; with Z

1 1

Z

ds D .s/

1 1

ds D C1; b1 s C b2 1=2

and Z dk D

0 1

.k .s//2 ds es

1=2 < 1:

Theorem 11.4. Let ' 2 D be such that H' is valid and t ! 't is continuous on R. /, then there exists a integral solution of (11.11)–(11.14) whenever Z 1C

0

s

e ds 1

1=2 ! X m

dk < 1:

kD1

11.3 Impulsive Evolution Inclusions with Infinite Delay and Multi-valued Jumps 11.3.1 Introduction In this section, we are concerned by the existence of mild solution of impulsive semi-linear functional differential inclusions with infinite delay and multi-valued jumps in a Banach space E: More precisely, we consider the following class of semi-linear impulsive differential inclusions: x0 .t/ 2 A.t/x.t/ C F.t; xt /;

t 2 J D Œ0; b; t ¤ tk ;

(11.15)

318

11 Functional Differential Inclusions with Multi-valued Jumps

ˇ xˇtDtk 2 Ik .x.tk //; k D 1; : : : ; m x.t/ D .t/;

t 2 .1; 0;

(11.16) (11.17)

where fA.t/ W t 2 Jg is a family of linear operators in Banach space E generating an evolution operator, F be a Carathéodory type multi-function from J B to the collection of all nonempty compact convex subset of E, B is the phase space defined axiomatically which contains the mapping from .1; 0 into E, 2 B, 0 D t0 < t1 < < tm < tmC1 D b, Ik W E ! P.E/, k D 1; : : : ; m are multi-valued maps with closed, bounded and convex values, x.tkC / D limh!0C x.tk C h/ and x.tk / D limh!0C x.tk h/ represent the right and left limits of x.t/ at t D tk . Finally P.E/ denotes the family of nonempty subsets of E:

11.3.2 Existence Results Definition 11.5. A function x 2 is said to be a mild solution of system (11.15)– (11.17) if there exists a function f 2 L1 .J; E/ such that f 2 F.t; xt / for a.e. t 2 J Rt P (i) x.t/ D T.t; 0/.0/ C 0 T.t; s/f .s/ds C 0
for a.e. t 2 JI

(11.5.4) There exists a function ˇ 2 L1 .J; RC / such that for all ˝ B, we have .F.t; D// ˇ.t/

sup

1s0

.˝.s//

for a.e. t 2 J;

where ˝.s/ D fx.s/I x 2 ˝g and is the Hausdorff MNC.

11.3 Impulsive Evolution Inclusions with Infinite Delay and Multi-valued Jumps

319

(11.5.5) There exist constants ak > 0; k D 1; : : : ; m such that kIk k ak ;

where Ik 2 Ik .x.tkC //:

Remark 11.6. Under conditions (11.5.1)–(11.5.3) for every piecewise continuous function v W J ! B the multi-function F.t; v.t// admits a Bochner integrable selection (see [144]). Let b D fx 2 W x0 D 0g : For any x 2 b we have kxkb D kxkB C sup kxk D sup kxk: 0sb

0sb

Thus .b ; k:kb / is a Banach space. We note that from assumptions (11.5.1) and (11.5.3) it follows that the superposition multi-operator SF1 W b ! P.L1 .J; E// defined by SF1 D ff 2 L1 .J; E/ W f .t/ 2 F.t; xt /;

a.e. t 2 Jg

is nonempty set (see [144]) and is weakly closed in the following sense. 1 Lemma 11.7. If we consider the sequence .xn / 2 b and ffn gC1 nD1 L .J; E/, 1 1 n 0 0 0 where fn 2 SF.:;x such that x ! x and f ! f then f 2 S . n F n/

Now we state and prove our main result. Theorem 11.8. Under assumptions (A) and (11.5.1)–(11.5.5), the problem (11.15)–(11.17) has at least one mild solution. Proof. To prove the existence of a mild solution for (11.15)–(11.17) we introduce the integral multi-operator N W b ! P.b /; defined as 8 Rt ˆ y.t/ D T.t; 0/.0/ C 0 T.t; s/f .s/ds ˆ
(11.18)

where SF1 and Ik 2 Ik .x/: It is clear that the integral multi-operator N is well defined and the set of all mild solution for the problem (11.15)–(11.17) on J is the set FixN D fx W x 2 N.x/g: We shall prove that N satisfies all the hypotheses of Lemma 1.40. The proof will be given in several steps.

320

11 Functional Differential Inclusions with Multi-valued Jumps

Step 1. Using in fact that the maps F and I has a convex values it easy to check that N has convex values. Step 2. N has closed graph. n C1 n n n n Let fxn gC1 nD1 , fz gnD1 , x ! x , z 2 N..x /; n 1/ and z ! z . Moreover, let C1 1 1 ffn gnD1 L .J; E/ an arbitrary sequence such that fn 2 SF for n 1. Hypothesis (11.5.3) implies that the set ffn gC1 nD1 integrably bounded and for a.e. t 2 J the set ffn .t/gC1 relatively compact, we can say that ffn gC1 nD1 nD1 is semi-compact C1 sequence. Consequently ffn gnD1 is weakly compact in L1 .J; E/, so we can assume w.l.g that fn * f . From Lemma 1.43 we know that the generalized Cauchy operator on the interval J, G W L1 .J; E/ ! C.J; E/; defined by

Z Gf .t/ D

t

0

T.t; s/f .s/ds;

t 2 J:

(11.19)

satisfies properties (G1) and (G2) on J. C1 Note that set ffn gC1 nD1 is also semi-compact and sequence .fn /nD1 weakly con 1 verges to f in L .J; E/. Therefore, by applying Lemma 1.44 for the generalized Cauchy operator G of (11.19) we have in C.J; E/ the convergence Gfn ! Gf . By means of (11.19) and (11.18), for all t 2 J we can write Z zn .t/ D T.t; 0/.0/ C

0

Z D T.t; 0/.0/ C

t

0

T.t; s/fn .s/ds C

X

T.t; tk /Ik .xn .tk //

0
T.t; s/fn ds C

D T.t; 0/.0/ C Gfn .t/ C

X

X

T.t; tk /Ik .xn .tk //

0
T.t; tk /Ik .xn .tk ///

0
where SF1 ; and Ik 2 Ik .x/. By applying Lemma 1.43, we deduce zn ! T.:; 0/.0/ C Gf C T.:; t/Ik .x .tk // in C.J; E/ and by using in fact that the operator SF1 is closed, we get f 2 SF1 . Consequently z .t/ ! T.t; 0/.0/ C Gf C T.t; t/Ik .x .tk //; therefore z 2 N.x /. Hence N is closed. With the same technique, we obtain that N has compact values. Step 3. We consider the MNC defined in the following way. For every bounded subset ˝ b

11.3 Impulsive Evolution Inclusions with Infinite Delay and Multi-valued Jumps

1 .˝/ D max .1 .˝/; mod C .˝//; ˝2.˝/

321

(11.20)

where .˝/ is the collection of all the denumerable subsets of ˝; 1 .˝/ D sup eLt .fx.t/ W x 2 ˝g/I

(11.21)

t2J

where mod C .˝/ is the modulus of equi-continuity of the set of functions ˝ given by the formula mod C .˝/ D lim sup max kx.t1 / x.t2 /kI ı!0 x2˝ jt1 t2 jı

and L > 0 is a positive real number chosen such that Z q WD M 2 sup

t

L.ts/

e 0

t2J

ˇ.s/ds C e

Lt

m X

(11.22)

! ck

<1

(11.23)

kD1

where M D sup.t;s/2 kT.t; s/k: From the Arzelá–Ascoli theorem, the measure 1 gives a nonsingular and regular MNC (see [144]). Let fyn gC1 nD1 be the denumerable set which achieves that maximum 1 .N.˝//, i.e.; C1 1 .N.˝// D .1 .fyn gC1 nD1 /; mod C .fyn gnD1 //:

Then there exists a set fxn gC1 nD1 ˝ such that yn 2 N.xn /, n 1. Then Z yn .t/ D T.t; 0/.0/ C

0

t

T.t; s/f .s/ds C

X

T.t; tk /Ik .x.tk //;

0
where f 2 SF1 and Ik 2 Ik .xn /, so that C1 1 .fyn gC1 nD1 / D 1 .fGfn gnD1 /:

We give an upper estimate for 1 .fyn gC1 nD1 /. Fixed t 2 J by using condition (11.5.4), for all s 2 Œ0; t we have C1 .ffn .s/gC1 nD1 / .F.s; fxn .s/gnD1 //

.fF.s; xn .s//gC1 nD1 / ˇ.s/ .fxn .s/gC1 nD1 / ˇ.s/eLs sup eLt .fxn .t/gC1 nD1 / t2J

D ˇ.s/e 1 .fxn gC1 nD1 /: Ls

(11.24)

322

11 Functional Differential Inclusions with Multi-valued Jumps

By using condition (11.5.3), the set ffn gC1 nD1 is integrably bounded. In fact, for every t 2 J, we have kfn .t/k kF.t; xn .t//k ˛.t/.1 C kxn .tk/: The integrably boundedness of ffn gC1 nD1 follows from the continuity of x in Jk and the boundedness of set fxn gC1 nD1 ˝. By applying Lemma 1.45, it follows that Z s / 2M ˇ.t/eLt .1 .fxn gC1 .fGfn .s/gC1 nD1 nD1 //dt 0

D

2M1 .fxn gC1 nD1 /

Z

s

0

ˇ.t/eLt :

Thus, we get C1 C1 1 .fxn gC1 nD1 / 1 .fyn gnD1 / D 1 .fGfn .s/gnD1 / Z s m X C1 Lt Lt D sup eLt 2M1 .fxn gC1 / ˇ.t/e M .fx g /e ck 1 n nD1 nD1 0

t2J

kD1

q1 .fxn gC1 nD1 /; (11.25)

and hence

1 .fxn gC1 nD1 /

1 .fxn .t/gC1 nD1 /

D 0, then

D 0; for every t 2 J: Consequently

1 .fyn gC1 nD1 / D 0: By using the last equality and hypotheses (11.5.3) and (11.5.4), we can prove that set ffn gC1 nD1 is semi-compact. Now, by applying Lemmas 1.43 and 1.44, we can conclude that set fGfn gC1 nD1 is relatively compact. The representation of yn given by (11.24) yields that set fyn gC1 nD1 is also relatively compact in b , therefore 1 .˝/ D .0; 0/. Then ˝ is a relatively compact set. Step 4. A priori bounds. We will demonstrate that the solutions set is a priori bounded. Indeed, let x 2 N. 1 Then there exists f 2 SF.:;x and Ik 2 Ik .x/ such that for every t 2 J we have t .:// kx.t/k D T.t; 0/.0/ C M.k.0/k C

Z

t

0

X

T.t; s/f .s/ds C kak k/ C M

M.k.0/k C

X 0
t 0

Z kak k/ C M

T.t; tk /Ik .x.tk //

0
Z

0
X

b 0

f .s/ds ˛.s/.1 C kxŒs k/ds:

11.3 Impulsive Evolution Inclusions with Infinite Delay and Multi-valued Jumps

323

Using condition (A1) we have Z

X

kx.t/k M.k.0/kC

kak k/CM

0
M.k.0/k C

˛.s/.1 C Nb kkB C Kb sup kx. /k/ds

0

0s

kak k/ C M.1 C Nb kkB /k˛kL1 .J/

kD1

Z C MKb

m X

t

t

0

˛.s/ sup kx. /kds: 0s

Since the last expression is a nondecreasing function of t, we have that sup kx. /k kM.k.0/k C

0t

kak k/ C M.1 C Nb kkB /k˛kL1 .J/

kD1

Z C MKb

m X

t 0

˛.s/ sup kx. /kds: 0s

Invoking Gronwall’s inequality, we get sup kx. /k eMKb k˛kL1 ;

0t

where

D M.k.0/k C

m X

kak k/ C M.1 C Nb kkB /k˛kL1 .J/ :

kD1

t u

11.3.3 An Example As an application of our results we consider the following impulsive partial functional differential equation of the form @ z.t; x/ @t

2

@ 2 a.t; x/ @x 2 z.t; x/ C

R0

1

P. /r.t; z.t C ; x//d;

x 2 Œ0; ; t 2 Œ0; b; t ¤ tk ;

(11.26)

z.tkC ; x/ z.tk ; x/ 2 Œbk jz.tk ; x/; bk jz.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m; z.t; 0/ D z.t; / D 0; z.t; x/ D .t; x/;

t 2 J WD Œ0; b;

1 < t 0; x 2 Œ0; ;

(11.27) (11.28) (11.29)

324

11 Functional Differential Inclusions with Multi-valued Jumps

where a.t; x/ is continuous function and uniformly Hölder continuous in t, bk > 0, k D 1; : : : ; m, 2 D, D D f W .1; 0 Œ0; ! RI is continuous everywhere except for a countablenumber of points at which .s /; .sC / exist with .s / D .s/g, 0 D t0 < t1 < t2 < < tm < tmC1 D b, z.tkC / D lim.h;x/!.0C ;x/ z.tk C h; x/, z.tk / D lim.h;x/!.0 ;x/ z.tk C h; x/, P W .1; 0 ! R a continuous function, r W R R ! Pcv;k .R/ a Carathéodory multi-valued map. Let y.t/.x/ D z.t; x/;

x 2 Œ0; ; t 2 J D Œ0; b;

Ik .y.tk //.x/ D Œbk jz.tk ; x/; bk jz.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m; R0 F.t; /.x/ D 1 P. /r.t; z.t C ; x//d ./.x/ D .; x/;

1 < t 0; x 2 Œ0; :

Consider E D L2 Œ0; and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0g: Then A.t/ generates an evolution system U.t; s/ satisfying assumptions (11.5.1) and (11.5.3). We can show that problem (11.26)–(11.29) is an abstract formulation of problem (11.15)–(11.17). Under suitable conditions, the problem (11.15)–(11.17) has at least one mild solution.

11.4 Impulsive Semi-linear Differential Evolution Inclusions with Non-convex Right-Hand Side 11.4.1 Introduction In this section, we shall be concerned by the existence of mild solution of impulsive semi-linear functional differential inclusions with infinite delay in a separable Banach space E. First, we consider the following class of semi-linear impulsive differential inclusions: x0 .t/ 2 A.t/x.t/ C F.t; xt /; t 2 J D Œ0; b; t ¤ tk ; ˇ xˇtDtk 2 Ik .x.tk //; k D 1; : : : ; m x.t/ D .t/;

t 2 .1; 0

(11.30) (11.31) (11.32)

where fA.t/ W t 2 Jg is a family of linear operators in Banach space E generating an evolution operator, F be a lower semi-continuous multi-function from J B to the collection of all nonempty closed compact subset of E, B is the phase space defined axiomatically which contains the mapping from .1; 0 into E, 2 B,

11.4 Impulsive Semi-linear Differential Evolution Inclusions: : :

325

0 D t0 < t1 < < tm < tmC1 D b, Ik W E ! P.E/, k D 1; : : : ; m are multi-valued maps with closed and bounded values, x.tkC / D limh!0C x.tk C h/ and x.tk / D limh!0 x.tk C h/ represent the right and left limits of x.t/ at t D tk . Finally P.E/ denotes the family of nonempty subsets of E. We mention that the model with multi-valued jump sizes may arise in a control problem where we want to control the jump sizes in order to achieve given objectives.

11.4.2 Existence Results In this section, we give our main existence result for problem (11.30)–(11.32). Before stating and proving this result, we give the definition of the mild solution. Definition 11.9. A function x 2 is said to be a mild solution of system (11.15)– (11.17) if there exist a function f 2 L1 .J; E/ such that f 2 F.t; xt / for a.e. t 2 J and Ik 2 Ik .x.tkC // Rt P (i) x.t/ D T.t; 0/.0/ C 0 T.t; s/f .s/ds C 0
for a.e. t 2 J; 8

2 BI

(11.9.4) There exists a function ˇ 2 L1 .J; RC / such that for all D B, we have .F.t; D// ˇ.t/

sup

1s0

.D.s// for a.e. t 2 J;

where, D.s/ D fx.s/I x 2 Dg and is the Hausdorff MNC. (11.9.5) There exist constants ak ; ck 0, k D 1; : : : ; m, such that kIk k ak kxk C bk ; where Ik 2 Ik .x.tkC //: .Ik .D// P ck .Ik .D//. with 1 M 0 0.

326

11 Functional Differential Inclusions with Multi-valued Jumps

Remark 11.10. Under conditions (11.9.1)–(11.9.3) for every for every piecewise continuous function v W Œ0; b ! B the multi-function F.t; v.t// admits a Bochner integrable selection (see [144]). Now we state and prove our main result. Theorem 11.11. Under assumptions (A) and (11.9.1)–(11.9.5), the problem (11.30)–(11.32) has at least one mild solution. We note that from assumptions (11.9.1) and (11.9.3) it follows that the superposition multi-operator SF1 W ! P.L1 .J; E//; defined by 1 D SF1 D ff 2 L1 .J; E/ W f .t/ 2 F.t; xt /; SF.:;x/

a.e. t 2 Jg

is nonempty set (see [144]). Proof. We break the proof into a sequence of steps. Step 1. The Mönch’s condition holds. Suppose that ˝ Br is countable and ˝ co.f0g [ N.˝// We will prove that ˝ is relatively compact. We consider the MNC defined in the following way. For every bounded subset ˝ 1 .˝/ D max .1 .D/; mod C .D//; D2.˝/

(11.33)

where .˝/ is the collection of all the denumerable subsets of ˝; 1 .D/ D sup eLt .fx.t/ W x 2 Dg/I

(11.34)

t2J

where mod C .D/ is the modulus of equi-continuity of the set of functions D given by the formula mod C .D/ D lim sup max kx.t1 / x.t2 /kI ı!0 x2D jt1 t2 jı

(11.35)

and L > 0 is a positive real number chosen so that Z q WD M 2 sup t2J

0

t

eL.ts/ ˇ.s/ds C eLt

m X kD0

! ck

<1

(11.36)

11.4 Impulsive Semi-linear Differential Evolution Inclusions: : :

327

where M D sup.t;s/2 kT.t; s/k: From the Arzelá–Ascoli theorem, the measure 1 give a nonsingular and regular MNC (see [144]). Let fyn gC1 nD1 be the denumerable set which achieves that maximum 1 .N.˝//, i.e.; C1 1 .N.˝// D .1 .fyn gC1 nD1 /; mod C .fyn gnD1 //:

Then there exists a set fxn gC1 nD1 ˝ such that yn 2 N.xn /, n 1. Then Z yn .t/ D T.t; 0/.0/ C

t

0

T.t; s/f .s/ds C

X

T.t; tk /Ik ;

(11.37)

0
where f 2 SF1 and Ik 2 Ik .x/, so that C1 1 .fyn gC1 nD1 / D 1 .fGfn gnD1 /:

We give an upper estimate for 1 .fyn gC1 nD1 /. Fixed t 2 J by using condition (11.9.4), for all s 2 Œ0; t we have C1 .ffn .s/gC1 nD1 / .F.s; fxn .s/gnD1 //

ˇ.s/ .fxn .s/gC1 nD1 / ˇ.s/eLs sup eLt .fxn .t/gC1 nD1 / t2J

D ˇ.s/e 1 .fxn gC1 nD1 /: Ls

By using condition (11.9.3), the set ffn gC1 nD1 is integrably bounded. In fact, for every t 2 J, we have kfn .t/k kF.t; xn .t//k ˛.t/: By applying Lemma 1.45, it follows that .fGfn .s/gC1 nD1 /

Z 2M

0

s

ˇ.t/eLt .1 .fxn gC1 nD1 //dt

D 2M1 .fxn gC1 nD1 /

Z 0

s

ˇ.t/eLt dt:

328

11 Functional Differential Inclusions with Multi-valued Jumps

Thus, we get C1 1 .fxn gC1 nD1 / 1 .fyn gnD1 / Lt

D sup e t2J

2M1 .fxn gC1 nD1 /

Z

t

ˇ.s/e ds C Ls

0

Lt M1 .fxn gC1 nD1 e

m X

ck

kD1

q1 .fxn gC1 nD1 /: (11.38) Therefore, we have that C1 C1 1 .fxn gC1 nD1 / 1 .˝/ 1 .f0g [ N.˝//1 .fyn gnD1 / q1 .fxn gnD1 /:

From (11.36), we obtain that C1 1 .fxn gC1 nD1 / D 1 .˝/ D 1 .fyn gnD1 /

Coming back to the definition of 1 , we can see C1 .fxn gC1 nD1 / D .fyn gnD1 / D 0:

By using the last equality and hypotheses (11.9.3) and (11.9.4) we can prove that set ffn gC1 nD1 is semi-compact. Now, by applying Lemmas 1.43 and 1.44, we can conclude that set fGfn gC1 nD1 is relatively compact in C.J; E/. The representation of yn given by (11.37) yields that set fyn gC1 nD1 is also relatively compact in C.J; E/; since 1 is a monotone, nonsingular, regular MNC, we have that 1 .˝/ 1 .co.f0g [ N.˝/// 1 .N.˝// D 1 .fyn gC1 nD1 / D .0; 0/: Therefore, ˝ is relatively compact. Step 2. It is clear that the superposition multioperator SF1 has closed and decomposable values. Following the lines of [144], we may verify that SF1 is l.s.c. Applying Lemma 1.46 to the restriction of SF1 on we obtain that there exists a continuous selection w W ! L1 .J; E/ We consider a map N W ! defined as Z t x.t/ D T.t; 0/.0/ C T.t; s/w.x/.s/ds: 0

Since the Cauchy operator is continuous, the map N is also continuous; therefore, it is a continuous selection of the integral multi-operator.

11.4 Impulsive Semi-linear Differential Evolution Inclusions: : :

329

Step 3. A priori bounds. We will demonstrate that the solutions set is a priori bounded. Indeed, let x 2 N1 and 2 .0; 1/. There exists f 2 SF1 and Ik 2 Ik .x/ such that for every t 2 J we have kx.t/k D T.t; 0/.0/ C

M k.0/k C kxk

Z

t 0

X

T.t; s/f .s/ds C

T.t; tk /Ik

0
m X

kak k C

kD1

m X

!

Z

kbk k C M

0

kD1

t

˛.s/ds;

hence, 1M

m X

! kak k kxk M k.0/k C k˛k C

kD1

m X

! kbk k :

kD1

Consequently kxk

P M.k.0/k C k˛k C m kD1 kbk k/ Pm D C: 1 M kD1 kak k

So, there exists N such that kxk ¤ N , set kxk < N g:

U D fx 2 W

From the choice of U there is no x 2 @U such that x D N1 x for some 2 .0; 1/. N due to the Mönch’s Theorem. Thus, we get a fixed point of N1 in U u t

11.4.3 An Example As an application of our results we consider the following impulsive partial functional differential equation of the form @ z.t; x/ @t

2

@ 2 a.t; x/ @x 2 z.t; x/ C

R0

1

P. /r.t; z.t C ; x//d;

x 2 Œ0; ; t 2 Œ0; b; t ¤ tk ; z.tkC ; x/

z.tk ; x/

2

(11.39)

Œbk jz.tk ; x/; bk jz.tk ; x/;

x 2 Œ0; ; k D 1; : : : ; m;

(11.40)

z.t; 0/ D z.t; / D 0;

(11.41)

z.t; x/ D .t; x/;

t 2 J WD Œ0; b;

1 < t 0; x 2 Œ0; ;

(11.42)

330

11 Functional Differential Inclusions with Multi-valued Jumps

where a.t; x/ is continuous function and uniformly Hölder continuous in t, bk > 0, k D 1; : : : ; m, 2 D. D D f W .1; 0 Œ0; ! RI is continuous everywhere except for a countable number of points at which .s /; .sC / exist with .s / D .s/g, 0 D t0 < t1 < t2 < < tm < tmC1 D b, z.tkC / D lim.h;x/!.0C ;x/ z.tk C h; x/, z.tk / D lim.h;x/!.0 ;x/ z.tk C h; x/, P W .1; 0 ! R a continuous function, r W R R ! Pcv;k .R/ a multi-valued map. Let y.t/.x/ D z.t; x/;

x 2 Œ0; ; t 2 J D Œ0; b;

Ik .y.tk //.x/ D Œbk jz.tk ; x/; bk jz.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m; R0 F.t; /.x/ D 1 P. /r.t; z.t C ; x//d ./.x/ D .; x/;

1 < t 0; x 2 Œ0; :

Consider E D L2 Œ0; and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0g: Then A.t/ generates an evolution system U.t; s/ satisfying assumption (11.9.1) and (11.9.3). We can show that problem (11.39)–(11.42) is an abstract formulation of problem (11.30)–(11.32). Under suitable conditions, the problem (11.30)–(11.32) has at least one mild solution.

11.5 Impulsive Evolution Inclusions with State-Dependent Delay and Multi-valued Jumps 11.5.1 Introduction In this section, we are concerned by the existence of mild solution of impulsive semi-linear functional differential inclusions with state-dependent delay and multivalued jumps in a Banach space E. More precisely, we consider the following class of semi-linear impulsive differential inclusions: x0 .t/ 2 A.t/x.t/ C F.t; x .t;xt / /; t 2 J D Œ0; b; t ¤ tk ; ˇ xˇtDtk 2 Ik .x.tk //; k D 1; : : : ; m x.t/ D .t/;

t 2 .1; 0;

(11.43) (11.44) (11.45)

where fA.t/ W t 2 Jg is a family of linear operators in Banach space E generating an evolution operator, F be a Carathéodory type multi-function from J B to the collection of all nonempty compact convex subset of E, B is the phase space, which

11.5 Impulsive Evolution Inclusions with State-Dependent Delay...

331

contains the mapping from .1; 0 into E, 2 B, 0 D t0 < t1 < < tm < tmC1 D b, Ik W E ! P.E/, k D 1; : : : ; m are multi-valued maps with closed, bounded and convex values, x.tkC / D limh!0C x.tk C h/ and x.tk / D limh!0C x.tk h/ represent the right and left limits of x.t/ at t D tk . W J B ! .1; b. Our goal here is to give existence results for the problem (11.43)–(11.45) without any compactness assumption. We prove existence and compactness of solutions set for problem (11.43)–(11.45), and we provide a conditions which guarantee the existence of a mild solution by using a fixed point theorem du to Mönch [162].

11.5.2 Existence Results for the Convex Case In this section we shall prove the existence of mild solutions of problem (11.43)–(11.45). We assume that the multi-valued nonlinearity of upper Carathéodory semi-continuous type satisfies a regularity condition expressed in terms of the measures of non-compactness. We apply the theory of condensing multi-valued maps to obtain global and compactness of the solutions set. We need the following definition in the sequel. Definition 11.12. A function x 2 is said to be a mild solution of system (11.15)– (11.17) if there exist a function f 2 L1 .J; E/ such that f 2 F.t; x .t;xt / / for a.e. t 2 J Rt P (i) x.t/ D T.t; 0/.0/ C 0 T.t; s/f .s/ds C 0
332

11 Functional Differential Inclusions with Multi-valued Jumps

(11.12.4) There exists a function ˇ 2 L1 .J; RC / such that for all ˝ B, we have .F.t; ˝// ˇ.t/

sup

1s0

.˝.s//

for a.e. t 2 J;

where, ˝.s/ D fx.s/I x 2 ˝g and is the Hausdorff MNC. (11.12.5) There exist constants ak ; ck > 0; k D 1; : : : ; m such that 1) kIk k ak ; where Ik 2 Ik .x.tkC //: 2) .Ik .D// ck .D/ for each bounded subset D of E. The next result is a consequence of the phase space axioms. Lemma 11.13 ([139], Lemma 2.1). If y W .1; b ! R is a function such that y0 D and yjJ 2 PC.J; R/, then kys kB .Mb C L /kkB C Kb supfky. /kI 2 Œ0; maxf0; sgg;

s 2 R. / [ J;

where L D

sup L .t/:

t2R. /

Remark 11.14. We remark that condition .H ) is satisfied by functions which are continuous and bounded. In fact, if the space B satisfies axiom C2 in [142] then there exists a constant L > 0 such that kkB L supfk./k W 2 .1; 0g for every 2 B that is continuous and bounded (see [142], Proposition 7.1.1) for details. Consequently, kt kB L

sup0 k./k kkB ; kkB

for every 2 B n f0g:

Remark 11.15. Under conditions (H) and (11.12.1)–(11.12.3) for every piecewise continuous function v W J ! B the multi-function F.t; v.t// admits a Bochner integrable selection (see [144]). Let b D fx 2 W x0 D 0g : For any x 2 b we have kxkb D kxkB C sup kxk D sup kxk: 0sb

0sb

Thus .b ; k:kb / is a Banach space. We note that from assumptions (11.12.1) and (11.12.3) it follows that the superposition multi-operator SF1 W b ! P.L1 .J; E// defined by SF1 D ff 2 L1 .J; E/ W f .t/ 2 F.t; x .t;xt / /;

a.e. t 2 Jg

is nonempty set (see [144]) and is weakly closed in the following sense.

11.5 Impulsive Evolution Inclusions with State-Dependent Delay...

333

1 Lemma 11.16. If we consider the sequence .xn / 2 b and ffn gC1 nD1 L .J; E/, 1 1 n 0 0 0 where fn 2 SF.:;xn n / such that x ! x and fn ! f then f 2 SF . .:;x: /

Now we state and prove our main result. Theorem 11.17. Under assumptions (A)–(H) and (11.12.1)–(11.12.5), the problem (11.43)–(11.45) has at least one mild solution. Proof. To prove the existence of a mild solution for (11.43)–(11.45) we introduce the integral multi-operator N W b ! P.b /; defined as 8 Rt ˆ y.t/ D T.t; 0/.0/ C 0 T.t; s/f .s/ds ˆ
(11.46)

where SF1 and Ik 2 Ik .x/. It is clear that the integral multioperator N is well defined and the set of all mild solution for the problem (11.43)–(11.45) on J is the set FixN D fx W x 2 N.x/g. We shall prove that the integral multioperator N satisfies all the hypotheses of Lemma 1.40. The proof will be given in several steps. Step 1. Using in fact that the maps F and I has a convex values it easy to check that N has convex values. Step 2. N has closed graph. n C1 n n n n Let fxn gC1 nD1 b , fz gnD1 , x ! x , z 2 N..x /; n 1/ and z ! z . Moreover, C1 1 1 let ffn gnD1 L .J; E/ an arbitrary sequence such that fn 2 SF for n 1. Hypothesis (11.12.3) implies that the set ffn gC1 nD1 integrably bounded and for a.e. C1 t 2 J the set ffn .t/gC1 nD1 relatively compact, we can say that ffn gnD1 is semi-compact C1 1 sequence. Consequently ffn gnD1 is weakly compact in L .J; E/, so we can assume that fn * f . From Lemma 1.43 we know that the generalized Cauchy operator on the interval J, G W L1 .J; E/ ! b ; defined by

Z Gf .t/ D

t

0

T.t; s/f .s/ds;

t2J

(11.47)

satisfies properties (G1) and (G2) on J. C1 Note that set ffn gC1 nD1 is also semi-compact and sequence .fn /nD1 weakly con 1 verges to f in L .J; E/. Therefore, by applying Lemma 1.44 for the generalized Cauchy operator G of (11.19) we have the convergence Gfn ! Gf . By means of (11.19) and (11.18), for all t 2 J we can write Z zn .t/ D T.t; 0/.0/ C

0

t

T.t; s/fn .s/ds C

X 0
T.t; tk /Ik .xn .tk //

334

11 Functional Differential Inclusions with Multi-valued Jumps

Z D T.t; 0/.0/ C

0

t

X

T.t; s/fn ds C

D T.t; 0/.0/ C Gfn .t/ C

X

T.t; tk /Ik .xn .tk //

0
T.t; tk /Ik .xn .tk ///;

0
where SF1 , and Ik 2 Ik .x/. By applying Lemma 1.43, we deduce zn ! T.:; 0/.0/ C Gf C T.:; t/Ik .x .tk // in b and by using in fact that the operator SF1 is closed, we get f 2 SF1 . Consequently z .t/ ! T.t; 0/.0/ C Gf C T.t; t/Ik .x .tk //; therefore z 2 N.x /. Hence N is closed. With the same technique, we obtain that N has compact values. Step 3. We consider the MNC defined in the following way. For every bounded subset ˝ b 1 .˝/ D max .1 .˝/; mod C .˝//; ˝2.˝/

(11.48)

where .˝/ is the collection of all the denumerable subsets of ˝; 1 .˝/ D sup eLt .fx.t/ W x 2 ˝g/I

(11.49)

t2J

where mod C .˝/ is the modulus of equi-continuity of the set of functions ˝ given by the formula mod C .˝/ D lim sup max kx.t1 / x.t2 /k; ı!0 x2˝ jt1 t2 jı

(11.50)

and L > 0 is a positive real number chosen such that Z q WD M 2 sup t2J

t

L.ts/

e 0

ˇ.s/ds C e

Lt

m X

! ck

<1

(11.51)

kD1

where M D sup.t;s/2 kT.t; s/k. From the Arzelá–Ascoli theorem, the measure 1 gives a nonsingular and regular MNC (see [144]). Let fyn gC1 nD1 be the denumerable set which achieves that maximum 1 .N.˝//, i.e.;

11.5 Impulsive Evolution Inclusions with State-Dependent Delay...

335

C1 1 .N.˝// D .1 .fyn gC1 nD1 /; mod C .fyn gnD1 //:

Then there exists a set fxn gC1 nD1 ˝ such that yn 2 N.xn /, n 1. Then Z yn .t/ D T.t; 0/.0/ C

0

t

T.t; s/f .s/ds C

X

T.t; tk /Ik .x.tk //;

(11.52)

0
where f 2 SF1 and Ik 2 Ik .xn /, so that C1 1 .fyn gC1 nD1 / D 1 .fGfn gnD1 /:

We give an upper estimate for 1 .fyn gC1 nD1 /. Fixed t 2 J by using condition (11.12.4), for all s 2 Œ0; t we have C1 .ffn .s/gC1 nD1 / .F.s; fxn .s/gnD1 //

.fF.s; xn .s//gC1 nD1 / ˇ.s/ .fxn .s/gC1 nD1 / ˇ.s/eLs sup eLt .fxn .t/gC1 nD1 / t2J

D ˇ.s/eLs 1 .fxn gC1 nD1 /: By using condition (11.12.3), the set ffn gC1 nD1 is integrably bounded. In fact, for every t 2 J, we have kfn .t/k kF.t; xn .t//k ˛.t/.1 C kxn .tk/: The integrably boundedness of ffn gC1 nD1 follows from the continuity of x in Jk and the boundedness of set fxn gC1 ˝. By applying Lemma 1.45, it follows that nD1 .fGfn .s/gC1 nD1 / 2M D

Z 0

s

ˇ.t/eLt .1 .fxn gC1 nD1 //dt

2M1 .fxn gC1 nD1 /

Z 0

s

ˇ.t/eLt :

336

11 Functional Differential Inclusions with Multi-valued Jumps

Thus, we get C1 C1 1 .fxn gC1 nD1 / 1 .fyn gnD1 / D 1 .fGfn .s/gnD1 / Z s m X C1 C1 Lt Lt Lt D sup e 2M1 .fxn gnD1 / ˇ.t/e M1 .fxn gnD1 /e ck 0

t2J

kD1

q1 .fxn gC1 nD1 /; (11.53)

C1 and hence 1 .fxn gC1 nD1 / D 0, then 1 .fxn .t/gnD1 / D 0, for every t 2 J. Consequently

1 .fyn gC1 nD1 / D 0: By using the last equality and hypotheses (11.12.3) and (11.12.4) we can prove that set ffn gC1 nD1 is semi-compact. Now, by applying Lemmas 1.43 and 1.44, we can conclude that set fGfn gC1 nD1 is relatively compact. The representation of yn given by (11.24) yields that set fyn gC1 nD1 is also relatively compact in b , therefore 1 .˝/ D .0; 0/. Then ˝ is a relatively compact set. Step 4. A priori bounds. We will demonstrate that the solutions set is a priori bounded. Indeed, let x 2 N. Then there exists f 2 SF1 and Ik 2 Ik .x/ such that for every t 2 J we have Z t X T.t; s/f .s/ds C T.t; tk /Ik .x.tk // kx.t/k D T.t; 0/.0/ C 0

M k.0/k C

m X

M k.0/k C

Z

ak C M

kD1 m X

0
! !

t 0

Z

ak C M

kD1

b 0

f .s/ds ˛.s/.1 C kxŒ .t;xt / k/ds:

Using Lemma 11.13, we have kx.t/k M k.0/k C

m X kD1

! ak C M

!

Z

t 0

˛.s/ 1 C .Mb C L /kkB

C Kb sup kx. /k ds 0s

M k.0/k C

! ak C M.1 C .Mb C L /kkB /k˛kL1 .J/

kD1

Z C MKb

m X

t 0

˛.s/ sup kx. /kds: 0s

11.5 Impulsive Evolution Inclusions with State-Dependent Delay...

337

Since the last expression is a nondecreasing function of t, we have that ! m X ak C M.1 C .Mb C L /kkB /k˛kL1 .J/ sup kx. /k M k.0/k C 0t

Z C MKb

t 0

kD1

˛.s/ sup kx. /kds: 0s

Invoking Gronwall’s inequality, we get sup kx. /k eMKb k˛kL1 Œ0;b ;

0b

where

D M k.0/k C

m X

! ak C M.1 C .Mb C L /kkB /k˛kL1 .J/ :

kD1

t u

11.5.3 Existence Results for the Non-convex Case This section is devoted to proving the existence of solutions for (11.43)–(11.45) with a non-convex valued right-hand side. Our result is based on Mönch’s fixed point theorem combined with a selection theorem due to Bressan and Colombo (see [86]). We will assume the following hypotheses: Let F be a multi-function defined from J B to the family of nonempty closed convex subsets of E such that (11.18.1) .t; x/ 7! F.:; x/ is L ˝ Bb -measurable (Bb is Borel measurable). (11.18.2) The multi-function F W .t; :/ ! Pk .E/ is lower semi-continuous for a.e. t 2 J; (11.18.3) there exists a function ˛ 2 L1 .J; RC / such that kF.t; /k ˛.t/;

for a.e. t 2 J; 8

2 BI

(11.18.4) There exists a function ˇ 2 L1 .J; RC / such that for all ˝ B, we have .F.t; ˝// ˇ.t/

sup

1s0

.˝.s//

for a.e. t 2 J;

where, ˝.s/ D fx.s/I x 2 ˝g and is the Hausdorff MNC, (11.18.5) There exist constants ak ; bk ; ck 0, k D 1; : : : ; m, such that 1) kIk k ak kxk C bk ; where Ik 2 Ik .x.tkC //: 2) .Ik .D// ck .D/ for each bounded subset D of E. Now we state and prove our main result.

338

11 Functional Differential Inclusions with Multi-valued Jumps

Theorem 11.18. Assume that (A)–(H) and (11.18.1)–(11.18.5) hold. If M

m X

ak < 1:

kD1

Then the problem (11.43)–(11.45) has at least one mild solution. Proof. We note that from assumptions (11.18.1) and (11.18.3) it follows that the superposition multi-operator SF1 W b ! P.L1 .J; E//; defined by SF1 D ff 2 L1 .J; E/ W f .t/ 2 F.t; x .t;xt / /;

a.e. t 2 Jg

is nonempty set (see [144]). Clearly, fixed points of the operator N are mild solutions of the problem (11.43)–(11.45). The proof will be given in several steps. Step 1. The Mönch’s condition holds. Suppose that ˝ Br is countable and ˝ co.f0g [ N.˝// We will prove that ˝ is relatively compact. We consider the MNC defined in (11.48) and L > 0 is a positive real number chosen such that Z q WD M 2 sup t2J

0

t

eL.ts/ ˇ.s/ds C eLt

m X

! ck

<1

(11.54)

kD1

where M D sup.t;s/2 kT.t; s/k. From the Arzelá–Ascoli theorem, the measure 1 give a nonsingular and regular MNC (see [144]). Let fyn gC1 nD1 be the denumerable set which achieves that maximum 1 .N.˝//, i.e.; C1 1 .N.˝// D .1 .fyn gC1 nD1 /; mod C .fyn gnD1 //:

Then there exists a set fxn gC1 nD1 ˝ such that yn 2 N.xn /, n 1. Then Z yn .t/ D T.t; 0/.0/ C

0

t

T.t; s/f .s/ds C

X 0
T.t; tk /Ik ;

(11.55)

11.5 Impulsive Evolution Inclusions with State-Dependent Delay...

339

where f 2 SF1 and Ik 2 Ik .xn /, so that C1 1 .fyn gC1 nD1 / D 1 .fGfn gnD1 /:

We give an upper estimate for 1 .fyn gC1 nD1 /. Fixed t 2 J by using condition (11.18.4), for all s 2 Œ0; t we have C1 .ffn .s/gC1 nD1 / .F.s; fxn .s/gnD1 //

ˇ.s/ .fxn .s/gC1 nD1 / ˇ.s/eLs sup eLt .fxn .t/gC1 nD1 / t2J

D ˇ.s/e 1 .fxn gC1 nD1 /: Ls

By using condition (11.18.3), the set ffn gC1 nD1 is integrably bounded. In fact, for every t 2 J, we have kfn .t/k kF.t; xn .t//k ˛.t/: By applying Lemma 1.45, it follows that .fGfn .s/gC1 nD1 / 2M D

Z

s

0

ˇ.t/eLt .1 .fxn gC1 nD1 //dt

2M1 .fxn gC1 nD1 /

Z 0

s

ˇ.t/eLt dt:

Thus, we get C1 1 .fxn gC1 nD1 / 1 .fyn gnD1 / Lt

D sup e t2J

2M1 .fxn gC1 nD1 /

Z

t 0

Lt ˇ.s/eLs ds C M1 .fxn gC1 nD1 /e

m X

ck

kD1

q1 .fxn gC1 nD1 /: (11.56) Therefore, we have that C1 C1 1 .fxn gC1 nD1 / 1 .˝/ 1 .f0g [ N.˝//1 .fyn gnD1 / q1 .fxn gnD1 /:

From (11.51), we obtain that C1 1 .fxn gC1 nD1 / D 1 .˝/ D 1 .fyn gnD1 /

340

11 Functional Differential Inclusions with Multi-valued Jumps

Coming back to the definition of 1 , we can see C1 .fxn gC1 nD1 / D .fyn gnD1 / D 0

By using the last equality and hypotheses (11.18.3) and (11.18.4) we can prove that set ffn gC1 nD1 is semi-compact. Now, by applying Lemmas 1.43 and 1.44, we can conclude that set fGfn gC1 nD1 is relatively compact. The representation of yn yields that set fyn gC1 nD1 is also relatively compact in b , since 1 is a monotone, nonsingular, regular MNC, we have that 1 .˝/ 1 .co.f0g [ N.˝/// 1 .N.˝// D 1 .fyn gC1 nD1 / D .0; 0/: Therefore, ˝ is relatively compact. Step 2. It is clear that the superposition multioperator SF1 has closed and decomposable values. Following the lines of [144], we may verify that SF1 is l.s.c.. Applying Lemma 1.46 to the restriction of SF1 on b we obtain that there exists a continuous selection w W b ! L1 .J; E/ We consider a map N W b ! b defined as Z x.t/ D T.t; 0/.0/ C

t

T.t; s/w.x/.s/ds 0

Since the Cauchy operator is continuous, the map N is also continuous, therefore, it is a continuous selection of the integral multi-operator. Step 3. A priori bounds. We will demonstrate that the solutions set is a priori bounded. Indeed, let x 2 N1 and 2 .0; 1/. There exists f 2 SF1 and Ik 2 Ik .x/ such that for every t 2 J we have kx.t/k D T.t; 0/.0/ C

M k.0/k C kxk

Z

t 0

m X kD1

X

T.t; s/f .s/ds C

ak C

m X

!

T.t; tk /Ik ;

0
Z

bk C M

kD1

t 0

˛.s/ds;

hence, 1M

m X kD1

! ak kxk M k.0/k C k˛kL1 C

m X kD1

! bk :

11.5 Impulsive Evolution Inclusions with State-Dependent Delay...

341

Consequently P M.k.0/k C k˛kL1 C m kD1 bk / P kxk D C: 1M m a kD1 k So, there exists N such that kxk ¤ N , set U D fx 2 b W

kxk < N g:

From the choice of U there is no x 2 @U such that x D Nx for some 2 .0; 1/: N due to the Mönch’s Theorem. u t Thus, we get a fixed point of N1 in U

11.5.4 An Example As an application of our results we consider the following impulsive partial functional differential equation of the form @2 @ z.t; x/ 2 a.t; x/ 2 z.t; x/ C m.t/b.t; z.t .z.t; 0///; x/; @t @x x 2 Œ0; ; t 2 Œ0; b; t ¤ tk ; z.tkC ; x/

z.tk ; x/

2

x 2 Œ0; ; k D 1; : : : ; m; z.t; 0/ D z.t; / D 0; z.t; x/ D .t; x/;

(11.57)

Œbk jz.tk ; x/; bk jz.tk ; x/; (11.58) t 2 J WD Œ0; b;

1 < t 0; x 2 Œ0; ;

(11.59) (11.60)

where a.t; x/ is continuous function and uniformly Hölder continuous in t, bk > 0, k D 1; : : : ; m, 2 D, D D f W .1; 0 Œ0; ! RI is continuous everywhere except for a countable number of points at which .s /; .sC / exist with .s / D .s/g, 0 D t0 < t1 < t2 < < tm < tmC1 D b, z.tkC / D lim.h;x/!.0C ;x/ z.tk C h; x/, z.tk / D lim.h;x/!.0 ;x/ z.tk C h; x/; b W R R ! Pcv;k .R/ a Carathéodory multivalued map, W R ! RC . Let y.t/.x/ D z.t; x/;

x 2 Œ0; ; t 2 J D Œ0; b;

Ik .y.tk //.x/ D Œbk jz.tk ; x/; bk jz.tk ; x/;

x 2 Œ0; ; k D 1; : : : ; m;

F.t; /.x/ D b.t/a.t; z.t .z.t; 0///; x/ ./.x/ D .; x/;

1 < t 0; x 2 Œ0; ;

.t; / D t ..0; 0//:

342

11 Functional Differential Inclusions with Multi-valued Jumps

Consider E D L2 Œ0; and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous; w00 2 E; w.0/ D w./ D 0g: Then A.t/ generates an evolution system U.t; s/ satisfying assumption (11.12.1) and (11.12.3). For the phase space, we choose B D B defined by B D 2 D W

lim e ./ exists

!1

with the norm kk D

e k./k:

sup 2.1;0

Notice that the phase space B satisfies axioms (A1) and (A3) (see [142] for more details). We can show that problem (11.57)–(11.60) is an abstract formulation of problem (11.43)–(11.45). Under suitable conditions, the problem (11.43)–(11.45) has at least one mild solution.

11.6 Controllability of Impulsive Differential Evolution Inclusions with Infinite Delay 11.6.1 Introduction In this section,we are concerned by a controllability problem for a system governed by a semi-linear functional differential inclusion in a separable Banach space E in the presence of impulse effects and infinite delay. x0 .t/ 2 A.t/x.t/ C F.t; xt / C .Bu/.t/; t 2 J D Œ0; b; t ¤ tk ; ˇ xˇtDtk 2 Ik .x.tk //; k D 1; : : : ; m x.t/ D .t/;

t 2 .1; 0:

(11.61) (11.62) (11.63)

Assuming that the compactness of the evolution operator generated by the linear part (11.61)–(11.61) is not required. Our aim here is to give global existence and controllability results for the above problem.

11.6 Controllability of Impulsive Differential Evolution Inclusions...

343

11.6.2 Existence and Controllability Results In this section, we shall establish sufficient conditions for the controllability of the first order functional semi-linear differential inclusions (11.61)–(11.63). We define what we mean by an mild solution of problem (11.61)–(11.63). Definition 11.19. A function x 2 is said to be a mild solution of the system (11.15)–(11.17) if there exist a function f 2 L1 .J; E/ such that f 2 F.t; xt / for a.e. t 2 J Rt P (i) x.t/ D T.t; 0/.0/ C 0 T.t; s/ Œ.Bx u/.s/ C f .s/ ds C 0
sup

1s0

.˝.s//

for a.e. t 2 J;

where, ˝.s/ D fx.s/I x 2 ˝g and is the Hausdorff MNC. (11.19.5) There exist constants ak ; k D 1; : : : ; m such that 1) kIk k ak ; where Ik 2 Ik .x.tkC //: 2) Ik are completely continuous. N for every piecewise Remark 11.20. Under conditions (11.19.1)–(11.19.3) and (A1) continuous function v W J ! B the multi-function F.t; v.t// admits a Bochner integrable selection (see [144]). Before stating and proving our main result in this section, we define controllability on the interval J:

344

11 Functional Differential Inclusions with Multi-valued Jumps

Definition 11.21. The system (11.61)–(11.63) is said to be controllable on the interval J, if for every x0 ; x1 2 E there exists a control u 2 L2 .J; U/, such that there exists a mild solution x.t/ of (11.61)–(11.63) satisfying x.b/ D x1 : Let b D fx 2 W x0 D 0g : For any x 2 b we have kxkb D kxkB C sup kxk D sup kxk: 0sb

0sb

Thus .b ; k:kb / is a Banach space . We note that from assumptions (11.19.1) and (11.19.3) it follows that the superposition multi-operator SF1 W b ! P.L1 .J; E// defined by SF1 D ff 2 L1 .J; E/ W f .t/ 2 F.t; xt /;

a.e. t 2 Jg

is nonempty set (see [144]) and is weakly closed in the following sense. 1 Lemma 11.22. If we consider the sequence .xn / 2 b and ffn gC1 nD1 L .J; E/, 1 1 n 0 0 0 where fn 2 SF.:;xn / such that x ! x and fn ! f then f 2 SF . :

Now, we are able to state and prove our main theorem: Theorem 11.23. Assume that hypotheses (11.19.1)–(11.19.5) hold. Moreover we suppose that (C1) B is a continuous operator from U to X and the linear operator W W L2 .J; U/ ! X, defined by Z b T.t; s/Bu.s/ ds; Wu D 0

has a bounded inverse operator W 1 W X ! L2 .J; U/=KerW such that kBk M1 and kW 1 k M2 , for some positive constants M1 ; M2 . (C2) There exists a function 2 L1 .J; RC / such that for all ˝ 2 Pb .E/, we have U .W 1 .˝/.t// .t/ E .˝/

for a.e. t 2 J;

Then the problem (11.61)–(11.63) has at least one mild solution. To prove the controllability of the problem. Using hypothesis (C1) for an arbitrary function x./ define the control ux .t/ D W

1

Z N h X x1 T.b; 0/.0/ T.b; tk /Ik .x.tk // kD1

b 0

i T.b; /f . / ds .t/;

11.6 Controllability of Impulsive Differential Evolution Inclusions...

345

where ux ./ 2 L2 .J; E/. We introduce the integral multi-operator N W b ! P.b /; defined as 8 Rt ˆ D T.t; 0/.0/ C 0 T.t; s/ Œf .s/ C Bux .s/ ds ˆ
t2J t 2 .1; 0; (11.64)

where SF1 and Ik 2 Ik .x/. It is clear that the integral multi-operator N is well defined and the set of all mild solution for the problem (11.61)–(11.63) on J is the set FixN D fx W x 2 N.x/g. Consider now the operator G W L1 .J; E/ ! C.J; E/ defined by .G f /.t/ D

Z

t 0

Z h T.t; s/BW 1 x1 T.b; 0/.0/

0

b

i T.b; /f . / ds .t/

(11.65)

Lemma 11.24 ([84]). The operator G satisfies the proprieties (G1’)–(G2). We shall prove that the integral multi-operator N satisfies all the hypotheses of Lemma 1.40. Proof. We break the proof into a sequence of steps. Step 1. Using the fact that the maps F and I has a convex values it easy to check that N has convex values. Step 2. N has closed graph. n C1 n n n n Let fxn gC1 nD1 b , fz gnD1 , x ! x , z 2 N..x /; n 1/ and z ! z . Moreover, C1 1 1 let ffn gnD1 L .J; E/ an arbitrary sequence such that fn 2 SF for n 1. Hypothesis (11.19.3) implies that the set ffn gC1 nD1 is integrably bounded and for a.e. t 2 J the set ffn .t/gC1 is relatively compact, we can say that ffn gC1 nD1 nD1 is semiC1 compact sequence. Consequently ffn gnD1 is weakly compact in L1 .J; E/, so we can assume that fn * f . From Lemma 1.43 we know that the generalized Cauchy operators, G; G W 1 L .J; E/ ! b ; defined by

Rt

t2J (11.66) 0 T.t; s/f .s/ds; h i Rt Rb .G f /.t/ D 0 T.t; s/BW 1 x1 T.b; 0/.0/ 0 T.b; /f . / ds .t/ Gf .t/ D

(11.67) satisfies properties (G1) and (G2) on J.

346

11 Functional Differential Inclusions with Multi-valued Jumps

C1 Note that set ffn gC1 nD1 is also semi-compact and sequence .fn /nD1 weakly con 1 verges to f in L .J; E/. Therefore, by applying Lemmas 1.44 and 11.67 we have the convergence Gfn ! Gf and G fn ! G f . By means (11.18), for all t 2 J we can write

zn .t/ D T.t; 0/.0/ C Gfn .t/ C G fn .t/ Z

t

T.t; s/BW 0

1

N X

!

X

T.b; tk /Ik .x.tk // C

T.t; tk /Ik .xn .tk ///;

0
kD1

(11.68)

where SF1 , Ik 2 Ik .x/ and By applying Lemma 1.43, we deduce zn .t/ D T.:; 0/.0/ C Gfn .:/ C G fn .:/ GBW 1 C

X

N X

! T.b; tk /Ik .xn .tk //

kD1

T.t; tk /Ik .x .tk /// n

0
zn ! T.:; 0/.0/ C Gf C G f GBW C

X

1

N X

!

T.b; tk /Ik .x .tk //

kD1

T.:; t/Ik .x .tk //

0
in b and by using in fact that the operator SF1 is closed, we get f 2 SF1 . Consequently

z .t/ D T.t; 0/.0/ C Gf C G f GBW C

X

1

N X

!

T.b; tk /Ik .x .tk //

kD1

T.t; s/Ik .x .tk //;

0
therefore z 2 N.x /. Hence N is closed. With the same technique, we obtain that N has compact values. Step 3. We consider the MNC defined in the following way. For every bounded subset ˝ b .˝/ D ..˝/; modC .˝//;

(11.69)

11.6 Controllability of Impulsive Differential Evolution Inclusions...

347

is the modulus of fiber non-compactness .˝/ D sup E .fx.t/ W x 2 ˝g/;

(11.70)

t2J

where ˝.t/ D fy.t/ W y 2 ˝g. modC .˝/ is the modulus of equi-continuity of the set of functions ˝ given by the formula modC .˝/ D lim sup max kx.t1 / x.t2 /kI ı!0 x2˝ jt1 t2 jı

and qN > 0 is a positive real number chosen such that Z b Z b N CM N 2M N1 qN WD M

.s/ds ˇ.s/ds < 1: 0

(11.71)

(11.72)

0

From the Arzelá–Ascoli theorem, the measure give a nonsingular and regular MNC , see [144]. Let ˝ b be a bounded subset such that .N.˝// .˝/:

(11.73)

For any t 2 Œ0; b we have 1 N.˝/.t/ T.:; 0/.0/ C .G C G / ı SF.˝/ :

From the boundedness of the operators fT.t; s/g0s;tb and B. Obviously there N M N 1 such that exist constants M; N M; kT.t; s/k. / M

for every 0 s; t b

N 1 M1 : kBk. / M

(11.74) (11.75)

We give an upper estimate for .N.˝//. By using (11.6.4) and (11.74)–(11.75) N .fT.t; s/f .s/g/ Mˇ.s/

sup

1s0

.˝Œs /

N Mˇ.s/.˝/: 1 Where f 2 SF.˝/ and ˝Œs D fxŒs W x 2 ˝g. Applying the Proposition 1.12, we have 1 N .G ı SF.˝/ .t// M.˝/

N M.˝/

Z Z

t 0 b 0

ˇ.s/ds (11.76) ˇ.s/ds:

348

11 Functional Differential Inclusions with Multi-valued Jumps

By using the condition (C2) and the estimates (11.74)–(11.76) we obtain

T.t; s/BW 1 Œx1 T.b; 0/.0/

NM N 1 .s/ M

Z

N 1 .˝/ N 2M M

Z

b

0

0

b

Z

b 0

T.t; /f . /d

T.t; /f . /d

ˇ.s/ds .s/

1 f 2 SF.˝/ . By Proposition 1.12, we have

Z 1 N 2M N 1 .˝/ G ı SF.˝/ .t/ M

b

0

Z ˇ.s/ds

0

b

.s/ds :

Thus, we get Z N CM N 2M N1 E .N.˝/.t// M

b

0

Z

.s/ds

b 0

ˇ.s/ds .˝/:

Then .N.˝// qN .˝/: Where qN is the constant in (11.72), consequently .˝/ D 0: By using the last equality and hypotheses (11.19.3) and (11.19.4) we can prove that set ffn gC1 nD1 is semi-compact. Now, by applying Lemmas 1.43 and 1.44, we can conclude that set f.G C G /fn gC1 nD1 is relatively compact. Therefore .˝/ D .0; 0/ and modC .˝/ D 0. Then ˝ is a relatively compact set. Step 4. A priori bounds. Let x 2 N.x/; 0 < 1, then we have kx.t/k T.t; 0/.0/ C C

X

Z

t 0

T.t; s/Œf .s/ C Bux .s/ds

T.t; tk /Ik .x.tk //

0
Mk.0/k C M

X

Z kIk k C M

0
D Mk.0/k C M

X 0
t

0

Z kIk k C M

0

t

Z kf .s/kds C MM1 kf .s/kds

0

t

kux .s/kds

11.6 Controllability of Impulsive Differential Evolution Inclusions...

Z C MM1 Z

b

0

t 0

349

N h X kW 1 x1 T.b; 0/.0/ T.b; tk /Ik .x.tk // kD1

i

T.b; /f . / ds .s/kds

Mk.0/k C M

Z

X

kIk k C M

t

0

0
kf .s/kds

N h X C MM1 kW 1 x1 T.b; 0/.0/ T.b; tk /Ik .x.tk //

Z

kD1 b

0

i

T.b; /f . / ds .s/kL1 .J;U/

Mk.0/k C M

X

Z kIk k C M

0

0
t

kf .s/kds

N h X p C MM1 bkW 1 x1 T.b; 0/.0/ T.b; tk /Ik .x.tk //

Z 0

kD1 b

i T.b; /f . / ds .s/kL2 .J;U/

Mk.0/k C M

m X

Z ak C M

kD1

0

t

kf .s/kds

" Z m X p ak C M C MM1 M2 b kx1 k C Mk.0/k C M

b

0

kD1

# kf . /k d

Using the hypothesis (11.19.3) and the condition (A1) we have kx.t/k Mk.0/k C M

m X

Z ak C M

kD1

p

t

0

˛.s/.1 C kxŒs k/ds

"

C MM1 M2 b kx1 k C Mk.0/k C M

m X kD1

Mk.0/k C M

m X kD1

Z ak C M

0

t

Z ak C M

b

0

# ˛.s/.1 C kxŒs k/ds

˛.s/.1 C Nb kkB C Kb sup kx./k/ds 0 s

"

Z m X p ak C M C MM1 M2 b kx1 k C Mk.0/k C M kD1

0

b

˛.s/.1 C Nb kkB

350

11 Functional Differential Inclusions with Multi-valued Jumps

# CKb sup kx./k/ds 0s

C C M

Z

t 0

p Z ˛.s/ sup kx./k/ds C M 2 M1 M2 b 0s

b 0

˛.s/ sup kx./k/ds; 0 s

where p P C1 D M.1 C MM1 M2 b/ m kD1 ak ; p C2 D M k.0/k C M1 M2 b/Œkx1 k C Mk.0/k ; p C3 D M .1 C Nb kkB // .1 C MM1 M2 b/k˛kL1 .J/ ; and C D C1 C C2 C C3 : Consider the function .t/ D sup0t kx. /k, so the function Z t ˛.s/.s/ds (11.77) v.t/ D 0

is nondecreasing and we have v 0 .t/ D ˛.t/.t/;

for a.e. t 2 J:

(11.78)

Applying the last inequality p v 0 .t/ ˛.t/.C C M 2 M1 M2 bv.b/ C Mv.t//;

(11.79)

Rt multiplying both sides of (11.79) by the function L.t/ D exp M 0 ˛.s/ds , we obtain p v 0 .t/L.t/ ˛.t/L.t/.C C M 2 M1 M2 bv.b/ C Mv.t//: It follows that p .v.t/L.t//0 ˛.t/L.t/.C C M 2 M1 M2 bv.b//:

(11.80)

Integrating from 0 to b both sides of (11.80) we get Z v.b/ exp M

b 0

Z p ˛.s/ds .C C M 2 M1 M2 bv.b//

b 0

˛.t/L.t/dt:

11.6 Controllability of Impulsive Differential Evolution Inclusions...

351

Thus `v.b/ C

Z

b

0

˛.t/L.t/dt;

where ` be a constant such that Z b p Z ` D exp M ˛.s/ds M 2 M1 M2 b 0

b

0

˛.t/L.t/dt > 0:

Consequently v.b/

C

Rb 0

˛.t/L.t/dt D C: `

(11.81)

Using the nondecreasing character of v, we obtain v.t/ C;

for all t 2 J:

Then p kx.t/k C C M 1 C MM1 M2 b C:

t u

11.6.3 An Example As an application of our results we consider the following impulsive partial functional differential equation of the form @2 @ z.t; x/ 2 a.t; x/ 2 z.t; x/ C m.t/ @t @x

Z

t 1

.t; x; s t/ds;

x 2 Œ0; ; t 2 Œ0; b; t ¤ tk ; z.tkC ; x/

(11.82)

z.tk ; x/ 2 Œbk jz.tk ; x/; bk jz.tk ; x/;

x 2 Œ0; ; k D 1; : : : ; m; z.t; 0/ D z.t; / D 0; z.t; x/ D .t; x/;

(11.83) t 2 J WD Œ0; b;

1 < t 0; x 2 Œ0; ;

(11.84) (11.85)

where a.t; x/ is continuous function and uniformly Hölder continuous in t, bk > 0, k D 1; : : : ; m, 2 D,

352

11 Functional Differential Inclusions with Multi-valued Jumps

D D f W .1; 0 Œ0; ! RI is continuous everywhere except for a countable number of points at which .s /; .sC / exist with .s / D .s/g, 0 D t0 < t1 < t2 < < tm < tmC1 D b, z.tkC / D lim.h;x/!.0C ;x/ z.tk C h; x/, z.tk / D lim.h;x/!.0 ;x/ z.tk C h; x/, b W R R ! Pcv;k .R/ a Carathéodory multivalued map, W R ! RC . Let y.t/.x/ D z.t; x/;

x 2 Œ0; ; t 2 J D Œ0; b;

Ik .y.tk //.x/ D Œbk jz.tk ; x/; bk jz.tk ; x/; x 2 Œ0; ; k D 1; : : : ; m; Rt F.t; /.x/ D m.t/ 1 .t; x; s t/ds ./.x/ D .; x/;

1 < t 0; x 2 Œ0; :

Consider E D L2 Œ0; and define A.t/ by A.t/w D a.t; x/w00 with domain D.A/ D fw 2 E W w; w0 are absolutely continuous, w00 2 E; w.0/ D w./ D 0g. Then A.t/ generates an evolution system U.t; s/ satisfying assumption (11.19.1) and (11.19.3). Assume that B W U ! Y, U J is a bounded linear operator and the operator Z b Wu D T.t; s/Bu.s/ ds; 0 1

has a bounded inverse operator W W E ! L2 .J; U/= ker W. ˚ For the phase space, we choose B D Bh defined by Bh D W .1; 0 ! R0 E W For a > 0 ./ is bounded and measurable function onŒa; 0; and 1 h.s/ sups0 j./jd where h W .1; 0 ! .0; C1/ is a continuous function with Z

0

lD

1

h.s/ds < C1;

endowed with the norm Z kkh D

0 1

h.s/ sup j./jd: s0

Notice that the phase space Bh is Banach space (see [142] for more details). We can show that problem (11.26)–(11.29) is an abstract formulation of problem (11.15)–(11.17). Under suitable conditions, the problem (11.61)–(11.63) has at least one mild solution.

11.7 Notes and Remarks The results of Chap. 11 are taken from Benchohra et al. [68, 70]. Other results may be found in [7].

Chapter 12

Functional Differential Equations and Inclusions with Delay

12.1 Introduction In this chapter, we shall prove the existence of solutions of some classes of functional differential equations and inclusions. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axis R: We will use some fixed point theorems combined with the semigroup theory.

12.2 Global Existence for Functional Differential Equations with State-Dependent Delay 12.2.1 Introduction In this section we will use Schauder’s fixed point theorem combined with the semigroup theory to have the existence of solutions of the following functional differential equation with state-dependent delay: y0 .t/ D Ay.t/ C f .t; y .t;yt / /; a.e. t 2 J WD RC

(12.1)

y.t/ D .t/; t 2 .1; 0;

(12.2)

where f W J B ! E is given function, A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J; B is the phase space to be specified later, 2 B, W J B ! R, and .E; j:j/ is a real Banach space.

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_12

353

354

12 Functional Differential Equations and Inclusions with Delay

12.2.2 Existence of Mild Solutions Now we give our main existence result for problem (12.1)–(12.2). Before starting and proving this result, we give the definition of the mild solution. Definition 12.1. We say that a continuous function y W .1; C1/ ! E is a mild solution of problem (12.1)–(12.2) if y.t/ D .t/; t 2 .1; 0 and the restriction of y.:/ to the interval RC is continuous and satisfies the following integral equation: Z y.t/ D T.t/.0/ C 0

t

T.t s/f .s; y .s;ys / /ds; t 2 J:

Set R. / D f .s; / W .s; / 2 J B; .s; / 0g: We always assume that W J B ! R is continuous. Additionally, we introduce the following hypothesis: .H / The function t ! t is continuous from R. / into B and there exists a continuous and bounded function L W R. / ! .0; 1/ such that kt k L .t/kk for every t 2 R. /: Remark 12.2. The condition .H / is frequently verified by functions continuous and bounded. For more details, see for instance [142]. Lemma 12.3 ([140], Lemma 2.4). If y W R ! E is a function such that y0 D , then kys kB .M C L /kkB C l supfjy. /j W 2 Œ0; maxf0; sgg; s 2 R. / [ J; where L D

sup L .t/.

t2R. /

Let us introduce the following hypotheses: (12.3.1) A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J which is compact for t > 0 in the Banach space E. Let M 0 D supfkTkB.E/ W t 0g: (12.3.2) The function f W J B ! E is Carathéodory. (12.3.3) There exists a continuous function k W J ! RC such that: jf .t; u/ f .t; v/j k.t/ku vkB ; t 2 J; u; v 2 B;

12.2 Global Existence for Functional Differential Equations with . . .

and k WD sup t2J

Z

t 0

k.s/ds < 1:

355

(12.3)

(12.3.4) The function t ! f .t; 0/ D f0 2 L1 .J; RC / with F D kf0 kL1 : Theorem 12.4. Assume that (12.3.1)–(12.3.4), .H / hold. If k M 0 l < 1; then the problem (12.1)–(12.2) has at least one mild solution on BC. Proof. Consider the operator N W BC ! BC defined by: 8 if t 2 .1; 0, ˆ < .t/; Z t .Ny/.t/ D ˆ : T.t/ .0/ C T.t s/ f .s; y .s;ys / / ds; if t 2 J: 0

Let x.:/ W R ! E be the function defined by: 8 < .t/; if t 2 .1; 0; x.t/ D : T.t/ .0/; if t 2 J; then x0 D : For each z 2 BC with z.0/ D 0, we denote by z the function 8 < 0; if t 2 .1; 0; z.t/ D : z.t/; if t 2 J: If y satisfies y.t/ D .Ny/.t/, we can decompose it as y.t/ D z.t/ C x.t/; t 2 J, which implies yt D zt C xt for every t 2 J and the function z.:/ satisfies Z

t

z.t/ D 0

T.t s/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds; t 2 J:

Set BC00 D fz 2 BC0 W z.0/ D 0g and let kzkBC00 D supfjz.t/j W t 2 Jg; z 2 BC00 : BC00 is a Banach space with the norm k:kBC00 : We define the operator A W BC00 ! BC00 by: Z A.z/.t/ D

0

t

T.t s/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds; t 2 J:

356

12 Functional Differential Equations and Inclusions with Delay

We shall show that the operator A satisfies all conditions of Schauder’s fixed point theorem. The operator A maps BC00 into BC00 ; indeed the map A.z/ is continuous on RC for any z 2 BC00 , and for each t 2 J we have jA.z/.t/j M 0 M0

Z Z

t 0 t 0

0

jf .s; z .s;zs Cxs / C x .s;zs Cxs / / f .s; 0/ C f .s; 0/jds jf .s; 0/jds C M 0

M F CM

0

Z 0

t

Z 0

t

k.s/kz .s;zs Cxs / C x .s;zs Cxs / kB ds

k.s/.ljz.s/j C .m C L C lM 0 H/kkB /ds:

Set C WD .m C L C lM 0 H/kkB : Then, we have jA.z/.t/j M 0 F C M 0 C

Z

t

k.s/ds C M 0

0

Z

t

ljz.s/jk.s/ds 0

M 0 F C M 0 Ck C M 0 lkzkBC00 k : Hence, A.z/ 2 BC00 : Moreover, let r > 0 be such that r

M 0 F C M 0 Ck ; 1 M 0 k l

and Br be the closed ball in BC00 centered at the origin and of radius r: Let z 2 Br and t 2 RC : Then jA.z/.t/j M 0 F C M 0 Ck C M 0 k lr: Thus kA.z/kBC00 r; which means that the operator A transforms the ball Br into itself. Now we prove that A W Br ! Br satisfies the assumptions of Schauder’s fixed theorem. The proof will be given in several steps. Step 1: A is continuous in Br . Let fzn g be a sequence such that zn ! z in Br : At the first, we study the convergence of the sequences .zn .s;zn / /n2 N; s 2 J: s

12.2 Global Existence for Functional Differential Equations with . . .

357

If s 2 J is such that .s; zs / > 0, then we have, kzn .s;zns / z .s;zs / kB kzn .s;zns / z .s;zns / kB C kz .s;zns / z .s;zs / kB Lkzn zkB C kz .s;zns / z .s;zs / kB ; which proves that zn .s;zn / ! z .s;zs / in B as n ! 1 for every s 2 J such that s .s; zs / > 0. Similarly, is .s; zs / < 0, we get n kzn .s;zns / z .s;zs / kB D k .s;z n / .s;zs / kB D 0 s

which also shows that zn .s;zn / ! z .s;zs / in B as n ! 1 for every s 2 J such that s .s; zs / < 0. Combining the pervious arguments, we can prove that zn .s;zs / ! for every s 2 J such that .s; zs / D 0: Finally, jA.zn /.t/ A.z/.t/j Z t M0 jf .s; zn .s;zns Cxs / C x .s;zns Cxs / / f .s; z .s;zs Cxs / C x .s;zs Cxs / /jds Z

M0

0

t 0

jf .s; z .s;zns Cxs / C x .s;zns Cxs / / f .s; z .s;zs Cxs / C x .s;zs Cxs / /jds:

Then by .12:3:2/ we have f .s; zn .s;zns Cxs / C x .s;zns Cxs / / ! f .s; z .s;zs Cxs / C x .s;zs Cxs / /; as n ! 1; and by the Lebesgue dominated convergence theorem we get, kA.zn / A.z/kBC00 ! 0; as n ! 1: Thus A is continuous. Step 2: A.Br / Br this is clear. Step 3: A.Br / is equi-continuous on every compact interval Œ0; b of RC for b > 0. Let 1 ; 2 2 Œ0; b with 2 > 1 ; we have: jA.z/. 2 / A.z/. 1 /j Z 1 kT. 2 s/ T. 1 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / /jds 0

Z

2

C Z 0

1

1

kT. 2 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / /jds

kT. 2 s/ T. 1 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / / f .s; 0/jds

Z

1

C 0

kT. 2 s/ T. 1 s/kB.E/ jf .s; 0/jds

358

12 Functional Differential Equations and Inclusions with Delay

Z

2

C Z

2

C Z

kT. 2 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / / f .s; 0/jds

1

kT. 2 s/kB.E/ jf .s; 0/jds

1

1

C 0

kT. 2 s/ T. 1 s/kB.E/ k.s/ds

Z

1

CrL 0

Z

1

C

kT. 2 s/ T. 1 s/kB.E/ jf .s; 0/jds

0

Z

2

CC

1

Z CrL Z C

2

1

2

1

kT. 2 s/ T. 1 s/kB.E/ k.s/ds

kT. 2 s/kB.E/ k.s/ds kT. 2 s/kB.E/ k.s/ds

kT. 2 s/kB.E/ jf .s; 0/jds:

When 2 ! 1 , the right-hand side of the above inequality tends to zero, since T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0 implies the continuity in the uniform operator topology (see [168]), this proves the equi-continuity. Step 4: A.Br /.t/ is relatively compact on every compact interval of t 2 Œ0; 1/. Let t 2 Œ0; b for b > 0 and let " be a real number satisfying 0 < " < t: For z 2 Br we define Z A" .z/.t/ D T."/ Note that the set Z t" 0

t" 0

T.t s "/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds:

T.t s "/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds W z 2 Br

is bounded. Since T.t/ is a compact operator for t > 0, the set, fA" .z/.t/ W z 2 Br g is precompact in E for every ", 0 < " < t. Moreover, for every z 2 Br we have

12.2 Global Existence for Functional Differential Equations with . . .

359

jA.z/.t/ A" .z/.t/j Z t T.t s/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds t"

M0 F " C M0 C

Z

t

k.s/ds C rM 0

Z

lk.s/ds;

t"

! 0 as

t t"

" ! 0:

Therefore, the set fA.z/.t/ W z 2 Br g is precompact, i.e., relatively compact. Step 5: A.Br / is equi-convergent. Let t 2 RC and z 2 Br , we have, Z t jf .s; z .s;zs Cxs / C x .s;zs Cxs / /jds jA.z/.t/j M 0 0

0

0

Z

M F CM C

t

0

Z

0

M0 F C M0 C

Z

t

k.s/ds C M r

Lk.s/ds 0

t

k.s/ds C M 0 rl

0

Z

t

k.s/ds: 0

Then by .12:3:4/; we have jA.z/.t/j ! M M 0 F C M 0 Ck C M 0 rlk ; as t ! C1: Hence, jA.z/.t/ A.z/.C1/j ! 0; as t ! C1: As a consequence of Steps 1–4, with Lemma 1.26, we can conclude that A W Br ! Br is continuous and compact. From Schauder’s theorem, we deduce that A has a fixed point z : Then y D z C x is a fixed point of the operators N; which is a mild solution of the problem (12.1)–(12.2). t u

12.2.3 An Example Consider the following functional partial differential equation Z Z 0 @2 @ z.t; x/ 2 z.t; x/ D et z s 1 .t/2 a. /jz.t; /j2 d ; x ds; @t @x 1 0 x 2 Œ0; ; t 2 RC z.t; 0/ D z.t; / D 0; t 2 RC ; z.; x// D z0 .; x/; t 2 .1; 0; x 2 Œ0; ;

(12.4) (12.5) (12.6)

360

12 Functional Differential Equations and Inclusions with Delay

where z0 © 0. Set Z f .t; /.x/ D

0

1

et .s; x/ds;

and Z .t; / D t 1 .t/2

0

2

a2 . /j .t; /j d ;

i W RC ! RC ; i D 1; 2 and a W R ! R are continuous functions. Take E D L2 Œ0; and define A W E ! E by A! D ! 00 with domain D.A/ D f! 2 E; !; ! 0 are absolutely continuous; ! 00 2 E; !.0/ D !./ D 0g: Then A! D

1 X

n2 .!; !n /!n ; ! 2 D.A/

nD1

q

2 sin ns; n D 1; 2; : : : is the orthogonal set of eigenvectors in where !n .s/ D A: It is well known (see [168]) that A is the infinitesimal generator of an analytic semigroup T.t/; t 0 in E and is given by

T.t/! D

1 X

exp.n2 t/.!; !n /!n ; ! 2 E:

nD1

Since the analytic semigroup T.t/ is compact, there exists a positive constant M such that kT.t/kB.E/ M: Let B D BCU.R ; E/ and 2 B; then .H /: The function f .t; /.x/ is Carathéodory, and jf .t;

1 /.x/

f .t;

2 /.x/j

et j

1 .t; x/

2 .t; x/j;

thus k.t/ D et , moreover we have k D sup

Z

t 0

es ds; t 2 RC D 1; f0 0:

12.3 Global Existence Results for Neutral Functional Differential Equations. . .

361

Then the problem (12.1)–(12.2) is an abstract formulation of the problem (12.4)– (12.6), and conditions (12.3.1)–(12.3.4), .H / are satisfied. Theorem 12.4 implies that the problem (12.4)–(12.6) has at least one mild solutions on BC:

12.3 Global Existence Results for Neutral Functional Differential Equations with State-Dependent Delay 12.3.1 Introduction In this section we prove the existence solutions of a functional differential equation. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axis R. We will use Schauder’s fixed point theorem combined with the semigroup theory to have the existence of solutions of the following functional differential equation with state-dependent delay: d Œy.t/ g.t; y .t;yt / / D AŒy.t/ g.t; y .t;yt / / C f .t; y .t;yt / /; a.e. t 2 J WD RC dt (12.7) y.t/ D .t/; t 2 .1; 0;

(12.8)

where f ; g W J B ! E are given functions, A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J; B is the phase space to be specified later, 2 B, W J B ! R, and .E; j:j/ is a real Banach space.

12.3.2 Existence of Mild Solutions Definition 12.5. We say that a continuous function y W .1; C1/ ! E is a mild solution of problem (11.15)–(11.16) if y.t/ D .t/; t 2 .1; 0 and the restriction of y.:/ to the interval RC is continuous and satisfies the following integral equation: Z t T.t s/f .s; y .s;ys / /ds; t 2 J: y.t/ D T.t/Œ.0/ g.0; .0// C g.t; y .t;yt / / C 0

Set R. / D f .s; / W .s; / 2 J B; .s; / 0g:

362

12 Functional Differential Equations and Inclusions with Delay

We always assume that W J B ! R is continuous. Additionally, we introduce the following hypothesis: .H / The function t ! t is continuous from R. / into B and there exists a continuous and bounded function L W R. / ! .0; 1/ such that kt k L .t/kk for every t 2 R. /: Remark 12.6. The condition .H / is frequently verified by functions continuous and bounded. Lemma 12.7 ([139]). If y W R ! E is a function such that y0 D , then kys kB .M C L /kkB C l supfjy. /j W 2 Œ0; maxf0; sgg; s 2 R. / [ J; where L D

sup L .t/.

t2R. /

Let us introduce the following hypotheses: (12.7.1) A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J which is compact for t > 0 in the Banach space E. Let M 0 D supfkTkB.E/ W t 0g: (12.7.2) The function f W J B ! E is Carathéodory. (12.7.3) There exists a continuous function k W J ! RC such that: jf .t; u/ f .t; v/j k.t/ku vkB ; t 2 J; u; v 2 B and k WD sup

Z

t2J

t 0

k.s/ds < 1:

(12.7.4) The function t ! f .t; 0/ D f0 2 L1 .J; RC / with F D kf0 kL1 : (12.7.5) The function g.t; / is continuous on J and there exists a constant kg > 0 such that jg.t; u/ g.t; v/j kg ku vkB ; for each; u; v 2 B and g WD sup jg.t; 0/j < 1: t2J

(12.7.6) For each t 2 J and any bounded set B B, the set fg.t; u/ W u 2 Bg is relatively compact in E (12.7.7) For any bounded set B B, the function ft ! g.t; yt / W y 2 Bg is equicontinuous on each compact interval of RC .

12.3 Global Existence Results for Neutral Functional Differential Equations. . .

363

Remark 12.8. By the condition .12:7:3/; .12:7:4/ we deduce that jf .t; y/j k.t/kukB C F ; t 2 J; u 2 B; and by .12:7:5/ we deduce that: jg.t; u/j kg kukB C g t 2 J; u 2 B: Theorem 12.9. Assume that (12.7.1)–(12.7.7) and .H / hold. If l.M 0 k C ˛1 / < 1; then the problem (12.1)–(12.2) has at least one mild solution on BC. Proof. Transform the problem (12.1)–(12.2) into a fixed point problem. Consider the operator N W BC ! BC defined by: 8 ˆ if t 2 .1; 0, ˆ .t/I ˆ ˆ ˆ < .Ny/.t/ D T.t/ Œ.0/ g.0; .0// ˆ Z t ˆ ˆ ˆ ˆ : Cg.t; y .t;yt / / C T.t s/ f .s; y .s;ys / / dsI if t 2 J: 0

Let x.:/ W R ! E be the function defined by: 8 < .t/I if t 2 .1; 0; x.t/ D : T.t/ .0/I if t 2 J; then x0 D : For each z 2 BC with z.0/ D 0, we denote by z the function 8 < 0I if t 2 .1; 0; z.t/ D : z.t/I if t 2 J: If y satisfies y.t/ D .Ny/.t/, we can decompose it as y.t/ D z.t/ C x.t/; t 2 J, which implies yt D zt C xt for every t 2 J and the function z.:/ satisfies z.t/ D g.t; z .t;zt Cxt / C x .t;zt Cxt / / T.t/g.0; .0// Z t C T.t s/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds; t 2 J: 0

Set BC00 D fz 2 BC0 W z.0/ D 0g and let kzkBC00 D supfjz.t/j W t 2 Jg; z 2 BC00 :

364

12 Functional Differential Equations and Inclusions with Delay

BC00 is a Banach space with the norm k:kBC00 : We define the operator A W BC00 ! BC00 by: A.z/.t/ D g.t; z .t;zt Cxt / C x .t;zt Cxt / / T.t/g.0; .0// Z t C T.t s/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds; t 2 J: 0

We shall show that the operator A satisfies all conditions of Schauder’s fixed point theorem. The operator A maps BC00 into BC00 ; indeed the map A.z/ is continuous on RC for any z 2 BC00 , and for each t 2 J we have jA.z/.t/j jg.t; z .t;zt Cxt / C x .t;zt Cxt / /j C M 0 jg.0; .0//j Z t CM 0 jf .s; z .s;zs Cxs / C x .s;zs Cxs / / f .s; 0/ C f .s; 0/jds 0

M 0 .kg kkB C g / C kg kz .t;zt Cxt / C x .t;zt Cxt / kB C g Z t Z t CM 0 jf .s; 0/jds C M 0 k.s/kz .s;zs Cxs / C x .s;zs Cxs / kB ds 0

0

0

M .kg kkB C g / C kg .ljz.t/j C .m C L C lM 0 H/kkB / C g Z t 0 0 CM F C M k.s/.ljz.s/j C .m C L C lM 0 H/kkB /ds: 0

Set C1 WD .m C L C lM 0 H/kkB : C2 WD M 0 .kg kkB C g / C kg .m C L C lM 0 H/kkB C g C M 0 F : Then, we have 0

jA.z/.t/j C2 C kg ljz.t/j C M C1

Z

t

k.s/ds C M

0

0

Z

t

ljz.s/jk.s/ds 0

C2 C kg lkzkBC00 C M 0 Ck C M 0 lkzkBC00 k : Hence, A.z/ 2 BC00 : Moreover, let r > 0 be such that r

C2 C M 0 Ck ; 1 l.M 0 k C ˛1 /

12.3 Global Existence Results for Neutral Functional Differential Equations. . .

365

and Br be the closed ball in BC00 centered at the origin and of radius r: Let z 2 Br and t 2 RC : Then jA.z/.t/j C2 C kg lr C M 0 Ck C M 0 k lr: Thus kA.z/kBC00 r; which means that the operator A transforms the ball Br into itself. Now we prove that A W Br ! Br satisfies the assumptions of Schauder’s fixed theorem. The proof will be given in several steps. Step 1: A is continuous in Br . Let fzn g be a sequence such that zn ! z in Br : At the first, we study the convergence of the sequences .zn .s;zn / /n2 N; s 2 J: s If s 2 J is such that .s; zs / > 0, then we have, kzn .s;zns / z .s;zs / kB kzn .s;zns / z .s;zns / kB C kz .s;zns / z .s;zs / kB lkzn zkBr C kz .s;zns / z .s;zs / kB ; which proves that zn .s;zn / ! z .s;zs / in B as n ! 1 for every s 2 J such that s .s; zs / > 0. Similarly, is .s; zs / < 0, we get n kzn .s;zns / z .s;zs / kB D k .s;z n / .s;zs / kB D 0 s

which also shows that zn .s;zn / ! z .s;zs / in B as n ! 1 for every s 2 J such that s .s; zs / < 0. Combining the pervious arguments, we can prove that zn .s;zs / ! for every s 2 J such that .s; zs / D 0: Finally, jA.zn /.t/ A.z/.t/j jg.t; zn .t;znt Cxt / C x .t;znt Cxt / / g.t; z .t;zt Cxt / C x .t;zt Cxt / /j Z t 0 CM jf .s; zn .s;zns Cxs / C x .s;zns Cxs / / f .s; z .s;zs Cxs / C x .s;zs Cxs / /jds 0

jg.t; zn .s;zns Cxs / C x .s;zns Cxs / / g.t; z .s;zs Cxs / C x .s;zs Cxs / /j Z t jf .s; z .s;zns Cxs / C x .s;zns Cxs / / f .s; z .s;zs Cxs / C x .s;zs Cxs / /jds: CM 0 0

Then by .12:7:2/; .12:7:5/ we have f .s; zn .s;zns Cxs / C x .s;zns Cxs / / ! f .s; z .s;zs Cxs / C x .s;zs Cxs / /; as n ! 1; g.t; zn .t;znt Cxt / C x .t;znt Cxt / / ! g.t; z .t;zt Cxt / C x .t;zt Cxt / /; as n ! 1;

366

12 Functional Differential Equations and Inclusions with Delay

and by the Lebesgue dominated convergence theorem we get, kA.zn / A.z/kBC00 ! 0; as n ! 1: Thus A is continuous. Step 2: A.Br / Br : This is clear. Step 3: A.Br / is equi-continuous on every compact interval Œ0; b of RC for b > 0. Let 1 ; 2 2 Œ0; b with 2 > 1 ; we have: jA.z/. 2 / A.z/. 1 /j jg. 2 ; z . 2 ;z 2 Cx 2 / C x . 2 ;z 2 Cx 2 / / g. 1 ; z . 1 ;z 1 Cx 1 / C x . 1 ;z 1 Cx 1 / /j CkT. 2 / T. 1 /kB.E/ jg.0; .0//j Z 1 C kT. 2 s/ T. 1 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / /jds Z

0

2

C

kT. 2 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / /jds

1

jg. 2 ; z . 2 ;z 2 Cx 2 / C x . 2 ;z 2 Cx 2 / / g. 1 ; z . 1 ;z 1 Cx 1 / C x . 1 ;z 1 Cx 1 / /j CkT. 2 / T. 1 /kB.E/ .kg kkB C g / Z 1 C kT. 2 s/ T. 1 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / / f .s; 0/jds Z

0

1

C Z

2

C Z

kT. 2 s/ T. 1 s/kB.E/ jf .s; 0/jds

0

kT. 2 s/kB.E/ jf .s; z .s;zs Cxs / C x .s;zs Cxs / / f .s; 0/jds

1

2

C

kT. 2 s/kB.E/ jf .s; 0/jds

1

kg jg. 2 ; z . 2 ;z 2 Cx 2 / C x . 2 ;z 2 Cx 2 / / g. 1 ; z . 1 ;z 1 Cx 1 / C x . 1 ;z 1 Cx 1 / /j CkT. 2 / T. 1 /kB.E/ .kg kkB C g / Z 1 CC1 kT. 2 s/ T. 1 s/kB.E/ k.s/ds Z CrL

0

1

0

Z

1

C 0

kT. 2 s/ T. 1 s/kB.E/ k.s/ds

kT. 2 s/ T. 1 s/kB.E/ jf .s; 0/jds

12.3 Global Existence Results for Neutral Functional Differential Equations. . .

Z CC1 Z

1

CrL Z C

2

2

1

2

1

367

kT. 2 s/kB.E/ k.s/ds kT. 2 s/kB.E/ k.s/ds

kT. 2 s/kB.E/ jf .s; 0/jds:

When 2 ! 1 , the right-hand side of the above inequality tends to zero. Since .12:7:7/ and T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0; implies the continuity in the uniform operator topology (see [168]), this proves the equi-continuity. Step 4: The set A.Br /.t/ is relatively compact on every compact interval of Œ0; 1/. Let t 2 Œ0; b for b > 0 and let " be a real number satisfying 0 < " < t: For z 2 Br we define A" .z/.t/ D g.t; z .t;zt Cxt / C x .t;zt Cxt / // T."/.T.t "/g.0; .0/// Z t" CT."/ T.t s "/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds: 0

Note that the set fg.t; z .t;zt Cxt / C x .t;zt Cxt / / T.t "/g.0; .0// Z t" C T.t s "/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds W z 2 Br g 0

is bounded. jg.t; z .t;zt Cxt / C x .t;zt Cxt / / T.t "/g.0; .0//: Z t" C T.t s "/f .s; z .s;zs Cxs / C x .s;zs Cxs / /dsj r 0

Since T.t/ is a compact operator for t > 0, and .12:7:6/ we have that the set, fA" .z/.t/ W z 2 Br g is precompact in E for every ", 0 < " < t. Moreover, for every z 2 Br we have jA.z/.t/ A" .z/.t/j Z t T.t s/f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds t"

368

12 Functional Differential Equations and Inclusions with Delay

0

0

M F "CM C

Z

t

k.s/ds C rM

t"

!0

0

Z

t

lk.s/ds; t"

" ! 0:

as

Therefore, the set fA.z/.t/ W z 2 Br g is precompact, i.e., relatively compact. Step 5: A.Br / is equi-convergent. Let t 2 RC and z 2 Br , we have, jA.z/.t/j jg.t; z .t;zt Cxt / C x .t;zt Cxt / /j C M 0 jg.0; .0//j Z t CM 0 jf .s; z .s;zs Cxs / C x .s;zs Cxs / /jds 0

0

C2 C kg lr C M C

Z

t

0

Z

t

k.s/ds C M rl

0

k.s/ds: 0

Then we have jA.z/.t/j ! C3 C2 C kg lr C M 0 Ck C M 0 lrk ; as t ! C1: Hence, jA.z/.t/ A.z/.C1/j ! 0; as t ! C1: As a consequence of Steps 1–5, we can conclude that A W Br ! Br is continuous and compact. From Schauder’s theorem, we deduce that A has a fixed point z : Then y D z C x is a fixed point of the operators N; which is a mild solution of the problem (12.7)–(12.8). t u

12.3.3 An Example Consider the following neutral functional partial differential equation: @2 @ Œz.t; x/ g.t; z.t .t; z.t; 0//; x// D 2 Œz.t; x/ g.t; z.t .t; z.t; 0//; x// @t @x f .t; z.t .t; z.t; 0//; x//; x 2 Œ0; ; t 2 RC

(12.9)

z.t; 0/ D z.t; / D 0; t 2 RC ;

(12.10)

z.; x/ D z0 .; x/; t 2 .1; 0; x 2 Œ0; ;

(12.11)

where f ; g is a given functions, and W R ! RC : Take E D L2 Œ0; and define A W E ! E by A! D ! 00 with domain D.A/ D f! 2 E; !; ! 0 are absolutely continuous; ! 00 2 E; !.0/ D !./ D 0g:

12.4 Global Existence Results for Functional Differential Inclusions with Delay

369

Then A! D

1 X

n2 .!; !n /!n ; ! 2 D.A/;

nD1

q

where !n .s/ D 2 sin ns; n D 1; 2; : : : is the orthogonal set of eigenvectors in A: It is well known that A is the infinitesimal generator of an analytic semigroup T.t/; t 0 in E and is given by T.t/! D

1 X

exp.n2 t/.!; !n /!n ; ! 2 E:

nD1

Since the analytic semigroup T.t/ is compact for t > 0, there exists a positive constant M such that kT.t/kB.E/ M: Let B D BCU.R ; E/ and 2 B; then .H /; where .t; '/ D t .'/: Hence, the problem (12.1)–(12.2) is an abstract formulation of the problem (12.9)–(12.11), and if the conditions (12.3.1)–(12.3.6), .H / are satisfied. Theorem 12.9 implies that the problem (12.9)–(12.11) has at least one mild solutions on BC:

12.4 Global Existence Results for Functional Differential Inclusions with Delay 12.4.1 Introduction In this section we are going to prove the existence of solutions of a class of semilinear functional evolution inclusion with delay. Our investigations will be situated in the Banach space of real continuous and bounded functions on the real half axis RC . We will use Bohnenblust–Karlin’s fixed theorem, combined with the Corduneanu’s compactness criteria. More precisely, we will consider the following problem y0 .t/ Ay.t/ 2 F.t; yt /; a.e. t 2 J WD RC

(12.12)

y.t/ D .t/; t 2 H;

(12.13)

where F W J C.H; E/ ! P.E/ is a multi-valued map with nonempty compact values, P.E/ is the family of all nonempty subsets of E; A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J; W H ! E is given continuous function, and .E; j:j/ is a real Banach space.

370

12 Functional Differential Equations and Inclusions with Delay

12.4.2 Existence of Mild Solutions Let us introduce the following hypotheses: (12.5.1) A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J which is compact for t > 0 in the Banach space E. Let M D supfkT.t/kB.E/ W t 0g: (12.5.2) The multi-function F W J C.H; E/ ! P.E/ is Carathéodory with compact and convex values. (12.5.3) There exists a continuous function k W J ! RC such that: Hd .F.t; u/; F.t; v// k.t/ku vk; for each t 2 J and for all u; v 2 C.H; E/ and d.0; F.t; 0// k.t/; with k WD sup t2J

Z

t 0

k.s/ds < 1:

(12.14)

Theorem 12.10. Assume that (12.5.1)–(12.5.3) hold. If k M < 1; then the problem (12.12)–(12.13) has at least one mild solution on BC: Proof. Consider the multi-valued operator N W BC ! P.BC/ defined by: 8 8 .t/; ˆ ˆ ˆ ˆ ˆ ˆ < < N.y/ WD h 2 BC W h.t/ D T.t/.0/ Z t ˆ ˆ ˆ ˆ ˆ ˆ :C : T.t s/ f .s/ ds; f 2 SF;y 0

9 if t 2 H; > > > = > > > if t 2 J: ; (12.15)

The operator N maps BC into BCI for any y 2 BC, and h 2 N.y/ and for each t 2 J, we have Z t jh.t/j Mkk C M jf .s/jds Z

0

t

Mkk C M Z

0 t

Mkk C M 0

.k.s/kys k C kF.s; 0/k/ds k.s/.kys k C 1/ds

Mkk C M.kykBC C 1/k WD c: Hence, h.t/ 2 BC:

12.4 Global Existence Results for Functional Differential Inclusions with Delay

371

Moreover, let r > 0 be such that r MkkCMk ; and Br be the closed ball in BC 1Mk centered at the origin and of radius r: Let y 2 Br and t 2 RC : Then, jh.t/j Mkk C Mk C Mk r: Thus, khkBC r; which means that the operator N transforms the ball Br into itself. Now we prove that N W Br ! Br satisfies the assumptions of Bohnenblust– Karlin’s fixed theorem. The proof will be given in several steps. Step 1: We shall show that the operator N is closed and convex. This will be given in two claims. Claim 1: N.y/ is closed for each y 2 Br : Let .hn /n0 2 N.y/ such that hn ! hQ in Br : Then for hn 2 Br there exists fn 2 SF;y such that: Z t hn .t/ D T.t/.0/ C T.t s/fn .s/ds: 0

Since F has compact and convex values and from hypotheses .12:5:2/; .12:5:3/, an application of Mazur’s theorem [185] implies that we may pass to a subsequence if necessary to get that fn converges to f 2 L1 .J; E/ and hence f 2 SF;y : Then for each t 2 J; Q D T.t/.0/ C hn .t/ ! h.t/

Z

t 0

T.t s/f .s/ds:

So, hQ 2 N.y/: Claim 2: N.y/ is convex for each y 2 Br : Let h1 ; h2 2 N.y/; the there exists f1 ; f2 2 SF;y such that, for each t 2 J we have: Z hi .t/ D T.t/.0/ C

t

0

T.t s/fi .s/ds; i D 1; 2:

Let 0 ı 1: Then, we have for each t 2 J: Z .ıh1 C .1 ı/h2 /.t/ D T.t/.0/ C

0

t

T.t s/Œıf1 .s/ C .1 ı/f2 .s/ds:

Since F.t; y/ is convex, one has ıh1 C .1 ı/h2 2 N.y/:

372

12 Functional Differential Equations and Inclusions with Delay

Step 2: N.Br / Br this is clear. Step 3: N.Br / is equi-continuous on every compact interval Œ0; b of RC for b > 0: Let 1 ; 2 2 Œ0; b with 2 > 1 ; we have jh. 2 / h. 1 /j kT. 2 s/ T. 1 s/kB.E/ kk Z 1 C kT. 2 s/ T. 1 s/kB.E/ jf .s/jds Z

0

2

C

1

kT. 2 s/kB.E/ jf .s/jds

kT. 2 s/ T. 1 s/kB.E/ kk Z 1 C kT. 2 s/ T. 1 s/kB.E/ .k.s/kys k C jF.s; 0/j/ds Z C

0

2

1

kT. 2 s/kB.E/ .k.s/kys k C jF.s; 0/j/ds

kT. 2 s/ T. 1 s/kB.E/ kk Z 1 C.r C 1/ kT. 2 s/ T. 1 s/kB.E/ k.s/ds Z C.r C 1/

0

2

1

kT. 2 s/kB.E/ k.s/ds:

When 2 ! 1 , the right-hand side of the above inequality tends to zero, since T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0; implies the continuity in the uniform operator topology (see [168]). This proves the equi-continuity. Step 4: N.Br / is relatively compact on every compact interval of RC : Let t 2 Œ0; b for b > 0 and let " be a real number satisfying 0 < " < t: For y 2 Br ; let h 2 N.y/; f 2 SF;y and define Z h" .t/ D T.t/.0/ C T."/ Note that the set

Z

t"

T.t/.0/ C 0

t" 0

T.t s "/f .s/ds:

T.t s "/f .s/ds W y 2 Br

is bounded. Z

t"

jT.t/.0/ C 0

T.t s "/f .s/dsj r:

12.4 Global Existence Results for Functional Differential Inclusions with Delay

373

Since T.t/ is a compact operator for t > 0, the set, H" .t/ D fh" .t/ W h" 2 N.y/; y 2 Br g is precompact in E for every ", 0 < " < t. Moreover, for every y 2 Br we have Z jh.t/ h" .t/j M

t

jf .s/jds

t"

Z M

t

.k.s/kys k C jF.s; 0j/ds

t"

M.1 C r/

Z

t

k.s/ds t"

! 0 as

" ! 0:

Therefore, the set H.t/ D fh.t/ W h 2 N.y/; y 2 Br g is precompact, i.e., relatively compact. Hence the set H.t/ D fh.t/ W h 2 N.Br /g is relatively compact. Step 5: N has closed graph. Let fyn g be a sequence such that yn ! y ; hn 2 N.yn / and hn ! h : We shall show that h 2 N.y /: hn 2 N.yn / means that there exists fn 2 SF;yn such that Z hn .t/ D T.t/ .0/ C

t 0

T.t s/ fn .s/ ds; t 2 J:

We must prove that there exists f Z h .t/ D T.t/ .0/ C

t 0

T.t s/ f .s/ ds; t 2 J:

Consider the linear and continuous operator K W L1 .J; E/ ! BC defined by Z t T.t s/v.s/ds: K.v/.t/ D 0

We have jK.fn /.t/ K.f /.t/j D j.hn .t/ T.t/ .0// .h .t/ T.t/ .0//j D jhn .t/ h .t/j khn h k1 ! 0; as n ! 1: From Lemma 1.11 it follows that K ı SF is a closed graph operator and from the definition of K has hn .t/ T.t/.0/ 2 K ı SF;yn :

374

12 Functional Differential Equations and Inclusions with Delay

As yn ! y and hn ! h , there exist f 2 SF;y such that: Z h .t/ T.t/.0/ D

t 0

T.t s/ f .s/:

Hence the multi-valued operator N has closed graph, which implies that it is upper semi-continuous. Step 6: N.Br / is equi-convergent. Let h 2 N.y/, there exists f 2 SF;y such that for each t 2 RC and y 2 Br we have Z t jh.t/j Mkk C M jf .s/jds 0

Z

Mkk C Mk C Mr

t

k.s/ds 0

Mkk C Mk C Mrk : Then, jh.t/j ! l Mkk C Mk .1 C r/; as t ! C1: Hence, jh.t/ h.C1/j ! 0; as t ! C1: As a consequence of Steps 1 6, and Lemma 1.26, we conclude from Bohnenblust–Karlin’s theorem that N has a fixed point y which is a mild solution of the problem (12.12)–(12.13). t u

12.4.3 An Example Consider the functional partial differential inclusion @2 @ z.t; x/ 2 z.t; x/ 2 F.t; z.t r; x//; x 2 Œ0; ; t 2 J WD RC ; @t @x

(12.16)

z.t; 0/ D z.t; / D 0; t 2 J;

(12.17)

z.t; x/ D .t/; t 2 H; x 2 Œ0; ;

(12.18)

where F is a given multi-valued map. Take E D L2 Œ0; and define A W E ! E by A! D ! 00 with domain D.A/ D f! 2 EI !; ! 0 are absolutely continuous; ! 00 2 E; !.0/ D !./ D 0g:

12.5 Global Existence Results for Functional Differential Inclusions with. . .

375

Then, A! D

1 X

n2 .!; !n /!n ; ! 2 D.A/

nD1

q

where !n .s/ D 2 sin ns; n D 1; 2; : : :, is the orthogonal set of eigenvectors in A: It is well known (see [168]) that A is the infinitesimal generator of an analytic semigroup T.t/; t 0 in E and is given by T.t/! D

1 X

exp.n2 t/.!; !n /!n ; ! 2 E:

nD1

Since the analytic semigroup T.t/ is compact for t > 0, there exists a positive constant M such that kT.t/kB.E/ M: Then the problem (12.12)–(12.13) is the abstract formulation of the problem (12.16)–(12.18). If conditions (12.5.1)–(12.5.3) are satisfied, Theorem 12.10 implies that the problem (12.16)–(12.18) has at least one global mild solution on BC:

12.5 Global Existence Results for Functional Differential Inclusions with State-Dependent Delay 12.5.1 Introduction In this section we are going to prove the existence of solutions of a functional differential inclusion. Our investigations will be situated in the Banach space of real functions which are defined, continuous, and bounded on the real axis R. We will use Bohnenblust–Karlin’s fixed theorem, combined with the Corduneanu’s compactness criteria. More precisely we will consider the following problem: y0 .t/ Ay.t/ 2 F.t; y .t;yt / /; a.e. t 2 J WD RC y.t/ D .t/; t 2 .1; 0;

(12.19) (12.20)

where F W J B ! P.E/ is a multi-valued map with nonempty compact values, P.E/ is the family of all nonempty subsets of E; A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J; and .E; j:j/ is a real Banach space. B is the phase space, 2 B, W J B ! R.

376

12 Functional Differential Equations and Inclusions with Delay

12.5.2 Existence of Mild Solutions Now we give our main existence result for problem (12.19)–(12.20). Before starting and proving this result, we give the definition of the mild solution. Definition 12.11. We say that a continuous function y W .1; C1/ ! E is a mild solution of problem (12.19)–(12.20) if y.t/ D .t/ for all t 2 .1; 0; and the restriction of y./ to the interval J is continuous and there exists f ./ 2 L1 .J; E/: f .t/ 2 F.t; y .t;yt / / a.e. in J such that y satisfies the following integral equation Z t y.t/ D T.t/.t/ T.t s/ f .s/ ds for each t 2 J: (12.21) 0

Set R. / D f .s; / W .s; / 2 J B; .s; / 0g: We always assume that W J B ! R is continuous. Additionally, we introduce the following hypothesis: .H / The function t ! t is continuous from R. / into B and there exists a continuous and bounded function L W R. / ! .0; 1/ such that kt k L .t/kk for every t 2 R. /: Remark 12.12. The condition .H /, is frequently verified by functions continuous and bounded. Let us introduce the following hypotheses: (12.11.1) A W D.A/ E ! E is the infinitesimal generator of a strongly continuous semigroup T.t/; t 2 J which is compact for t > 0 in the Banach space E. Let M 0 D supfkTkB.E/ W t 0g: (12.11.2) The multi-function F W J B ! P.E/ is Carathéodory with compact and convex values. (12.11.3) There exists a continuous function k W J ! RC such that: Hd .F.t; u/; F.t; v// k.t/ ku vkB for each t 2 J and for all u; v 2 B and d.0; F.t; 0// k.t/ with k WD sup t2J

Z

t 0

k.s/ds < 1:

(12.22)

12.5 Global Existence Results for Functional Differential Inclusions with. . .

377

Theorem 12.13. Assume that (12.11.1)–(12.11.3),.H / hold. If k M 0 L < 1; then the problem (12.19)–(12.20) has at least one mild solution on BC. Proof. Consider the multi-valued operator N W BC ! P.BC/ defined by:

N.y/ WD

8 ˆ < ˆ :

h 2 BC W h.t/ D

8 ˆ < .t/; ˆ : T.t/ .0/ C

9 if t 2 .1; 0; > =

Z

t

0

T.t s/ f .s/ ds;

if t 2 J;

> ;

where f 2 SF;y .s;ys / : Let x./ W R ! E be the function defined by: x.t/ D

8 < .t/;

if t 2 .1; 0;

: T.t/ .0/; if t 2 J:

Then x0 D : For each z 2 BC with z.0/ D 0, we denote by z the function z.t/ D

8 < 0;

if t 2 .1; 0;

: z.t/; if t 2 J;

if y./ satisfies (12.21), we can decompose it as y.t/ D z.t/ C x.t/; t 2 J, which implies yt D zt C xt for every t 2 J and the function z./ satisfies Z

t

z.t/ D 0

T.t s/f .s/ds; t 2 J;

where f 2 SF;z .s;zs Cxs / Cx .s;zs Cxs / : Set BC00 D fz 2 BC0 W z.0/ D 0g and let kzkBC00 D supfjz.t/j W t 2 Jg; z 2 BC00 : BC00 is a Banach space with the norm k kBC00 :

We define the operator A W BC00 ! P.BC00 / by: 8 ˆ <

8 ˆ < 0; 0 A.z/ WD h 2 BC0 W h.t/ D Z t ˆ ˆ : : T.t s/ f .s/ ds; 0

9 if t 0; = > ; if t 2 J; >

378

12 Functional Differential Equations and Inclusions with Delay

where f 2 SF;z .s;zs Cxs / Cx .s;zs Cxs / :

The operator A maps BC00 into BC00 ; indeed the map A.z/ is continuous on RC for any z 2 BC00 , h 2 A.z/ and for each t 2 J we have jh.t/j M 0

Z

t

0

M

0

Z

t

0

M0

Z

t

jf .s/jds .k.s/kz .s;zs Cxs / C x .s;zs Cxs / kB C jF.s; 0/j/ds k.s/ds C M 0

0

M 0 k C M 0

Z

t

k.s/.Ljz.s/j C .M C L C LM 0 H/kkB /ds

0

Z

t 0

k.s/.Ljz.s/j C .M C L C LM 0 H/kkB /ds:

Set C WD .M C L C LM 0 H/kkB : Then, we have jh.t/j M 0 k C M 0 C

Z

t

k.s/ds C M 0

0

Z

t

Ljz.s/jk.s/ds 0

M 0 k C M 0 Ck C M 0 LkzkBC00 k : Hence, A.z/ 2 BC00 : Moreover, let r > 0 be such that r

M 0 k C M 0 Ck ; 1 M 0 k L

and Br be the closed ball in BC00 centered at the origin and of radius r: Let z 2 Br and t 2 RC : Then jh.t/j M 0 k C M 0 Ck C M 0 k Lr: Thus khkBC00 r; which means that the operator A transforms the ball Br into itself. Now we prove that A W Br ! P.Br / satisfies the assumptions of Bohnenblust– Karlin’s fixed theorem. The proof will be given in several steps.

12.5 Global Existence Results for Functional Differential Inclusions with. . .

379

Step 1: We shall show that the operator A is closed and convex. This will be given in several claims. Claim 1: A.z/ is closed for each z 2 Br : Let .hn /n0 2 A.z/ such that hn ! hQ in Br : Then for hn 2 Br there exists fn 2 SF;z .s;zs Cxs / Cx .s;zs Cxs / such that for each t 2 J; Z hn .t/ D

0

t

T.t s/fn .s/ds:

Using the fact that F has compact values and from hypotheses .12:11:2/; .12:11:3/ we may pass a subsequence if necessary to get that fn converges to f 2 L1 .J; E/ and hence f 2 SF;z .s;zs Cxs / Cx .s;zs Cxs / : Then for each t 2 J; Q D hn .t/ ! h.t/

Z

t 0

T.t s/f .s/ds:

So, hQ 2 A.z/: Claim 2: A.z/ is convex for each z 2 Br : Let h1 ; h2 2 A.z/; the there exists f1 ; f2 2 SF;z .s;zs Cxs / Cx .s;zs Cxs / such that, for each t 2 J we have: Z t T.t s/fi .s/ds; i D 1; 2: hi .t/ D 0

Let 0 ı 1: Then, we have for each t 2 J: Z t T.t s/Œıf1 .s/ C .1 ı/f2 .s/ds: .ıh1 C .1 ı/h2 /.t/ D 0

Since F has convex values, one has ıh1 C .1 ı/h2 2 A.z/ Step 2: A.Br / Br this is clear. Step 3: A.Br / is equi-continuous on every compact interval Œ0; b of RC for b > 0. Let 1 ; 2 2 Œ0; b; h 2 A.z/ with 2 > 1 ; we have: jh. 2 / h. 1 /j Z 1 kT. 2 s/ T. 1 s/kB.E/ jf .s/jds 0

C

Z

2

1

kT. 2 s/kB.E/ jf .s/jds

380

12 Functional Differential Equations and Inclusions with Delay

Z

1

0

Z

C Z C

kT. 2 s/ T. 1 s/kB.E/ .k.s/kz .s;zs Cxs / C x .s;zs Cxs / kB C jF.s; 0/j/ds

2

1

0

Z

kT. 2 s/ T. 1 s/kB.E/ k.s/ds

Z

CrL C

kT. 2 s/kB.E/ .k.s/kz .s;zs Cxs / C x .s;zs Cxs / kB C jF.s; 0/j/ds

1

1 0

1

kT. 2 s/ T. 1 s/kB.E/ k.s/ds

0

Z

CC

2

1

Z CrL Z C

kT. 2 s/ T. 1 s/kB.E/ k.s/ds

2

1

2

1

kT. 2 s/kB.E/ k.s/ds kT. 2 s/kB.E/ k.s/ds

kT. 2 s/kB.E/ k.s/ds:

When 2 ! 2 , the right-hand side of the above inequality tends to zero, since T.t/ is a strongly continuous operator and the compactness of T.t/ for t > 0 implies the continuity in the uniform operator topology (see [168]), this proves the equi-continuity. Step 4: A.Br / is relatively compact on every compact interval of Œ0; 1/. Let t 2 Œ0; b for b > 0 and let " be a real number satisfying 0 < " < t: For z 2 Br we define Z h" .t/ D T."/

t" 0

T.t s "/f .s/ds:

Note that the set Z

t"

0

T.t s "/f .s/ds W z 2 Br

is bounded. ˇZ ˇ ˇ ˇ

t" 0

ˇ ˇ T.t s "/f .s/dsˇˇ r:

Since T.t/ is a compact operator for t > 0, the set, fh" .t/ W z 2 Br g

12.5 Global Existence Results for Functional Differential Inclusions with. . .

381

is precompact in E for every ", 0 < " < t. Moreover, for every z 2 Br we have jh.t/ h" .t/j Z t 0 M jf .s/jds t"

M0

Z

t

k.s/ds C M 0 C

t"

!0

Z

t

k.s/ds C rM 0

t"

as

Z

t

Lk.s/ds; t"

" ! 0:

Therefore, the set fh.t/ W z 2 Br g is precompact, i.e., relatively compact. Step 5: A has closed graph. Let fzn g be a sequence such that zn ! z ; hn 2 A.zn / and hn ! h : We shall show that h 2 A.z /: hn 2 A.zn / means that there exists fn 2 SF;zn .s;zn Cx / Cx .s;zns Cxs / s s such that Z t hn .t/ D T.t s/ fn .s/ ds; 0

we must prove that there exists f Z t h .t/ D T.t s/ f .s/ ds: 0

Consider the linear and continuous operator K W L1 .J; E/ ! Br defined by Z t K.v/.t/ D T.t s/v.s/ds: 0

we have jK.fn /.t/ K.f /.t/j D jhn .t/ h .t/j khn h k1 ! 0; as n ! 1 From Lemma 2:2 it follows that K ı SF is a closed graph operator and from the definition of K has hn .t/ 2 K ı SF;zn .s;zn Cx / Cx .s;zns Cxs / : s

s

As zn ! z and hn ! h , there exist f 2 SF;z

.s;z s Cxs /

Z h .t/ D

t 0

Cx .s;z Cxs /

T.t s/ f .s/ds:

Hence the multi-valued operator A is upper semi-continuous.

such that:

382

12 Functional Differential Equations and Inclusions with Delay

Step 6: A.Br / is equi-convergent. Let z 2 Br , we have, for h 2 A.z/: jh.t/j M 0

Z

t 0

jf .s/jds

M 0 k C M 0 C

Z

t

k.s/ds C M 0 r

Z

0

0

0

Z

M k CM C

t

Lk.s/ds 0

t

0

Z

t

k.s/ds C M rL

0

k.s/ds: 0

Then by (12.22), we have jh.t/j ! l M 0 k .1 C C C rL/; as t ! C1: Hence, jh.t/ h.C1/j ! 0; as t ! C1: As a consequence of Steps 1–4, with Lemma 1.26, we can conclude that A W Br ! P.Br / is continuous and compact. From Bohnenblust–Karlin’s fixed theorem, we deduce that A has a fixed point z : Then y D z C x is a fixed point of the operators N; which is a mild solution of the problem (12.19)–(12.20). t u

12.5.3 An Example Consider the following functional partial differential equation @ z.t; x/ @t

@2 z.t; x/ @x2

2 F.t; z.t .t; z.t; 0//; x//

x 2 Œ0; ; t 2 RC

(12.23)

z.t; 0/ D z.t; / D 0; t 2 RC ;

(12.24)

z.; x/ D z0 .; x/; t 2 .1; 0; x 2 Œ0; ;

(12.25)

where F is a given multi-valued map, and W R ! RC is continuous. Take E D L2 Œ0; and define A W E ! E by A! D ! 00 with domain D.A/ D f! 2 E; !; ! 0 are absolutely continuous; ! 00 2 E; !.0/ D !./ D 0g: Then A! D

1 X nD1

n2 .!; !n /!n ; ! 2 D.A/

12.6 Notes and Remarks

383

q where !n .s/ D 2 sin ns; n D 1; 2; : : : is the orthogonal set of eigenvectors in A: It is well known (see [168]) that A is the infinitesimal generator of an analytic semigroup T.t/; t 0 in E and is given by T.t/! D

1 X

exp.n2 t/.!; !n /!n ; ! 2 E:

nD1

Since the analytic semigroup T.t/ is compact, there exists a positive constant M such that kT.t/kB.E/ M:

12.6 Notes and Remarks The results of Chap. 12 are taken from [2, 5, 46–49, 70]. Other results may be found in [139, 171].

Chapter 13

Second Order Functional Differential Equations with Delay

13.1 Introduction In this chapter, we present some existence of global mild solutions for some classes of second order semi-linear functional equations with delay.

13.2 Global Existence Results of Second Order Functional Differential Equations with Delay 13.2.1 Introduction In this section we provide sufficient conditions for the existence of global mild solutions for two classes of second order semi-linear functional equations with delay. Our investigations will be situated in the Banach space of real continuous and bounded functions on the real half axis RC . First, we will consider the following problem y00 .t/ D Ay.t/ C f .t; yt /I a.e. t 2 J WD RC 0

y.t/ D .t/I t 2 H; y .0/ D ';

(13.1) (13.2)

where f W J C.H; E/ ! E is given function, A W D.A/ E ! E is the infinitesimal generator of a strongly continuous cosine family of bounded linear

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7_13

385

386

13 Second Order Functional Differential Equations with Delay

operators .C.t//t2R ; on E; W H ! E is given continuous function, and .E; j:j/ is a real Banach space. Later, we consider the following problem y00 .t/ D Ay.t/ C f .t; y .t;yt / /I a.e. t 2 J WD RC 0

y.t/ D .t/ 2 BI y .0/ D ';

(13.3) (13.4)

where f W J B ! E is given function, A W D.A/ E ! E as in problem (11.25)– (7.1), 2 B, W J B ! R, and .E; j:j/ is a real Banach space. The main results are based upon Schauder’s fixed theorem combined with the family of cosine operators. Our purpose in this section is to consider a simultaneous generalization of the classical second order abstract Cauchy problem studied by Travis and Weeb in [179, 180]. Additionally, we observe that the ideas and techniques in this section permit the reformulation of the problems studied in [38, 67] to the context of partial second order differential equations.

13.2.2 Existing Result for the Finite Delay Case In this section by BC WD BC.Œr; C1// we denote the Banach space of all bounded and continuous functions from Œr; C1/ into R equipped with the standard norm kykBC D

jy.t/j:

sup t2Œr;C1/

Now we give our main existence result for problem (13.1)–(13.2). Before starting and proving this result, we give the definition of a mild solution. Definition 13.1. We say that a continuous function y W Œr; C1/ ! E is a mild solution of problem (11.25)–(7.1) if y.t/ D .t/; t 2 H; y.:/ and y0 .0/ D '; and Z t C.t s/f .s; ys /ds; t 2 J: y.t/ D C.t/.0/ C S.t/' C 0

Let us introduce the following hypotheses: (13.1.1) C.t/ is compact for t > 0 in the Banach space E. Let M D supfkCkB.E/ W t 0g; and M 0 D supfkSkB.E/ W t 0g: (13.1.2) The function f W J C.H; E/ ! E is Carathéodory. (13.1.3) There exists a continuous function k W J ! RC such that: jf .t; u/ f .t; v/j k.t/ku vk; t 2 J; u; v 2 C.H; E/ and

Z

t

k WD sup t2J

0

k.s/ds < 1:

13.2 Global Existence Results of Second Order Functional Differential. . .

387

(13.1.4) The function t ! f .t; 0/ D f0 2 L1 .J; RC / with F D kf0 kL1 : (13.1.5) For each bounded B BC and t 2 J the set: Z

t

fC.t/.0/ C S.t/' C 0

C.t s/f .s; yt /ds W y 2 Bg

is relatively compact in E: Theorem 13.2. Assume that (13.1.1)–(13.1.5) hold. If K M < 1; then the problem (13.1)–(13.2) has at least one mild solution on BC. Proof. Let the operator: N W BC ! BC be defined by:

.Ny/.t/ D

8 ˆ < .t/;

if t 2 H,

ˆ : C.t/ .0/ C S.t/' C

Z

t

0

C.t s/ f .s; ys / ds; if t 2 J:

The operator N maps BC into BCI indeed the map N.y/ is continuous on Œr; C1/ for any y 2 BC, and for each t 2 J, we have j.Ny/.t/j Mkk C M 0 k'k C M 0

Z Z

t 0 t

Mkk C M k'k C M 0

jf .s; ys / f .s; 0/ C f .s; 0/jds Z jf .s; 0/jds C M

Mkk C M 0 k'k C MF C M

Z

t

0

t 0

k.s/kys kds

k.s/kys kds

Mkk C M 0 k'k C MF C MkykBC k WD c: Let C D Mkk C M 0 k'k: Hence, N.y/ 2 BC: ; and Br be the closed ball in BC Moreover, let r > 0 be such that r CCMF 1Mk centered at the origin and of radius r: Let y 2 Br and t 2 RC : Then, j.Ny/.t/j C C MF C Mk r: Thus, kN.y/kBC r; which means that the operator N transforms the ball Br into itself.

388

13 Second Order Functional Differential Equations with Delay

Now we prove that N W Br ! Br satisfies the assumptions of Schauder’s fixed theorem. The proof will be given in several steps. Step 1: N is continuous in Br . Let fyn g be a sequence such that yn ! y in Br : We have Z j.Nyn /.t/ .Ny/.t/j M

0

t

jf .s; ysn / f .s; ys /jds:

Then by (13.1.2) we have f .s; ysn / ! f .s; ys /; as n ! 1; for a.e. s 2 J, and by the Lebesgue dominated convergence theorem we have k.Nyn / .Ny/kBC ! 0; as n ! 1: Thus, N is continuous. Step 2: N.Br / Br this is clear. Step 3: N.Br / is equi-continuous on every compact interval Œ0; b of RC for b > 0. Let 1 ; 2 2 Œ0; b with 2 > 1 ; we have jN.y/. 2 / N.y/. 1 /j kC. 2 s/ C. 1 s/kB.E/ kk C kS. 2 s/ S. 1 s/kB.E/ k'k Z 1 C kC. 2 s/ C. 1 s/kB.E/ jf .s; ys /jds Z

0

2

C

1

kC. 2 s/kB.E/ jf .s; ys /jds

kC. 2 s/ C. 1 s/kB.E/ kk C kS. 2 s/ S. 1 s/kB.E/ k'k Z 1 C kC. 2 s/ C. 1 s/kB.E/ jf .s; ys / f .s; 0/ C f .s; 0/jds Z

0

2

C

1

kC. 2 s/kB.E/ jf .s; ys / f .s; 0/ C f .s; 0/jds

kC. 2 s/ C. 1 s/kB.E/ kk C kS. 2 s/ S. 1 s/kB.E/ k'k Z 1 Cr kC. 2 s/ C. 1 s/kB.E/ k.s/ds 0

1

Z C 0

Z

2

Cr Z C

kC. 2 s/ C. 1 s/kB.E/ jf .s; 0/jds

1

2

1

kC. 2 s/kB.E/ k.s/ds

kC. 2 s/kB.E/ jf .s; 0/jds:

13.2 Global Existence Results of Second Order Functional Differential. . .

389

When 2 ! 2 , the right-hand side of the above inequality tends to zero, since C.t/; S.t/ are a strongly continuous operator and the compactness of C.t/; S.t/ for t > 0; implies the continuity in the uniform operator topology (see [179, 180]). This proves the equi-continuity. Step 4:N.Br / is relatively compact on every compact interval of Œ0; 1/ by .13:1:5/: Step 5: N.Br / is equi-convergent. Let y 2 Br , we have: Z

0

t

j.Ny/.t/j Mkk C M k'k C M C C MF C Mr

Z

0

jf .s; ys /jds

t

k.s/ds 0

C C MF C Mr

Z

t

k.s/ds: 0

Then j.Ny/.t/j ! C1 C C MF C Mk r; as t ! C1: Hence, j.Ny/.t/ .Ny/.C1/j ! 0; as t ! C1: As a consequence of Steps 1–5, with Lemma 1.26, we can conclude that N W Br ! Br is continuous and compact. From Schauder’s theorem, we deduce that N has a fixed point y which is a mild solution of the problem (13.1)–(13.2). t u

13.2.3 Existing Results for the State-Dependent Delay Case In this section by BC WD BC.R/ we denote the Banach space of all bounded and continuous functions from R into E equipped with the standard norm kykBC D sup jy.t/j: t2R

Finally, by BC0 WD BC0 .RC / we denote the Banach space of all bounded and continuous functions from RC into E equipped with the standard norm kykBC0 D sup jy.t/j: t2RC

390

13 Second Order Functional Differential Equations with Delay

Now we give our main existence result for problem (13.3)–(13.4). Before starting and proving this result, we give the definition of a mild solution. Definition 13.3. We say that a continuous function y W .1; C1/ ! E is a mild solution of problem (11.15)–(11.16) if y.t/ D .t/; t 2 .1; 0; y.:/ is continuously differentiable and y0 .0/ D ' and Z

t

y.t/ D C.t/.0/ C S.t/' C 0

C.t s/f .s; y .t;yt / /ds; t 2 J:

Set R. / D f .s; / W .s; / 2 J B; .s; / 0g: We always assume that W J B ! R is continuous. Additionally, we introduce the following hypothesis: .H / The function t ! t is continuous from R. / into B and there exists a continuous and bounded function L W R. / ! .0; 1/ such that kt k L .t/kk for every t 2 R. /: Remark 13.4. The condition .H / is frequently verified by functions continuous and bounded. Let us introduce the following hypotheses: (13.3.1) C.t/; S.t/ are compact for t > 0 in the Banach space E. Let M D supfkCkB.E/ W t 0g; and M 0 D supfkSkB.E/ W t 0g: (13.3.2) The function f W J B ! E is Carathéodory. (13.3.3) There exists a continuous function k W J ! RC such that: jf .t; u/ f .t; v/j k.t/ku vk; t 2 J; u; v 2 B and k WD sup t2J

Z

t 0

k.s/ds < 1:

(13.3.4) The function t ! f .t; 0/ D f0 2 L1 .J; RC / with F D kf0 kL1 : (13.3.5) For each bounded B BC0 and t 2 J the set: Z fS.t/' C 0

is relatively compact in E:

t

C.t s/f .s; y .t;yt / /ds W y 2 Bg

13.2 Global Existence Results of Second Order Functional Differential. . .

391

Theorem 13.5. Assume that (13.3.1)–(13.3.5),.H / hold. If K Ml < 1; then the problem (13.3)–(13.4) has at least one mild solution on BC. Proof. Consider the operator: N W BC ! BC define by:

.Ny/.t/ D

8 ˆ < .t/;

if t 2 .1; 0,

ˆ : C.t/ .0/ C S.t/' C

Z 0

t

C.t s/ f .s; y .t;yt / / ds; if t 2 J:

Let x.:/ W R ! E be the function defined by: x.t/ D

8 < .t/I

if t 2 .1; 0;

: C.t/ .0/I if t 2 J;

then x0 D : For each z 2 BC with z.0/ D 0; y0 .0/ D ' D z0 .0/ D '1 ; we denote by z the function z.t/ D

8 < 0I

if t 2 .1; 0;

: z.t/I if t 2 J:

If y satisfies y.t/ D .Ny/.t/, we can decompose it as y.t/ D z.t/ C x.t/; t 2 J, which implies yt D zt C xt for every t 2 J and the function z.:/ satisfies Z z.t/ D S.t/'1 C

t

0

C.t s/ f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds; t 2 J:

Set BC00 D fz 2 BC0 W z.0/ D 0g and let kzkBC00 D supfjz.t/j W t 2 Jg; z 2 BC00 : BC00 is a Banach space with the norm k:kBC00 : We define the operator A W BC00 ! BC00 by: Z A.z/.t/ D S.t/'1 C

t 0

C.t s/ f .s; z .s;zs Cxs / C x .s;zs Cxs / /ds; t 2 J:

We shall show that the operator A satisfies all conditions of Schauder’s fixed point theorem. The operator A maps BC00 into BC00 ; indeed the map A.z/ is continuous on RC for any z 2 BC00 , and for each t 2 J we have

392

13 Second Order Functional Differential Equations with Delay

Z

0

jA.z/.t/j M k'1 k C M M 0 k'1 k C M

t

0

Z

t

0

jf .s; z .s;zs Cxs / C x .s;zs Cxs / / f .s; 0/ C f .s; 0/jds Z jf .s; 0/jds C M

M 0 k'1 k C MF C M

Z

t 0

t

0

k.s/kz .s;zs Cxs / C x .s;zs Cxs / kB ds

k.s/.ljz.s/j C .m C L C lMH/kkB /ds:

Let C D .m C L C lMH/kkB : Then, we have: 0

jA.z/.t/j M k'1 k C MF C MC

Z

t

Z k.s/ds C Ml

0

t

k.s/jz.s/jds 0

M 0 k'1 k C MF C MCk C MlkzkBC00 k : Hence, A.z/ 2 BC00 : 0 CMCk Moreover, let r > 0 be such that r M k'1 kCMF ; and Br be the closed ball 1Mlk 0 in BC0 centered at the origin and of radius r: Let y 2 Br and t 2 RC : Then, jA.z/.t/j M 0 k'1 k C MF C MCk C Mlk r: Thus, kA.z/kBC00 r; which means that the operator N transforms the ball Br into itself. Now we prove that A W Br ! Br satisfies the assumptions of Schauder’s fixed theorem. The proof will be given in several steps. Step 1: A is continuous in Br . Let fzn g be a sequence such that zn ! z in Br : At the first, we study the convergence of the sequences .zn .s;zn / /n2 N; s 2 J: s If s 2 J is such that .s; zs / > 0, then we have, kzn .s;zns / z .s;zs / kB kzn .s;zns / z .s;zns / kB C kz .s;zns / z .s;zs / kB lkzn zkBr C kz .s;zns / z .s;zs / kB ; which proves that zn .s;zn / ! z .s;zs / in B as n ! 1 for every s 2 J such that s .s; zs / > 0. Similarly, is .s; zs / < 0, we get n kzn .s;zns / z .s;zs / kB D k .s;z n / .s;zs / kB D 0 s

13.2 Global Existence Results of Second Order Functional Differential. . .

393

which also shows that zn .s;zn / ! z .s;zs / in B as n ! 1 for every s 2 J such that s .s; zs / < 0. Combining the pervious arguments, we can prove that zn .s;zs / ! for every s 2 J such that .s; zs / D 0: Finally, jA.zn /.t/ A.z/.t/j Z t jf .s; zn .s;zns Cxs / C x .s;zns Cxs / / f .s; z .s;zs Cxs / C x .s;zs Cxs / /jds: M 0

Then by (13.3.2) we have f .s; zn .s;zns Cxs / C x .s;zns Cxs / / ! f .s; z .s;zs Cxs / C x .s;zs Cxs / /; as n ! 1; and by the Lebesgue dominated convergence theorem we get, kA.zn / A.z/kBC00 ! 0; as n ! 1: Thus A is continuous. Step 2: A.Br / Br this is clear. Step 3: A.Br / is equi-continuous on every compact interval Œ0; b of RC for b > 0. Let 1 ; 2 2 Œ0; b with 2 > 1 ; we have jA.z/. 2 / A.z/. 1 /j kS. 2 s/ S. 1 s/kB.E/ k'1 k Z 1 C kC. 2 s/ C. 1 s/kB.E/ jf .s; zn .s;zns Cxs / C x .s;zns Cxs / /jds Z

0

2

C

1

kC. 2 s/kB.E/ jf .s; zn .s;zns Cxs / C x .s;zns Cxs / /jds

kS. 2 s/ S. 1 s/kB.E/ k'1 k Z 1 C kC. 2 s/ C. 1 s/kB.E/ jf .s; zn .s;zns Cxs / C x .s;zns Cxs / / f .s; 0/jds Z

0

1

C Z C Z C

0

2

1

2

1

kC. 2 s/ C. 1 s/kB.E/ f .s; 0/jds kC. 2 s/kB.E/ jf .s; zn .s;zns Cxs / C x .s;zns Cxs / / f .s; 0/jds kC. 2 s/kB.E/ jf .s; 0/jds

kS. 2 s/ S. 1 s/kB.E/ k'1 k Z 1 CC kC. 2 s/ C. 1 s/kB.E/ k.s/ds 0

394

13 Second Order Functional Differential Equations with Delay

Z

1

Clr 0

1

Z C

kC. 2 s/ C. 1 s/kB.E/ jf .s; 0/jds

0

Z

2

CC

1

Z

2

Clr Z C

1

2

1

kC. 2 s/ C. 1 s/kB.E/ k.s/ds

kC. 2 s/kB.E/ k.s/ds kC. 2 s/kB.E/ k.s/ds

kC. 2 s/kB.E/ jf .s; 0/jds:

When 2 ! 2 , the right-hand side of the above inequality tends to zero, since C.t/ are a strongly continuous operator and the compactness of C.t/ for t > 0 implies the continuity in the uniform operator topology (see [179, 180]). This proves the equi-continuity. Step 4: N.Br / is relatively compact on every compact interval of Œ0; 1/. This is satisfied from (13.1.5). Step 5: N.Br / is equi-convergent. Let y 2 Br , we have: 0

jA.z/.t/j M k'1 k C M

Z

t 0

jf .s; zn .s;zns Cxs / C x .s;zns Cxs / /jds

M 0 k'1 k C MF C MCk C Mrl

Z

t

k.s/ds: 0

Then jA.z/.t/j ! C1 M 0 k'1 k C MF C Mk .C C lr/; as t ! C1: Hence, jA.z/.t/ A.z/.C1/j ! 0; as t ! C1: As a consequence of Steps 1–5, with Lemma 1.26, we can conclude that A W Br ! Br is continuous and compact. we deduce that A has a fixed point z : Then y D z C x is a fixed point of the operators N; which is a mild solution of the problem (13.3)–(13.4). t u

13.2 Global Existence Results of Second Order Functional Differential. . .

395

13.2.4 Examples Example 1. Consider the functional partial differential equation of second order: @2 z.t; x/ @t2

D

@2 z.t; x/ @x2

C f .t; z.t r; x//; x 2 Œ0; ; t 2 J WD RC ;

z.t; 0/ D z.t; / D 0; t 2 RC ; z.t; x/ D .t/;

@z.0;x/ @t

D w.x/; t 2 H; x 2 Œ0; ;

(13.5) (13.6) (13.7)

where f is a given map. Take E D L2 Œ0; and define A W E ! E by A! D ! 00 with domain D.A/ D f! 2 EI !; ! 0 are absolutely continuous; ! 00 2 E; !.0/ D !./ D 0g: It is well known that A is the infinitesimal generator of a strongly continuous cosine function .C.t//t2R on E, respectively. Moreover, A has discrete spectrum, the eigenvalues are n2 ; n 2 N with corresponding normalized eigenvectors zn . / WD 1 . 2 / 2 sin n ; and the following properties hold: (a) fzn W n 2 Ng is an orthonormal basis of E: P 2 (b) If y 2 E; then Ay D 1 n < y; zn > zn : P1 nD1 (c) For y 2 E; C.t/y D nD1 cos.nt/ < y; zn > zn ; and the associated sine family is S.t/y D

1 X sin.nt/ nD1

n

< y; zn > zn

which implies that the operator S.t/ is compact for all t > 0 and that kC.t/k D kS.t/k 1; for all t 0: (d) If ˚ denotes the group of translations on E defined b ˚.t/y./ D yQ . C t/ where yQ is the extension of y with period 2, then C.t/ D 12 .˚.t/ C ˚.t//I A D B2 ; where B is the infinitesimal generator of the group ˚ on X D fy 2 H 1 .0; / W y.0/ D x./ D 0g: Then the problem (13.1)–(13.2) is an abstract formulation of the problem (13.5)– (13.7). If conditions (13.1.1)–(13.1.5) are satisfied. Theorem 13.2 implies that the problem (13.5)–(13.7) has at least one mild solution on BC: Example 2. Take E D L2 Œ0; I B D C0 L2 .g; E/ and define A W E ! E by A! D ! 00 with domain D.A/ D f! 2 EI !; ! 0 are absolutely continuous; ! 00 2 E; !.0/ D !./ D 0g:

396

13 Second Order Functional Differential Equations with Delay

It is well known that A is the infinitesimal generator of a strongly continuous cosine function .C.t//t2R on E, respectively. Moreover, A has discrete spectrum, the eigenvalues are n2 ; n 2 N with corresponding normalized eigenvectors zn . / WD 1 . 2 / 2 sin n ; and the following properties hold: (a) fzn W n 2 Ng is an orthonormal basis of E: P 2 (b) If y 2 E; then Ay D P 1 nD1 n < y; zn > zn : (c) For y 2 E; C.t/y D 1 nD1 cos.nt/ < y; zn > zn ; and the associated sine family is S.t/y D

1 X sin.nt/ nD1

n

< y; zn > zn

which implies that the operator S.t/ is compact for all t > 0 and that kC.t/k D kS.t/k 1 for all t 2 R: (d) If ˚ denotes the group of translations on E defined by ˚.t/y./ D yQ . C t/; where yQ is the extension of y with period 2, then C.t/ D

1 .˚.t/ C ˚.t//; A D B2 ; 2

where B is the infinitesimal generator of the group ˚ on X D fy 2 H 1 .0; / W y.0/ D x./ D 0g: Consider the functional partial differential equation of second order: @2 @2 z.t; x/ D 2 z.t; x/ C 2 @t @x

Z

0 1

a.s t/z.s 1 .t/ 2 .kz.t/k/; x/ds;

x 2 Œ0; ; t 2 J WD RC ;

(13.8)

z.t; 0/ D z.t; / D 0; t 2 RC ;

(13.9)

z.t; x/ D .t/;

@z.0; x/ D !.x/; t 2 H; x 2 Œ0; ; @t

(13.10)

where W Œ0; 1/ ! Œ0; 1/; aI R ! R be continuous, and Lf D Z 0 2i a .s/ ds 12 < 1: Under these conditions, we define the function 1 g.s/ f W J B ! E; W J B ! R by Z 0 f .t; /.x/ D a.s/ .s; x/ds; 1

.s; / D s 1 .s/ 2 .k .0/k; we have kf .t; :/kB Lf :

13.3 Notes and Remarks

397

Then the problem (13.3)–(13.4) is an abstract formulation of the problem (13.8)– (13.10). If conditions (13.1.1)–(13.1.5) are satisfied. Theorem 12.4 implies that the problem (13.8)–(13.10) has at least one mild solution on BC:

13.3 Notes and Remarks The results of Chap. 13 are taken from Alaidarous et al. [21, 22] and Benchohra et al. [55–57, 82, 83]. Other results may be found in [111, 166].

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Index

A Arzelá-Ascoli, 58, 96, 150, 172, 195, 209, 230, 239, 252, 270, 286, 313, 321, 327, 334, 338, 347

B Banach space, 1, 2, 5, 7, 9–13, 53, 56, 62, 143, 156, 166, 182, 191, 192, 203, 224–226, 228, 237, 246, 261, 262, 265, 278, 305, 306, 317, 319, 324, 330, 332, 342, 344, 353, 355, 361, 362, 364, 369, 370, 375, 377, 386, 389–391 Bochner integrable, 1, 2, 319, 326, 332, 343

C Carathéodory, 2, 5, 114, 128, 144, 226, 263, 279, 294, 318, 330, 341, 352, 354, 360, 362, 370, 376, 386, 390 Closed graph operator, 6, 151, 173, 271, 287, 373, 381 Compact, 5, 11, 48, 54, 56, 58, 59, 63, 65, 69, 79, 84, 85, 87, 94–97, 102, 107, 108, 114, 117, 118, 120, 125, 128, 132, 133, 144, 146, 147, 149, 150, 159, 163, 166, 167, 169, 172, 177, 188, 192, 195, 196, 200, 202, 203, 206, 209, 225, 230, 231, 239, 245, 252, 261, 263, 267, 270, 277, 279, 282, 283, 286, 292, 293, 303, 305, 310, 311, 314, 318, 320, 322, 324, 326, 328, 330, 334, 336, 338, 340, 345, 358–360, 362, 366–369, 371–373, 375, 376, 379–382, 386, 387, 389, 390, 393–396

Condensing multi-function, 14, 331 Controllability, 18, 24, 53, 54, 62, 63, 68, 75, 76, 83, 84, 92, 156, 165, 182, 218, 342–344 Convex, 5, 13, 114, 118, 125, 128, 133, 141, 143, 144, 146, 147, 157, 159, 166, 167, 169, 170, 183, 185, 188, 262, 263, 267, 268, 277–279, 282–284, 292, 311, 318, 320, 325, 330, 333, 337, 345, 370, 371, 376, 379

D Densely defined operator, 7, 8, 192, 262

E Equi-continuous, 11, 57, 77, 85, 96, 147, 148, 169, 170, 178, 194, 207, 229, 238, 250, 267, 269, 283, 284, 310, 311, 357, 362, 366, 372, 379, 388, 393 Evolution, 7, 17, 18, 22–24, 29–32, 36–39, 42–44, 47, 48, 53, 60–62, 64, 67–69, 74–78, 82–85, 87, 91–94, 99, 100, 105, 106, 112–115, 118–121, 124, 125, 127, 129, 133–136, 140–142, 318, 324, 330, 342, 352, 369 Extrapolation method, 278, 292, 293, 304

F Finite delay, 17, 93, 100, 113, 293, 386 Fixed point, 6, 13, 14, 19, 21, 23, 26, 29, 31, 33, 37, 39, 45, 49, 52, 55, 56, 60, 64, 65, 67, 70, 71, 74, 79, 82, 87, 91, 95, 101, 102,

© Springer International Publishing Switzerland 2015 M. Benchohra, S. Abbas, Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics 39, DOI 10.1007/978-3-319-17768-7

407

408 Fixed point (Cont.) 104, 108, 110, 115, 118, 121, 124, 129, 130, 136, 137, 140, 153, 158, 177, 191, 198, 203, 212, 214, 223, 228, 234, 236, 237, 243, 246, 249, 256, 264, 265, 274, 280, 289, 295, 296, 309, 337, 341, 353, 359, 363, 368, 374, 382, 394 Fréchet space, 2, 11, 14, 17, 19, 25, 33, 48, 69, 84, 85, 94, 101, 107, 114, 129, 295, 296 G Global existence, 342, 353, 361, 369, 375, 385 Gronwall, 177, 214, 323, 337 H Hausdorff, 318, 325, 332, 337, 343 Hille-Yosida, 9, 167, 188, 204, 205, 224, 235, 247, 258, 292, 307, 316 I Impulses, 192 Infinite delay, 47, 53, 62, 68, 75, 84, 93, 106, 127, 135, 317, 324, 342 Infinitesimal generator, 8, 144, 163, 192, 202, 226, 244, 263, 277, 353, 354, 360–362, 369, 375, 376, 383, 385, 395, 396 Integral equation, 18, 23, 25, 31, 33, 37, 39, 44, 48, 54, 63, 69, 76, 84, 94, 100, 107, 114, 120, 128, 135, 144, 155, 157, 181, 192, 218, 263, 279, 281, 293, 306, 354, 361, 376 Integral solution, 165–168, 175–177, 179–183, 187, 203, 206, 212, 214–218, 223, 225, 234, 236, 245, 246, 248, 256, 259, 261, 305, 306, 308, 317 Integro-differential, 17, 32, 36–39, 42, 44 IVP: Initial Value Problems, 145, 154–156, 158, 168, 264, 274–276, 280, 289–291, 294, 295 L Laplacian operator, 304 Lebesgue dominated convergence theorem, 56, 194, 207, 228, 237, 250, 357, 366, 388, 393 Lebesgue integrable, 1, 30, 43, 75, 105, 243, 257 Lipschitz, 9, 83, 112, 125, 142, 175, 212 Lower semi-continuous, 324, 325, 337 M Maximum, 321, 327, 334, 338

Index Mild solution, 17–19, 21–26, 29–33, 35–39, 42, 44, 47–49, 53, 54, 60, 62–64, 68–70, 75, 76, 82–84, 87, 91–94, 100, 101, 104, 106–108, 110, 112–115, 118, 119, 121, 124, 125, 129, 133, 134, 136, 140, 142, 144–146, 153–155, 157, 161, 191, 193, 198, 199, 201, 225–227, 234, 243, 245, 263, 264, 266, 274, 278–280, 282, 289, 293, 294, 303, 305, 315, 317, 319, 324, 325, 330, 331, 333, 338, 343–345, 352, 354, 355, 359, 361, 368, 370, 374, 376, 377, 382, 385, 387, 389–391, 394, 395, 397 MNC: Measure of Non-Compactness, 14, 15, 318, 320, 321, 325–328, 332, 334, 337, 338, 340, 343, 346, 347 Multi-valued jumps, 305, 317, 325, 330 N Non-densely defined operator, 165, 203, 234, 246, 278, 293, 306 Nonlocal condition, 22, 30, 37, 44, 180 P Phase space, 3, 4, 17, 47, 61, 68, 82, 91, 111, 127, 134, 141, 143, 243, 244, 247, 257, 258, 262, 278, 305, 306, 308, 316, 318, 324, 330, 332, 342, 352, 353, 361, 375 S Selection, 15, 115, 117, 121, 123, 129, 132, 136, 140, 318, 319, 326, 328, 331, 332, 337, 340, 343 Semi-compact set, 15, 320, 322, 328, 333, 336, 340, 345, 346, 348 Semigroup, 7–9 Solution set, 14, 177 State-dependent delay, 245, 305, 330, 353, 361, 389 Strongly continuous semigroup, 10, 17, 84, 113, 127, 144, 163, 188, 192, 202, 224, 226, 244, 258, 263, 277, 293, 304, 353, 354, 360–362, 369, 375, 376, 383 T Topological structure, 165 U Uniqueness, 17, 22, 24, 30, 32, 37, 38, 48, 200 Upper semi-continuous, 176, 178, 179, 185, 318, 331, 343