Evolution Inclusions and Variation Inequalities for Earth Data Processing I: Operator Inclusions and Variation Inequalities for Earth Data Processing (Advances in Mechanics and Mathematics 24)

Advances in Mechanics and Mathematics Volume 24

Series Editors: David Y. Gao, Virginia Polytechnic Institute and State University Ray W. Ogden, University of Glasgow Romesh C. Batra, Virginia Polytechnic Institute and State University Advisory Board: Ivar Ekeland, University of British Columbia Tim Healey, Cornell University Kumbakonom Rajagopal, Texas A&M University ´ Tudor Ratiu, Ecole Polytechnique F´ed´erale David J. Steigmann, University of California, Berkeley

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Mikhail Z. Zgurovsky • Valery S. Mel’nik Pavlo O. Kasyanov

Evolution Inclusions and Variation Inequalities for Earth Data Processing I Operator Inclusions and Variation Inequalities for Earth Data Processing

ABC

Dr. Mikhail Z. Zgurovsky Pavlo O. Kasyanov Valery S. Mel’nik National Technical University of Ukraine “Kyiv Polytechnic Institute” Institute for Applied System Analysis National Academy of Sciences of Ukraine 37, Peremogy Ave. 03056 Kyiv Ukraine [email protected]

ISSN 1571-8689 e-ISSN 1876-9896 ISBN 978-3-642-13836-2 e-ISBN 978-3-642-13837-9 DOI 10.1007/978-3-642-13837-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010936816 c Springer-Verlag Berlin Heidelberg 2011  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

The necessity of taking into account non-linear effects, memory effects, sharpening conditions, semipenetration etc has arisen in recent years. It is caused by intensification of processes in applied chemistry, petrochemistry, transportation of energy carriers, physics, energetics, mechanics, economics and in other fields of technology and industry. When modeling such phenomena we are faced with nonlinear boundary value problems for partial differential equations with multivalued or discontinuous right-hand side, variational inequalities (evolutional as well as stationary), an evolutional problem on manifolds (either with or without boundary), paired equations, cascading systems etc. Interpreting a concept of derivative properly we can treat all these objects as operator or differential-operator inclusions in Banach spaces and study them by the help of theory of multivalued maps of pseudo-monotone type. The given book arose from seminars and lecture courses on multi-valued and non-linear analysis and their geophysical application. These courses were delivered for rather different categories of learners in National Technical University of Ukraine “Kiev Polytechnic Institute”, Kiev National Taras Shevchenko University, Second University of Naples, University of Salerno etc. during 10 years. The book is addressed to a wide circle of mathematical as well as engineering readers. It is unnecessary to tell that the pioneering works of such authors as V.I. Ivanenko, J.-L. Lions, V.V. Obukhovskii, N. Panagiotopoulos, N.S. Papageoriou, who created and developed the theory of mentioned problems, exerted the powerful influence on this book. We are thankful to V.O. Gorban, I.N. Gorban, N.V. Gorban, V.I. Ivanenko, A.N. Novikov, V.I. Obuchovskii, N.A. Perestyuk, A.E. Shishkov for useful remarks. We are grateful to the many students who have attended our lectures while we were developing the notes for this volume. We want to express our gratitude to O.P. Kogut and N.V. Zadoyanchuk for the exceptional diligence when preparing the electronic version of the book and friendly help in linguistic issues. We want to express the special gratitude to Kathleen Cass and Olena L. Poptsova for a technical support of our book.

v

vi

Preface

Finally, we express our gratitude to editors of the “Springer” Publishing House who worked with our book and everybody who took part in preparation of the manuscript. Beforehand we apologize to people whose works were missed inadvertently when making the references. We will be grateful to readers for any remarks and corrections. Kyiv, Ukraine August 2010

Milkhail Z. Zgurovsky Valery S. Mel’nik Pavlo O. Kasyanov

Contents

1

Preliminary Results . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 1 1.1 The Main Results from Multivalued Mapping Theory .. . . . . . . . . .. . . . . . . 2 1.2 Classes of Multivalued Maps .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 29 1.3 Subdifferentials in Infinite-Dimensional Spaces . . . . . . . . . . . . . . . . .. . . . . . . 95 1.4 Minimax Inequalities in Finite-Dimensional Spaces .. . . . . . . . . . . .. . . . . . .125 References . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .136

2

Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .139 2.1 Strong Solutions for Parameterized Operator Inclusions.. . . . . . . .. . . . . . .139 2.2 Parameterized Operator Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .145 2.3 Variation Inequalities in Banach and Frechet Spaces . . . . . . . . . . . .. . . . . . .158 2.4 The Penalty Method for Multivariation Inequalities . . . . . . . . . . . . .. . . . . . .162 2.5 Nonlinear Operators Equations of the Hammerstein Type. System of Operators Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .190 2.6 Nonlinear Non-coercive Operator Equations and Their Normalization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .201 2.7 Some Example Connected with Membranes Theory .. . . . . . . . . . . .. . . . . . .238 References . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .242

Index . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .245

vii

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Acronyms

for a.e. l.s.c. N -s.b.v. N -sub-b.v. r.c. r.l.s.c. r.s.c. r.u.s.c. s.b.v. s.m. sub-m. sub-b.v. u.h.c. u.s.c. u.s.b.v. V -s.b.v. w.l.s.c.

For almost each Lower semicontinuous N -semibounded variation N -subbounded variation Radial continuous Radial lower semicontinuous Radial semicontinuous Radial upper semicontinuous Semibounded variation Semimonotone Submonotone Subbounded variation Upper hemicontinuous Upper semicontinuous Uniform semibounded variation V -semibounded variation Weakly lower semicontinuous

ix

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Introduction

System analysis is a unique ground where almost all mathematical disciplines are closely related and widely applied. Besides, while solving difficult system problems, necessity to create new mathematical directions which in their turn simulate the development of classical chapters of mathematics arises. Here, approaches developed to solve system problems quite often give new methods to grasp the problems of classical mathematics. Motivated by various applications the theory of partial differential equations and inclusions is now a well-developed branch of mathematics attracting both mathematicians and experts in other fields. Having used some additional a priori estimations we establish the smoothness properties of such solutions. The book deals with the solvability for nonlinear operator problems and with differential operator inclusions in infinite dimensional spaces with multivalued maps of w0 -pseudomonotone type. When investigating partial differential equations, inclusions and evolution variation inequalities describing different processes in hydromechanics, geophysics and other fields of continuum and quantum mechanics and physics the following scheme is frequently used: at first we construct approximate solutions of the problem, after that we establish a priori estimations for the given solutions and then we prove that there exists such convergent sequence of the approximate solutions which limit is some exact solution of the original problem. Let us cite some examples. At first we consider some problem connected with the analysis of ecological state of the atmosphere in physical anomalies conditions. The character of diffusion and transfer of harmful impurities in the atmosphere are determined by factors of natural and anthropogenic character. The natural factors include the stability level of the atmosphere, its meteorological state, such as wind direction and velocity, temperature, humidity, air pressure and others. The anthropogenic factors are connected with the impact on the atmosphere due to industrial or other activities of a man. Such factors are the intensity of atmospheric pollution determined by the volume, physical, and chemical properties of impurities discharged into the atmosphere. The anthropogenic factor is also the human impact on the atmospheric meteorological state, which may even lead to climatic changes. We will divide the factors determining the ecological state of the atmosphere into ordinary or classical, and anomalous or nonclassical. The influence of ordinary factors results in monotonous (nonjump, two-directional, recoverable) changes in xi

xii

Introduction

the atmospheric ecological state, see for example [Ma82]. Under the influence of anomalous factors, the atmosphere behavior is characterized by jumps or unilateral, nonrecoverable state changes, see for example [DuLi76]. There are known natural and anthropogenic anomalies which result in the atmosphere’s anomalous behavior, such as inverse stratification of the atmosphere, heat island, processes with restrictions of inequalities typical in the upper or lower level of impurity concentration. Let us characterize these anomalies, see for example [ZgMel04, ZgNov96]. This is a meteorological factor of natural origin. It occurs when the temperature rises with the altitude (see Fig. 1a). Inversion holds the impurity in the vertical layer and weakens the turbulent exchange within it (see Fig. 1b). Thus, inversion promotes an increased accumulation of the impurities near to the ground layer. Such phenomena are especially dangerous in valleys, low lands and other places with weak air circulation. This heat anomaly occurs over large industrial centers. It is of anthropogenic origin. The essence of this anomaly is in an excess of the atmospheric air temperature within industrial centers over the air temperature in their vicinities. The heat island weakens turbulent exchange in the horizontal layer and holds an impurity in the area of the center. The heat island does not impede penetration of impurities from the bordering vicinities (see Fig. 2). Thus, this class of anomalies promotes the accumulation of impurities within the boundaries of industrial centers. It is especially dangerous during summer at night in places with weak air circulation. Restrictions on the upper level of impurity concentration in the atmosphere are determined by physical and chemical properties within the impurity. For example, when some threshold concentration level of a finely disperse impurity in the atmosphere exceeded – u  umax , its particles are crystallized and precipitated by gravity (see Fig. 3). Thus, here is a nonlinear effect of the atmospheric purification. This process is activated when some threshold concentration level is overcome. Restrictions on the lower level of impurity concentration in the atmosphere are also determined by physical and chemical properties of this impurity. An example of this anomaly is the process of washing-out a gaseous impurity by precipitation. There exists the lower threshold value of gaseous impurity concentration – u  umin at which the action of precipitation on the concentration level becomes indistinguishable (see Fig. 3). Classical problems of atmospheric diffusion and transfer are stated in the form of variation principles. If there are no additional requirements to these problems solution, they are reduced to the boundary problems for mathematical physics equations, see for example [OmSe89]. Anomalous problems of atmospheric diffusion differ from the ordinary or classical problems by the presence of additional restrictions on the states. These restrictions are written in the form of inequalities and characterize nonlinear effects of unilateral conductivity of the boundary and other peculiarities. Thus, according for these peculiarities at the stage of a variation statement of a mathematical physics problem results in variation inequalities introduced and developed, see for example [DuLi76].

Introduction

xiii

Fig. 1 Temperature profile (a) and anomalous behavior effect of atmosphere (b)

Let u be impurity concentration in atmosphere, determined on the open set ˝ of space R3 with the smooth bound B D BL [ BR [ BU [ BD [ BF [ BT and time interval .0; tf / for tf < 1, Q D ˝  .0; tf /, ˙ D B  .0; tf /, is the solution of the inequality, see for example [DuLi76] ; v  u/ C .A./u; v  u/ C ˚j .v/  ˚j .u/ . @u @t  .f; v  u/; j D 1; 2; 8 v 2 H 1 .˝/ D V; ujt D0 D u0 in ˝

(1)

xiv

Introduction

Fig. 2 Accumulation of impurities as a result of heat island effect

Fig. 3 Restrictions on the upper and lower level of impurity concentration in the atmosphere

where .f; g/ is the action of functional f 2 .H 1 .˝// on the element g 2 H 1 .˝/, operator A./ W V ! V  , is defined by bilinear form: .A./u; / D

3 Z h X i D1 ˝

k.z/

@u @ @u i  ci .z/  @z  @zi @zi @zi

Z d.z/ud z; 8  2 H 1 .˝/ ˝

Introduction

xv

if .f; g/ R2 L2 .˝/, operation .f; g/ coincides with inner product in L2 .˝/, i.e., .f; g/ D f .z/g.z/d zI k.z/ – coefficient of turbulent diffusion; z D .z1 ; z2 ; z3 /I ˝

ci .z/ (for i D 1; 2; 3) be the wind auxiliary parameters z1 ; z2 ; z3 , respectively; K P d.z/ be the impurity absorption coefficient. Variable f .t; z/ D qi .t/ı.z  zi /, i D1

is the force function of the process; qi .t/ be the sources function, operating in the sub-spaces ˝i 2 ˝, i D 1; 2; : : : ; K, K – the number of external action points; ı.z  zi / – characteristic function. Let us set up that parameters  D fk.z/; ci .z/; d.z/g, functional ˚j , and the force function of the process f have the technological restrictions. So, we obtain the solvability problem for evolution variation inequality (1) Let us point out that in mathematical theory that there occur such objects is too little investigated or nearly uninvestigated in mathematics. These are operator inclusions, variation inequalities with multivalued maps, differential-operator inclusions, evolution variation inequalities with multimaps, systems containing operator equalities, inequalities and inclusions of different types etc. For all these classes of objects the original mathematical apparatus is being developed. Thereupon all results obtained are new ones even for those cases that have been rather actively investigated in mathematical literature. Specifically, several J.-L. Lions’s problems from his nonlinear boundary problems theory were solved. So in the present book the new mathematical apparatus is developed, which makes it possible to investigate problems for a wide class of nonlinear objects with distributed parameters, also here authors offer an axiomatic study of different classes of nonlinear maps in Cartesian product of spaces (fcontrol spaceg  fstate spaceg), examine their relations and cite a great number of examples. Using these constructions authors prove Theorems on solvability and about dependence on parameter for solutions of nonlinear operator equations, operator inclusions, variation inequalities, different-type system of equations and inclusions, differential operator equations, evolution inclusions and parabolic variation inequalities. These results are of great significance in their own right and have not been illuminated in mathematical literature before. To illustrate the distinction features of the main object of our consideration let us cite one more motivational example. Let ˝ be a bounded domain in Rn with a regular boundary @˝. Let us consider the following optimization problem for a quasilinear elliptic system: ! ˇ ˇ ˇ @y ˇp2 @y ˇ u .x/ ˇˇ D f .x/ ; x 2 ˝; @xi ˇ @xi

n X @ @xi i D1

y j@˝ D 0

Z ˝

jy .x/jp dx C  kukL1 .˝/ ! inf

(2) (3) (4)

xvi

Introduction

where p  2,  > 0, f 2 Lq .˝/,

1 p

C

1 q

D 1,

u 2 U D fv 2 L2 .˝/ j0 < ˛  v .x/  ˇ for a.e. x 2 ˝ g : For a fixed u 2 U the boundary problem (2), (3) has a unique generalized solution ˇ   ˇ @y 2 Lp .˝/ ; i D 1; : : : ; n ; y 2 Wp1 .˝/ D y 2 Lp .˝/ ˇˇ @xi which continuously (within the correspondent topology) depends on u 2 U and f 2 Lq .˝/. However problem (2)–(4) has no solution, namely a good problem of the differential equations theory is bad one for the optimization theory. On the other hand let us consider ! ˇ ˇ n X @ ˇˇ @y ˇˇp2 @y D f .x/ ; x 2 ˝; (5) @xi ˇ @xi ˇ @xi i D1

ˇ @y ˇˇ D u./;  2 @˝; @vA ˇ@˝

(6)

where f 2 Lq .˝/ ;

u 2 U  Lq .@˝/ ;

ˇ n ˇ X ˇ @y ˇp2 @y @y ˇ ˇ D i : ˇ @x ˇ @A @xi i i D1

It is required to minimize a functional Z ˝

jy .x/jp dx C ı kukLq .@˝/ ! inf;

ı>0

(7)

on the solutions of (5), (6). The pair .uI y/ 2 U  Wp1 .˝/, satisfying (5), (6), refers to be admissible, and a problem (5)–(7) refers to be regular if the set of admissible pairs is nonempty. We remark that from the point of view of the partial differential equations theory problem (5), (6) is a “bad” one since it is not coercive. At the same time the optimization problem (5)–(7) is a “good” one since it is regular, coercive and also has a solution. To prove its regularity it is sufficient to consider (5) with the boundary condition (3). Thus (3), (5) has a unique solution y0 2 Wp1 .˝/, and acting on it by the operator @v@A j@˝ we receive an element

@y0 u0 D @v j@˝ 2 Lq .@˝/. Hence, the pair .u0 I y0 / is admissible for (5)–(7). These A examples show basic distinctions of the purposes of the partial differential equations theory and the optimization theory for the objects described by the partial differential equations.

Introduction

xvii

The boundary problem (5), (6) generates a nonlinear map A W U  X ! X  by the rule hA .u; y/ ; wi D

ˇ Z n Z ˇ X ˇ @y ˇp2 @y @w ˇ ˇ dx C u ./ w ./ d; u 2 Lq .@˝/ ; ˇ ˇ @xi @xi ˝ @xi @˝ i D1

y; w 2 Wp1 .˝/ D X: Let us consider the parameterized multivalued map 

B W X ! 2X ; B .y/ D fA .u; y/ ju 2 U g : With its help (5), (6) can be represented as the operator inclusion B .y/ 3 f:

(8)

Thus, to prove regularity of the optimization problem (5)–(7) it is sufficient to establish a resolvability of the operator inclusion (8). When describing series of real processes in chemistry, physics, biology, ecology, economics or geophysics there may arise a necessity to investigate the situation when an equation or a system modeling the correspondent process has series of features, namely: instability, break phenomenon, noncoercivity, nonuniqueness of solutions and bifurcation phenomena, changes of model’s structure depending on control functions, etc. Such distributed systems are called nonlinear singular ones. When investigating singular systems, difficulties in trans-calculating arise and approximation solution algorithm must contain regularization procedure. Using optimal control methods for nonlinear infinite-dimension systems we develop a new fundamental approach to study singular boundary problems for partial differential equations. This approach is based on replacement of initial illposed problem by auxiliary (associated) optimal control problem [ZgMel04]. On this way authors managed to construct stable solving algorithms for Navier–Stokes equations, Benard systems, etc. Therefore the methods of optimal control theory represent a mathematical apparatus for effective research of ill-posed equations of mathematical physics. Let us cite some more examples. Let ˝  Rn be a bounded domain with a smooth boundary v > 0. For y W .0; T /˝ ! Rn let us consider the n-dimensional Navier–Stokes problem X @y @y  vy C yi D f .t; x/  rp; @t @xi n

(9)

i D1

y j˙

div y D 0; X D 0; D .0; T /  @˝; yjt D0 D y0 .x/;

(10) (11) (12)

xviii

Introduction

where f is a inhomogeneity function, p is a pressure. Auxiliary extremal problem: X @y @y  vy C ui D f .t; x/  rp: @t @xi n

(13)

i D1

Here the function u W .0; T /  ˝ ! Rn belongs to the class of solutions of (9)– (12) W . For each u 2 W problem (10)–(13) has a unique generalized solution. On solutions of (13) the following extremal problem is posed J.u/ D ku  y.u/k2W ! inf :

(14)

Under natural conditions existence of a solution of (10)–(14) is proved using methods of optimal control theory. If for some u J.u/ D 0 , u D y there exists a solution for (9)–(12). The next example concerns evolution inequalities on cones. Let us consider a positive solution problem for evolution inequalities



y 0 C A.y/  f; y.0/ D y0  0;

(15)

y  0;

(16)

y 2 X , A W X ! X . This problem is nontrivial even for evolution equations. In order of modeling and prognostication of large social-economic systems we remark that one of directions was construction of network models with associate memory (something like Hopfield neuronets) for a wide class of processes: social, economic, political, natural ones. Among recent achievements on this way there can be mentioned the invention of multivalued solutions behavior for such models, opening perspectives for better understanding of the occurrence of different progress scenarios for large social systems. Multivalued solutions of models for social systems open perspectives for wide applications. So, there arise problems studying dependence on parameter of models behavior in multivalued case, occurrence of limit cycles, etc., and especially attractors in the case of multivalued maps which are actively investigated due to the grounds laid before. To solve these problems we need nearly all arsenal of system analysis methods: nonlinear functional analysis, dynamic systems theory, differential-operator and discrete equations theory. One more direction of research related to modeling of movement (of transport and pedestrians) for a large group of objects. As one of approaches to solving such kind of problems the cellular automaton method was chosen, where movement is analyzed within its discrete approximation (where the space is divided into regular blocs of cells) and we consider transitions of an object moving from one cell to other. Mathematically this problem looks like investigation of discrete dynamic systems (determined or stochastic ones). There was made a graph modeling under certain conditions using models adapted for real geometry of traffic current and jams, and also problems about discounting the movement prognosis by members of the motion

Introduction

xix

were set. Such problems again bring us to necessity of further mathematical investigations from the point of multivalued maps and also differential-operator inclusions and optimization theory. Motivated by various applications the theory of partial differential equations and inclusions is now a well-developed branch of mathematics attracting both mathematicians and experts in other fields. The book deals with the solvability of nonlinear operator, differential-operator problems and evolution inclusions in infinite-dimensional spaces with multivalued maps of “pseudomonotone” type. Many important applied problems (for example for processes of diffusion of oil in porous mediums, for processes of atmospheric diffusion and transfer, etc.) can be reduced to so-called problems with one-sided boundary conditions or to variation inequalities, which also generate differential-operator inclusions. Let us consider the simplest example of this kind [Li69]: it is required to find a solution of the equation M y D f in a domain ˝ with a boundary  , such that on  the following conditions @u @u u  0;  0; u D 0: @n @n are satisfied. The generalized solution of such problem does not satisfy the integrated identity (as, for example, in a Dirichlet problem) but it satisfies some integrated inequality which is called a variation inequality. The theory of variation inequalities as a new section of the partial differential equations theory arose in 60-s of the last century from the Signorini problem of the elasticity theory, which is completely investigated in G.Fichera’s work [Fi64]. In this work fundamentals of the variation inequalities theory were laid. Then the investigation of variation inequalities was continued in papers of J.-L. Lions [Li69], G. Stampacchia [St64] and their pupils. In particular an abstract statement of problems that could be reduced to such inequalities was considered. At the same time the Russian school of mathematicians laid the basis of the differential inclusions theory. A great number of applications generated the development of the theory of the differential equations with discontinuous right parts which generate differential inclusions in finite-dimensional spaces. A great deal of problems in mechanics, electrical engineering and automatic control theory that are described by the given objects were considered in A.A.Andronov’s work [AnViHa59]. Wide use of different switches (relays) requires the developed constructions of the given theory. The basic directions of the theory of the differential equations with discontinuous right-hand side and theories of differential inclusions in finite-dimensional spaces were stated in A.F.Filippov’s monograph [FiAr88]. The theory of multivalued maps as a new section of nonlinear analysis has arisen at the intersection of two mathematical sciences, namely the variation inequalities theory and the theory of differential-operator inclusions, which in particular, are generated by variation inequalities. Fundamentals of the given theory were laid in the papers of B.M. Pshenichniy, M.M. Vainberg, V.F. Dem’yanov, L.V. Vasiliev, A.M. Rubinov, A.D. Ioffe, V.M.Tihomirov, A.A. Chikrii, I.Ekeland, J.-P. Aubin, H.Frankowska and others in [Au79, AuFr90, DemVas81, IoTi79, AuEk84, Chi97, Li69, Psh80]. One of the main classes of multivalued maps is subdifferential maps,

xx

Introduction

their basic properties (for Banach spaces) were investigated in [AuEk84, Chi97, DemVas81, IoTi79]. Significant progress in study of nonlinear boundary problems for partial differential equations and inclusions is caused by profound development of methods of the nonlinear analysis which can be applied to different sections of mathematics. At the present time it became natural to reduce the given problems to nonlinear operator or differential–operator equations and inclusions in functional spaces. When using such approach the results for concrete systems with contributed parameters turn out as a corollary of operator existence Theorems. Convergence of approximate solutions to an exact solution of the differentialoperator equation or inclusion is frequently proved on the basis of a monotony or a pseudomonotony of corresponding operator. If the given property of an initial operator takes place then it is possible to prove convergence of the approximated solutions within weaker a priori estimations than it is demanded when using embedding Theorems. The monotony concept has been introduced in papers of Vainberg, Kachurovsky, Minty, Sarantonello and others. Significant generalization to monotonicity has given H.Brezis [Br68]. Namely, Brezis refers operator A W X ! X  to be pseudomonotone if (a) The operator A is bounded (b) From un ! u weakly in X and from lim hA.un /; un  uiX  0

n!1

it follows, that lim hA.un /; un  viX  hA.u/; u  viX

8v 2 X:

n!1

In applications, as a pseudomonotone operator the sum of radially continuous monotone bounded operator and strongly continuous operator was considered [GaGrZa74]. Concrete examples of pseudomonotone operators were obtained by extension of elliptic differential operators when only their summands complying with highest derivatives satisfied the monotony property [Li69]. In papers of J.-L. Lions, H. Gajewski, K. Groger, R K. Zacharias [Li69, GaGrZa74] the main results of solvability theory for abstract operator equations and differential operator equations that are monotone or pseudomonotone in Brezis sense are set out. Also the application of the given Theorems to the concrete equations of mathematical physics, and in particular, to free boundary problems. The theory of monotone operators in reflexive Banach spaces is one of the major areas of nonlinear functional analysis. Its basis is so-called variation methods, and since 60-s of the last century the theory intensively has been developed in tight interaction with the theory of convex functions and the theory of partial differential equations. The papers of F. Browder and P. Hess [Br77, BrHe72] became classical ones in the given direction of investigations. In particular in F. Browder and P. Hess work [BrHe72] the class of generalized pseudomonotone operators that enveloped

Introduction

xxi

a class of monotone mappings was introduced. Let W be some normalized space continuously embedded in the normalized space Y . The strict multivalued map A W  Y ! 2Y is called generalized pseudomonotone on W if for each pair of sequences fyn gn1  W and fdn gn1  Y  such that dn 2 A.yn /, yn ! y weakly in W , dn ! d weakly star in Y  , from the inequality lim hdn ; yn iY  hd; yiY

n!1

it follows that d 2 A.y/ and hdn ; yn iY ! hd; yiY . The grave disadvantage of the given theory is the fact that in common case it is impossible to prove a closure within the sum of pseudomonotone (in the classical sense) maps (the given statement is problematic). This disadvantage becomes more substantial on investigating differential-operator inclusions and evolution variation inequalities when we necessarily consider the sum of the classical monotone mapping and the subdifferential (on Gateaux or on Clarke) for multivalued map which is generalized pseudomonotone [CaVyLeMo04]. Here we realize I.V. Skrypnik’s idea of passing to subsequences in classical definitions [Sk94]. This enable us to consider essentially wider class of 0 -pseudomonotone maps, closed within the sum of maps, that appeared problematic for classical definitions [ZgMel02]. In V.S. Mel’nik and P.O. Kasyanov papers [KaMe05a, KaMeTo06] there was introduced the class of w0 -pseudomonotone maps which includes, in particular, a class of generalized pseudomonotone multivalued operators and also it is closed as for sum ming. A strict multivalued map A W Y ! 2Y with the nonempty, convex, bounded, closed values is called w0 -pseudomonotone (0 -pseudomonotone on W ), if for any sequence fyn gn0  W such that yn ! y0 weakly in W , dn ! d0 weakly star in Y  as n ! C1, where dn 2 A.yn / 8n  1, from inequality lim hdn ; yn  y0 iY  0

n!1

the existence of such subsequences fynk gk1 from fyn gn1 and fdnk gk1 from fdn gn1 for which lim hdnk ; ynk  wiY  ŒA.y0 /; y0  w

8w 2 Y

k!1

is follows. Here we have to prove a resolvability for differential-operator inclusions with 0 -pseudomonotone on D.L/ multivalued maps in Banach spaces: Lu C A.u/ C B.u/ 3 f; 



u 2 D.L/;

(17)

where A W X1 ! 2X1 , B W X2 ! 2X2 are multivalued maps of D.L/0 pseudomonotone type with nonempty, convex, closed, bounded values, X1 , X2 are Banach spaces continuously embedded in some Hausdorff linear topological space, X D X1 \ X2 , L W D.L/  X ! X  is linear, monotone, closed, densely defined operator with a linear definitional domain D.L/.

xxii

Introduction 

Let us remark that any multivalued map A W Y ! 2Y , naturally generates upper and, accordingly, lower form: ŒA.y/; !C D sup hd; wiY ; d 2A.y/

ŒA.y/; ! D

inf hd; wiY ;

d 2A.y/

y; ! 2 X:

Properties of the given objects have been investigated in M.Z. Zgurovsky and V.S. Mel’nik works [Me97, ZgMel99, ZgMel00, ZgMel02]. Thus, together with the classical coercivity condition for operator A: hA.y/; yiY ! C1; kykY

kykY ! C1;

as

which ensures the important a priori estimations, arises C-coercivity (and, accordingly, -coercivity): ŒA.y/; yC./ ! C1; kykY

as

kykY ! C1:

C-coercivity is much weaker condition than -coercivity. When investigating multivalued maps of w0 -pseudomonotone type it was found out that even for subdifferentials of convex lower semicontinuous functionals the boundness condition is not natural [KaMe05b]. Thus it was necessary to introduce an adequate relaxation of the boundness condition which would have enveloped at least a class of monotone multivalued maps. In the paper [IvMel88] the following  definition was introduced: multivalued map A W Y ! 2Y satisfies Property .˘ / if for any bounded set B  Y , any y0 2 Y , for some k > 0 and for some selector d (d.y/ 2 A.y/ 8y 2 B), for which hd.y/; y  y0 iY  k

for all y 2 B;

there exists such C > 0 that kd.y/kY   C

for all y 2 B:

Recent development of the monotony method in the theory of differentialoperator inclusions and evolution variation inequalities [CaMo03, DeMiPa03, NaPa95, CaVyLeMo04, Pa85, Pa86, Pa87a, Pa87b, Pa88a, Pa88b, Pa94, PaYa06] ensures resolvability of the given objects under the conditions of -coercivity, boundness and the generalized pseudomonotony (it is necessary to notice, that the proof is not constructive). With relation to applications it would be actual to relax some conditions of multivalued maps in a problem (17) replacing -coercivity by C-coercivity, boundness by Condition .˘ / and pseudomonotony in classical sense or generalized pseudomonotony by w0 -pseudomonotony. At the last time the operator, differential–operator equations, inclusions and evolution variation inequalities are being studied intensively enough by

Introduction

xxiii

many authors: Zgurovsky M.Z., Mel’nik V.S., Kasyanov P.O., Vakulenko A.N., Solonoukha O.M., Borisovich Yu. G., Slusarchuk V.Yu., Gelman B.D., Kamenskii M.I., Mishkis A.D., Obuchovskii V.V., Gorodnij M.F., Kogut P.I., Kovalevsky O.A., Nicolosi Fr., Chajkovs’kyj A.V., Ansari Q.H., Khan Z., Yao J.-C., Aubin J.-P., Frankowska H., E.P. Avgerinos, N.S. Papageorgiou, Barbu V., Benchohra M., Ntouyas S.K., S. Carl, D. Motreanu, Hu S. and others in [AnKh03, AnYao00, Au79, AuEk84, AuFr90, AvPa96, AzZg83, AzZg84, AzZg86, AzZgKo88, AzSe86, Bar76, BeNt01, BoYuGeMiOb90, BrSt94, Br68, Br72, BrNiSt72, Br77, BrHe72, CaLaMeVa03, CaMo03, CaVyLeMo04, Chi97, ChTan96,CoCha88,CoObZe00,DemVas81,DeMiPa03,GaPa05,GaSm02,GlLiTr81, GiGiLa03,GoNiOb97,GoCh02,HuPa97,HuPa00,IoTi79,IvMel88,Kam94,KaOb91, KaOb93,KaObZe96,KapKas03a,KapKas03b,KaMeVa03,KaMeVa07,Ka05,Ka05, KaMePi08, KaMeVa08, KaMe05a, KaMe05b, KaMe05c, KaMe06, KaMeTo05, KaMeTo06, KaMePi06, KaMePi06, KaMeYa07, KaMa01, Ki91, Ko96, Ko99, Ko01, Ko01, KoMu01, Kon01, KoNi97, KoNi99, KoNi99, LaOp71, LyKo83, MaNgSt96, Ma82,Me84,Me97,Me98,Me00,Me06,MeKo98,MeSo97,MeVa98,MeVa00,Mi95, Mo67, Nag99, NaPa95, NiJiJu98, NiObZe94, Ob92, Pa85, Pa85, Pa85, Pa86, Pa87a, Pa87b,Pa88a,Pa88b,Pa94,PaYa06,PeKaZa08,Sh97,Sli06,Tol92a,Tol92b,VaMel98, VaMel99, VaMel00, ZadKas06, ZadKas07, ZgMel99, ZgMel00, ZgMel02, ZgMel04, ZgNov96, ZgMel04, ZgPa07] etc. By analogy with the differential-operator equations at least four approaches are well-known: Faedo–Galerkin method, elliptic regularization, the theory of semigroups, difference approximations. Extension of these approaches on evolution inclusions encounters a series of basic difficulties. For differential-operator inclusions the method of semigroups is realized in works of A.A. Tolstonogov, Yu.I. Umansky [Tol92a, Tol92b] and V. Barbu [Bar76]. The method of finite differences for the first time was extended on evolution inclusions and variation inequalities in the work of P.O. Kasyanov, V.S. Melnik and L. Toskano [KaMeTo05]. Method of singular perturbations (H. Brezis [Br72] and Yu.A. Dubinsky [DubYu65]) and Faedo–Galerkin method for differentialoperator inclusions for w0 -pseudomonotone multivalued maps have not been still systematically investigated. It is required to prove the singular perturbations method and Faedo–Galerkin method for differential-operator inclusions with w0 -pseudomonotone multivalued maps in Banach spaces. One of the most important classes of multivalued maps of w0 -pseudomonotone type is a class of subdifferential maps. Variation inequalities with convex lower semicontinuous main-functional ' [ZgMel00] are one of the sources generating operator inclusions. In Banach spaces a subdifferential @'./ of convex lower semicontinuous main-functional has a series of the important properties [AuEk84,Psh80, Chi97] which are dominant when researching variation inequalities. But in locally convex spaces similar properties were not investigated. With respect to applications there appears a problem of study the properties for subdifferential maps in locally convex spaces. If we apply the compactness method to research evolution equations and inclusions of (17) type in irreflexive Banach and Frechet spaces a priori estimations are required. Classical compactness Theorems and Theorems about continuous

xxiv

Introduction

embedding for special spaces of distributions with integrable derivatives [GaGrZa74, Li69] are valid for reflexive spaces only. For wide class of special irreflexive spaces of distributions with integrable derivatives the given problem has not been solved. In the monograph authors developed the methodology of formalization and investigation for system analysis problems by the instrumentality of interdisciplinary models. In Chap. 1 there were considered main classes of multivalued maps defined in space product. There a number of important properties were introduced, specifically those for subdifferential maps in locally-convex spaces. Also in this chapter authors investigate minimax inequalities in finite-dimensional spaces. As a corollary a new multivalued analogue of acute-angle Lemma, which special case had been stated and proved for -coercive maps in [AuEk84], was obtained here. Chapter 2 is devoted to parameterized operator inclusions and variation inequalities in infinite-dimensional spaces with multivalued maps of 0 -pseudomonotone type. The multivalued penalty method for variation inequalities on convex closed sets was developed here. Also there were considered systems of Hammerstein type and systems of operator inclusions. In Chap. 3 some special classes of distribution spaces with integrated derivatives were investigated. Authors prove embedding and approximation Theorems. The basis theory was developed. Also in this chapter classes of multivalued maps of w -pseudomonotone type were studied. Chapter 4 is devoted to solvability problems for differential operator inclusions of I and II orders in infinite-dimensional spaces, to solution properties, to their approximations, to parameter dependence of resolving operator, to extremal solutions. There were considered the following methods: Faedo–Galerkin method, singular perturbations method, Dubinskii method. Also authors considered classes of noncoercive problems. We consider evolution variation inequalities in Chap. 5. The solvability Theorems and the dependence of solutions on parameters have been proved, the classes of noncoercive problems have been considered. The multivalued penalty method for the evolution variation inequalities has been introduced. The authors proved solvability for the evolution variation inequalities on convex, closed with empty interior set with W0 -pseudomonotone operator and obtained important a priori estimations. In this book we also examine processes connected with harmful impurities expansion in the atmosphere, unilateral processes of diffusion of petroleum in porous mediums, etc. The obtained results allows also to consider the different type of applications from nonlinear boundary hydrodynamic problems to the finding of positive solutions of differential-operator equations and inclusions, from the dynamics of extremal solutions of 3D Navier–Stokes to the free boundary problems. To understand the material of the given monograph the reader should have a certain degree of university program knowledge of functional analysis and recept about the foundations of operator equations and differential-operator equations theory [GaGrZa74, Li69]. For scientific associates, graduate and post-graduate students of universities which specialize in the fields of “applied mathematics” and “informatics”.

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Introduction [Psh80] Pshenichniy BN (1980) Convex analysis and extremal problems [in Russian]. Nauka, Moscow [RepSem98] Repovs D, Semenov PV (1998) Continuous Selections of multivalued mappings. Kluwer, Dordrecht [Sch71] Schaefer H (1971) Topological vector spaces. Springer, New York [Sh97] Showalter R (1997) Monotone operators in Banach space and nonlinear partial differential equations. American Mathematical Society, Providence, RI [Si33] Signorini A (1933) Sopra alcune questioni di Elastostatica Atti Soc Ital per il Progresso della Sci, Roma [Sk94] Skrypnik IV (1994) Methods for analysis of nonlinear elliptic boundary value problems. American Mathematical Society, Providence, RI [Sli06] Sliusarchuk VYu (2006) Diffirential equations in Banach spaces. National Univ Vodn Gosp i Prirodokor, Rivne [St64] Stampacchia G (1964) Formes bilineaires coercitives sur les ensembles convexes. CR Acad Sci Paris 258:4413–4416 [Tem75] Temam R (1975) A non-linear eigenvalue problem: The shape at equilibrium of a confined plasma. Avch Ration Mech Anal doi:10.1007/BF00281469 [Tol92a] Tolstonogov AA (1992) Solutions of evolution inclusions. I Siberian Math J 33. doi:10.1007/BF00970899 [Tol92b] Tolstonogov AA, Umanskii YuA (1992) On solutions of evolution inclusions. II. Siberian Math J 33. doi:10.1007/BF00971135 [VaMel98] Vakulenko AN, Mel’nik VS (1998) On a class of operator inclusions in Banach spaces. Reports NAS of Ukraine 8:20–25 [VaMel99] Vakulenko AN, Mel’nik VS (1999) Solvability and properties of solutions of one class of operator inclusions in Banach spaces. Naukovi visti NTUU ”KPI” 3:105–112 [VaMel00] Vakulenko AN, Mel’nik VS (2000) On topological method in operator inclusions whith densely defined mappings in Banach spaces. Nonlinear Boundary Value Probl 10:125–142 [ZadKas06] Zadoyanchuk NV, Kasyanov PO (2006) On solvabbility for the second order nonlinear differential–operator equations with the noncoercive w pseudomonotone type operators. Reports NAS of Ukraine 12:15–19 [ZadKas07] Zadoyanchuk NV, Kasyanov PO (2007) Faedo–Galerkin method for nonlinear second-order evolution equations with Volterra operators. Nonlinear Oscillations doi:10.1007/s11072-007-0016-y [ZgMel99] Zgurovsky MZ, Mel’nik VS (1999) Nonlinear analysis and control of infinitedimentional systems [in Russian]. Naukova dumka, Kiev [ZgMel00] Zgurovskii MZ, Mel’nik VS (2000) Penalty method for variational inequalities with multivalued mappings. Part I. Cybern Syst Anal. doi:10.1007/ BF02667060 [ZgMel02] Zgurovskii MZ, Mel’nik VS (2002) The Ki-Fan inequality and operator inclusions in Banach spaces. Cybern Syst Anal. doi:10.1023/A:1016391328367 [ZgMel04] Zgurovsky MZ, Mel’nik VS (2004) Nonlinear analysis and control of physical processes and fields. Springer, Berlin [ZgNov96] Zgurovsky MZ, Novikov AN (1996) Analysis and control of unilateral physical processes [in Russian]. Naukova Dumka, Kiev [ZgMel04] Zgurovsky MZ, Mel’nik VS, Novikov AN (2004) Applied methods of analysis and control by nonlinear processes and fields [in Russian]. Naukova Dumka, Kiev [ZgPa07] Zgurovsky MZ, Pankratova ND (2007) System analysis: theory and applications. Springer, Berlin

Chapter 1

Preliminary Results

Abstract In this chapter we state basic data on about the multivalued theory. Such maps will be needed further when studying mathematical models of geophysical processes and fields of different nature. We start with an idea of multivalued map and functional-topological properties connected with it. Then we give main properties of classes of extensions and multivalued Nemytskii maps for differential operators from mathematical models of Earth Data Processing. The special attention is given to classes of pseudo-monotone multivalued maps. There are also given ideas and main properties of a local subdifferential and Clarke’s subdifferential for Lipschitzian functional in a locally convex space. In the conclusion of the chapter we cite properties of minimax inequalities in finite-dimensional spaces. Particularly, it is given the correct proof of the generalized acute angle lemma. When writing this chapter we used many well-known guidance. Among them: Aubin and Ekeland (Applied nonlinear analysis, Wiley-Interscience, New York, 1984); Aubin and Frankowska (Set-valued analysis, Birkhauser, Boston, 1990); Brezis (J Math Pures Appl 51:377–406, 1972); Chikrii (Conflict-controlled processes, Kluver, Boston, 1997); Clarke (Optimization and nonsmooth analysis, SIAM, Philadelphia, 1990); Dem’yanov and Vasiliev (Non-differentiated optimization [in Russian]. Nauka, Moscow, 1981); Dubinskii (Mat Sb 67:609–642, 1965); Gajewski et al. (Nichtlineare operatorgleichungen und operatordifferentialgleichungen, Akademie, Berlin, 1974); Hu and Papageorgiou (Handbook of multivalued analysis, vol I: Theory, Kluwer, Dordrecht, 1997); Hu and Papageorgiou (Handbook of multivalued analysis. vol II: Applications, Kluwer, Dordrecht, 2000); Ioffe and Tihomirov (Theory of extremal problems, North Holland, Amsterdam, 1979); Ivanenko and Mel’nik (Variational methods in control problems for systems with distributed parameters [in Russian], Naukova dumka, Kiev, 1988); Lions (Quelques methodes de resolution des problemes aux limites non lineaires, Dunod Gauthier-Villars, Paris, 1969); Showalter (Monotone operators in banach space and nonlinear partial differential equations, American Mathematical Society, Providence, 1997); Skrypnik (Methods for analysis of nonlinear elliptic boundary value problems. American Mathematical Society, Providence, 1994); Vainberg (Variation methods and method of monotone operators, Wiley, New York, 1973); Zgurovsky and Mel’nik (Nonlinear analysis and control of physical processes and fields, Springer, Berlin, 2004).

M.Z. Zgurovsky et al., Evolution Inclusions and Variation Inequalities for Earth Data Processing I, Advances in Mechanics and Mathematics 24, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-13837-9 1, 

1

2

1 Preliminary Results

1.1 The Main Results from Multivalued Mapping Theory Let X and Y be some sets. The multivalued mapping from X to Y is called the correspondence F comparing to each point x 2 X the set F .x/  Y which is called the image of the point x at the mapping F or the value of F at the point x. The multivalued mapping F is proper if there exists at least one element x 2 X such that F .x/ 6D ;. In this case Dom F , fx 2 X W F .x/ 6D ;g is an effective set of the mapping F . We will say that the multivalued mapping F from X to Y is strict if Dom F D X . It was possible not to introduce the new notion by substituting the multivalued mapping F W X  Y with the singlevalued one f W X ! 2Y of the set X to the set 2Y of all subsets of the set Y . However, in specific situations the set Y may have additional structure, for example, geometric, topological or algebraic, and the mapping F is often connected with this particular structure. The re-statement of the obtained results in terms of singlevalued mappings results in detection of the similar structure on the set 2Y , which, as a rule, leads to considerable difficulties and is less obvious than the use of the multivalued mappings language. Thus, if Y is a vector space then 2Y is not such any longer, since 2Y is not an Abelian group with respect to the arithmetic sum of the sets. Everywhere below the multivalued mappings F from X in Y will be indicated by the symbol F W Dom F  X ! 2Y

and F W X ! 2Y .F W X  Y /; if Dom F D X:

Moreover, if Y is a topological space then C.Y / is a collection of all nonempty closed subsets, K.Y / is a set of all nonempty compacts in Y . Let D  Y . The small (complete) pre-image of D at the mapping F W X ! 2Y is called the set 1 .D/ D fy 2 X W F .y/  Dg FM

.F 1 .D/ D fx 2 X W F .x/ \ D 6D ;g/: Let us point out some elementary properties of the pre-images: T 1 T 1 T 1 Vi , then FM .V / D FM .Vi /, F 1 .V / D F .V / ˘1: If Y  V D iS 2I i 2I iS 2I S 1 1 ˘ 2: If Y  V D Vi , then FM .Vi /  FM .V /. F 1 .V / D F 1 .V / i 2I

i 2I

i 2I

Note that in the first relationship “˘ 2” as compared to the singlevalued case, the equality is not fulfilled. Let X be a topological space. On C.X / some natural topologies are defined transforming it into a topological space. To introduce them we will need the notion of convergence of set sequences. Consider the generalized sequences (or net), which fix the index set A and the directed system F of (nonempty) subsets A. Remember that the system of nonempty subsets F of the set A is called directed, if for any two elements D1 ; D2 2 F 9 D3 2 F such that D3  D1 \ D2 . The function defined on the set A is called generalized sequence (or net). Let X be a topological space, ' be

1.1 The Main Results from Multivalued Mapping Theory

3

a sequence, the elements of which are subsets of the space X and in this sequence the set X˛ corresponds to the index ˛, i.e., ' D fX˛ ; ˛ 2 Ag. Definition 1.1. The set of points x of the space X is called the upper limit Lim ' F

of the sequence ' on the directed system F if it satisfies the following condition: for any neighborhood O.x/ of the point x in the space X and any element F 2 F there exists ˇ 2 F (ˇ  F ) for which Xˇ \ O.x/ 6D ;. Definition 1.2. The set of all points x 2 X is called a lower limit of the sequence ' on the centered system F and denoted by Lim ' if for any neighborhood O.x/ of F

the point x in the space X there exists F 2 F for which X˛ \ O.x/ 6D ; 8 ˛ 2 F (˛  F ). Let us remember that the system of subsets of the set A is called centered, if the intersection of any finite number of its elements is nonempty. It is easily seen that any directed system is centered but not otherwise. Remark 1.1. If A is a set of natural numbers then F presents a collection of sets expected as fn; n C 1; : : :g. And here x belongs to the upper limit of the sequence fXn g1 nD1 , if any neighborhood of the point x is intersecting with infinite number of the sets Xn . Similarly, x belongs to the lower limit of the sequence fXn g1 nD1 , if any neighborhood O.x/ is intersecting with all Xn which starting from the some one. If X is metric space with the metric then x 2 Lim Xn is equivalent to the exisn!1

tence of the sequence fxn g, xn 2 Xn such that lim .x; Xn / D 0 (or xn ! x n!1

in X ). Similarly, x 2 Lim Xn is equivalent 9fxkn g such that k1 < k2 <    , n!1 x D lim xkn and xkn 2 Xkn 8n  1. n!1

It is obvious that Lim '  Lim '. F

F

Let us give the main rules of actions: (1) If X˛0  X˛ for any ˛ 2 A then Lim X˛0  Lim X˛ and Lim X˛0  Lim X˛ ; F

F

F

F

(2) If X0  X and X˛ D X0 for any ˛ 2 A, then Lim X˛ D Lim X˛ D ŒX0 X ; F

F

(3) If A0  X and F0 D fF \ A0 ; F 2 Fg consists of nonempty subsets then Lim X˛  Lim X˛ and Lim X˛  Lim X˛ ; 0 F

F

F0

F

(4) Lim X˛ D LimŒX˛ X D ŒLim X˛ X ; F

F

F

F

F

F

Lim X˛ D LimŒX˛ X D ŒLim X˛ X ; ˇ

(5) Let B be an arbitrary set, fX˛ W ˛ 2 A; ˇ 2 Bg be a family of subsets from the topological space X then ˚T ˇ  T ˚T ˇ  T Lim X˛  Lim X˛ˇ , Lim X˛  Lim X˛ˇ , F

S

ˇ

ˇ Lim X˛ F ˇ

ˇ

F

F

ˇ

ˇ

F

˚S ˇ  S ˚S ˇ  ˇ  Lim X˛ , Lim X˛  Lim X˛ ; F

ˇ

ˇ

F

F

ˇ

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1 Preliminary Results

(5’) If B consists of two elements then ˚  Lim X˛ˇ1 [ X˛ˇ2 D Lim X˛ˇ1 [ Lim X˛ˇ2 ; F

F

F

(6) If A D N is a set of natural numbers then 1  T S S S S Lim Xn D XnCi X  Xn D ŒXn X Lim Xn , n i D0 n n    Lim X  Y Lim X  Lim Yn , n n n   Lim Xn  Yn D Lim Xn  Lim Yn , \Xn  Lim Xn . Definition 1.3. The sequence of sets is called converging to the set X0 if its upper and lower limits coincide. And we write that X0 D Lim X˛ . F

The following properties are valid: (7) If for any ˛ 2 A there exists the set F 2 F, such that X˛  Xˇ for any  F S ˇ2 (ˇ  F ), then the sequence fX˛ W ˛ 2 Ag converges and Lim X˛ D X˛ X ; F

˛

any (8) If for any ˛ 2 A there exists F 2 F, such that Xˇ  X˛ forT  ˇ 2 F (ˇ  F ), then the sequence fX˛ W ˛ 2 Ag converges and Lim X˛ D X˛ X . F

˛

The Vietoris Topology. This topology on C.X / corresponds to the notion of the ˚set sequences convergence. First of all, let us note that C.X / n C.X n U0 / D G 2 C.X / W G \ U0 6D ; . Leopold Vietoris (4 June 1891 – 9 April 2002) was an Austrian mathematician and a World War I veteran who gained additional fame by becoming a supercentenarian (unusual especially for a male). He was born in Radkersburg and died in Innsbruck. He was known for his contributions to topology and other fields of mathematics, his interest in mathematical history and for being a keen alpinist. Vietoris attended the University of Vienna, where he earned his Ph.D in 1920. Let U0 ; U1 ; : : : ; Un  X . Suppose O.U0 ; U1 ; : : : ; Un / D C.U0 / \ fC.X / n C.X n U1 /g \    \ fC.X / n C.X n Un /g, i.e., the set G lies in O.U0 ; U1 ; : : : ; Un / when G 2 C.X /, G  U0 and G \ Ui 6D ;, i D 1; : : : ; n. Moreover, the sets O.U0 ; U1 ; : : : ; Un / where U0 ; U1 ; : : : ; Un are open, form the basis of topology on C.X / which is called the Vietoris topology or exponential topology. If the set U runs all open sets of the space X then the sets C.U / and C.X / n C.X n U / form an open subbase in the space C.X /. Upper Semifinite Topology. The open basis of this topology is formed by the sets C.U /, when U runs all nonempty open sets of the space X . The upper semifinite topology is weaker than the Vietoris topology and C.X / in this topology is not T1 -space. Lower Semifinite Topology. Its open subbasis is formed by the sets C.X / n C.X n U / when U runs the collection of all nonempty open sets of the space X . This topology is weaker than the Vietoris topology and C.X / in this case is not

1.1 The Main Results from Multivalued Mapping Theory

5

T1 -space. It is the weakest topology in which the sets C.K/ (where K is closed set in X ) are closed. Let X be a metric space with the metric . Indicate by Cr .X / the collection of nonempty closed bounded subsets in X . The value dist.G1 I G2 / D sup inf .x; y/ x2G1 y2G2

is called a deviation of the set G1 from the set G2 . Suppose O" .x/ D fy 2 X W .x; y/ < "g;

O" .G/ D

[

O" .x/:

x2G

The distance distH .G1 I G2 / between the sets G1 ; G2 2 Cr .X / is accepted to be the following number .G1 I G2 / distH .G1 I G2 / D maxfdist.G1 I G2 /; dist.G2 I G1 /g D b D inf f" W "  0; G1  O" .G2 /; G2  O" .G1 /g: The functional distH .; / W Cr .X /  Cr .X / ! R is nonnegative and symmetric on the pair of its arguments and is nondegenerate (i.e., distH .G1 I G2 / D 0 ) G1 D G2 ) and satisfies the triangle inequality. Thus, distH sets the metric on Cr .X / which is called the Hausdorff metric. In this case from the completeness of the space X it follows the completeness of Cr .X /. Then (if it is not otherwise specified) we will assume that X and Y are arbitrary topological spaces and the mappings F W X ! 2Y are strict. Definition 1.4. The multivalued mapping F W X ! 2Y is lower semicontinuous at the point y0 2 X if F .y0 /  Lim F .y/ and lower semicontinuous if it is lower y!y0

semicontinuous at each point of the space X . Note that this definition of the lower semicontinuity of mapping F is equivalent to the following: for any 0 2 F .y0 / and any neighborhood V of the point 0 there exists the neighborhood U of the point y0 such that F .y/ \ V 6D ; 8y 2 U . Proposition 1.1. The multivalued mapping F W X ! 2Y is lower semicontinuous at the point x0 2 X if and only if for each net fx˛ I ˛ 2 Ag converging to x0 in X and an arbitrary 0 2 F .x0 / there exists the net f˛ I ˛ 2 Ag such that ˛ 2 F .x˛ / and ˛ ! 0 in Y . Proof. Necessity. Let F be lower semicontinuous at the point x0 2 X and x˛ ! x0 in X . Let us consider the arbitrary 0 2 F .x0 / and its neighborhood V in Y . Then there may be found the neighborhood U of the point x0 such that F .x/ \ V 6D ; 8x 2 U . In addition to it the net fx˛ I ˛ 2 Ag converges to x0 , that is why 9 ˛0 such that x˛ 2 U 8˛  ˛0 . Since F .x˛ / \ V 6D ;, then choosing ˛ 2 F .x˛ / \ V 8 ˛  ˛0 we obtain the net f˛ I ˛ 2 Ag which almost all lies in V . Because V is arbitrary we have ˛ ! 0 in Y . Sufficiency. Let the conditions of the Proposition be fulfilled, but the mapping F is not lower semicontinuous at the point x0 . Then there may be found 0 2 F .x0 /

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1 Preliminary Results

and its neighborhood V in Y such that in any neighborhood U of the point x0 there exists the element xU such that F .xU / \ V 6D ;. Let us consider the net fxU I U 2 ˝x0 g where ˝x0 is a directed set of all neighborhoods of the point x0 from the space X . It can be easily understood that this net converges to the point x0 , since it almost all lies in the arbitrary neighborhood of the point x0 . Nevertheless, any net U 2 F .xU / possesses the property that U … V and, therefore, it cannot converge to the point 0 , but it condition of the Proposition. Proposition 1.2. The following conditions are equivalent: (a) (b) (c) (d)

F W X ! 2Y is lower semicontinuous; 1 .B/ is closed in X ; For any closed B  Y a small pre-image FM For any open V  Y a complete pre-image F 1 .V / is open in X ; For any A  X F .A/  F .A/.

Proof. Let us prove only that “a” , “d”. Consider “a” ) “d”. Let F be lower semicontinuous and x0 be an arbitrary adherent point of the set A. Let us show that each point 0 2 F .x0 / is an cluster point of the set F .A/. Let V be an arbitrary neighborhood of 0 . Then by lower semicontinuity there may be found such a neighborhood U of the point x0 such that F .x 0 / \ V 6D ; 8 x 0 2 U and since x0 2 A then 9 x 0 2 A \ U and  0 2 F .x 0 / \ V , i.e.,  0 belongs to F .A/ and V simultaneously. Thus, F .x0 /  F .A/ for any point x0 2 A or F .A/  F .A/. Now let us prove “d” ) “a”. Consider the arbitrary closed set B in Y and 1 .B/ is closed in X and due to the Proposiprove that a small pre-image FM 1 .B/ then tion “b” is equivalent to “a”. Let x0 be an arbitrary point from FM 1 1 1 F .x0 /  F .FM .B//  F .FM .B//. It is obvious that F .FM .B//  B, 1 1 F .FM .B//  B D B, thus, F .x0 /  B, i.e., x0 2 FM .B/. Consequently, 1 1 FM .B/  FM .B/, i.e., small pre-image is closed. Proposition 1.3. The multivalued mapping F W X ! 2Y is lower semicontinuous if the complete pre-images of open sets included in some topology subbasis of the space Y are open in X . Proof. Let ˛ be some subbasis of topology of the space Y , ˇ be a topology base generated by this generators system. Any V 2 ˇ can be represented in the form n n T T Vi , Vi 2 ˛. Then F 1 .V / D F 1 .Vi / where F 1 .Vi / are open sets V D i D1

i D1

from X in accordanceS with the property “˘1”. Now S let W be an arbitrary open set from Y , hence, W D Vi , Vi 2 ˇ and F 1 .W / D F 1 .Vi / is open as a union i

i

of the open sets in accordance with “˘ 2”. It remains to use Proposition 1.2. Definition 1.5. The mapping F W X ! 2Y is !-upper semicontinuous at the point y0 2 X if F .y0 /  Lim F .y/. F is !-upper semicontinuous if it is !-upper y!y0

semicontinuous at each point of the space X .

1.1 The Main Results from Multivalued Mapping Theory

7

Definition 1.6. The mapping F W X ! 2Y is upper semicontinuous at the point y0 2 X if for any neighborhood V of the set F .y0 / there exists a neighborhood U of the point y0 such that F .U /  V . F is upper semicontinuous if it is upper semicontinuous at each point of the space. Proposition 1.4. Let F W X ! 2Y be lower (upper) semicontinuous mapping and X0  X . Then F jX0 is lower (upper) semicontinuous. Proposition 1.5. The following conditions are equivalent: (a) The mapping F W X ! 2Y is upper semicontinuous; 1 (b) For any open set V  Y the small pre-image FM .V / is open in X ; (c) For any closed set B  Y the complete pre-image F 1 .V / is closed in X . Remark 1.2. Let f W X ! Y be a singlevalued mapping of the topological space X into the topological space Y . It is upper semicontinuous as a multivalued mapping if and only if when it is continuous as singlevalued one. Suppose F W Y ! 2X (F .y/ D f 1 .y/). This mapping is upper semicontinuous if f is closed. Proposition 1.6. Let X be a compact topological space, F W X ! 2Y be an upper semicontinuous mapping with compact values. Then F .X / is a compact in Y . Proof. Let U be an arbitrary covering of the set F .X / by the open in Y sets and U  is a collection of all finite subsets from U . Consider the arbitrary ı 2 U  and 1 .Vı / is open set from X . For any suppose that Vı D [ı is open set in Y . Then FM x 2 X the set F .x/ is compact X

[ ı2U 

1 1 FM .Vı /  FM

 [

Vı



ı2U 

Since the space X is compact, it is possible to choose the finite subcovering 1 1 1 fFM .Vı1 /I    I FM .Vın /g from the open covering fFM .Vı /I ı 2 U  g. Theren S ıi is the finite subcovering of U . fore, i D1

Proposition 1.7. Let the mapping F W X ! 2Y be a upper (lower) semicontinuous and F .x/ be a connected set for any x 2 A  X . Moreover, let A be a connected set in X . Then F .A/ is the connected subset in Y . Proof. Let us remember that the set A is connected in X if there do not exist two nonempty separated sets V1 and V2 such that A1 D A \ V1 6D ; and A2 D A \ V2 6D ;. Assume the opposite, i.e., F .A/ D U1 [ U2 and Ui are nonempty open sets without generic points, i.e., U1 and U2 are separated (U1 \U2 D 1 ;, U1 \ U2 D ;). Then due to upper semicontinuity W1 D FM .U1 / and 1 W2 D FM .U2 / are nonempty and open, and since F .W1 /  U1 , F .W2 /  U2 then W1 \W2 D ;. Then since the values of F are connected, we have A D W1 [W2 . The obtained contradiction proves the Proposition for the upper semicontinuous mappings. If F is lower semicontinuous then we must consider the closed sets U1 and U2 similarly.

8

1 Preliminary Results

Proposition 1.8. The mapping F W X ! 2Y is upper semicontinuous at the point x0 2 X if and only if when for an arbitrary net fx˛ I ˛ 2 Ag converging to the point x0 in X , the set F .x0 / attracts fF .x˛ /g, i.e., for any neighborhood V .F .x0 // of the set F .x0 / there exists ˛0 such that V .F .x˛ //  V .F .x0 // 8 ˛  ˛0 . Proof. Necessity. Let F be upper semicontinuous at the point x0 2 X then for the arbitrary neighborhood V .F .x0 // in Y there exists the neighborhood U.x0 / of the point x0 in X such that F .U..x0 //  V .F .x0 //. Let the net x˛ ! x0 in X then 9 ˛0 such that x˛ 2 U.x0 / 8˛  ˛0 , i.e., F .x˛ /  V .F .x0 // 8˛  ˛0 . The sufficiency can be proved by contradiction. Suppose graph F D f.xI y/ W x 2 X; y 2 F .x/g: The multivalued mapping F is completely characterized by its graph, otherwise, if G is nonempty subset in X  Y then the multivalued mapping F corresponds to it: y 2 F .x/ , .xI y/ 2 G. And here Dom F is a projection of the graph F on X , S S F .x/ D F .x/ is a projection of the graph F on Y . and Im F , x2X

x2Dom F

The inverse mapping F 1 W Y ! 2X of the multivalued mapping F is determined in accordance with the rule x 2 F 1 .y/ , y 2 F .x/ or x 2 F 1 .y/ , .xI y/ 2 graph F . Thus, Dom.F 1 / D Im F , Im.F 1 / D Dom F and graph F 1 D f.yI x/ 2 Y  X W .xI y/ 2 graph F g: Proposition 1.9. For the multivalued mapping F W X ! 2Y the following conditions are equivalent: (a) graph F is closed; (b) For arbitrary x 2 X; y 2 Y such that y … F .x/ there exist the neighborhoods U.x/  X and V .y/  Y for which F .U.x// \ V .y/ D ;; (c) For arbitrary nets fx˛ g  X , fy˛ g  Y such that x˛ ! x in X , y˛ 2 F .x˛ /, y˛ ! y in Y ) y 2 F .x/. Remark 1.3. The closed mapping is necessarily close-valued. Proposition 1.10. If Y is regular then each upper semicontinuous mapping F W X ! C.Y / is closed. Proof. Consider the nets fx˛ I ˛ 2 Ag, fy˛ I ˛ 2 Ag such that x˛ ! x0 in X , y˛ 2 F .x˛ / and y˛ ! y0 in Y , and show that y0 2 F .x0 /. For the arbitrary neighborhood V .F .x0 //  Y of the set F .x0 / there exists the index ˛0 such that y˛ 2 V .F .x0 //

8˛  ˛0

from which it may be concluded that y0 2 V .F .x0 //. The set F .x0 / is closed, and the space Y is regular, that is why because of the arbitrariness of V we have y0 2 F .x0 /.

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9

Remark 1.4. If F W X ! K.Y / then in Proposition 1.10 it is sufficient that the space Y will be Hausdorff instead of regularity. Let us remember that the mapping F W X ! 2Y is called upper semicompact if out of x˛ ! x in X from the net ˛ 2 F .x˛ / it is possible to separate the subnet  !  in Y and here  2 F .x/. Proposition 1.11. Each upper semicompact mapping F W X ! 2Y is upper semicontinuous. Proof. Consider the point x0 2 X at which F is not upper semicontinuous. Then there exists the neighborhood V of the set F .x0 / such that in any neighborhood U of the point x0 2 X there exist xU such that F .xU / 6 V . Consider the nets fxU I U 2 ˝x0 , fU I U 2 F .xU /g where ˝x0 is a directed set of all neighborhoods of the point x0 . It is obvious that the net fxU g converges to the point x0 , and since the operator F is upper semicompact then it is possible to extract such a subnet fU g (let us maintain for it the same indication) that U ! 0 in Y and here 0 2 F .x0 /. Proposition 1.12. Let F W X ! K.Y / and graph F be locally compact. Then F is upper semicontinuous. Proposition 1.13. Let F W X ! K.Y / be upper semicontinuous, X and Y are metric spaces. Then for any sequence xn ! x0 in X , yn 2 F .xn / it is possible to extract the subsequence fym g such that ym ! y0 2 F .x0 /. Proof. Consider the set X0 D f0g [ f2n ; n D 1; 2; : : :g and the singlevalued mapping f W X0 ! X .f .2n / D xn ; f .0/ D x0 /: It is obvious that the mapping f is continuous and due to Remark 1.2 and Proposition 1.2 the mapping G D F ı f W X0 ! 2Y is upper semicontinuous. It means the mapping G0 W X0 ! 2Y



G0 .2n / D yn /

 and G0 .0/ D G.0/ D F .x0 / :

Under condition of the Proposition the set G0 .x0 / is compact, therefore, the set fyn I n D 1; 2; : : :g has the cluster point y0 . Therefore, it is possible to choose the subsequence fynk g converging to the point y0 . Thus, y2

1 \ ˚

 ynk I k D m; m C 1; : : : [ F .x0 / D F .x0 /:

mD1

Remark 1.5. Let X and Y be Hausdorff spaces and F W X ! 2Y be an upper semicompact mapping. Then F is closed. The validity of this Proposition results from the fact that X  Y is a Hausdorff space in Tikhonov topology and, consequently, any compact (in this case graph F ) is a closed set.

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1 Preliminary Results

Proposition 1.14. If the mapping F W X ! C.Y / is closed then F .A/ 2 C.Y / 8 A 2 K.X /. Proposition 1.15. Let F W X ! K.Y / be an upper semicontinuous mapping. Then F .A/ 2 K.Y / 8 A 2 K.X /. Definition 1.7. The mapping F W X ! 2Y is called quasiopen if Dom.intF / D X (.int F /.x/ D int F .x/) and graph.intF / is open in X  Y . It turns out that if Y is a metric space then the mapping F W X ! 2Y is upper Y semicontinuous if and  only if when for any " > 0 the mapping F" W X ! 2 F" .x/ D U" .F .x// is quasiopen. if and only Proposition 1.16. The mapping F W X ! 2Y is lower semicontinuous  if F W X ! C.Y / is lower semicontinuous F .x/ D F .x/ . Definition 1.8. The multivalued mapping F W X ! 2Y is called continuous if it is upper semicontinuous and lower semicontinuous. Proposition 1.17. The mapping F W X ! C.Y / (being considered as a singlevalued mapping from X into C.Y / with Vietoris topology) is continuous if and only if when it is upper semicontinuous and lower semicontinuous. The important class of continuous multivalued mappings is the class of Lipschitzean mappings. Point out that if X and Y are metric spaces and F W X ! K.Y / then F is upper semicontinuous at x0 2 X if and only if when 8 " > 0 9 ı > 0 such that F .x/  B" .F .x0 // 8 x 2 Bı .x0 /. If F .x0 / is not compact then upper semicontinuity does not follow from the above Proposition [AuFr90]. Definition 1.9. The mapping F W X ! 2Y (X; Y are metric spaces) is called Lipschitzean in the neighborhood of x0 2 X if there exists the neighborhood U.x0 / of the point x0 2 X and c > 0 such that F .x/  Bcd.x;y/ .F .y//

8x; y 2 U.x0 /:

(1.1)

The mapping is locally Lipschitzean if it is Lipschitzean in the neighborhood of each point x 2 X and it is Lipschitzean if 9 c > 0 such that F .x/  Bcd.x;y/ .F .y//

8 x; y 2 X:

Here d is a metric in X . Definition 1.10. The multivalued mapping F from the metric space X into the metric space Y is upper locally Lipschitzean at x0 2 X if there exists the neighborhood U.x0 / of the point x0 2 X and c > 0 such that F .x/  Bcd.x;y/ .F .x0 //

8x 2 U.x0 /:

Consider some operations on multivalued mappings.

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11

Proposition 1.18. 1. Let fFi ; i 2 I g be a family of lower semicontinuous mapS pings at the point x0 2 X . Then the mapping F .x/ D Fi .x/ is lower i 2I

semicontinuous at the point x0 2 X ; 2. Let F1 ; F2 W X ! 2Y be upper semicontinuous mappings at the point x0 2 X . Then F .x/ D F1 .x/ [ F2 .x/ is upper semicontinuous at the point x0 2 X ; 3. Let F1 ; F2 W X ! C.Y / be closed mappings. Then the mapping F D F1 [ F2 W X ! C.Y / is closed. Proposition 1.19. 1. Let F1 ; F2 W X ! C.Y / be upper semicontinuous mappings and  the space Y be normal. Then the mapping F D F1 \ F2 W X ! C.Y / F .x/ D F1 .x/ \ F2 .x/ 6D ; 8 x 2 X is upper semicontinuous. 2. If F1 W X ! C.Y / is closed, F2 W X ! K.Y / is upper semicontinuous and F1 .x/ \ F2 .x/ 6D ; 8 x 2 X then the mapping F1 \ F2 W X ! K.Y / is upper semicontinuous. T Fj .x/ 6D ; 3. Let the multivalued mappings Fj W X ! C.Y / be closed and j 2I T Fj is closed. 8 x 2 X . Then the mapping F D j 2I

4. If the space Y is Hausdorff, Fj W X ! K.Y / is the upper T semicontinuous T Fj .x/ 6D ; 8 x 2 X then F D Fj W X ! K.Y / mapping 8j 2 I and j 2I

j 2I

is upper semicontinuous. 5. Let F1 W X ! 2Y be lower semicontinuous at the point x0 2 X and the mapping F2 W X ! 2Y be quasiopen at x0 2 X and F1 .x0 / \ F2 .x0 /  F1 .x0 / \ .int F2 .x0 //: Then F D F1 \ F2 is lower semicontinuous at the point x0 2 X . Proof. 1. Let us consider an arbitrary open neighborhood V .F .x0 // of the set F .x0 /. We must find such a neighborhood U.x0 / of the point x0 such that F .x/  V .F .x0 //

8 x 2 U.x0 /:

The sets Wi D Fi .x0 / n V .F .x0 //; i D 1; 2 are closed in the space Y and do not intersect. Since the space Y is normal, there may be found nonintersecting open neighborhoods Zi of the sets Wi . For Vi D Zi [V .F .x0 // we have Fi .x0 /  Vi . Thus, Vi is an open neighborhood of the set Fi .x0 / and due to the mapping’s upper semicontinuity there exists the neighborhood of the point x0 such that Fi .x/  Vi 8x 2 Ui .x0 /. Suppose U.x0 / D U1 .x0 / \ U2 .x0 /, then for x 2 U.x0 / we have     F .x/ D F1 .x/ \ F2 .x/  V1 \ V2 D Z1 [ V .F .x0 // \ Z2 [ V .F .x0 // D .Z1 \ Z2 / [ V .F .x0 // D V .F .x0 //: The other statements can be proved similarly.

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1 Preliminary Results

Corollary 1.1. The closed mapping F2 from X into the compact space Y is upper semicontinuous. Proposition 1.20. Let X; Y; Z be topological spaces, F W X ! 2Y , G W Y ! 2Z be a upper semicontinuous mapping (lower semicontinuous). Then S the mapping H D G ı F W X ! 2Z determined by the equality H.x/ D G./ is upper 2F .x/

semicontinuous (lower semicontinuous). Proof. The validity of the Proposition immediately follows from the relationships  1  1 1 HM GM .B/ ; .B/ D FM

  H 1 .B/ D F 1 G 1 .B/

8B 2 2Z :

Remark 1.6. If F W X ! K.Y / is upper semicontinuous and G W Y ! C.Z/ is closed then H D G ı F W X ! C.Z/ is a closed mapping. Besides, if Y is a Hausdorff space and F W X ! K.Y / is a closed mapping then H is closed. Y Proposition 1.21. Let Y be a topological vector  space. If F1 ; F2 W X ! 2 is Y lower semicontinuous then F1 C F2 W X ! 2 .F1 C F2 /.x/ D F1 .x/ C F2 .x/ is lower semicontinuous. If F1 ; F2 W X ! K.Y / is upper semicontinuous then F1 C F2 W X ! K.Y / is upper semicontinuous.

Proposition 1.22. Let f W X ! R be a continuous function. If the mapping F W X ! 2Y is lower semicontinuous (F W X ! K.Y / is upper semicontinuous) then the mapping f ı F W X ! 2Y (respectively f ı F W X ! K.Y /) possesses the same property where .f ı F /.x/ D f .x/F .x/. Let F W X ! 2Y be a multivalued mapping of an arbitrary set X into the topological space .Y I /. Obviously, if in some topology the mapping F on X is upper semicontinuous (lower semicontinuous) then it has the same property with respect to a stronger topology on X . That is why it is interesting to find the weakest topology on the set X , with respect to which the mapping F will be upper semicontinuous (respectively, lower semicontinuous). Let us remark that the family of the subsets from the set X consisting of F 1 .V / where V 2 , is the weakest topology on X , with respect to which F 1 is lower semicontinuous. Besides, the family consisting of FM .V / where V 2  forms the base of the weakest topology on X , with respect to which F is upper semicontinuous. Yi Let the multivalued ˚  mapping Fi W X ! 2 corresponds to each i 2 I , where X is a set and .Y I / i 2I is a family of topological spaces. Let us denote i D  ˚ 1 S Fi .Vi /; Vi 2 i . Then

i is a prebase of the topology  on X , which is i 2I

called a complete initial topology. The topology  is the weakest topology on X , with respect to which all Fi are lower semicontinuous. Similarly a small initial topology M on X (i.e., the weakest topology on X , with respect to which all Fi are upper semicontinuous) is constructed. Let us point out that M   .

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13

Proposition 1.23. Let Z be a topological space, G W Z ! 2X be a mapping from Z into X with the topology  . The mapping G is lower semicontinuous if for each i 2 I the composition Fi ı G is lower semicontinuous. Proof. The necessity is evident. Let us prove the sufficiency.S Let any Fi ıG be lower semicontinuous and V be an arbitrary set from the prebase i , i.e., V belongs to i

some i0  and, thus, V D Fi1 .U /, where U 2 i0 . It is obvious that 0 .U // D .Fi0 ı G/1 .U / G 1 .V / D G 1 .Fi1 0 is an open set by the lower semicontinuity of Fi0 ı G. From this in virtue of Proposition 1.11 and arbitrariness of V we conclude that G is lower semicontinuous. Let us introduce the final topology. Let X be an arbitrary set, fYi ; i g be some family of topological spaces and for each i 2 I the multivalued mapping Fi W Yi ! 2X is specified. Let us consider the family of all subsets U  X such that Fi1 .U / is an open set in Yi 8i 2 I . It is obvious that satisfies all axioms of 1 the topology on X (for FiM .U / it is not true). Proposition 1.24. The multivalued mapping G from the topological space X with the final topology into the topological space Y is lower semicontinuous if and only if for any i 2 I the composition G ı Fi W Yi ! 2Z is lower semicontinuous. Corollary 1.2. If Fi W Yi ! X are singlevalued mappings and X is endowed with the final topology then G W X ! 2Z is upper semicontinuous if and only if when the composition G ı Fi is upper semicontinuous 8i 2 I . Let us consider the Cartesian product of multivalued mappings. Let Z D X  Y be a product of topological spaces X; Y , ˘X W Z ! X , ˘Y W Z ! Y are canonical projections defined by the formulas ˘X .xI y/ x 2 X;

˘Y .xI y/ y 2 Y:

As it is known, ˘Y ; ˘X are continuous open mappings (but not closed). Let E be some topological space and F W E ! 2XY be a multivalued mapping with the components FX W E ! 2X , FY W E ! 2Y (FX D ˘X ı F , FY D ˘Y ı F ). Proposition 1.25. The mapping F W E ! 2XY is lower semicontinuous if and only if when each component FX ; FY is lower semicontinuous. Proof. The necessity is obvious, since the projection operation is continuous. Let us consider the sufficiency. Let FX and FY be lower semicontinuous. Prove that F 1 .U  V / is an open set for any open U and V in X and Y respectively. Really, since U  V D .U  X / \ .V  Y / D ˘X1 .U / \ ˘Y1 .V /;

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1 Preliminary Results

then   F 1 .U  V / D F 1 ˘X1 .U / \ ˘Y1 .V /     D F 1 ˘X1 .U / \ F 1 ˘Y1 .V / D FX1 .U / \ FY1 .V / is an open set as an intersection of open sets. The set U  V forms the base of the topology on X  Y and in virtue of Proposition 1.3 F is a lower semicontinuous mapping. Proposition 1.26. The mapping F W E ! 2Z is closed, if each component FX ; FY is closed. If, in addition, F W E ! K.Z/, graphF is locally compact and the spaces Y; X are regular then the mentioned condition is necessary too. Proof. Sufficiency. Let FX and FY be some closed sets in E  X and E  Y respectively. In the space E  X  Y we consider the Cartesian product of the operators .I  ˘X / W E  X  Y ! E  X;

.I  ˘Y / W E  X  Y ! E  Y;

operating in accordance with the rule .I  ˘X /.eI xI y/ .eI x/; .I  ˘Y /.eI xI y/ .eI y/ 8.eI xI y/ 2 E  X  Y: It is clear that the operators .I ˘X /; .I ˘Y / are continuous and open, that is why .I  ˘X /1 .graphFX / and .I  ˘Y /1 .graphFY / are closed sets in E  X  Y , therefore,     graphF D .I  ˘X /1 .graphFX / \ .I  ˘Y /1 .graphFY / is a closed set. Necessity. Let F W E ! K.Z/ be a closed locally compact mapping. By virtue of Proposition 1.12 the mapping F is upper semicontinuous, and it means that each of the mappings FX and FY is upper semicontinuous as well. And since the spaces Y and X are regular, then the mappings FX W E ! K.X / and FY W E ! K.Y / are closed (Proposition 1.10). Remark 1.7. If F W X ! 2Y Z is an upper semicontinuous mapping then each component FY and FZ is upper semicontinuous. The converse Proposition is not true. Proposition 1.27. If FY W X ! K.Y / and FZ W X ! K.Z/ is an upper semicontinuous mapping then the mapping F W X ! K.Y  Z/ is also upper semicontinuous. In applications, starting from the given multivalued mappings F1 W E ! 2X , F2 W E ! 2Y we can construct the mapping F W E ! 2XY such that its components

1.1 The Main Results from Multivalued Mapping Theory

15

coincide with F1 and F2 , i.e., F1 D FX , F2 D FY . Such mapping ˚  F .X / D .1 I 2 / 2 X  Y W 1 2 F1 .x/; 2 2 F2 .x/ ;   given by the formula F .x/ D F1 .x/I F2 .x/ is called a diagonal product of the mappings F1 and F2 and denoted by F D F1 4F2 . From the definition of the mapping F it follows that the diagram F2

F1

2X

 E ! 2Y - ˘X # F % ˘Y 2XY

is commutative, i.e., F1 D ˘X ı F , F2 D ˘Y ı F . Remark 1.8. It is obvious that for a diagonal product of two (finite number) of mappings Propositions 1.25–1.27 are valid. Let F1 W X1 ! 2Y1 , F2 W X2 ! 2Y2 , then the mapping F W X1  X2 ! 2Y1 Y2 defined by the formula   F .x1 ; x2 / D F1 .x1 /I F2 .x2 /  ˚ D .1 I 2 / 2 Y1  Y2 W 1 2 F1 .x1 /; 2 2 F2 .x2 / is called a Cartesian product of F1 and F2 and it is denoted by F D F1  F2 . If ˘Yi W Y1  Y2 ! Yi and ˘Xi W X1  X2 ! Xi (i D 1; 2) are canonical projections, then the following diagram is commutative: X2 # F2 2Y2

˘X2



FY2

.

˘Y2



X1  X2 #F 2Y1 Y2

˘X1

! FY1

&

˘Y1

!

X1 # F1 2Y1 ;

i.e., FY1 D F1 ı ˘X1 D ˘Y1 ı F , FY2 D F2 ı ˘X2 D ˘Y2 ı F . From the commutativity of this diagram it is clear that the Cartesian product F1  F2 can be considered as a diagonal product of the mappings FY1 and FY2 , i.e., F1  F2 D FY1 4FY2 . Proposition 1.28. The Cartesian product F D F1 F2 of the multivalued mappings F1 and F2 is lower semicontinuous if F1 and F2 are lower semicontinuous. Proof. The sufficiency directly results from Remark 1.8 and from Proposition 1.25, because and FY2 of the mapping F D FY1 4FY2 are lower semicontinuous. Necessity. Let the mapping F be lower semicontinuous then the components FY1 and FY2 are lower semicontinuous too, from which it follows that F1 and F2 are

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1 Preliminary Results

lower semicontinuous. Really, let V be an arbitrary open set in Y1 then FY1 .V / D 1   ˘X11 F11 .V / is an open set in X1  X2 . But since the projection mapping ˘X1 is open then the image   .V / D F11 .V / \ ˘X1 .X1  X2 / D F11 .V / \ X1 D F11 .V / ˘X1 FY1 1 open in X1 too. The lower semicontinuity of F2 is proved similarly. Remark 1.9. From Remark 1.7 we may conclude that from the upper semicontinuity for F D F1  F2 it follows the upper semicontinuity F1 and F2 . Proposition 1.29. 1. The mapping F D F1  F2 is closed, if F1 and F2 are closed too. 2. If F D F1  F2 W X1  X2 ! K.Y1  Y2 / is a closed mapping, graphF is locally compact set, and Y1 and Y2 are regular spaces, then F1 and F2 are closed mappings. Proof. 1. Let F1 and F2 are closed mappings, therefore, graphF1 and graphF2 are closed sets in X1  Y1 and in X2  Y2 respectively. Let us prove that FY1 and FY2 are closed mappings. If the spaces X1 ; X2 are Hausdorff then this Proposition immediately follows from Remark 1.6. That is why for completion of the proof it is enough to use Proposition 1.26. In the general situation let us consider the arbitrary nets f˛ g  X1  X2 and fy˛ g  Y1 such that ˛ !  in X1  X2 and y˛ ! y in Y1 . Then ˘X1 ˛ ! ˘X1  in X1 , and since the mapping F1 is closed, then y 2 F1 .˘X / D FY1 ./, i.e., in virtue of Proposition 1.9 FY1 is a closed mapping. Analogously we can prove that FY2 is closed. 2. In virtue of Remark 1.9 the mappings F1 W X ! K.Y1 /, F2 W X ! K.Y2 / are upper semicontinuous. Since the mapping F is upper semicontinuous, and since the spaces Y1 and Y2 are regular, then F1 and F2 are closed (Proposition 1.10). Proposition 1.30. Let F1 W X1 ! K.Y1 /, F2 W X2 ! K.Y2 / are upper semicontinuous mappings. Then F D F1  F2 W X1  X2 ! K.Y1  Y2 / is upper semicontinuous too. Proof. It is obvious that the mappings FY1 W X1  X2 ! K.Y1 /, FY2 W X1  X2 ! K.Y2 / are upper semicontinuous. In virtue of Proposition 1.27 it follows that F D F1  F2 is upper semicontinuous. Proposition 1.31. Let each of the mappings F W X ! C.Y /, G W X ! C.Y / be upper semicontinuous and lower semicontinuous and the space Y be Hausdorff. Then the set A D fx 2 X W F .y/ D G.y/g is closed in X . Let us consider one class of the multivalued mappings operating in Banach spaces. Let X1 , X2 , X3 are Banach spaces and X D X1  X2  X3 . Let us remark that Bi .i D 1; 2; 3/ be the space of linear functionals that divide points in Xi . Let .Xi I Bi / be the weakest topology on Xi in which all functionals from Bi are

1.1 The Main Results from Multivalued Mapping Theory

17

continuous. The sequence fyn g  Xi is -weakly converges to the element y in Xi  (yn ! y), if it converges in .Xi I Bi /-topology on Xi . Further we will consider two cases: (a) Xi D Bi (then -week topology characterizes by weakly star convergence); (b) Bi D Xi (then -weak convergence coincides with weak convergence). Besides, in Cartesian product X is possible the combination of cases “a” and “b”. For example X1 D B1 ; X2 D B2 ; B3 D X2 . 

The sequence fyn g  Xi is  -weakly converges to y in X (yn ! y), if it converges either -weakly, or strongly (i.e., either weakly, or weakly star, or strongly). 3 Q Similarly we may determine the  -weak convergence on X D Xi as  -weak i D1

convergence of components. In future such properties for sets, like  -weak closeness and -weak compactness, we will understand in the sequence sense. Definition 1.11. The multivalued mapping R W domR  X1 ! 2X is: (1)  -upper weakly semicompact if the graph of the mapping R .graphR/ is  -weakly semicompact set, i.e., from an arbitrary sequence fun I n g  graphR 



it is possible to extract such subsequence fum I m g, that um ! u; m !  for some u 2 domR and  2 R.u/; 

(2)  -weakly closed if graphR is -weakly closed, i.e., from domR 3 un ! u, 

R.un / 3 n !  it follows that u 2 domR.u/ and  2 R.u/; 



(3)  -weakly semiclosed if from domR 3 un ! u 2 domR, R.un / 3 n !  it follows that  2 R.u/; (4)  -weakly semicompact for arbitrary sequence fun I n g  graphR, such that 

fun g is bounded in X1 there is a subsequence fum I m g for which um ! u; 

m ! , in particular, u 2 domR and  2 R.u/; 

(5) Completely  -weakly semicompact, if for any domR 3 un ! u 2 domR and 

8n 2 R.un / there exist a subsequence fm g such that m !  2 R.u/; (6)  -weakly compact if for any  -weakly closed subset U  domR the set R.U / is  -weakly compact in X ; 

(7)  -lower weakly semicompact if for arbitrary sequence domR 3 un ! u 2 domR and 8 2 R.u/ there are subsequences fum g, m 2 R.um / such that 

m ! . X Definition 1.12. Let i D 1; 7.SThe multivalued mapping S R W domR  X1 ! 2 satisfies Property is , if R D R˛ (i.e., graphR D graphR˛ ), where 8˛ 2 I ˛2I

˛2I

the mapping R˛ W X1 ! 2X satisfies the corresponding property “i” from Definition 1.11.

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1 Preliminary Results

Proposition 1.32. The following implications are true: 2.2s / H) 3.3s / (H 5.5s / .1/

% 1.1s / H) 4.4s / (H 6.6s / Besides, if domR domR˛ then 5s ) 7. Definition 1.13. Banach space X0 satisfies Condition ./, if X0 either reflexive or it is adjoint with some separable normalized space. Proposition 1.33. Let the spaces Xi ; i D 1; 2; 3; satisfy Condition ./ and . / D ./ D . /. Herewith next Propositions are true: (a) If domR (respectively domR˛ 8˛ 2 I ) is weakly star closed set, then the implication (1) on the diagram from the Propositionˇ is reversible; (b) If U is bounded weakly star closed set, then for RˇU the implication 3.3s / ) 1.1s / is true. Let X be a Hausdorff topological space, Y be a Hausdorff vector space. As Y  we will denote the topologically adjoint space to Y . For each nonempty subset KY we consider its upper support function K W Y  ! R [ fC1g which has the following properties: (1ı / K ./ is convex lower semicontinuous positively homogeneous function; (2ı / K ./ D coK ./ (3ı / The barrier cone b.K/ D f 2 Y  jK ./ < C1g of the set K is a convex set; (4ı / coK D fy 2 Y jhy; iY  K ./ 8 2 Y  g. Let X be a Banach space and Kv .X / be a family of all nonempty convex compact sets in X . The family Kv .X / with respect to arithmetic sum of sets and multiplication on positive scalars is semilinear metric space (with respect to Hausdorff’s metric). Let ˝ be measurable space with the measure . Let as consider some map F W ˝ ! Kv .X /. According to aforesaid we can naturally define the Bochner integral. Definition 1.14. The multivalued mapping F W ˝ ! 2X is called measurable if for 1 every open V  X FM .V / is a measurable set, or, equivalently, for every closed 1 U  X F .U / is a measurable set. The multivalued mapping F W ˝ ! K.X / is called strongly measurable if almost everywhere on ˝ it is a pointwise limit of piecewise maps sequence. It is well known that for strongly measurable maps in Banach spaces the analogues of Lusin’s C -property and Pettis’s Theorem hold true.

1.1 The Main Results from Multivalued Mapping Theory

19

The collection of all Bochner integrals of all summable selectors is called the Aumann integral for multivalued map F W ˝ ! 2X , namely 8 9 Z
˝

Let X , Y be Banach sets, ˝ be locally compact topological space with the measure . Let us assume that  is the measure of dissipation (namely for every -measurable ˝0  ˝ there can be found -measurable subset ˝1  ˝0 such that .˝1 / D 12 .˝0 /). Let us consider the family F .!; / W X ! 2Y of multivalued maps which satisfy the following conditions (the modification of Caratheodory conditions): (a) For almost each ! 2 ˝ F .!; / W X ! 2Y is upper semicontinuous mapping; (b) 8x 2 X the mapping ˝ 3 ! ! F .!; x/ 2 2Y is measurable. Definition 1.15. Let us set F W ˝  X ! 2Y . The multivalued mapping F which corresponds to each multivalued map G W ˝ ! 2X the multivalued map S W˝ ! 2Y by the following rule: S.!/ D F.G/.!/ D F .!; G.!// is called the multivalued superposition operator or multivalued Nemitsky operator. Remark 1.10. The Caratheodory conditions “a” and “b” are not sufficient for Nemitsky operator F to convert measurable maps into measurable [Pa87]. The role of Krasnoselsky Theorem for nonlinear analysis and evolutionary equations theory is well-known. Therefore let us consider some its multivalued analogues which are of interest for differential-operator inclusions. Proposition 1.34. Let X , Y be some Banach spaces, F W ˝  X ! 2Y be some multivalued mapping which satisfy Caratheodory conditions “a” and “b”, and ˝ be a measurable space with the measure . Let xn 2 Lp1 .˝; X / and yn 2 Lp2 .˝; Y / .pi  1; i D 1; 2; n  1/ be such that for every " > 0 and almost each ! 2 ˝ there exists N D N."; !/ such that d.fxn .!/I yn .!/gI graphF .!; //  " 8n  N

(1.2)

Moreover, if xn ! x strongly in Lp1 .˝; X / and yn ! y weakly in Lp2 .˝; Y / then y 2 co F.x/ or y.!/ 2 coF .!; x.!// a.e. in ˝. Proof. Due to the Mazur’s Theorem, for any n  1 the weak limit y belongs to the weak closure of the convex hull for fym gmn and there can be found some convex combination ! 1 1 X X m m m zn D an  0; an ym an D 1 ; mDn

mDn

20

1 Preliminary Results

anm are equal zero excepting perhaps finite number of elements, such that ky  zn kLp2 .˝I Y /  1=n. Hence we may assume that zn .!/ ! y.!/ and xn .!/ ! x.!/ a.e. in ˝. Let ! 2 ˝ be such that zn .!/ ! y.!/ in Y , xn .!/ ! x.!/ in X and (1.2) holds true. For fixed " > 0 and  2 Y  from the condition “a” we have: 9ı > 0 such that if kw  x.!/kX  2ı then .F .!; w/; /  .F .!; x.!//; / C "=2: Due to the condition (1.1) with each ı > 0 there is N  1 such that 8n  N 9.wn ; n / 2 graphF .w; /; kwn  xn .!/kX  ı; kn  zn .!/kY  :

(1.3)

Besides 9N1  N such that kxn .!/  x.!/kX  ı

8n  N1 :

Then in virtue of (1.3) we have 8n  N1 hzn .!/; iY  hn ; iY  C ı  .F .!; wn /; p/ C ıkkY  : Further since kwn  x.!/kX  2ı then .F .!; wn /; /  .F .!; wn /; / C "=2: Selecting ı 2 .0; "=2kkY  / from the previous inequality we obtain hzn .!/; iY  .F .!; x.!//; / C "

8n  N1 :

By passing to the limit as n ! 1 and " ! 0 we have hy.!/; iY  .F .!; x.!//; /

8 2 Y  :

Hence and from the properties of the support function we have y 2 coF.x/ or y.!/ 2 coF .!; x.!// a.e. in ˝. The Proposition is proved. Corollary 1.3. If mapping F W ˝  X ! 2Y has closed and convex values. Then in virtue of conditions of Proposition 1.34 y.!/ 2 F .!; x.!// a.e. in ˝. Corollary 1.4. Let the mapping F W ˝  X ! 2Y satisfies the Caratheodory conditions “a”, “b” and, besides, there exists x 2 Lp1 .˝I X / and measurable mapping l W ˝ ! 2Y such that I.!/  F .!; x.!// and l  Lp2 .˝I Y /. Then the closure of graphF in Lp1 .˝I X /  Lp2 .˝I Y / with the strong topology for Lp1 .˝I X / and with the weak topology for Lp2 .˝I Y / belongs to graph.coF /.

1.1 The Main Results from Multivalued Mapping Theory

21

Remark 1.11. Let X1 ; X2 ; : : : ; Xn ; Y be some Banach spaces, ˝ be a measurable set and the mapping F W ˝  .X1  X2  : : :  Xn / ! 2Y satisfies the following properties: n X p =r kF .!; h1 ; : : : ; hn /kC  g.!/ C khi kXii ; (1.4) i D1

where p1 ; p2 ; : : : ; pn ; r  1; g 2 Lr .˝/ for any xi 2 Lpi .˝I Xi / G.!/ D F .!; x1 ; x2 ; : : : ; xn / is a measurable mapping. Then the Nemitsky operator F .x/.!/ D F .!; x1 .!/; x2 .!/; : : : ; xn .!// bounded values in the space n Y

Lpi .˝I Xi / in Lr .˝I Y /:

i D1

Proof. Using the inequality (1.4) we have " kF .!; x1 .!/; x2 .!/; : : : ; xn .!//krC

 g.!/ C

n X

#r jxi .!/j

pi =r

i D1

"

 .n C 1/

.r1/

jg.!/j C r

n X

# jxi .!/j

pi

:

i D1

Since the function G./ W ˝ ! 2X is measurable and kG.!/krC  '.!/ where ' 2 Lp1 .˝/, then there exists the Aumann–Bochner integral, namely Z kF .!; x1 .!/; x2 .!/; : : : ; xn .!//krC d! < 1; ˝

therefore F .x/ 2 2Lr .˝I Y / . Corollary 1.5. Let the conditions of Proposition 1.34 hold true and xn ! x strongly in Lp1 .˝I X /, yn ! y strongly in Lp2 .˝I X / and the property (1.2) is satisfied. Then y 2 F .x/ or y.!/ 2 F .!; x.!// for a.e. ! 2 ˝. Corollary 1.6. Let the mapping F W ˝X ! 2Y satisfies conditions of Remark 1.11, where X D X1  X2  : : :  Xn ; 1 < r < 1; Y is a reflexive space. If, besides, for a.e. ! 2 ˝ the mapping F .!; / W X ! 2Y is closed with respect to the strong topology in X and the weak one in Y then the Nemitsky operator F W Z D Lp1 .˝I X1 /  : : :  Lpn .˝I Xn / ! 2Lr .˝I Y / is completely -weakly semicompact where  -weak convergence on Z coincides with the strong one, and it coincides on Lr .˝I Y / with the weak one.

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1 Preliminary Results

Proof. Let n D 1; p1 D p and a subsequence yn ! y strongly in Lp .˝I X /. Then due to Remark 1.11 kF .yk /kC  const and from the arbitrary sequence n 2 F .yk / there can be extracted a subsequence fm g such that m !  weakly in Lr .˝I Y /. We will assume that for almost each ! 2 ˝ ym .!/ ! y.!/ in X and since m .!/ 2 F .!; ym .!// a.e. in ˝, due to the inequality (1.4), the sequence fm .!/g is bounded in X for a.e. ! 2 ˝. Therefore up to a subsequence k .!/ ! .!/ weakly in Y for a.e. ! 2 ˝, moreover .!/ 2 F .!; y.!//. Since k !  weakly in Lr .˝I Y / then  D . The Corollary is proved. Definition 1.16. The class of objects Ob.K/ together with the morphisms class Mor.K/ D

[

Mor.X I Y /

X;Y 2Ob.K/

and the composition law ı make the category K if the following properties (category axioms) hold true: (k1 ) For A 2 Mor.X I Y /; B 2 Mor.Y I Z/ and D 2 Mor.ZI U / D ı .B ı A/ D .D ı B/ ı A is true; (k2 ) 8X 2 Ob.K/ 91X 2 Mor.X I X / ) 8A 2 Mor.X I Y / and 8B 2 Mor.ZI X / A ı 1X D A; 1X ı B D B. Let us consider the category Km (or multi), where Ob.Km / is a class of all linear topological spaces, Mor.Km / is a class of all multivalued maps, and Mor.X I Y / D fA W X ! Y g. It can be easily checked that all category axioms hold true. Let us denote by co a semicovariant functor .co.A1 ı A2 /  coA1 ı coA2 / acting from the category Km in Km by the following rule: 8X 2 Ob.Km /

coX D X;

8A 2 Mor.X I Y / coA W X ! 2Y

..coA.y/ D coA.y//;

where bar means the closure in the due topology. Similarly the functor co is defined. Now let us consider the duality of vector and affine processes. Adjoint maps to the convex map were investigated by Rokafellar, Pshenichny, Aubin, Morduchovich, Beresnev and others. Vector and affine processes are the special cases of convex processes. However accounting of the vector structure of maps allows to obtain new helpful properties for them. It is known that in nonHausdorff spaces none of linear operators has closed extension. Moreover its closure is a vector process. Let X , Y be locally convex spaces, X  , Y  be their topologically adjoint spaces, h; iX W X   X ! R be a canonical form. Space X  with .X  ; X /-topology we will denote by X , space X with the topology .X  ; X / by X! , and let P .Y / be the collection of all nonempty subsets of the space Y .

1.1 The Main Results from Multivalued Mapping Theory

23

Definition 1.17. Mapping  W D./  X ! P .Y / is called a vector process if: (a) D./ is a linear manifold; (b) .x1 / C .x2 /  .x1 C x2 / 8x1 ; x2 2 D./; (c) ˛.x/ D .˛x/ 8˛ 2 R .˛ ¤ 0/; 8x 2 D./. Definition 1.18. Mapping  W D./  X ! P .Y / is called an affine process if: (a) D./ is an affine manifold; (b) ˛1 .x1 / C ˛2 .x2 /  .˛1 x1 C ˛2 x2 / 8x1 ; x2 2 D./ and 8˛1 ; ˛2 2 R .˛1 C ˛2 D 1/. The following Proposition is valid. Proposition 1.35. Mapping  W D./  X ! P .Y / is a vector (affine) process if and only if the graph graph of the map  is a linear (affine) manifold. The proof follows from the definitions. Corollary 1.7. Let  W D./  X ! P .Y / be a linear (affine) process. Then .y/; .D.// are linear (affine) manifolds. Definition 1.19. Mapping  W D. /  Y  ! P .X  / defined by the relation .x  ; y  / 2 graph , .x  ; y  / 2 .graph/? ; is called adjoint with the vector process . Here M ? is an annihilator of the set M . We remark that D. / D fy 2 Y  j9x  2 X  ; .x  ; y  / 2 .graph/? g and

 .y  / D fx  2 X  j.x  ; y  / 2 .graph/? g:

Theorem 1.1. Let  W D./  X ! P .Y / be a vector process. Then the adjoint mapping  W D. /  Y  ! P .X  / is a vector process closed in X  Y . Proof. Since the annihilator is a linear set then graph is a linear manifold. Therefore due to Proposition 1.35  is a vector process. Let us prove that it is closed. A space adjoint with X  Y is identified with X   Y  . Here the canonical bilinear form is defined be the formula h.x  ; y  /; .x; y/i D hx  ; xiX C hy  ; yiY : The mapping æ W X  Y ! Y  X .æ.x  ; y  / D .y  ; x  //

24

1 Preliminary Results

is isomorphism. Let us show that the polar line .graph/0 of the set graph coincides with æ.graph /. The embedding æ.graph /  .graph/0 is obvious. Let us prove the reverse application. We assume the opposite, namely that there exists   D .x  ; y  / 2 .graph/0 such that   D .x  ; y  / … .graph /. This means that there can be found .x; y/ 2 graph for which hx  ; xiX C hy  ; yiY D ˛ ¤ 0: Without any restrictions we assume that ˛ > 0. Then for big enough ˇ > 0 such that ˛ˇ > 1 we have hx  ; ˇxiX C hy  ; ˇyiY > 1: Since ˇy 2 .ˇx/ then .x  ; y  / … .graph/0 that contradicts to the assumption. Hence the equality (1.5) .graph/0 D æ.graph / is proved. But the polar line is a closed set in X Y , and therefore the set graph is closed too. The Theorem is proved. Theorem 1.2. Let  W D. /  Y  ! P .X  / be a process adjoint with a vector b is a process adjoint with  with respect to .Y  ; Y /-topology process , and   b is minimal closed extension of . on Y . Then  b is closed in X!  Y! . Remark Proof. The process  is closed in X  Y and  b and .x/   b 8x 2 D./. Let æ W! X  Y be the adjoint that D./  D./ mapping to æ. Similarly to (1.5) we state the validity of the equality b D .graph /0 : æ .graph/

(1.6)

b are closed the equality (1.6) is equivalent to the Due to the fact that  and  following one: b D .æ.graph //0 : (1.7) graph Indeed, we have b D graph: b .æ.graph //0 D æ1 Œ.graph /0  D æ1 .æ .graph// From the other hand let (1.7) hold true. Then due to Theorem 2.3 [91, p. 164] b D æ1 Œ.graph /0  D graph, b hence æ .graph/ b D we obtain æ .graph/  0 .graph / and required equivalency is proved. Hence in virtue of (1.5) and (1.7) b Due to the Theorem on bipolar line graph b we obtain that .graph/00 D graph. b is a closure of graph in X!  Y! . Therefore,  is the minimal closed extension of . The Theorem is proved. Theorem 1.3. Let  be the adjoint mapping to the vector process . The manifold D. / is dense in Y is and only if  is a single valued mapping with closed

1.1 The Main Results from Multivalued Mapping Theory

25

b to  is single valued and is a minimal closed extension. Here the dual mapping  extension of the operator . Proof. The necessity. Let D. / be dense in Y . Topologically adjoint space to Y is Y , hence the dual operator to  is closed in X!  Y! . Here the single valuedness b and  immediately follows from the density of D. /. It is obvious also that of  b is a closed extension of . Similarly b  D./ and j b D./ D , namely  D./ b is a minimal closed extension to the proof of Theorem 1.2 it can be shown that  of . b  /  D. / then to simplify the proof let us The sufficiency. Since D. assume that  is a closed operator. Let us consider y 2 .D. //0 . Then .0I y/ 2 .graph /0 and .yI 0/ 2 æ.graph /0 . If graph is closed then due to the Theorem on bipolar line Œæ..graph/0 /0 D .graph/00 D graph and to .0I y/ 2 graph it follows that y D 0. The Theorem is proved. Remark 1.12. If the operator  is densely defined then all Propositions of Theorem 1.3 adjust with [Sch71, p. 199]. Definition 1.20. Let  W D./  X ! P .Y / be affine process and  D .x; y/ 2 graph. The mapping  W D. /  Y  ! P .X  /, defined by the relation .x  ; y  / 2 graph , .x  ; y  / 2 .graph  /? ; ia called locally adjoint (dual) mapping to the affine process . Remark 1.13. In accordance to the previous results we conclude that the locally adjoint mapping  to the affine process is closed in X  Y 8 2 graph. Theorem 1.4. Let  W D. /  Y  ! P .X  / be locally adjoint process to b be the process dual to  the affine process  W D./  X ! P .Y / and  

b W D./ b X ! with respect to .Y  I Y /-topology on Y  . The affine mapping  b b P .Y / such that graph D graph C  is minimal closed extension of . b is closed. Besides, b closed in X!  Y! since the mapping  Proof. Operator  b b D./  D./ and .x/  .x/. Indeed, the first inclusion follows from the b  /  .D./  x/, and therefore, D./ b D D. b  / C x  D./. fact that D. b  graph  . Similarly to the proof of The second one follows from graph b D .graph  /00 , namely graph b is Theorem 1.2 we state the equality graph b C  is a closure of graph a closure of graph   in X!  Y! , and hence graph b and therefore  is a minimal closed extension of . The Theorem is proved. Remark 1.14. 1. It is clear that all Propositions of Theorem 1.3 can be extended on the case of locally dual maps to affine processes.

26

1 Preliminary Results

2. Let us state some continuous properties of maps adjoint to vector processes. Let us consider an arbitrary multivalued mapping F W D.F /  X ! P .Y / operating in topological vector sets X , Y . Together with the mapping F we will consider the support function ŒF .y/; hC D sup hd; hiY where h 2 Y  . d 2F .y/

Definition 1.21. The mapping F is hemicontinuous (h.c.) in y0 2 D.F /, if the support function D.F / 3 y 7! ŒF .y/; hC is continuous in Y0 8h 2 Y  . The mapping F is upper hemicontinuous (lower semicontinuous) in y0 , if the support function is upper semicontinuous (lower semicontinuous) in y0 . The mapping F is called h.c. (u.h.c. or l.h.c.), if it is h.c. (u.h.c. or l.h.c.) in all points of its effective set. Proposition 1.36. Every continuous in y0 2 D.F / mapping is h.c. in y0 . Proof. As it is well known [AuFr90], the mapping F is u.h.c. in y0 . Therefore to prove the Proposition we must show that F is l.h.c., namely from lower semicontinuity of F in y0 lower hemicontinuity in this point follows. Let x0 2 F .y0 / be an arbitrary element. Then for any open neighborhood U .x0 / of x0 there can be found an open neighborhood O.y0 / of the element y0 , such that F .y/ \ U .x0 / ¤ ;. This means that for any net D.F / 3 y ! y0 and 8x0 2 F .y0 / there can be found a net fx g; x 2 F .y /, such that x ! x0 . But then lim infŒF .y ; h/C  

limhx ; hiY D hx0 ; hiY 8h 2 Y  ; 8x0 2 F .y0 /. Therefore due to the 

arbitrariness of x0 2 F .y0 / we obtain lim infŒF .y /; hC  ŒF .y0 /; hC 

and this proves lower hemicontinuity. The Proposition is proved. Proposition 1.37. Let X and Y be normalized spaces,  W D./  X ! P .Y / be a vector process,  be its adjoint process. Then  is the hemicontinuous mapping in all points y  2 D. / with respect to the topology induced from Y . Let us consider the composition of vector and affine processes. Let X , Y , Z be locally convex spaces,  W D./  X ! P .Y /;

F W D.F /  Y ! P .Z/

be vector processes. Let us consider the mapping G D F ı  W D.G/  X ! P .Z/

.G.y/ D F ..y//

1.1 The Main Results from Multivalued Mapping Theory

namely

[

G.y/ D

27

F .//:

2.y/\D.F /

Since graphG D graph.F ı / D f.xI z/ 2 X  Z j 9y 2 Y; .xI y/ 2 graph; .yI z/ 2 graphF g then, obviously, G is a vector process. For G naturally the dual process is introduced G  W D.F  /  Z  ! P .X  /. Theorem 1.5. If there can be found an element .x; y; z/, such that either .x; y/ 2 int graph; .y; z/ 2 graphF or .x; y/ 2 graph; .y; z/ 2 int graphF then the following equality is valid G  D   F  namely G  .Z  / D

[ f. .y  /; y  / j y  2 D.y  / \ F  .z /g

Now let  W D./  X ! P .Y /; F W D.F /  Y ! P .Z/ be affine processes. Then G D F ı  is an affine process. Theorem 1.6. Let all conditions of Theorem 1.5 are fulfilled and .x0 I z0 / 2   D .x0 I y0 / ı F.x , where graphG. Then the following relation is valid G.x 0 I z0 / 0 I y0 / y0 is an arbitrary element from Y , such that .x0 ; y0 / 2 graph; .y0 I z0 / 2  graphF , and .x0 I y0 / ; F.x are locally adjoint processes to  and F respec0 I y0 / tively. Proof. The proof represents the detailing of [Sch71, Theorem 3.2.7] with use of Theorem 1.5. The Theorem is proved. We give a typical example which represents the interest to the investigated problems. Let ˝ be a bounded domain in Rn . In the space X D L1 .˝/ with the norm kykL1 .˝/ D vrai max jy.x/j let us consider a linear differential expression x2˝

of 2m-th order .y/.x/ D

X

a˛ .x/D ˛ y.x/;

j˛j2m

where x 2 ˝ D ˛ y.x/ D

@j˛j y.x/ ˛ ˛ , @x1 1 :::@xn n

a˛ ./ are smooth functions.

˛ D .˛1 ; ˛2 ; : : : ; ˛n /, ˛i > 0, j˛j D

P i

˛i ,

28

1 Preliminary Results

In this case when X D Y D L1 .˝/ the operator  is defined on the set D./ D C02m .˝/ of 2m times continuously differentiable finite functions. Obviously, the definitional domain D./ of the operator  is not dense in X , therefore the dual mapping  W D. /  X  ! P .X  / will be a vector process operating in the space .L1 .˝// . It is closed in .X  ; X /-topology on .L1 .˝// and all previous Propositions are valid for it. Let us consider a dynamic system, generated by an evolutionary inclusion with vector (affine) processes. Let us have in a Banach set X an evolutionary inclusion dx 2 Ax dt

(1.8)

with a vector process A W D.A/  X ! P .X /. The solution of the inclusion on the segment Œ0; T  is a function x.t/ which satisfies the following conditions: (a) The values of the function x.t/ belong to D.A/ for all t 2 Œ0; T ; (b) In each point t of the segment Œ0; T  there exists a strong derivative x 0 .t/ of the function x.t/; (c) The inclusion (1.8) is valid for all t 2 Œ0; T . As Cauchy problem on the segment Œ0; T  we understand a problem of searching a solution of the inclusion on Œ0; T  which satisfies the initial condition x.0/ D x0 2 D.A/:

(1.9)

The family of the multivalued maps G.t; / depending on the parameter t .0  t < 1/ are called m-semiflow if G.0; / D I and G.t1 C t2 ; x/  G.t1 ; G.t2 ; x//

8t1 ; t2  0:

We show that multivalued maps G.t; / generated by solutions of problem (1.8)–(1.9) form the m-semiflow, consisting of vector processes. We assume that G.t; x0 / D fx.t/jx.0/ D x0 ; x.t/ satisfies (1.8)g. Let us prove that G.t; / W D.A/ ! P .D.A// is the m-semiflow. At first we make sure that G.t; / is a vector process. Let x1 .t/ and x2 .t/ satisfy the inclusion (1.8) with initial conditions x01 and x02 respectively. Then d.x1 C x2 / 2 A.x1 / C A.x2 /  A.x1 C x2 /; dt

.x1 C x2 /.0/ D x01 C x02 :

Therefore, x1 C x2 is a solution of the inclusion (1.8) with the initial condition x01 C x02 . Hence we have G.t; x01 / C G.t; x02 /  G.t; x01 C x02 /:

1.2 Classes of Multivalued Maps

29

Similarly the property ˛G.t; x0 / D G.t; ˛x0 / 8˛ ¤ 0 can be proved, namely G.t; / is the vector process. Let w.t/ D x.t C /. Then the function w.t/ 2 G.t C ; x0 / satisfies the inclusion (1.8) and the initial condition w.0/ D x. /. The function w1 .t/ D G.t; G. ; x0 // is also a solution of the inclusion (1.8) with the initial condition w1 .0/ 2 G. ; x0 /. Therefore, G.t C ; x0 /  G.t; G. ; x0 //

8x0 2 D.A/;

and it means that the maps G W RC  D.A/ ! P .D.A// form the m-semiflow. Similar constructions are valid for affine processes too. Remark 1.15. Nowadays the theory of m-semiflows develops actively in connection with the study of attractors of evolutionary inclusions and nonlinear ill-posed boundary problems for partial differential equations.

1.2 Classes of Multivalued Maps Let X be a Banach space, X  be its topologically adjoint, h; iX W X   X ! R 

be the canonical duality between X and X  ; 2X be a family of all subsets of the  space X  , let A W X ! 2X be the multivalued map, graphA D f.I y/ 2 X   X j 2 A.y/ g ; DomA D fy 2 X jA.y/ ¤ ;g : The multivalued map A is called strict if DomA D X: Together with every multivalued map A we consider its upper ŒA.y/; C D sup hd; iX d 2A.y/

and lower ŒA.y/;  D

inf hd; iX

d 2A.y/

support functions, where y;  2 X: Let also kA.y/kC D sup kd kX  ; d 2A.y/

kA.y/k D

inf kd kX  ;

d 2A.y/

k;kC D k;k D 0:



For arbitrary sets C; D 2 2X we set dist.C; D/ D sup inf ke  d kX  ; e2C d 2D

dH .C; D/ D max fdist.C; D/; dist.D; C /g :

30

1 Preliminary Results

Then, obviously, kA.y/kC D dH .A.y/; 0/ D dist.A.y/; 0/;

kA.y/k D dist.0; A.y//:



Together with the operator A W X ! 2X let us consider the following maps coA W X ! 2X







co A W X ! 2X ;

and

defined by relations .coA/.y/ D co.A.y//





and .co A.y// Dco .A.y//



respectively, where co .A.y// is the weak star closure of the convex hull co.A.y// for the set A.y/ in the space X  . Besides for every G  X .coA/.G/ D

[

.coA/.y/;



.co A/.G/ D

y2G

[



.co A/.y/:

y2G w



Further we will denote the strong, weak and weak star convergence by !; !; ! or !, *, * respectively. As Cv .X  / we consider the family of all nonempty convex closed bounded subsets from X  (Fig. 1.1). Proposition 1.38. Let A; B; C W X  X  . Then for all y; v; v1 ; v2 2 X the following statements take place: 1. The functional X 3 u ! ŒA.y/; uC is convex, positively homogeneous and lower semicontinuous; 2. ŒA.y/; v1 C v2 C  ŒA.y/; v1 C C ŒA.y/; v2 C , ŒA.y/; v1 C v2   ŒA.y/; v1  C ŒA.y/; v2  , ŒA.y/; v1 C v2 C  ŒA.y/; v1 C C ŒA.y/; v2  , ŒA.y/; v1 C v2   ŒA.y/; v1 C C ŒA.y/; v2  ; 3. ŒA.y/ C B.y/; vC D ŒA.y/; vC C ŒB.y/; vC , ŒA.y/ C B.y/; v D ŒA.y/; v C ŒB.y/; v ; 4. ŒA.y/; vC  kA.y/kC kvkX , ŒA.y/; v  kA.y/k kvkX ; 



5. k co A .y/ kC D ,  kA.y/kC , k co A .y/ k D kA.y/k    ŒA .y/ ; vC D co A .y/ ; v ; ŒA .y/ ; v D co A .y/ ; v I C

6. kA.y/  B.y/kC  j kA.y/kC  kB.y/k j, kA.y/  B.y/k  kA.y/k  kB.y/kC ; 

7. d 2 co A.y/

,

8! 2 X ŒA.y/; !C  hd; wiX ;



1.2 Classes of Multivalued Maps

31

Fig. 1.1 The monotone multivalued map

8. dH .A.y/; B.y//  jkA.y/kC  kB.y/kC j, dH .A.y/; B.y//  jkA.y/kC  kB.y/kC j, where dH is Hausdorff metric; 9. dist.A.y/ C B.y/; C.y//  dist.A.y/; C.y// C dist.B.y/; 0/; dist.C.y/; A.y/ C B.y//  dist.C.y/; A.y// C dist.0; B.y//; dH .A.y/ C B.y/; C.y//  dH .A.y/; C.y// C dH .B.y/; 0/I 10. For any D  X  and bounded E 2 Cv .X  / 

dist.D; E/ D dist.co D; E/: Remark 1.16. Together with the forms Œ; C , Œ;  let us consider the following ones aC .y; !/ D ŒŒA.y/; !C D sup jhd; wij; d 2A.y/

a .y; !/ D ŒŒA.y/; ! D inf jhd; wij d 2A.y/

8y; ! 2 X:

Hence it is obvious that ŒA.y/; !C  jŒA.y/; !C j  ŒŒA.y/; !C  kA.y/kC k!kX ; ŒA.y/; !  jŒA.y/; ! j  ŒŒA.y/; !  kA.y/k k!kX : Proof. The Properties 1, 2, 4, 6, 7 can be proved directly. The Property 3 is wellknown. We consider the Property 5. We can easily see that 

k co A.y/kC  kcoA.y/kC  kA.y/kC ; 

and now we prove the inverse inequality. For an arbitrary f 2 co A.y/ there exists a sequence fn 2 coA.y/ such that fn ! f weakly star in X  and and due to Banach–Steinhaus Theorem we have kcoA.y/kC  lim kfn kX   kf kX  : n!1

32

1 Preliminary Results 

Since the last inequality is valid for all f 2 co A.y/ then 

kcoA.y/kC D k co A.y/kC : Now we show that kcoA.y/kC  kA.y/kC . Let f 2 coA.y/ be arbitrary, then n P ˛i D 1), g1 ; : : : ; gn (gi 2 A.y/) such for n  1 there exist ˛1 ; : : : ; ˛n (˛i  0, that f D

i D1

n P i D1

˛i gi . Hence

kf kX  

n X

˛i kgi kX  

i D1

n X

˛i kA.y/kC D kA.y/kC :

i D1

From here and from arbitrariness of f 2 coA.y/ we obtain the required inequality which proves the first equality in 5. We prove the second one now. Let us introduce the map f W A.y/  B 1 ! R; defined by the following equality: f .d; / D hd; iX , where B 1 is a unit closed ball 

in the space X with its center at zero point. Let f . / D f . ; /, then f .p/ D Œco A.y/; p  C and 

domf D fp 2 X jŒco A.y/; p  C < C1g:   S domf . Indeed, 0N 2 domf0 , B1  domf and We remark that 0N 2 int 2B1

the function f satisfies the conditions of nonsymmetric nonsymmetric minimax Theorem [Me84]. Therefore, inf

sup f .d; / D sup

d 2A.y/ 2B

inf f .d; /;

2B 1 d 2A.y/

1

and the required statement immediately follows. 

Proposition 1.39. The inclusion d 2 co A.y/ holds true if and only if one of the following relations takes place: either ŒA.y/; vC  hd; viX 8v 2 X; or ŒA.y/; v  hd; viX 8v 2 X: 

Proof. Necessity Let d 2 .co A/.y/; then    hd; viX 2 h; viX j 2 .co A/.y/ ;

1.2 Classes of Multivalued Maps

33

hence, in virtue of Proposition 1.38 h  i hd; viX  .co A/.y/; v D ŒA.y/; v 

and similarly h  i hd; viX  ŒA.y/; vC D .co A/.y/; v

C

8v 2 X: 



Sufficiency. Let ŒA.y/; v  hd; viX 8v 2 X and d … .co A/.y/: The set

.co A/.y/ is convex and closed in .X  I X /-topology of the space X  , therefore due to separation Theorem [Ru73] there can be found v0 2 X that hh; v0 iX < hd; v0 iX



8h 2 .co A/.y/;

or ŒA.y/; v0 C < hd; v0 iX , and the observed results are contradictory to to the conditions of the Proposition. The Proposition is proved. Proposition 1.40. Let D  X and a. ;  / W D  X ! R. For each y 2 D the functional X 3 w 7! a.y; w/ is positively homogeneous, convex and lower semi continuous if and only if there exists the multivalued map A W X ! 2X such that D.A/ D D and a.y; w/ D ŒA.y/; wC

8y 2 D.A/; w 2 X:



Proof. Let A W D.A/  X ! 2X . Then for any y 2 D.A/ the functional X 3 v 7! a.y; v/ D ŒA.y/; vC is positively homogeneous and semiadditive due to Proposition 1.38. Hence it is convex. Its lower semicontinuity is obvious. Now let X 3 v 7! a.y; v/ be positively homogeneous, convex and lower semicontinuous functional for every y 2 D  X . Since a.y; 0/ D 0 then it is a pointwise limit of the continuous linear functionals set. We will denote this set by A.y/  X  . Therefore a.y; v/ D ŒA.y/; vC . It will be easy to check the following Propositions: Proposition 1.41. The functional k  kC W Cv .X  / ! RC satisfies the following properties: 1. f0g D A , kAkC D 0 2. k˛AkC D j˛kjAkC ; 8˛ 2 R; A 2 Cv .X  / 3. kA C BkC  kAkC C kBkC 8A; B 2 Cv .X  /

34

1 Preliminary Results

Proposition 1.42. The functional k  k W Cv .X  / ! RC satisfies the following properties: 1. 0 2 A , kAk D 0 2. k˛Ak D j˛jkAk ; 8˛ 2 R; A 2 Cv .X  / 3. kA C Bk  kAk C kBk 8A; B 2 Cv .X  / Let us consider the new class of multivalued maps of pseudomonotone type. As before, let X be a Banach space, X  be its topologically adjoint, h; iX W X   X ! R be the duality form.  We remind that the multivalued map A W D.A/  X ! 2X is called the monotone one if hd1  d2 ; y1  y2 iX  0 8y1 ; y2 2 D.A/

8d1 2 A.y1 /; 8d2 2 A.y2 /:

Using the above mentioned brackets it is easy to see that the multivalued operator  A W D.A/  X ! 2X is monotone if and only if ŒA.y1 /; y1  y2   ŒA.y2 /; y1  y2 C

8y1 ; y2 2 D.A/:

In addition to the common monotony of multivalued maps we are interested in the following concepts (Fig. 1.2):  N -monotony, namely

ŒA.y1 /; y1  y2   ŒA.y2 /; y1  y2 

8y1 ; y2 2 D.A/I

 V -monotony, namely

ŒA.y1 /; y1  y2 C  ŒA.y2 /; y1  y2 C

8y1 ; y2 2 D.A/I

 w-monotony, namely

ŒA.y1 /; y1  y2 C  ŒA.y2 /; y1  y2 

Fig. 1.2 The N -monotone multivalued map

8y1 ; y2 2 D.A/:

1.2 Classes of Multivalued Maps

35

Definition 1.22. Let D.A/ be some subset. The multivalued map A W D.A/   X ! 2X is called:  Weakly +(-)-coercive, if for each f 2 X  there exists R > 0 such that

ŒA.y/; yC./  hf; yiX

8y 2 X \ D.A/ W kykX D R:

 +(-)-coercive, if

ŒA.y/; yC./ ! C1 kykX

as

kykX ! C1;

y 2 D.A/I

 Uniformly +(–)-coercive if for some c > 0

ŒA.y/; yC./  ckA.y/kC./ ! C1 kykX

as kykX ! C1;

y 2 D.A/I

 Bounded if for any L > 0 there exists l > 0 such that kA.y/kC  l 8y 2 D.A/

kykX  L;

 Locally bounded, if for an arbitrary fixed y 2 D.A/ there exist constants m > 0

and M > 0 such that kA./kC  M when ky  kX  m,  2 D.A/;

 Finite-dimensionally locally bounded, if for any finite-dimensional space F  X

the contraction of A on F \ D.A/ is locally bounded;

 d-closed if from the fact that D.A/ 3 yn ! y 2 D.A/ strongly in X it follows

that lim ŒA.yn /; '  ŒA.y/; '

8 ' 2 XI

n!1

Definition 1.23. We will say that the real two-variable function C W RC  RC ! R belongs to the class ˚0 if C.r1 I  / W RC ! R is a continuous function for every r1  0. Moreover,

1 C.r1 I r2 / ! 0 as ! 0 C

8r1 ; r2  0:

Definition 1.24. We say that a continuous function C W RC  RC ! R belongs to the class ˚1 , if C.r; 0/ 0 (Fig. 1.3). Let k k0X is some (semi)norm on X , compact with respect to k kX on X ; C 2˚0 .

Fig. 1.3 The V -monotone multivalued map

36

1 Preliminary Results

Definition 1.25. A strict multivalued map A W X  X  is called:  Radial lower semicontinuous (r.l.s.c.) if 8y,  2 X

lim ŒA.y C t/; C  ŒA.y/;  I

t !0C

 Radial upper semicontinuous (r.u.s.c.) if the real function

Œ0; 1 3 t ! ŒA.y C t/; C is upper semicontinuous at the point t D 0 for any y,  2 X ;  Radial semicontinuous (r.s.c.) if 8y,  2 X

lim ŒA.y  t/; C  ŒA.y/;  I

t !0C

 Radial continuous (r.c.) if the real function

Œ0; 1 3 t ! ŒA.y C t/;  is continuous at the point t D 0 from the right for any y,  2 X ;  (upper) hemicontinuous (u.h.c.) if the function

X 3 x 7! ŒA.x/; yC is u.s.c. on X for any y 2 X ;  The operator with semibounded variation on X (s.b.v.) if 8R  0 8y1 ; y2 2 X W

ky1 kX  R, ky2 kX  R the following inequality holds true:

ŒA.y1 /; y1  y2   ŒA.y2 /; y1  y2 C  C.RI ky1  y2 k0X /I  The operator with N -semibounded variation on X (with N -s.b.v.) if 8R  0

8y1 ; y2 2 X W ky1 kX  R, ky2 kX  R the following inequality holds true: ŒA.y1 /; y1  y2   ŒA.y2 /; y1  y2   C.RI ky1  y2 k0X /I

 The operator with V -semibounded variation on X (with V -s.b.v.) if 8R  0

8y1 ; y2 2 X W ky1 kX  R, ky2 kX  R the following inequality holds true: ŒA.y1 /; y1  y2 C  ŒA.y2 /; y1  y2 C  C.RI ky1  y2 k0X /I

 Semimonotone (s.m.) if 8R > 0 for arbitrary yi 2 X such that kyi kX  R;

i D 1; 2; the inequality is valid

ŒA.y1 /; y1  y 2   ŒA.y2 /; y1  y2 C  C.RI ky1  y2 kX /; where C 2 ˚0 I

(1.10)

1.2 Classes of Multivalued Maps

37

 Submonotone (sub-m.) if in the inequality (1.10) C 2 ˚1 I  The operator with subbounded variation (sub-b.v.) if in the definition of the

operator with s.b.v. C 2 ˚1 :

 The operator with N -subbounded variation (N -sub-b.v.) if in the definition of

the operator with N -s.b.v. C 2 ˚1 :

 -pseudomonotone on X if for any sequence fyn gn0  X such that yn * y0

in X as n ! C1 from the inequality

lim hdn ; yn  y0 iX  0;

(1.11)

n!1 

where dn 2 co A.yn / 8n  1 the existence of subsequences fynk gk1 from fyn gn1 and fdnk gk1 from fdn gn1 follows for which the next relation holds true: lim hdnk ; ynk  wiX  ŒA.y0 /; y0  w 8w 2 X I (1.12) k!1

 0 -pseudomonotone on X , if for any sequence fyn gn0  X such that yn *y0 

in X , dn * d0 in X  as n ! C1 where dn 2 co A.yn / 8n  1 from the inequality (1.11) the existence of subsequences fynk gk1 from fyn gn1 and fdnk gk1 from fdn gn1 follows for which the relation (1.12) holds true. The above mentioned multivalued map satisfies:  Condition ./C./ if for each bounded set D in X there exists c 2 R such that ŒA.v/; vC./  ckvkX

N 8v 2 D n f0g:

 Condition .˘ / if for any k > 0, any bounded set B  X , any y0 2 X and for

some selector d 2 A for which

hd.y/; y  y0 iX  k

for all y 2 B;

(1.13)

there exists C > 0 such that kd.y/kX   C

for all y 2 B:

Remark 1.17. The idea of up to a subsequence in the definition of a singlevalued pseudomonotone operator was proposed by Skrypnik I.V. in his work [Sk94]. Remark 1.18. Obviously, the following implications are valid: “A is u.h.c.” ) “A is r.u.s.c.” ) “A is r.s.c.”. In papers [MeSo97, BrHe72] it is proved that for semimonotone maps these implications are convertible. 

Remark 1.19. Let A W X ! 2X . Then the following implications are valid: “A is u.h.c.” ) “A is r.u.s.c.” ) “A is r.l.s.c.”

38

1 Preliminary Results

Remark 1.20. At an analysis and control of different geophysical and socioeconomical processes it is often appears such problem: at a mathematical modeling of effects related to friction and viscosity, quantum effects, a description of different nature waves the existing “gap” between rather high degree of the mathematical theory of analysis and control for non-linear processes and fields and practice of its using in applied scientific investigations make us require rather stringent conditions for interaction functions. These conditions related to linearity, monotony, smoothness, continuity and can substantially have an influence on the adequacy of mathematical model. Let us consider for example stationary diffusion process. Its mathematical model has the next form:  y ˇ C f .y/ D g.x/ in ˝; (1.14) y ˇ@˝ D 0; here n  2; ˝  Rn is a bounded domain with a rather smooth boundary, g W ˝!R are rather regular functions, f W R ! R is an interaction function, y W ˝ ! R is an unknown function. It is well known that if f is a rather smooth function and satisfies for example the next condition of no more than polynomial growth: 9p > 1;

9c > 0 W

jf .s/j  c.1 C jsjp1 /

8s 2 R;

(1.15)

then problem (1.14) has a unique rather regular solution. Let us consider the case when f is continuous and external forces are nonregular (for example g 2 L2 .˝/). Then, as a rule, we consider the generalized setting of problem (1.14): A.y/ C B.y/ D g

(1.16)

here A W V1 ! V1 is an energetic extension of operator “”, B W V2 ! V2 is the Nemytskii operator for F; V1 D H01 .˝/ is a real Sobolev space, V2 D Lp .˝/; V1 D H 1 .˝/; V2 D Lq .˝/; q is the conjugated index. A solution of problem (1.16) in the class V D V1 \V2 refers to be the generalized solution of problem (1.14). To prove the existence of solutions for problem (1.14) as a rule we need to add supplementary “signed condition” for an interaction function f , for example, 9˛; ˇ > 0 W

f .s/s  ˛jsjp  ˇ

8s 2 R:

(1.17)

But we do not succeed in proving the uniqueness of the solution of such problem in the general case. We remark also that different conditions for parameters of problem (1.14) provide corresponding conditions for generated mappings A and B. Problem (1.16) is usually investigated in more general case: A .y/ D g;

(1.18)

1.2 Classes of Multivalued Maps

39

here A W V ! V  is such that A .y/ D A.y/ C B.y/ 8y 2 V; V D V1 \ V2 ; V  D V1 C V2 : Solutions of problem (1.18) are also searched in the class V: Note that: If f is monotone nondecreasing function then the corresponding extension for A is monotone; If f is globally Lipschitzian function and p  2 then A is the operator with s.b.v.; If p > 1 such that V1  V2 with compact embedding, then A is pseudomonotone operator on W ; In the general case A satisfies Property Sk on W . Note that A is the Volterra type operator. In cases when the continuity of the interaction function f have an influence on the adequacy of mathematical model fundamentally then problem (1.14) is reduced to such problem:  y ˇ C F .y/ 3 g.x/ in ˝; (1.19) y ˇ@˝ D 0; here F .s/ D Œf .s/; f .s/; f .s/ D lim f .t/; f .s/ D lim f .t/; s 2 R; t !s

t !s

ˇ Œa; b D f˛a C .1  ˛/b ˇ˛ 2 Œ0; 1g; 1 < a < b < C1: A solution of such differential-operator inclusion A .y/ 3 g;

(1.20)

is usually thought to be the generalized solution of problem (1.19). Here A W V ! Cv .V  /; A .u/ D A.u/ C B.u/; u 2 V; A W V1 ! V1 is the energetic extension of “” in H01 .˝/; B W V2 ! Cv .V2 / is the Nemytskii operator for F : ˇ B.v/ D fz 2 V2 ˇz.x/ 2 F .v.x// for a.e. x 2 ˝g;

v 2 V2 :

Similarly if f is monotone nondecreasing function then A is monotone operator, if V1  V2 with compact embeddings then A is pseudomonotone operator on V; in the general case A satisfies Property Sk on V . At that the next condition 9˛; N ˇN W

sup t  s  ˛jsj N p  ˇN t 2F .s/

8s 2 R

40

1 Preliminary Results

Fig. 1.4 The w-monotone multivalued map

provides the condition of C-coercivity and the next condition f .s/  s  ˛jsj N p  ˇN

9˛; N ˇN W

8s 2 R

provides the condition of -coercivity. Taking into account all variety of classes of mathematical models for different nature geophysical processes and fields we propose rather general approach to investigation of them in this book. Further we will study classes of mathematical models in terms of general properties of generated mappings like A (Fig. 1.4). Proposition 1.43. If a multivalued operator A W X  X  satisfies Condition (˘ ) then it satisfies Condition ./C as well. Proof. We will prove this Proposition by contradiction. Let D  X be a bounded N ŒA.vc /; vc C  ckvc kX  0. set such that for every c > 0 there exists vc 2 Dnf0g: Then due to Condition .˘ / we have sup kA.vc /kC DW d < C1: c>0

Therefore, ckvc kX  ŒA.vc /; vc C  kA.vc /kC kvc kX  d kvc kX N and .d  c/kvc kX  0 for any c > 0. This contradicts to the condition vc ¤ 0. Definition 1.26. The operator L W D.L/  X ! X  is called  monotone if for all y1 ; y2 2 D.L/

hLy1  Ly2 ; y1  y2 iX  0I  maximal monotone on D.L/ if it is monotone and from

hw  Lu; v  uiX  0 for each u 2 D.L/ it follows that v 2 D.L/ and Lv D w.

1.2 Classes of Multivalued Maps

41

Remark 1.21. Let A W X  X  be +-coercive multivalued map, let another multivalued map F W X  X  satisfy the following monotony condition ŒF .y1 /; y1  y2 C  ŒF .y2 /; y1  y2 

8y1 ; y2 2 X:

Then A C F W X  X  is the +-coercive map. Indeed, let for each y 2 X ŒF .0/; y  kF .0/k kykX : Then due to Proposition 1.38 ŒA.y/ C F .y/; yC D ŒA.y/; yC C ŒF .y/; yC  ŒA.y/; yC C ŒF .0/; y  ŒA.y/; yC  kF .0/k kykX : It is clear that kF .0/k < C1. Therefore C-coercivity of A C F is obvious (Fig. 1.5). Let G  X be some nonempty subset. 

Definition 1.27. We say that the map A W X ! 2X satisfies Condition: w

(a) ˛.G/.˛ 0 .G// if for an arbitrary sequence fyn g  G from yn ! y in X and from the inequality lim ŒA.yn /; yn  y  0 (1.21) n!1

the existence of subsequence fynk g  fyn g such that ynk ! y in X .kynk kX ! kykX / follows; w



w

(b) ˛0 .G/.˛00 .G// if from G 3 yn ! y in X , .co A/.yn / 3 dn ! 0 in X  and from

(1.21) it follows that 9fynk g  fyn g such that ynk ! y in X 

yn ! kyk I k

X

X



w

(c) ˛1 .G/ if from G 3 yn ! y in X; .co A/.yn / 3 dn in X  and from the inequality lim hdn ; yn  yiX  0

n!1

Fig. 1.5 The “”-coercive multivalued map

(1.22)

42

1 Preliminary Results

 ˚  the existence of ˝ ˛ subsequences fynk g  fyn g, dnk  fdn g for which dnk ! d and dnk ; ynk X ! hd; yiX follows; (d) ˛2 .G/ if in (c) instead of (1.22) the relation (1.21) holds true; w (e) ˛3 .G/.˛30 .G// if from G 3 yn ! y in X and from validity of the inequality 

(1.22) where dn 2 .co A/.yn / there can be isolated  the subsequence fynk g  fyn g such that ynk ! y in X ynk X ! kykX . Remark 1.22. For singlevalued maps the properties ˛.G/; ˛0 .G/ were introduced by Skripnik I.V. [Sk94] and they were investigated by Browder F., Petrishin V, Skripnik I.V. and others. A related property .S /C was introduced by Browder F. [BrHe72]. Condition ˛1 .G/ represents the extension of generalized pseudomonotone maps class [BrHe72]. In [Sk94] the conditions were cited that allowed elliptic divergent quasilinear 2m-th order operator to satisfy Condition ˛.G/. 

Let us consider the duality map J W X ! 2X , namely o n J.y/ D  2 X  j h; yiX D kk2X  D kyk2X

y 2 X:

Proposition 1.44. The duality map satisfies Conditions ˛ 0 .G/, ˛00 .G/, ˛1 .G/, ˛30 .G/. Moreover, if the spaces X; X  are uniformly convex then it satisfies Conditions ˛00 .G/; ˛.G/; ˛20 .G/; ˛3 .G/ as well. Proof. Property ˛ 0 .G/ is arranged in the work [VaMe99]. Property ˛00 .G/ is proved similarly. Let us consider Condition ˛1 .G/. It is well-known [AuEk84] that DomJ D X , namely, the duality map is strict and the set J.x/ is bounded, convex, 

w

weakly star closed in X  : Let G 3 yn ! y in X and dn 2 J.yn / D co.yn / such  that dn ! d; lim hdn ; yn  yiX  0: n!1 Then lim hdn ; yn iX D lim fhdn ; yi X C hdn ; yn  yiX g  hd; yiX ;

n!1

n!1

or lim hdn  ; yn  wiX  hd  ; y  wiX 8w 2 X; 8 2 J.w/:

n!1

However since the operator J is maximal monotone [Br72] then d 2 J.y/ and hence, hd; yiX D kyk2X  lim hdn ; yn iX  lim hdn ; yn iX  hd; yiX ; n!1

n!1

i.e., lim hdn ; yn iX D lim hdn ; yn iX D hd; yiX :

n!1

n!1

The last relation proves Property ˛1 .G/:

1.2 Classes of Multivalued Maps

43 w

We pass to Property ˛30 .G/: Let G 3 yn ! y in X and dn 2 J.yn / such that lim hdn ; yn  yiX  0: In virtue of boundness of the map J we can isolate a n!1 ˚   subsequence dnk  fdn g such that dnk ! d in X  : In this case lim

nk !1

˝ ˛ ˝ ˝ ˛ ˛ dnk ; ynk X D lim dnk ; y X C lim dnk ; ynk  y X  hd; yiX nk !1

nk !1

and therefore,

2 ˝ ˛ kyk2X  lim ynk X D lim dnk ; ynk X  hd; yiX D kyk2X : nk !1

nk !1

The last equality is valid due to maximal monotony of the map J , and Property ˛30 .G/ is proved. At last, let the spaces X and X  be uniformly convex. In this case J is singlevalued and from the convergence of corresponding norms the strong convergence follows. The Proposition is proved. 

Proposition 1.45. Let the map A0 W X ! 2X satisfy Condition ˛.G/ (˛ 0 .G/  respectively), and let A1 W X ! 2X be the map with sub-b.v. (N -sub-b.v.) and compact values. Then the map A D A0 C A1 satisfies Condition ˛.G/ (˛ 0 .G/ respectively). For the map A0 which satisfies Condition, ˛3 .G/ (˛30 .G/ respectively) the map A D A0 C A1 has the same property if A0 or A1 are bounded-valued. w

Proof. Let G 3 yn ! y in X and lim ŒA.yn /; yn  y D lim fŒA0 .yn /; yn  y C ŒA1 .yn /; yn  y g  0:

n!1

n!1

 ˚ Passing to a subsequence, if necessary, ynk  fyn g, from the last relation we obtain the validity of one of the following inequalities: lim



nk !1

A0 .ynk /; ynk  y

 

 0;

lim

nk !1

  A1 .ynk /; ynk  y   0:

If the second inequality holds true then in virtue of subboundness of variation we have lim

nk !1



A1 .ynk /; ynk  y

 

 lim

nk !1



A1 .y/; ynk  y

 C

0  lim C.RI ynk  y X /  0; nk !1

  i.e., A1 .ynk /; ynk  y  ! 0 and we come to the inequality lim

nk !1



A0 .ynk /; ynk  y

 

 0;

44

1 Preliminary Results

which guarantees the strong convergence of ynk to y. Now let A1 be an operator with N -sub-b.v., then acting similarly to the previous case we obtain the relation lim

nk !1

0     A1 .ynk /; ynk  y   lim f A1 .y/; ynk  y   C.RI ynk y X / g nk !1

 lim

nk !1



A1 .y/; ynk  y





:

If A1 .y/ is a compact in X  , then the function X 3 w 7! ŒA1 .y/; w w

is weakly lower semicontinuous. Indeed, let wn ! w in X then 8n  1 there exists dn 2 A1 .y/ such that hdn ; wn iX D ŒA1 .y/; wn  ;  ˚ and due to compactness of A1 .y/ we can isolate a subsequence dnk  fdn g such w

that dnk ! d in X  , therefore ˝

˛

! hd; yiX ; lim ŒA1 .y/; wn  D lim hdn ; wn iX n!1 n!1 ˝ ˛ D lim dnk ; wnk X D hd; wiX  ŒA1 .y/; w :

dnk ; ynk

X

nk !1

  Hence we obtain A1 .ynk /; ynk  y  ! 0: Now we prove the second part of the Proposition. Let A0 satisfy Condition ˛.G/, w G 3 yn ! y in X and lim hdn ; yn  yiX  0; n!1



where dn 2 .co A/.yn / (Fig. 1.6). Lemma 1.1. Let one of the operators A0 , A1 be bounded-valued. Then 





.co.A0 C A1 //.y/ D .co A0 /.y/ C .co A1 /.y/

Fig. 1.6 The “C”-coercive multivalued map, but not “”-coercive

8y 2 X:

1.2 Classes of Multivalued Maps

45

Proof. It is well known that .coA/.y/ D .coA0 /.y/ C .coA1 /.y/; therefore







.co A/.y/  .co A0 /.y/ C .co A1 /.y/; 

hence we should prove the inverse embedding. Let  2 .co A/.y/ then there w exists n 2 .coA/.y/; such that n !  in X  . But then n D n0 C n00 where n0 2 .coA0 /.y/, n00 2 .coA1 /.y/, and since one of the maps (for example, A1 ) is 

w

bounded-valued, then n00k !  00 . Hence n0 k !  0 and 



 D  0 C  00 2 .co A0 /.y/ C .co A1 /.y/: The Lemma is proved. 



In virtue of Lemma 1.1 dn D dn0 C dn00 ; where dn0 2 .co A0 /.yn /, dn00 2

.co A1 /.yn / and passing to subsequences, if necessary, we have one of two relations: ˝ ˛ lim dn0 ; yn  y X  0;

˝ ˛ lim dn00 ; yn  y X  0:

n!1

n!1

If the first one holds true then the Proposition is proved, and if the second one is valid then in virtue of subboundness of variation we obtain ˝

dn00 ; yn  y

˛ X

 ŒA1 .yn /; yn  y ! 0

and we come back to the first case. Let F  X , G  X  be some subsets. 

Definition 1.28. The map A W X ! 2X is called: (a) .F I G/-pseudomonotone if from w

X  F 3 yn ! y

in X

and from the inequality (1.21) the existence of a subsequence fynk g  fyn g such that lim

nk !1



A.ynk /; ynk  

 

 ŒA.y/; y   8 2 G  X

follows; w



(b) .F I G/ -pseudomonotone if from F 3 yn ! y in X , dn 2 .co A/.yn / and  ˚ from the inequality (1.22) there can be found subsequences fynk g  fyn g, dnk 

46

1 Preliminary Results

The condition (k)+ The condition (P )

The boundedness condition

Fig. 1.7 Some properties for strict multivalued map in Banach space

fdn g such that lim

nk !1

˝ ˛ dnk ; ynk  w X  ŒA.y/; y  w 8w 2 G  X I w

(c) .F I G/w -pseudomonotone if from F 3 yn ! y in X and from the inequality 

(1.21) the existence of subsequences fynk g  fyn g and dnk 2 .co A/.ynk / such that lim

nk !1

˝

˛ dnk ; ynk  w X  ŒA.y/; y  w 8w 2 G  X

follows. .X I X /-pseudomonotone (.X I X / -pseudomonotone, .X I X /w -pseudomonotone) map we will call pseudomonotone (-pseudomonotone, w-pseudomonotone) (Fig. 1.7). Remark 1.23. Obviously, the following implications are valid a) ) b) ) c): Besides, if w-pseudomonotone map is bounded-valued then it is -pseudomonotone one. Some examples of pseudomonotone maps were cited in [BrHe72, MeSo97]. Analyzing the proof of Proposition 1.45 we obtain the following result. 

Proposition 1.46. Let the map A0 W X ! 2X satisfy Condition ˛3 .G/.˛30 .G//  and let A1 W X ! 2X be .GI X / -pseudomonotone map and one of the maps be bounded-valued. Then the map A D A0 C A1 inherit the properties of A0 .

1.2 Classes of Multivalued Maps

47 

Proposition 1.47. Let A0 W X ! 2X be r.s.c. operator with s.b.v., let A1 W X !  2X be a -pseudomonotone operator and let one of them be bonded-valued. Then A D A0 C A1 is -pseudomonotone. 

w

Proof. Let us assume that yn ! y and the inequality (1.22) where dn 2 .co A/.yn / is valid. Then due to Lemma 1.1, dn D dn0 C dn00 ;





dn0 2 .co A0 /.yn /;

dn00 2 .co A1 /.yn /;

and from (1.22) we obtain (at least for subsequences) either

˝ ˛ lim dn0 ; yn  y X  0;

or

n!1

˝ ˛ lim dn00 ; yn  y X  0:

n!1

In the case when the second relation holds true (isolating subsequences, if necessary) in virtue of -pseudomonotony of the map A1 we have ˝ ˛ lim dn00 ; yn  h X  ŒA1 .y/; y  h 8h 2 X;

(1.23)

n!1

hence hdn0 ; yn  yiX ! 0, and therefore, ˝ ˛ lim dn0 ; yn  y X  lim hdn ; yn  yiX  lim hdn00 ; yn  yiX  0:

n!1

n!1

n!1

Using semiboundness of variation of the map A0 we conclude that hdn0 ; yn  yiX ! 0. For arbitrary h 2 X and 2 Œ0; 1 we set . / D y C .h  y/, then 8 2 Œ0; 1 ˛ ˝ 0 dn ; yn  . / X  ŒA0 .y n /; yn  . /  ŒA0 .. //; yn  . /C  C.RI kyn  . /k0X /; therefore, ˝ ˛

lim dn0 ; yn  h X  ŒA0 .. //; y  hC  C.RI ky  hk0X /: n!1

Dividing the last inequality on > 0 and passing to the limit as ! 0C we obtain in virtue of r.s.c. and due to the properties C 2 ˚0 : ˝ ˛ lim dn0 ; yn  h X  lim ŒA0 .. //; y  hC  ŒA0 .y/; y  h 8h 2 X:

n!1

Since and

!0C

˝ 0 ˛ ˝ ˛ ˝ ˛ dn ; yn  h X D dn0 ; yn  y X C dn0 ; yn  h X ˝

dn0 ; yn  y

˛ X

! 0;

48

1 Preliminary Results

then

˝ ˛ lim dn0 ; yn  h X  ŒA0 .y/; y  h 8h 2 X:

n!1

From here and from the inequality (1.23) we have ˝ ˝ ˛ ˛ lim hdn ; yn  hiX  lim dn0 ; yn  h X C lim dn00 ; yn  h X

n!1

n!1

n!1

 ŒA0 .y/; y  h C ŒA1 .y/; y  h D ŒA.y/; y  h 8h 2 X: The Proposition is proved. Remark 1.24. Remark that: (a) If one of the operators satisfies Condition ˛.G/ or ˛3 .G/ then in Proposition 1.47 semiboundness of variation of the map A0 can be replaced by semimonotony; (b) If the conditions of Proposition 1.47 hold true then A0 is -pseudomonotone map. 

Proposition 1.48. Every -pseudomonotone map W X ! 2X has Property A , w namely from yn ! y in X and from the inequality lim hdn ; yn iX  hd; yiX ;

n!1 





where dn 2 .co A/.yn / and dn ! d in X  it follows that d 2 .co A/.y/. Proof. Let the map A W X ! 2X 



w

be -pseudomonotone, yn ! y in X;



.co A/.yn / 3 dn ! d X  and lim hdn ; yn iX  hd; yiX :

n!1

Then lim hdn ; yn  yiX D lim hdn ; yn iX  lim hdn ; yiX  0

n!1

n!1

n!1

and ˚ in  virtue of -pseudomonotony we can isolate subsequences fynk g  fyn g, dnk  fdn g such that hd; y  hiX  lim

nk !1

˝

˛ dnk ; ynk  h X  ŒA.y/; y  h

8h 2 X:



From here due to Proposition 1.39 we conclude that d 2 .co A/.y/. 



Definition 1.29. The map A W X ! 2X is demiclosed if graph co A is closed in X   X with respect to the strong convergence in X and the weak star convergence in X  (Fig. 1.8).

1.2 Classes of Multivalued Maps

49

Fig. 1.8 The multivalued map with semibounded variation



Proposition 1.49. The map A W X ! 2X satisfying Condition A is demiclosed. The proof is trivial. Remark 1.25. The following statements hold true: (a) The sum of two .F I G/-pseudomonotone operators (operators that satisfy Conditions ˛.G/; ˛ 0 .G/) is .F I G/-pseudomonotone operator (the operator satisfying Conditions ˛.G/; ˛ 0 .G/ respectively) as well; (b) The sum of -pseudomonotone operators (demiclosed, w-pseudomonotone, satisfying Conditions ˛1 .G/, ˛10 .G/, ˛2 .G/, ˛2 .G/, ˛3 .G/, ˛30 .G/) inherits the correspondent properties if one of the summands is bounded-valued. Let Y be a reflexive Banach space. 

Proposition 1.50. Every map A W X ! 2X that is upper semicontinuous at the point x0 2 DomA with respect to the weakly star topology in X  is u.r.s.c. at this point. The proof is trivial. 

Proposition 1.51. Let X be reflexive, A W X ! 2X be r.u.s.c. map and let for h 2 X , x0 2 DomA the sequence y 2 A .x0 C h/ be weakly star convergent in the space X  to y0 as ! 0C. Then y0 2 coA .x0 /. 

Remark 1.26. Let A D A1 C A2 where A1 W X ! 2X is a monotone map and A2 W X 7! X  is a locally Lipschitzean operator, namely kA .y1 /  A.y2 /kX   k.R/ ky1  y2 kX ;

if

kyi kX  R

.i D 1; 2/ ;

k W RC ! RC is a continuous nondecreasing function. Then A the is semimonotone map. Moreover if A2 W Y ! Y  is locally Lipschitzean operator and the space X is compactly embedded in Banach space Y then A D A1 C A2 is the map with s.b.v. Also it is clear that all constructions in the previous definition are certain generalizations of the monotone maps and for them the following implications are valid: “A is the operator with s.b.v.” ) “A is the operator with sub-b.v.” ) “A is the semimonotone map” ) “A is the submonotone map”. In the most prevailing cases C W RC  RC ! R is the function increasing by the second variable. Here semiboundness of variation property for the map A is stronger than semimonotony property, and subboundness of variation is stronger than submonotony.

50

1 Preliminary Results

Fig. 1.9 The multivalued map with N -semibounded variation

Fig. 1.10 The multivalued map with V -semibounded variation

The next four lemmas order classes of multivalued maps with semibounded variation (s.b.v) and classes of locally bounded maps with bounded values which satisfy Property .˘ / and Property ./C . The obtained statements allow us to retract additional hypotheses for a supplementary boundedness of the generated multivalued maps with s.b.v. in applications when investigating differential-operators inclusions and multivariational inequalities which describe mathematical models for nonlinear geophysical processes and fields which contain partial differential equations with discontinuous or multivalued relationship between determinative parameters of the problem. These results are the fundamental generalization of well-known corresponding results for single-valued maps of monotone type (see for example [GaGrZa74, IvMe88, ZgMe99, ZgMe04] and references there) (Figs. 1.9 and 1.10). 

Lemma 1.2. Every strict multivalued operator A W X ! 2X with semibounded variation has bounded values, namely A W X  X  . Proof. Remark that for each y 2 X X 3 ! ! ŒA.y/; !C 2 R [ fC1g;

X 3 ! ! ŒA.y/; ! 2 R [ f1g:

Therefore, due to the definition of semibounded variation on .Y; X / we obtain that for all ! 2 Y , for some R D R.!; y/ > 0 ŒA.y/; !C  ŒA.y C !/; ! C CA .RI k!k0X / < C1: From the last relation in virtue of Banach–Steinhaus Theorem we have that kA.y/kC < C1 for each y 2 X . The Lemma is proved.

1.2 Classes of Multivalued Maps

51

Lemma 1.3. The multivalued operator A W X  X  with semibounded variation is locally bounded. Proof. We will prove this statement by contradiction. If A is not locally bounded then for some y 2 X there exists a sequence fyn gn1  X such that yn ! y in X and kA.yn /kC ! C1 as n ! C1. We assume that ˛n D 1 C kA.yn /kC kyn  ykX for each n  1. Then due to Proposition 1.38, 8 ! 2 X for some R > 0 we have n o ˛n1 ŒA.yn /; !C  ˛n1 ŒA.yn /; yn  yC C ŒA.yn /; ! C y  yn C n  ˛n1 ŒA.yn /; yn  yC C ŒA.y C !/; y C !  yn C o CCA .RI kyn  y  !k0X / : Since the sequence f˛n1 g is bounded and kyn  y  !k0X ! k!k0X (due to the assumption kk0X  kkkX for all y 2 X ), in virtue of Proposition 1.38, we have 8n  1

n ˛n1 ŒA.yn /; !C  ˛n1 CA .RI kyn  y  !k0X /

o CkA.y C !/kC  ky C !  yn kX C 1  N1 ;

where N1 does not depend on n  1. Therefore, ˇ ˇ sup ˇ˛n1 ŒA.yn /; !C ˇ < 1 8 ! 2 X:

n1

Hence, due to Banach–Steinhaus Theorem there exists N > 0 such that kA.yn /kC  N˛n D N .1 C kA.yn /kC  kyn  ykX /

8n  1:

Choosing n0  1 such that N kyn  yk  1=2 8n  n0 we obtain that for each n  n0 kA.yn /kC  2N , and this fact contradicts to the assumption. Hence the local boundness is proved. The Lemma is proved. Lemma 1.4. Every strict submonotone (semimonotone with sub-b.v.) operator A W  X 7! 2X is locally bounded. Proof. We will prove the Lemma by contradiction. Let for some y 2 X there can be found yn 2 X such that yn ! y strongly in X and kA.yn /kC ! 1. Let us set ˛n D 1 C kA.yn /kC kyn  ykX 

52

1 Preliminary Results

In virtue of Proposition 1.38 8w 2 X we have ŒA.yn /; w C ŒA.yn /; yn  yC  ŒA.y C w/; yn  y  wC C .RI kyn  y  wkX / ; hence ˚ ˛n1 ŒA.yn /; wC  ˛n1 ŒA.y C w/; y  w  yn   CC .RI kyn  y  wkX / C ŒA.yn /; y  yn   ˛n1 C .RI kyn  y  wkX / C ˛n1 kA.yCw/k kyCwyn kX C˛n1 kA.yn /k kyn  ykX  1C˛n1 C .RI kyn ywkX / C˛n1 kA.y C w/k ky C w  yn kX :

 ˚ Since the sequence ˛n1 is bounded and kyn  y  wkX ! kwkX ; in virtue of continuity of the function C we obtain the following estimation ˛n1 ŒA.yn /; wC  NO .w/

8w 2 X:

(1.24)

Hence for any selector d 2 A ˇ ˇ 1 ˇ˛ hd.yn /; wiX ˇ  NO .w/ n

and due to the Banach–Steinhaus Theorem kd.yn /kX   N.1 C kA.yn /kC kyn  ykX /: Since the right side of the last inequality does not depend on d 2 A then choosing n0 from the condition N kyn  ykX 

1 2

8n  n0 ;

we find kA.yn /kC  N2 and this fact contradicts to assumptions. The Lemma is proved. Remark 1.27. The statement of Lemma 1.4 holds true for N -submonotone maps, namely when instead of (1.10) the weaker inequality holds true ŒA.y1 /; y1  y2   ŒA.y2 /; y1  y2   C .RI ky1  y2 k/ ; C 2 ˚1 ; and the map A is bounded-valued.

1.2 Classes of Multivalued Maps

53

Proof. The difference lies only in obtaining the estimation (1.24). We have ˚ ˛n1 ŒA.yn /; wC  ˛n1 ŒA.y C w/; w C y  yn C C C .RI kyn  y  wkX /  C ŒA.yn /; y  yn   ˛n1 C .RI kyn  y  wkX / C kA.y C w/kC kyn  y  wkX C kA.yn /k kyn  ykX  1 C ˛n1 C .RI kyn  y  wkX / C˛n1 kA.y C w/kC kyn  y  wkX :

Lemma 1.5. The multivalued operator A W X  X  with semibounded variation satisfies Property .˘ /. Proof. In virtue of local boundness of A there exist " > 0 and M" > 0 such that kA./kC  M" 8 2 B W kkX  ", B  X is a bounded set. This means that for some R  " (Fig. 1.11) 1 1 hd.y/; iX  sup fŒA.y/;   yC C hd.y/; yiX g " kkX " kkX " " o 1n ŒA./;   y C hd.y/; yiX C CA .RI ky  k0X /  sup kkX " " o 1n kA./kC  k  ykX C hd.y/; yiX C CA .RI ky  k0X /  sup kkX " "

kd.y/kX  D sup

 where l D

1 ."M" C k2 M" C k1 C l/ D C; " sup

sup CA .RI ky  k0X / < C1 since C.RI  / W RC ! R is

kykX k2 kkX "

the continuous function and k k0X is continuous with respect to k  kX on X . The Lemma is proved. Nonlinear mappings determined on dense subsets of the Banach space X are considered by many authors [Du65,Pa85,ZgMe04]. Yu.A. Dubinskii has introduced an important class of operators of a semi-bounded variation, the properties of which we used. The modifications necessary for this notion in future are given below.

Fig. 1.11 The weakly “”-coercive multivalued map

54

1 Preliminary Results

Let ˚ be a set of continuous functions C.I / W RC  RC ! R such that

1 C.r1 I r2 / ! 0 as ! C0

8r1 ; r2  0I

M , N be normalized spaces. Definition 1.30. The operator  W D./  X ! X  is called the operator with .M I N /-semi-bounded variation if 8y1 ; y2 2 D./ \ ER the following inequality is true h.y1 /  .y2 /; y1  y2 iX  C .RI ky1  y2 k0N / where ER is a ball in M of the radius R; C 2 ˚, k  k0N is a compact with respect to k  kN seminorm. If M D N D X then .M I N /-semi-bounded variation coincides with semibounded variation. In some cases which are not specified separately, we suppose that the seminorm k  k0N is continuous in respect to k  kN . ı

ı

Example 1.1. Let ˝ be the area in Rn , X DWpm .˝/ and the operator A WWpm .˝/ ! Wpm .˝/ has the form A.y/ D



.1/j˛j D ˛ A˛ .x; D ˇ y/ C B˛ .x; D y/

X j˛jDm

C

X

.1/jıj D ı Tı .x; D  y/;

jˇj D m; jj  m; j j  m

jıjm1

where the functions A˛ .x; ˇ /, B˛ .x;  /, Tı .x;  / satisfy the following conditions:

P (1) jA˛ .x; ˛ /j  K jˇ jp1 C 1 ; p  2; jˇ jDm  P (2) The operator A0 .y/ D .1/j˛j D ˛ A˛ .x; D ˇ y/ is strongly elliptic, i.e., j˛jDm

hA0 .y1 /  A0 .y2 /; y1  y2 iX  a0 ky1  y2 kpı

Wpm .˝/

I

(3) The functions B˛ .x;  /, Tı .x;  / are differentiable on  ;  and jB˛ .x;  /j  K

X j jm

jTı .x;  /j  K

X

jjm

j jq C 1 ;

j jq C 1 ;

1.2 Classes of Multivalued Maps

55

jB˛ .x;  /j  K

X

q1 j j C1 ;

j jm1

jTı .x;  /j  K

X



j j

q1

C1 ;

q < p  1;

jjm

and the mappings i W D.i /  X ! X  are determined by functions of the following form .1 y/.x/ D shy.x/;

.2 y/.x/ D ln y.x/;

.3 y/.x/ D lg y.x/;

.4 y/.x/ D 10y.x/ ;

.5 y/.x/ D expfy.x/g;

.6 y/.x/ D y.x/ expfy.x/g:

Under the above conditions the operators Ai D A C i W D.i /  X ! X  (i D 1; : : : ; 6) are the operators with semi-bounded variation where D.i / D f 2 X W i ./ 2 X  g. It is obvious that D.i / (i D 1; 4  6) is dense in X . Example 1.2. Suppose that in Example 1.1 m D 2 and the operator  W D./  X ! X  is of the following form: .y/ D y.x/

@3 y @y @2 y C 2 ; @x1 @x12 @x13

ı

where X D Wp2 .˝/\ Wp1 .˝/ It is known [Li69, J.L. Lions, Chap. 4, problem 10.4] 3n this operator does not map from X to X  and, consequently, is that for p < nC3 not the “subordinate” to the operator A. Let us show that  is defined on the subset dense in X and A C  is the operator of .M I N /-s.b.v. where   N D X \ Pq2 ; Pq2 D y 2 Lq2 .˝/ W yx0 1 2 Lq2 .˝/ ; kykPq2 D kykLq2 .˝/ C kyx0 1 kLq2 .˝/ ;

2 N D Wpq .˝/:

It should be pointed out that the choice of the spaces M and N is far from optimal and in this direction more “fine” results may be obtained, however, for our purposes the presented classes are quite sufficient. Let us consider 8y1 ; y2 2 D.˝/ Z 

@  @2 y2  @  @2 y1  y1 2  y2 2 @x1 @x1 @x1 @x1 ˝ Z     @y1 @2 y1 @y2 @2 y2  y1  y2 dx   y1  y2 dx C 2 2 @x1 @x1 @x1 @x1

h.y1 /  .y2 /; y1  y2 iX D

˝

56

1 Preliminary Results

Z



Z 2  @2 y1 @2 y2 @y2  @ y1 @2 y2 y dx C   y  1 2 @x12 @x12 @x1 @x12 @x12 ˝ ˝

@y2 @y1  y2 dx  @x1 @x1



Z Z @2 y1 @y1 @y2 @2 y2 @y1 @y2 dx  y1 2 dx   D y2 2 @x1 @x1 @x1 @x1 @x1 @x1 ˝ ˝ 2



Z @ y1 @2 y2 @y1 @y2 D y2 dx   @x1 @x1 @x12 @x12 ˝

Z   @2 y2 @y1 @y2 C dx D I1 C I2 : y1  y2  @x12 @x1 @x1

D

˝

Using the inequalities of Young and H¨older with the corresponding indexes we will have

ˇ ˇ

@2 y1 @2 y2 p ˇI1 C I2 ˇ  " ky1  y2 kp

C  Lp .˝/ p @x12 @x12 Lp .˝/

@y1 @y2 q

@2 y2 q ı q



C ky2 kLq .˝/ C  2 q @x12 Lq2 .˝/ @x1 @x1 Lpq .˝/

@y1 @y2 q "

 ky1  y2 kpW 2 .˝/ C K.ı; R/  ; p p @x1 @x1 Lpq .˝/ where kyi kPq2  R, i D 1; 2. As it results from [ZgMe04] A W X ! X  is the operator with semi-bounded variation then due to condition 2 out of the last inequality for the sufficiently small " > 0 we obtain 8y1 ; y2 2 D./ hA .y1 /  A .y2 /; y1  y2 iX  a00 ky1  y2 kpW m .˝/ p   CA RI ky1  y2 k0Wpm .˝/

@y1 @y2 q

 ; K.ı; R/ @x1 @x1 Lpq .˝/

where A .y/ D A.y/ C .y/, k  k0W m .˝/ is the compact norm with respect to p k  kWpm .˝/ and with respect to k  kN , since N is included in X continuously. Example 1.3. Let the fixed numbers m > 1, mi > 1, i D 1; n, ˝ be a bounded area in Rn ,  be a part of its boundary @˝, the elements of the square matrix A D kaij .x/k satisfy the conditions aij 2 Lmi .˝/, i; j D 1; n. Let us indicate by ı

1 1 .˝/ where C0;

.˝/ is the set of the continuous H m;m . / a completeness of C0;

functions in ˝ which are equal to zero on  and have the partial derivatives yx0 i

1.2 Classes of Multivalued Maps

57

bounded in accordance with (on) the norm kyk ı

H m;m .˝/

X

D kykLm .˝/ C

1i n

kAi rykLmi .˝/

where Ai ry is the i -th component of the vector Ary. ı

As well known [GaGrZa74], the space H m;m . / is the separable and reflexive. Let us consider the functions Li0 .x; y; p/, i D 1; n and L00 .x; y; p/ which satisfy Caratheodory conditions on ˝  R  Rn , i.e., they are measurable on x 2 ˝ for each .y; p/ 2 R  Rn and continuous on .y; p/ 2 R  Rn at a.a x 2 ˝. Moreover, 8 .y; p/ 2 R  Rn and a.e. x 2 ˝ jLi0 .x; y; p/j  1

X n

0 0 jpk jmk =mi C jyjmk =mi C 'i .x/ ;

kD1

jL00 .x; y; p/j  2

X n

i D 1; : : : ; n;



0

jpk jmk =mi C jyjm1 C '0 .x/

kD1

where 'i 2 Lm0i .˝/, '0 2 Lm0 .˝/; 1=mi C 1=m0i D 1=m C 1=m0 D 1. ı  ı Let the operator A WH m;m . / ! H m;m . / is determined by the form Z hA .y/; i D

 0  .L .x; y; Ary/; Ar / C L00 .x; y; Ary/ dx

˝ 1 8 y; 2 C0;

.˝/. The mapping A is bounded and continuous [ZgMe04]. 0 Suppose that L .x; y; p/ D A .L 0 .x; y; Ap/, L0 .x; y; p/ D L00 .x; y; Ap/ C @b i b i .x/pi where b i 2 C.˝/, 2 L1 .˝/, i D 1; : : : ; n,  is the internal normal, @xi 1 .˝/ b.x/ D b i .x/i . Let X be a Banach space obtained by the supplement of C0;

R  1=2 2 by the norm kykX D kyke jbjy ds where  D fx 2 @˝ W X C kykL2 .˝/ C

A 6D 0g if b 6 0 in ˝ and kykX D kyke X if b 0 in ˝. 1 .˝/ with accordance of the norm Let us indicate Y is a completeness of C0;

kykY D kykX C

n X

kb i yxi kL2 .˝/ :

i D1

e where each inclusion is topological dense and It is evident that Y  X  X e  L2 .˝/  X   Y  . Determine the operators if b 6 0 then Y  X  X A W X ! X  and  W X ! Y  by the formulas

58

1 Preliminary Results

Z hA .y/; i D ˝

   L 0 .x; y; Ary/; Ar C L 0 0 .x; y; Ary/ dx Z

h.y/; i D 

 y b i /xi dx 

˝

Z b i i y ds

y; 2 X

1 y; 2 C0;

.˝/;  2 L .X; Y  /:

10

If A W X ! X  is the operator with semi-bounded variation and  W X ! Y  is the non-negative definite operator on V D f 2 X W  2 X  g then A C  W V  X ! X  is the operator with semi-bounded variation. The next theorem orders classes of radially upper semicontinuous semimonotone multivalued maps and classes of upper semicontinuous multivalued maps. These results are the fundamental generalization of well-known corresponding results for single-valued maps of monotone type (see for example [GaGrZa74, HuPa97, HuPa00, IvMe88, Me97, ZgMe99, ZgMe04] and references there). 

Theorem 1.7. Let A W X ! 2X be a strict r.u.s.c. semimonotone map with convex weakly star compact images. Then A is upper semicontinuous from X with the strong topology in X  , provided with the topology .X  ; X /. Proof. The validity of the Theorem will follow from Castaing results [AuEk84, p.132] if we prove upper hemicontinuity for the map A. Lemma 1.6. Under the conditions of the Theorem the map A is u.h.c. Proof. Let yn ! y strongly in X , then for an arbitrary v 2 X and t 2 Œ0; 1 we have ŒA.yn /; yn  y  tv  ŒA.y C tv/; yn  y  tvC  C.RI kyn  y  tvkX /: The function X 3 v 7! ŒA.y/; vC is lower semicontinuous 8y 2 X , hence, lim ŒA.yn /; yn  y  tv  ŒA.y C tv/; tvC  C.RI tkvkX /

n!1

 ŒA.yCtv/; tv C.RI t kvkX /:

(1.25)

In virtue of Proposition 1.38 ŒA.yn /; yn  y  tv  ŒA.yn /; tv C ŒA.yn /; yn  yC :

(1.26)

Due to Lemma 1.4 the operator A is locally bounded, specifically, for large enough n there can be found k > 0 such that kA.yn /kC  k: Therefore, lim ŒA.yn /; yn  yC  lim kA.yn /kC kyn  ykX D 0;

n!1

n!1

1.2 Classes of Multivalued Maps

59

and from semimonotony the relations follow lim ŒA.yn /; yn  yC  lim ŒA.yn /; yn  y

n!1

n!1

 lim ŒA.yn /; yn  yC  0; n!1

hence, ŒA.yn /; yn  yC ! 0. In this case from (1.25), (1.26) we obtain C.RI t kvkX / C ŒA.y C tv/; tv  lim ŒA.yn /; yn  y  tv n!1

 lim ŒA.yn /; tv : n!1

We divide the last inequality on t and in virtue of radial upper semicontinuity of the map A we pass to the limit as t ! 0C ŒA.y/; v  lim ŒA.yn /; v ; n!1

hence in virtue of arbitrariness of v 2 X we obtain the required property. The Lemma is proved. The Theorem is proved. Proposition 1.52. Let X be a reflexive Banach space, Dom A be a closed convex  body, A W X 7! 2X be a radial lower semicontinuous operator with semibounded variation satisfying the following condition: G/ if Dom A 3 yn ! y weakly in X and ŒA.yn /; yn  y ! 0; then lim ŒA.yn /; y  yn   0:

n!1

Then A is the .X I Dom A/-pseudomonotone map. Proof. Let Dom A 3 yn ! y weakly in X and the estimation (1.21) be valid. Then from the s.b.v. property we conclude that lim ŒA.yn /; yn  y D 0

n!1

and hence (Condition G), lim ŒA.yn /; y  yn   0:

(1.27)

n!1

For an arbitrary h 2 Dom A and 2 Œ0; 1 we set . / D y C .h  y/. Then 8 2 Œ0; 1

60

1 Preliminary Results

ŒA.yn /; yn  . /  ŒA.. //; yn  . /C  C.RI kyn  . /k0X /:

(1.28)

In virtue of Proposition 1.38 we have ŒA.yn /; yn  yC  ŒA.yn /; yn  . / C ŒA.yn /; h  yC ; hence, taking into account (1.28)

ŒA.yn /; yh  ŒA.yn /; yn y CŒA.. //; yn . /C C.RI kyn  . /k0X /: Passing to the limit in the last inequality as n ! 1 and using (1.27) we obtain

lim ŒA.yn /; y  h  ŒA.. //; y  hC  C.RI ky  hk0X /: n!1

We divide the obtained inequality on and pass to the limit as ! 0 W lim ŒA.yn /; y  h  lim ŒA.. //; y  hC  ŒA.y/; y  h 8h 2 Dom A:

n!1

!0C

Since ŒA.yn /; yn h  ŒA.yn /; yn y CŒA.yn /; y h and ŒA.yn /; yn y ! 0; we have lim ŒA.yn /; yn  h  lim ŒA.yn /; y  h  ŒA.y/; y  h 8h 2 Dom A:

n!1

n!1

The Proposition is proved. Corollary 1.8. Let us replace Condition G in Proposition 1.52 with Condition ˛.Dom A/. Then A is the .X I Dom A/-pseudomonotone map. Proof. Let Dom A 3 yn ! y weakly in X and lim ŒA.yn /; yn  y  0:

n!1

Then yn ! y strongly in X; y 2 DomA and ŒA.yn /; yn  y ! 0: Due to Lemma 1.4 the map A is locally bounded, and hence the map coA; is locally bounded. Therefore 9dn 2 coA.yn / such that ŒA.yn /; y  yn  D ŒcoA.yn /; yn  y

D hdn ; yn  yiX   kdn kX  kyn  yk0X :

1.2 Classes of Multivalued Maps

61

Hence,

lim ŒA.yn /; y  yn   0 n!1

i.e., Condition G holds true. Remark 1.28. Corollary 1.8 remains valid if we replace semiboundness of variation of the operator A with semimonotony. 

Proposition 1.53. Let A0 W X ! 2X be a pseudomonotone operator and let a  map A1 W X ! 2X satisfy the conditions of Proposition 1.52 (or the same of Corollary 1.8). Then A D A0 C A1 is a .F I G/-pseudomonotone map where F D Dom A0 \ Dom A1 ; G D Dom A: Proof. Let F 3 yn ! y weakly in X and the estimation (1.21) is valid. Then (at least up to a subsequence) either (Fig. 1.12) lim ŒA0 .yn /; yn  y  0;

(1.29)

lim ŒA1 .yn /; yn  y  0:

(1.30)

n!1

or n!1

If (1.29) is valid, in virtue of pseudomonotony we have ŒA0 .yn /; yn  y ! 0; and hence, lim ŒA1 .yn /; yn  y  lim fŒA.yn /; yn  y  ŒA0 .yn /; yn  y g  0:

n!1

n!1

Then due to Proposition 1.52 the operator A1 is .X I Dom A1 /-pseudomonotone, i.e., lim ŒA1 .yn /; yn    ŒA1 .y/; y   8 2 Dom A1 : n!1

Therefore lim ŒA.yn /; yn    lim ŒA0 .yn /; yn   C lim ŒA1 .yn /; yn  

n!1

n!1

n!1

 ŒA.y/; y   8 2 DomA1 :

Fig. 1.12 The weakly “C”-coercive, but not weakly “”-coercive multivalued map

62

1 Preliminary Results

Now let the relation (1.30) take place. Then y 2 Dom Dom A1 and due to the conditions of the we have ŒA1 .yn /; yn  y ! 0; hence lim ŒA0 .yn /; yn  y D lim fŒA.yn /; yn  y  ŒA1 .yn /; yn  y g  0:

n!1

n!1

The Proposition is proved. 

Proposition 1.54. Let X be a reflexive Banach space, A W X ! 2X be a map with closed convex values and let one of the following conditions hold true: (1) (2) (3) (4)

A is a strict u.h.c. map; A is a strict, semimonotone, r.u.s.c.; A is a pseudomonotone map and Dom A is a closed set; A is a strict, semimonotone, r.l.s.c. map.

Then the operator A is demiclosed. Proof. Condition 1 is fulfilled. In the case when 2 is fulfilled the validity of the Proposition immediately follows from Lemma 1.6. Let us consider condition 3. Let DomA 3 yn ! y strongly in X; A.yn / 3 dn ! d weakly in X  . Then y 2 DomA and lim ŒA.yn /; yn  y  lim hdn ; yn  yiX  0;

n!1

n!1

and hence, in virtue of pseudomonotony 8 2 X ŒA.y/; y    lim ŒA.yn /; yn    lim hdn ; yn  iX D hd; y  iX ; n!1

n!1

whence due to the separation Theorem [Ru73] d 2 A.y/: Now we consider condition 4. Let graphA 3 .yn I dn /; yn ! y strongly in X; dn ! d weakly in X  , then lim ŒA.yn /; yn  y  0

n!1

and with respect to semimonotony ŒA.yn /; yn  y ! 0: The operator A is locally bounded (Lemma 1.4) and hence there exists n 2 A.yn / such that either ŒA.yn /; y  yn  D h n ; y  yn iX

1.2 Classes of Multivalued Maps

63

or lim ŒA.yn /; y  yn    lim k n kX  kyn  ykX D 0: n!1

n!1

For an arbitrary h 2 X and 2 Œ0; 1 we set .t/ D y C .h  y/: In virtue of Proposition 1.38 ŒA.yn /; yn  yC  ŒA.yn /; yn  . / C ŒA.yn /; h  yC ; whence

ŒA.yn /; y  h  ŒA.yn /; y  yn  C ŒA.. //; yn  . /C C .RI kyn  . /kX  / ; where C 2 ˚0 : In the last equality we pass to the limit as n ! 1:

lim ŒA.yn /; y  h  ŒA.. //; y  hC  C.RI ky  hkX /: n!1

Let us divide both parts of the obtained relation on and pass to the limit as ! 0C taking into account r.l.s.c. of the operator A (Figs. 1.13 and 1.14): lim ŒA.yn /; y  h  lim ŒA.. //; y  hC  ŒA.y/; y  h :

n!1

!0C

Hence lim ŒA.yn /; yn  h  lim ŒA.yn /; y  h C lim ŒA.yn /; yn  y

n!1

n!1

n!1

 ŒA.y/; y  h

8h 2 X:

Therefore d 2 coA.y/: Let V be some reflexive Banach space, V  be its topologically adjoint, h; iV W V   V ! R be the pairing.

Fig. 1.13 r.s.c., but not r.c., r.u.s.c., u.h.c. multivalued map

64

1 Preliminary Results

Fig. 1.14 r.l.s.c., but not r.c., r.u.s.c., u.h.c. multivalued map



Definition 1.31. An operator AW V ! 2V is called s -pseudomonotone if from yn * y in V and the inequality lim hdn ; yn  yiV 6 0;

n!1

where dn 2 coA.yn / it follows that there can be isolated subsequences fym g  fyn g, fdm g  fdn g and a selector d of the map coA such that lim hdm ; ym  wiV > hd.y/; y  wiV 8 w 2 V:

m!1

Remark 1.29. Every s -pseudomonotone map is -pseudomonotone. 

We remind that the map AW V ! 2V is called generalized pseudomonotone if from fwn I yn g  graph coA; yn * y in V , dn * d in V  and lim hdn ; yn  yiV 6 0

n!1

it follows that d 2 coA.y/ and that for some subsequence fdm I ym g we have hdm ; ym iV ! hd; yiV : 

Proposition 1.55. For a bounded map AW V ! 2V the following implications are valid: “A is demiclosed and has Property ˛.V /” ) “A is generalized pseudomonotone” ) “A is s -pseudomonotone”. Proof. Let fdn I yn g  graph coA; dn * d in V  , yn * y in V , lim hdn ; yn  yiV 6 0;

n!1

therefore, lim ŒA.yn /; yn  y 6 0:

n!1

Then there can be found a subsequence fym g such that ym ! y in V . Hence, hdm ; ym iV ! hd; yiV and due to demicloseness d 2 coA.y/. The first implication is proved.

1.2 Classes of Multivalued Maps

65

Let us prove the second one. Let yn * y in V , dn 2 coA.yn / and lim hdn ; yn  yiV 6 0:

n!1

In virtue of boundness of the map coA there can be found a subsequence dm 2 coA.ym / such that dm * d , and due to generalized pseudomonotony hdm ; ym iV ! hd; yiV ; and moreover d 2 coA.y/. But then lim hdm ; ym  iV D hd; y  iV

m!1

8  2 V:

The Proposition is proved. 

Proposition 1.56. Let A; BW V ! 2V be s -pseudomonotone maps and one of them be bounded valued. Then A C B is s -pseudomonotone map. 

Proposition 1.57. Let AW V ! 2V satisfy Condition ˛1 .V / and from .dn I yn / 2 graph coA, yn * y in V , dn * d in V  and hdn ; yn iV ! hd; yiV it follows that d 2 coA.y/. Then A is s -pseudomonotone map. 

Definition 1.32. A map AW V ! 2V is called s-radial semicontinuous (s-r.s.c.) if there can be found a selector d of the operator coA such that 8y; w 2 V;

lim ŒA.y  tw/; wC > hd.y/; wiV :

t !0

The last relation is equivalent to 8y; w 2 V;

lim ŒA.y C tw/; w 6 hd.y/; wiV :

t !0C

Proposition 1.58. Let AW V ! 2V pseudomonotone.



be s-r.s.c. map with s.b.v. Then A is s -

The proof is quite similar to the one of Proposition 1.47. Now again let X be a Banach space. Lemma 1.7. Let 'W X ! R be a convex lower semicontinuous function, dom 'DX .  Then a subdifferential map @'W X ! 2X is a s -pseudomonotone map. Proof. Let yn * y in X , dn 2 @'.yn / and lim hdn ; yn  yiX 6 0;

n!1

66

1 Preliminary Results

then in virtue of monotony of @' hdn ; yn  yiX ! 0: The next statements orders some classes of multivalued maps with semibounded variation maps, classes of sub-bounded variation maps, classes of semimonotone maps and locally bounded maps that satisfy Condition (˘ ). These results are the fundamental generalization of well-known corresponding results for single-valued maps of monotone type (see for example [GaGrZa74, IvMe88, ZgMe99, ZgMe04] and references there). 

Proposition 1.59. Let an operator BW X ! 2X be s.m. (sub-m., with s.b.v., with sub-b.v.). Then Property (˘ ) holds true. Proof. It is known (Lemma 1.4), that under the conditions of Proposition 1.59 the operator B is locally bounded, in particular 9 " > 0 and M" > 0 such that kB.w C h/kC 6 M" as kwkX 6 ". Then, using semiboundness of variation of the map (for the last condition the proof is similar), we have kd.y/kX  D

1 1n hd.y/; iX 6 sup hd.y/; y  hiX kkX 6 " " kkX 6 " " sup

CC.RI ky    hk 0X / C kB. C h/kC .k C hkX C kykX / 6

o

1 .l C l0 C M" k C hkX C M" kykX / 6 M: "

The Proposition is proved. So there exists l > 0, for which hdn ; yn  yiX 6 l, and hence due to Proposition 1.59, kdn kX  6 ı. Therefore we can assume that dn * d in X  , and lim hdn ; yn  wiX D lim hdn ; yn iX  hd; yiX D hd; y  wiX 8 w 2 X; n!1

n!1

or 0 6 lim hdn  ; yn  wiX 6 hd  ; y  wiX 8  2 @'.w/; 8 w 2 X: (1.31) n!1

The next propositions concern generalization of the classical Minty lemma for multi-valued maps of 0 -pseudomonotone type. 

Proposition 1.60. Every r.s.c. map BW X ! 2X with s.b.v. satisfies the condition b): from the inequality hf; y  wiX > ŒB.w/; y  wC  C.RI ky  wkX / we obtain f 2 co B.y/.

(1.32)

1.2 Classes of Multivalued Maps

67

Proposition 1.61. Let BW X  X  be a locally bounded u.h.c. operator with s.b.v. and the inequality (1.32) is valid 8 w 2 D where D is a dense set in X . Then f 2 co B.y/. Remark 1.30. Propositions 1.60, 1.61 are valid for semimonotone maps as well. Proof. Let us prove Proposition 1.60. Substituting in (1.32) w D y  th where t > 0; h 2 X we find 1 hf; hiX > ŒB.y  th/; hC  C.RI tkhk0X /; t whence after passing to the limit as t ! 0C and taking into account radial semicontinuity and the properties of the function C we obtain hf; hiX > ŒB.y/; h 8 h 2 X:

(1.33)

In virtue of Proposition 1.39 the obtained relation is equivalent to the inclusion f 2 co  B.y/: The Proposition is proved. Proof. Let us prove Proposition 1.61. For each w 2 X Let us consider a sequence wn 2 D such that wn ! w in X , hf; y  wn iX > ŒB.wn /; y  wn C  C.RI ky  wn k 0X / > hdn ; y  wn iX  C.RI ky  wn k 0X /; where dn 2 co  B.wn /. From boundness of the map B the boundness of the sequence fdn g follows, so we will assume that dn * d in X  . Since the map B is demiclosed (see Propositions 1.48 and 1.49), then hf; y  wiX > ŒB.w/; y  wC  C.RI ky  wk 0X /:

(1.34)

In virtue of u.h.c. of the map B, in particular, as w D y C th; t > 0, we have lim ŒB.y C th/; hC 6 ŒB.y/; hC 8 h 2 X;

t !0C

or lim ŒB.y  th/; h > ŒB.y/; h :

t !0C

Then from the relation (1.34) we obtain hf; hiX > ŒB.y/; h 8 h 2 X; that due to Proposition 1.39 is equivalent to the inclusion f 2 co  B.y/: The Proposition is proved.

68

1 Preliminary Results

Let us continue to prove the Lemma. It is clear that there can be found l > 0 such that hdn ; yn  yiX 6 l. Then due to Proposition 1.59 kdn kX  6 ı and with no restrictions we may assume that dn * d in X  where 8 w 2 X we have lim hdn ; yn  wiX D lim hdn ; yn iX  hd; wiX D hd; y  wiX ; n!1

n!1

or 0 6 lim hdn  ; yn  wiX 6 hd  ; y  wiX 8  2 @'.w/: n!1

Therefore, hd; y  wiX > Œ@'.w/; y  wC 8 w 2 X and due to Proposition 1.60 d 2 @'.y/ that proves the Lemma. Remark 1.31. Analyzing the proof of Lemma 1.7, taking into account Propositions 1.59, 1.60 we come to the following Proposition: every r.s.c. operator B W  X ! 2X with s.b.v. is s -pseudomonotone (Fig. 1.15). w



Proof. Let us consider an arbitrary sequence yn ! y in X , dn 2 co B .yn / such that lim hdn ; yn  yiX  0: n!1

Due to the semiboundness of variation we conclude that hdn ; yn  yiX ! 0; hence there can be found l > 0 for which hdn ; yn  yiX  l: Using Proposition 1.59 we have kdn kX   ı w

and we will assume that dn ! d in X  and moreover 8w 2 X we obtain lim hdn ; yn  wiX D lim .hdn ; yn iX  hdn ; wiX / D hdn ; y  wiX :

n!1

Fig. 1.15 r.u.s.c. multivalued map

n!1

1.2 Classes of Multivalued Maps

69



In the same time 8 2 co B .w/ either   C RI ky  wk0X  lim hdn  ; yn  wiX D hdn  ; y  wiX ; n!1

is valid or   hd; y  wiX  ŒB .w/ ; y  wC  C RI ky  wk0X 8w 2 X: 

Now we must only use Proposition 1.60, from which we conclude that d 2 co B .y/. Therefore, lim hdn ; yn  wiX D hd; y  wiX 8w 2 X

n!1

and Remark 1.31 is proved (Fig. 1.16). 

Proposition 1.62. Let A W X ! 2X be a r.s.c. (s-r.s.c.) operator with s.b.v., and  let an operator B W X ! 2X satisfy the following conditions: 



(a) A map co B W X ! 2X is compact, i.e., it images bounded sets into precompact ones; 

(b) A graph co B is closed in X   X with respect to the weakly star topology in X  . Then the map A D A C B is -pseudomonotone (s -pseudomonotone respectively). w

Proof. Let yn ! y in X and lim hdn ; yn  xiX  0;

n!1 







where dn 2 co A .yn / : Since the map coB is bounded then co A D co A C co B, 



therefore dn D dn0 C dn00 , where dn0 2 co A .yn / ; dn00 2 co B .yn /. Then passing to the subsequence, if necessary, we obtain one of two relations: ˝ ˛ lim dn0 ; yn  y X  0;

n!1

Fig. 1.16 r.c. multivalued map

˝ ˛ lim dn00 ; yn  y X  0:

n!1

70

1 Preliminary Results

If the first one is valid then in virtue of s.b.v. we have lim hdn00 ; yn  yiX  lim ŒA .yn / ; yn  y

n!1

n!1

   lim ŒA .y/ ; yn  yC  lim C RI kyn  yk0X  0; n!1

n!1

namely

hdn0 ; yn  yiX  0;

and we come to the second relation: lim hdn00 ; yn  yiX  0:

n!1

Here lim hdn00 ; yn  yiX  lim ŒB .yn / ; yn  y D lim hgn ; yn  yiX ;

n!1

n!1

n!1



where gn 2 co B .yn /. In virtue of the condition a of Proposition 1.62 there can be found a sequence fgm g  fgn g such that gn ! g in X  , lim hgn ; yn  yiX D lim hgm ; ym  yiX D 0: n!1

n!1



Then due to the condition b, g 2 co B .y/. Further since hgm ; ym  yiX ! 0, then lim hdn00 ; yn  yiX  0, namely n!1

˝

dn00 ; ym  y

˛ X

! 0:

Therefore, 8w 2 X we have lim hdn ; yn  wiX  lim hdn0 ; yn  wiX C lim hdn00 ; yn  wiX ;

n!1

n!1

n!1

lim hdn0 ; yn  wiX  lim ŒA.w/; yn  w  lim C.RI kyn  wk0X /

n!1

n!1

n!1

 ŒA.w/; y  wC  C.RI ky  wk0X / Since hdn0 ; yn  yiX ! 0, we obtain t lim hdn0 ; y  hiX D lim hdn0 ; yn  yiX C lim hdn ; y  wiX n!1

n!1

n!1

 tŒA.y  t.y  h//; y  hC  C.RI tky  hk0X /

1.2 Classes of Multivalued Maps

71

Dividing the obtained relation on t, passing to the limit as t ! 0C; and taking into account r.s.c. we have ˝ ˝ ˝ ˛ ˛ ˛ lim dn0 ; yn  h D lim dn0 ; yn  y X C lim dn0 ; yn  h X n!1

n!1

n!1

 lim ŒA .yt .yh// ; y  hC  ŒA .y/ ; y  h 8h2X: t !0C

On the other hand from the sequence fdn00 g there can be isolated a subsequence 00 fdm g such that 00 ; ym  yiX ; lim hdn00 ; yn  yiX D lim hdm m!1

n!1

00 dm ! dm in X  :



Here d 00 2 co B .y/, lim hdn ; yn  hiX  hd 00 ; y  hiX C ŒA .y/ ; y  h

n!1

 ŒA .y/ ; y  h

8h 2 X:

The case of s -pseudomonotone maps is similarly proved. 

Remark 1.32. Under the conditions of Proposition 1.62 the map A W X ! 2X is pseudomonotone. 

Proposition 1.63. Let A W X ! 2X be a -pseudomonotone (s -pseudomonotone  respectively) map, and let an operator B W X ! 2X satisfy satisfy all requirements of Proposition 1.62. Then operator A D A C B is -pseudomonotone (s -pseudomonotone respectively) map. Proof. Analyzing the proof of Proposition 1.62, it is sufficient to notice that the operator B is s -pseudomonotone (and hence it is -pseudomonotone) and bounded. It is left to use Proposition 1.56. 

Remark 1.33. Let A; B W X ! 2X be -pseudomonotone (s -pseudomonotone respectively) maps and Dom .A \ B/ ¤ ;. Then the map A \ B is -pseudomonotone (s -pseudomonotone respectively) on Dom .A \ B/. At the same time for the map A [ B these properties do not hold true. 

Definition 1.33. The operator A W X ! 2X is called -pseudomonotone if for w yn ! y and lim hdn ; yn  yiX  0; (1.35) n!1

where dn 2 A .yn / there can be found subsequences fym g, fdm g such that lim hdm ; ym  wiX  ŒA .y/ ; y  w 8w 2 X;

m!1

72

1 Preliminary Results

and we call it s -pseudomonotone if from fulfillment of the condition (1.35) it fol

lows that there can be found subsequences fym g, fdm g and d .y/ 2 co A .y/ such that lim hdm ; ym  wiX  hd .y/ ; y  wiX 8w 2 X: m!1

Remark 1.34. It is clear that the following implications are valid: “A is -pseudomonotone map” ) “A is -pseudomonotone map”; “A is s pseudomonotone map” ) “A is s -pseudomonotone map”. 

Proposition 1.64. Let A; B W X ! 2X be -pseudomonotone (s -pseudomonotone respectively) maps. Then the map A [ B has the same property. 

Proposition 1.65. An arbitrary -coercive multivalued map A W X ! 2X is Ccoercive. For monotone maps the inverse statement remains valid. Proof. The right side of the given Proposition is a direct corollary of the upper Œ; C and the lower Œ;  forms definition. We pass to the second part of the Proposition.  Let A W X ! 2X be a monotone C-coercive map. Its -coercivity follows from the estimations: 2ŒA.y/; y  12 y 2ŒA. 12 y/; y  12 yC ŒA.y/; y D  kykX kykX kykX 1 1 ŒA. 2 y/; 2 yC D ! C1 as kykX ! 1: k 12 ykX The Proposition is proved. The next proposition and corollary show that C-coercivity (weaker condition) provides -coercivity, the uniform -coercivity and the uniform C-coercivity (stronger conditions) for monotone multi-valued maps, particularly, for subdifferentials of convex lower semi-continuous functionals. These conditions are used for the investigation of evolutional multi-variational inequalities and differentialoperators inclusions and allows us to use the weaker condition of C-coercivity instead corresponding stronger conditions of coercivity without loss of supplementary estimations. Let us remark that similar properties for subdifferentials of a convex functional were considered in the book [Sh97]. Proposition 1.66. Every monotone C-coercive multivalued map is -coercive, uniformly -coercive and uniformly C-coercive. 

Proof. Let A W X ! 2X be monotone , C(–)-coercive map. We prove that A is uniformly C(–)-coercive. From Lemma 1.3 it follows that there exists a ball BNr D fy 2 X j kykX  rg

1.2 Classes of Multivalued Maps

73

and a constant c1 > 0 such that kA.!/kC  c1

8! 2 BN r :

Therefore for any y 2 X kA.y/kC D

sup

sup

d.y/2A.y/ !2B r

1 1 hd.y/; !iX D sup ŒA.y/; !C r r !2B r



1 sup fŒA.y/; !yC C ŒA.y/; yC g r !2B r



˚  1 sup ŒA.!/; !yC C ŒA.y/; yC r !2B r

1 fc1 .r C kykX / C ŒA.y/; yC g r c1 1 D ŒA.y/; yC C c1 C kykX I r r 1 D inf sup hd.y/; !iX d.y/2A.y/ !2B r r 

kA.y/k

D 

inf

sup

d.y/2A.y/ !2B

inf

r

sup

d.y/2A.y/ !2B

r

1 fhd.y/; !  yiX C hd.y/; yiX g r 1 fhd.!/; !  yiX C hd.y/; yiX g r

1  inf fhd.!/; yiX C c1 .r C kykX /g r d.y/2A.y/ c1 1 D ŒA.y/; y C c1 C kykX ; r r

namely 8y 2 X So as c D

r 2

kA.y/kC./ 

1 c1 ŒA.y/; yC./ C c1 C kykX : r r

> 0, uniform C./-coercivity for A follows from the estimations:

ŒA.y/; yC./  12 ŒA.y/; yC./  ŒA.y/; yC./  ckA.y/kC./  kykX kykX D

ŒA.y/; y  12 y  kykX



ŒA. 12 y/; y  12 yC  kykX

D

ŒA. 12 y/; 12 yC 2k 12 ykX as



rc1 2



rc1 2

rc1 2

c1 kykX 2



c1 kykX 2

c1 rc1 ! C1  2 2kykX

kykX ! 1:

To complete the proof it is sufficient to apply Proposition 1.65.



c1 kykX 2

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1 Preliminary Results

Corollary 1.9. Let ' W X ! R be convex lower semicontinuous functional such that '.y/ ! C1 as kykX ! 1: kykX Then its subdifferential map @'.y/ D fp 2 X  j hp; !  yiX  '.!/  '.y/

8! 2 X g ¤ ;;

y2X

is C-coercive and hence -coercive, uniformly -coercive and uniformly C-coercive. Proof. Due to monotony of the map @' W X  X  and to Propositions 1.65 and 1.66, it is sufficient to prove its +-coercivity. It follows from the estimations: 1 1 kyk1 X Œ@'.y/; yC  kykX '.y/  kykX '.0/ ! C1 as kykX ! C1:

The Corollary is proved. The next proposition orders classes of 0 -pseudomonotone multivalued maps that satisfy Property .˘ / and classes of subdifferentials of convex functionals. The obtained properties are used in the work when investigating multivalued inequalities more than once. Proposition 1.67. Let a function ' W X ! R be convex, lower semicontinuous on X . Then the multivalued map B D @' W X ! Cv .X  / is 0 -pseudomonotone on X and it satisfies Condition (˘ ). Proof. (a) Condition .˘ /. Let k > 0 and a bounded set B  X be arbitrary fixed. Then for all y 2 B and for all d.y/ 2 @'.y/ the inequality hd.y/; y  y0 iX  k is valid. Let u 2 X be an arbitrary fixed and hence we have hd.y/; uiX D hd.y/; u  yiX C hd.y/; yiX  '.u/  '.y/ C k  '.u/  inf '.y/ C k const < C1; y2B

since every convex, lower semicontinuous functional is bounded below on any bounded set. Therefore due to Banach–Steinhaus Theorem there exists N D N.y0 ; k; B/ such that kd.y/kX   N for every y 2 B; (b) 0 -pseudomonotony on X . Let yn * y0 in X , @'.yn / 3 dn * d in X  and the inequality (1.11) is valid. Then due to monotony of @' for any d0 2 @'.y0 / and for any n  1 we have hdn ; yn  y0 iX D hdn  d0 ; yn  y0 iX C hd0 ; yn  y0 iX  hd0 ; yn  y0 iX : Therefore, lim hdn ; yn  y0 iX  lim hd0 ; yn  y0 iX D 0: n!C1

n!C1

1.2 Classes of Multivalued Maps

75

In virtue of the last inequality and the inequality (1.11) we have lim hdn ; yn  y0 iX D 0:

n!C1

Hence for any w 2 X lim hdn ; yn  wiX  lim hdn ; yn  y0 iX n!C1

n!C1

C lim hdn ; y0  wiX D hd0 ; y0  wiX : (1.36) n!C1

on the other hand we have hd0 ; w  y0 iX  lim hdn ; w  yn iX  '.w/ n!C1

 lim '.yn /  '.w/  '.y0 /;

(1.37)

n!C1

since every convex, lower semicontinuous functional is weakly lower semicontinuous. From (1.37) it follows that d0 2 @'.y0 /. Hence due to (1.36) we obtain the inequality (1.12) as B D @' on X . The Proposition is proved. Lemma 1.8. Let X be a reflexive Banach space. Then every -pseudomonotone on X map A W X ! Cv .X  / is 0 -pseudomonotone on X . For bounded maps the inverse statement is valid as well. Proof. The first statement of the Lemma directly follows from Definition 1.25. Let us prove the inverse statement. Let A W X  X  be 0 -pseudomonotone on X map, yn * y in X , the inequality (1.11) holds true with dn 2 A.yn /. Boundedness of the map A implies boundness of the sequence fdn gn1 in X  . Hence due to Banach– Alaoglu Theorem there exist subsequences fdnk gk1  fdn gn1 and fynk gk1  fyn gn1 such that dnk * d in X  and besides ˝ ˛ lim dnk ; ynk  v X  lim hdn ; yn  viX  0:

k!1

n!1

Since the operator A is 0 -pseudomonotone on X then there exist subsequences fynkm gm1  fynk gk1 and fdnkm gm1  fdnk gk1 for which the inequality (1.12) holds true. This proves the Lemma. Remark 1.35. For classical definitions (without passing to subsequences) the previous statement is problematic. In the paper of Browder F. and Hess P. [BrHe72] the new class of generalized pseudomonotone operators was introduced. Definition 1.34. An operator A W X ! Cv .X  / is called generalized pseudomonotone on X if for each pair of sequences fyn gn1  X and fdn gn1  X  such that dn 2 A.yn /, yn * y in X , dn * d in X  from the inequality

76

1 Preliminary Results

lim hdn ; yn iX  hd; yiX

(1.38)

n!1

the inclusion d 2 A.y/ and the convergence hdn ; yn iX ! hd; yiX follow. Proposition 1.68. Every generalized pseudomonotone on X 0 -pseudomonotone on X .

operator is

Proof. Let yn * y in X , A.yn / 3 dn * d in X  and (1.38) holds true. Hence, lim hdn ; yn  yiX  lim hdn ; yn iX C lim hdn ; yiX

n!1

n!1

n!1

 hd; yiX C hd; yiX D 0: From generalized pseudomonotony we have that hdn ; yn iX ! hd; yiX ;

d 2 A.y/:

Hence due to Proposition 1.39, lim hdn ; yn  viX D hd; y  viX  ŒA.y/; y  v

8 v 2 X:

n!1

The Proposition is proved. The inverse Proposition is not valid. Proposition 1.69. Let A W X  X  be 0 -pseudomonotone operator. Then the fol

lowing property takes place: from yn ! y weakly in X , co A.yn / 3 dn ! d weakly star in X  and from the inequality (1.11) the existence of subsequences fym g  fyn g and fdm g  fdn g such that hdm ; ym iX ! hd; yiX ; 

with d 2 co A.y/ follows. Proof. Let fyn g; fdn g be required sequences. Hence we can isolate subsequences fym g; fdm g such that the inequality (1.38) will hold true. Fixing ! D y in the last statement we obtain hdm ; ym  yiX ! 0 or hdm ; ym iX ! hd; yiX , hd; y  viX D lim hdm ; ym  viX  ŒA.y/; y  v 8 v 2 X: m!1



Therefore due to Proposition 1.39 we obtain that d 2 co A.y/. The Proposition is proved. Proposition 1.70. Let A D A0 C A1 W X  X  where A0 W X  X  is a monotone map and an operator A1 W X  X  has the following properties:

1.2 Classes of Multivalued Maps

77

(1) There exists a linear normalized space Z in which X is compactly and densely embedded; (2) The operator A1 W Z  Z  is singlevalued and locally polynomial, namely P 8R > 0 there exists n D n.R/ and a polynomial function PR .t/ D 0<˛n

˛ .R/t ˛ with continuous multipliers ˛ .R/  0 such that the following estimation holds true .Z  /

kA1 .y1 /  A1 .y2 /kC

 PR .ky1  y2 kZ / 8 kyi kZ  R; i D 1; 2:

Then A is the operator with semibounded variation. Proposition 1.71. Let in the previous Proposition the operator A0 W X  X  is N -monotone and instead of the condition (2) we have: (2’) The (multivalued) map A1 W Z  Z  is locally polynomial in the following sense: 8R > 0 there exists n D n.R/ and a polynomial PR .t/ for which dist .A1 .y1 /; A1 .y2 //  PR .ky1  y2 kZ / 8 kyi kZ  R; i D 1; 2:

(1.39)

Then A D A0 C A1 is the operator with N -semibounded variation on X . Proof. We give the proof of Proposition 1.71. In the case of Proposition 1.70 argumentation is the same. Since for any y1 ; y2 2 X ŒA0 .y1 / ; y1  y2   ŒA0 .y2 / ; y1  y2  ; we must obtain the estimation of ŒA1 .y1 / ; y1  y2   ŒA1 .y2 / ; y1  y2  . For any d1 2 A1 .y1 / ; d2 2 A1 .y2 / we find hd2 ; y1  y2 iX  hd1 ; y1  y2 iX D hd2 ; y1  y2 iZ  hd1 ; y1  y2 iZ  kd1  d2 kZ  ky1  y2 kZ ; therefore ŒA1 .y2 / ; y1  y2   ŒA1 .y1 / ; y1  y2   dist .A1 .y1 / ; A1 .y2 // ky1  y2 kZ : O RD hence from the estimation (1.39) when kyi kZ  R .i D 1; 2/ (kyi kX  R, O R.R/ respectively) we obtain   O ky1  y2 k0X ; ŒA1 .y1 / ; y1  y2   ŒA1 .y2 / ; y1  y2   C RI where k  k0X D k  kZ ; C .R; t/ D PR .t/ t. It is easy to check that C 2 ˚. New classes of 0 -pseudomonotone maps are defined in the next proposition. It is fundamentally used when validating an existence of a solution for the second order differential-operator inclusion by the singular perturbations method [PeKaZa08, ZaKa07].

78

1 Preliminary Results

Proposition 1.72. Let one of the following conditions hold true: (1) A W X  X  is r.l.s.c. operator with s.b.v.; (2) A W X  X  is r.u.s.c. operator with N -s.b.v. and compact values in X  . Then A is 0 -pseudomonotone on X map. 

Proof. Let yn * y in X , co A .yn / 3 dn * d in X  and (1.11) holds true. Since A is the operator with s.b.v. then for each v 2 X   hdn ; yn  viX  ŒA .yn / ; yn  v  ŒA .v/ ; yn  vC  C RI kyn  vk0X : The function X 3 w ! ŒA .v/ ; wC is convex and l.s.c. and in virtue of Corollary 1.10 it is w.l.s.c. Therefore due to continuity of C by the second variable, substituting v D y to the last inequality after passing to the limit and taking into account (1.11) we obtain: hdn ; yn  yiX ! 0 as n ! 1:

(1.40)

For any h 2 X and 2 Œ0; 1 we set ! D h C .1  / y. Then   hdn ; yn  ! iX  ŒA .! / ; yn  ! C  C RI kyn  ! k0X ; or after passing to the lower limit:  

lim hdn ; y  hiX  ŒA .w / ; y  hC  C RI ky  hk0X : n!1

Dividing the last inequality on and passing to the limit as ! C0 in virtue of radial lower semicontinuity of the operator A and the continuity of the function C we obtain that for each h 2 X  1  C RI ky  hk0X !0C

lim hdn ; y  hiX  lim ŒA .! / ; y  hC C lim

n!1

!0C

 ŒA .y/ ; y  h : Since (1.40) holds true and C 2 ˚0 then lim hdn ; yn  hiX D lim hdn ; y  hiX  ŒA .y/ ; y  h

n!1

8h 2 X;

n!1

that proves the first statement of the Proposition. Now let us look on the basic moment of the second statement. Due to N -semiboundness of variation of the operator A we conclude that

1.2 Classes of Multivalued Maps

79

lim hdn ; yn  viX  lim ŒA .yn / ; yn  v

n!1

n!1

   lim ŒA .v/ ; yn  v  C RI ky  vk0X

(1.41)

n!1

Let us obtain the estimation for the first member in the right side of (1.41). We prove that the function X 3 h 7! ŒA .v/ ; h is weakly lower semicontinuous 

8v 2 X . Let zn ! z weakly in X then for any n D 1; 2; : : : 9n 2 co A .v/ such that ŒA .v/ ; zn  D hn ; zn iX : From the sequence fn I zn g we isolate a subsequence fm I zm g such that lim ŒA .v/ ; zn  D lim hn ; zn iX D lim hm ; zm iX

n!1

n!1

m!1



and due to compactness of the set co A .v/ we find that m !  strongly in X  with 

 2 co A .v/. Therefore lim ŒA .v/ ; zn  D lim hm ; zm iX D h; ziX D ŒA .v/ ; z ; n!1

n!1

and this relation proves weak lower semicontinuity of the function h 7! ŒA .v/ ; h : Hence from (1.41) we have lim hdn ; yn  viX  lim ŒA .yn / ; yn  v n!1    ŒA .v/ ; y  v  C RI ky  vk0X :

n!1

Then replacing v with y in the last inequality we have hdn ; yn  yiX ! 0, therefore   lim hdn ; yn  viX  ŒA .v/ ; v  w  C RI ky  vk0X

8v 2 X:

n!1

Replacing in the last inequality v with tw C .1  t/ y where w 2 X , t 2 Œ0; 1 then, dividing the result on t and passing to the limit as t ! 0C, in virtue of radial upper semicontinuity we have lim hdn ; yn  wiX  ŒA .y/ ; y  w

n!1

The Proposition is proved.

8w 2 X:

80

1 Preliminary Results

Now let X be a Banach space such that X D X1 \ X2 where X1 , X2 is the interpolation pair of reflexive Banach spaces [Tr78] which satisfies X1 \ X2

is dense in

X1 ; X2 :

(1.42) 



Definition 1.35. A pair of multivalued maps A W X1 ! 2X1 and B W X2 ! 2X2 is called s-mutually bounded if for each M > 0 and a bounded set B  X there exists a constant K.M / > 0 such that from kykX  M

and hd1 .y/; yiX1 C hd2 .y/; yiX2  M

8y 2 B

it follows that either kd1 .y/kX1  K.M /;

or

kd2 .y/kX2  K.M / 8y 2 B

for some selectors d1 2 A and d2 2 B. 



Remark 1.36. It is clear that if one of the maps A W X1 ! 2X1 or B W X2 ! 2X2 is bounded then the pair (AI B) is s-mutually bounded. The next three lemmas concern completeness of classes of 0 -pseudomonotone C-coercive multi-valued maps which satisfy Property .˘ / with respect to the sum. These statements are used when studying solutions of anisotropic evolutional inclusions and multi-variational inequalities. Lemma 1.9. Let A W X1  X1 and B W X2  X2 be some multivalued +(–)coercive maps which satisfy Condition ./. Then the multivalued map C WD A C B W X  X  is C(–)-coercive too. Proof. We will prove the Proposition by contradiction. Let 9fyn gn1 X W kyn kX D kyn kX1 C kyn kX2 ! C1 as n ! C1, but ŒC.yn /; yn C./ < C1: kyn kX n1 sup

(1.43)

The case 1. kyn kX1 ! C1 as n ! 1, kyn kX2  c 8n  1. A .r/ WD

inf

kvkX1 Dr

ŒA.v/; vC./ ; kvkX1

B .r/ WD

inf

kwkX2 Dr

ŒB.w/; wC./ ; kwkX2

r > 0:

We remark that A .r/ ! C1; B .r/ ! C1 as r ! C1. Then 8n  1 kyn k1 X1 ŒA.yn /; yn C./  A .kyn kX1 /kyn kX1

1.2 Classes of Multivalued Maps

and

81

ŒA.yn /; yn C./ kyn kX1  A .kyn kX1 / ! C1 kyn kX kyn kX

as kyn kX1 ! C1 and kyn kX2  c. Due to Condition ./C./ for each n  1 ŒB.yn /; yn C./ kyn kX2 kyn kX2  B .kyn kX2 /  c1 !0 kyn kX kyn kX kyn kX

as n ! 1;

where c1 2 R is a constant, as in Condition ./C./ with D D fy 2 X2 j kykX2  cg: It is clear that ŒA.yn /; yn C./ ŒB.yn /; yn C./ ŒC.yn /; yn C./ D C ! C1 kyn kX kyn kX kyn kX

as

n ! 1:

This fact contradicts to (1.43). The case 2. The case when kyn kX1  c 8n  1 and kyn kX2 ! 1 as n ! C1 can be checked in the same manner. The case 3. Let us consider the situation when kyn kX1 ! C1 and kyn kX2 ! C1 as n ! C1. Then ŒC.yn /; yn C./ kyn kX1  A .kyn kX1 / kyn kX kyn kX1 C kyn kX2 n1

C 1 > sup

CB .kyn kX2 / kyn kX1 > kyn kX kyn kX1 example kyn kX

kyn kX2 kyn kX

It is clear that 8n  1

0 and

restrictions, for

! 0 then

kyn kX2 : (1.44) kyn kX1 C kyn kX2

> 0 and moreover even if one of the

kyn kX2 kyn kX1 D1 ! 1: kyn kX kyn kX We have the contradiction to (1.44). Let X D X1 \ X2 where .X1 ; k  kX1 / and .X2 ; k  kX2 / are reflexive Banach spaces such that for some linear topological space Y Xi  Y continuously. Then the adjoint X  D X1 C X2  . For f 2 X  and v 2 X we assume hf; viX D hf1 ; viX1 C hf2 ; viX2 ; where f D f1 C f2 , f1 2 X1  , f2 2 X2  .

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1 Preliminary Results

Lemma 1.10. Let A W X1  X1 and B W X2  X2 be strict multivalued maps satisfying Condition .˘ /. Then the pair .AI B/ is s-mutually bounded and the multivalued map C WD A C B W X  X  satisfies Condition .˘ /. Proof. We show that the strict multivalued map C WD A C B W X  X  satisfies Condition .˘ /. Let k > 0, D  X be a bounded set, y0 2 X and for some selector d 2 C, (1.45) hd.y/; y  y0 iX  k for all y 2 B: Then for some selectors d1 2 A and d2 2 B: d D d1 C d2 from (1.45) it follows that the sets ˇ   ˇ k D1 D y 2 B ˇˇ hd1 .y/; y  y0 iX1  ; 2 ˇ   ˇ k ˇ D2 D y 2 B ˇ hd2 .y/; y  y0 iX2  2 together cover the whole set D. Therefore there exists C1 > 0, such that kd1 .y/kX1  C1

for each y 2 B1 ;

kd2 .y/kX2  C1

for each y 2 B2 ;

Hence taking into account (1.45) there exists C2 > 0: hd1 .y/; y  y0 iX1  C2 ;

8y 2 B2

hd2 .y/; y  y0 iX1  C2

8y 2 B1 :

Since B D B1 [ B2 for C3 D maxfC1 I C2 g kd1 .y/kX1  C3 ;

kd2 .y/kX2  C3

for all

y2B

and kd.y/kX1  2C3

for each y 2 B:

Property .˘ / for C is checked; s-mutual boundness for the pair .AI B/ follows from Property .˘ / for C . The Lemma is proved. Lemma 1.11. Let A W X1 ! Cv .X1 / and B W X2 ! Cv .X2 / be s-mutually bounded, 0 -pseudomonotone on X1 and X2 respectively multivalued maps. Then C WD A C B W X ! Cv .X  / is 0 -pseudomonotone on X multivalued map. Remark 1.37. If the pair (AI B) is not s-mutually bounded then the Proposition holds true for -pseudomonotone (on X1 and on X2 respectively) maps only. Proof. At first we check that 8y 2 X C.y/ 2 Cv .X  /. Convexity of C.y/ follows from convexity of A.y/ and B.y/. Due to [Ru73, Theorem 3.13, p.79] it is sufficient

1.2 Classes of Multivalued Maps

83

to show weak completeness of C.y/. Let g be a cluster point of C.y/ in the topology .X  I X  / D .X  I X / (the space X is reflexive). Then 9fgm gm1  C.y/ W

gm ! g

weakly in

X

as

m ! 1:

Hence taking into account boundness of values for the maps A and B, Banach– Alaoglu Theorem and the norm definition in X  D X1 C X2 we can assume that for any m  1 there exist vm 2 A.y/ and wm 2 B.y/: vm C wm D gm and up to a subsequence vm * v in X1 and wm * w in X2 for some v 2 A.y/ and w 2 B.y/. Hence g D v C w 2 C.y/. Therefore weak completeness of the set C.y/ in X  is proved. Now let yn * y0 in X (hence it follows that yn * y0 in X1 and yn * y0 in X2 ), C.yn / 3 d.yn / * d0 in X  and the inequality (1.11) is valid. Hence for some dA .yn / 2 A.yn /

and dB .yn / 2 B.yn / W

dA .yn / C dB .yn / D d.yn /:

Due to s-mutual boundness of the pair (AI B) and to the estimation hd.yn /; yn iX D hdA .yn /CdB .yn /; yn iX D hdA .yn /; yn iX1 ChdB .yn /; yn iX2  k we have either kdA .yn /kX1  C or kdB .yn /kX2  C . Then in virtue of reflexivity of X1 and X2 up to a subsequence (if necessary) we have: dA .yn / * d00 in X1

and dB .yn / * d000 in X2 :

(1.46)

Due to the inequality (1.11) we have: lim hdB .yn /; yn  y0 iX2 C lim hdA .yn /; yn  y0 iX1

n!1

n!1

 lim hd.yn /; yn  y0 iX  0; n!1

or symmetrically lim hdA .yn /; yn  y0 iX1 C lim hdB .yn /; yn  y0 iX2

n!1

n!1

 lim hd.yn /; yn  y0 iX  0: n!1

Let us consider the last inequality. Obviously there exists a sequence fym gm  fyn gn1 such that 0  lim hdB .yn /; yn  y0 iX2 C lim hdA .yn /; yn  y0 iX1 n!1

n!1

 lim hdB .ym /; ym  y0 iX2 C lim hdA .ym /; ym  y0 iX1 : m!1

m!1

(1.47)

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1 Preliminary Results

Hence we have: or

lim hdA .ym /; ym  y0 iX1  0;

or

m!1

lim hdB .ym /; ym  y0 iX2  0:

m!1

Without any loss of generality we assume that lim hdA .ym /; ym  y0 iX1  0:

m!1

Then from (1.46) and due to 0 -pseudomonotony of A on X1 there exists fymk gk1 from fym gm such that lim hdA .ymk /; ymk  viX1  ŒA.y0 /; y0  v

k!1

8v 2 X1 \ X1 :

(1.48)

Setting in the last inequality v D y0 we obtain that hdA .ymk /; ymk  y0 iX1 ! 0

as k ! C1:

Then in virtue of (1.47), lim hdB .ymk /; ymk  y0 iX2  0:

k!1

Due to 0 -pseudomonotony of B on X2 up to a subsequence fym0k g  fymk gk1 we have: lim hdB .ym0k /; ym0k  wiX2  ŒB.y0 /; y0  w

8w 2 X2 \ X2 :

(1.49)

k!1

At last from the relations (1.48) and (1.49) we obtain: lim hd.ym0k /; ym0k  xiX  lim hdA .ym0k /; ym0k  xiX1

k!1

k!1

C lim hdB .ym0k /; ym0k  xiX2 k!1

 ŒA.y0 /; y0  x C ŒB.y0 /; y0  x D ŒC.y0 /; y0  x

8x 2 X \ X:

The Lemma is proved. Let Y be Banach space and here X is reflexively and Y continuously and densely included into X , X  is the space which is topologically dual to X . Out of continuity and density of inclusion of Y ! X results continuity of inclusion of X   Y  and this inclusion is dense in case of Y reflectivity. Further we will indicate h; iY is canonic bilinear form for Y  Y  and for any fixed g 2 X  h; giX jY D h; giY . Definition 1.36. The operator A W D.A/  X ! Y  possesses Property . / if the following conditions are satisfied:

1.2 Classes of Multivalued Maps

85

(1 ) The operator A is weakly star continuous, i.e., it transforms any weakly converging sequence into weakly star converging one; (2 ) If yn ! y (yn ; y 2 Y \ D.A/) weakly in X then hA.yn /; yn iY ! hA.y/; yiY : Definition 1.37. The operator A W D.A/  X ! Y  is called the operator of variational calculus on D.A/, if it may be presented in the form A.y/ b A.y; y/, where the operator b A W .D.A/  X /  X ! Y  has the following properties: (a) For the arbitrary  2 D.A/ b A.; / W X ! Y  is a radially continuous operator along Y and 8y;  2 Y \ D.A/, kykX  R, kkX  R hb A.y; y/  b A.y; /; y  iY  C.RI ky  k0X /I b / is weakly pre-compact, i.e., the mapping (b) 8 2 X the operator y ! A.y; b A.; / W X ! Y  is pre-compact in respect to weak topology in X and weakly star topology in Y (in such a way from any weakly converging sequence yn ! y (yn ; y 2 Y \ D.A/) in X it is possible to extract such a sequence fym g that b A.ym ; / ! ~ weakly star in Y  ; (c) From Y \ D.A/ 3 yn ! y 2 Y \ D.A/ weakly in X and hb A.yn ; yn /  b A.yn ; / ! b A.y; / weakly star A.yn ; y/; yn  yi ! 0 it follows that 8 2 X b in Y  ; (d) If yn ! y (yn ; y 2 Y \ D.A/) weakly in X and b A.yn ; / ! ~ weakly star in A.yn ; /; yn iY ! h~; yiY : Y  then hb Proposition 1.73. The following implication is true: “A is an operator of variational calculus” ) “A is a pseudomonotone operator”. Proof. Assume that yn ! y (yn ; y 2 Y \ D.A/) weakly and the inequality lim hA.yn /; yn  yiY  0

n!C1

(1.50)

is valid. Because of condition “b” of Definition 1.37 it may be considered that b A.yn ; /; yn iY ! A.yn ; / ! ~ weakly star in Y  and it means that (condition “d”) hb A.yn ; y/; yn  yiY D 0. From this and (1.50) we h~; yiY . Therefore, lim hb n!1 conclude that lim hb A.yn ; yn /  b A.yn ; y/; yn  yiY D 0:

n!1

However, because of the condition “a” lim hb A.yn ; yn /  b A.yn ; y/; yn  yiY   lim C.RI kyn  yk0X / D 0;

n!1

n!1

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1 Preliminary Results

which together with the previous inequality gives hb A.yn ; yn /  b A.yn ; y/; yn  yiY ! 0: A.y; / weakly star in Y  Then, in accordance with the condition “b”, b A.yn ; / ! b for all  2 X , it means that hA.yn ; /; yn  yiY ! 0. Thus, O n ; y/; yn  yiY  C.RI kyn  yk0X / ! 0 hb A.yn ; yn /; yn  yiY  hA.y and taking into consideration (1.50) hb A.yn ; yn /; yn  yiY ! 0:

(1.51)

For any fixed  2 Y suppose !. / D y C .  y/; 2 .0; 1/: Since 8 2 Y 8 2 .0; 1/ hb A.yn ; yn /  b A.yn /; !. //; yn  !. /iY  C.RI kyn  !. /k0X / then A.yn ; yn /; yn  yiY C hb A.yn ; !. //; y  iY

hb A.yn ; yn /; y  iY   hb C hb A.yn ; !. //; yn  yiY  C.RI kyn  !. /k0X /: Thus, by dividing both parts of the latter inequality by and taking into account (1.51) 1 lim hb A.yn ; yn /; yn  iY  hb A.y; !.. //; y  iY  C.RI k .y  /k0X /:

n!1 Passing in the last inequality to the limit at ! C0 taking into consideration the property “a” we obtain lim hb A.yn ; yn /; yn  iY  hb A.y; y/; y  iY

8 2 Y:

n!1

The statement is proved. Proposition 1.74. Let the operator A W D.A/  X ! Y  be presented in the form A.y/ D A1 .y/ C A2 .y/ where A1 W X ! Y  is the operator of variational calculus and the operator A2 W D.A/  X ! Y  has the property . /. Then A is the operator of variational calculus on D.A/ where A.y/ b A.y; y/; b A1 W X  X ! Y  ; The proof is evident.

b A.y; / D b A1 .y; / C A2 .y/;   b A W D.A/  X  X 2 Y  :

1.2 Classes of Multivalued Maps

87

Remark 1.38. Above we assumed the space X is reflective. It is possible to get rid of this requirement by considering the operators A W D.A/  X ! Y  where X is an arbitrary Banach space. The results given earlier can be easily expanded on this case. An example of the operator of variational calculus on D.A/. Let ˝ be the restricted area in n-measured Euclidean space Rn with the boundary @˝, x D .x1 ; : : : ; xn / 2 ˝, ˛ D .˛1 ; : : : ; ˛n / be multi-index, ˛i  0,



X @ ˛1 @ ˛n ˛ j˛j D ˛i ; D y.x/ D  y.x/; @x1 @xn D k y D fD ˛ yj j˛j D kg: Let M; N indicate the number of various multi-indexes ˛; ˇ of the length m and not more than m1 respectively. Suppose that x 2 ˝, D f ˇ j jˇj  m1g 2 RN ,  D f˛ j j˛j  mg 2 RM the functions A˛ .x; ; /, .j˛j  m/ and Bˇ .x; /, .jˇj  m  1/ are defined and meet the following conditions: 1. A˛ .x; ; /, j˛j  m and Bˇ .x; /, jˇj  m  1 are measured with respect to x 2 ˝ as any 2 RN and  2 RM , they are continuous along  and at almost all x 2 ˝ and the following inequality is fulfilled   jA˛ .x; ; /j  C k kp1 C kkp1 C g.x/ ; where 1 < p < 1; C D const; g 2 Lq .˝/; p 1 C q 1 D 1; 2. For all 2 RN and a.a. x 2 ˝ X

  A˛ .x; ; 1 /  A˛ .x; ; 2 / 1˛  2˛ > 0

81 6D 2 I

j˛jDm

3. For all bounded 2 RN and a.a. x 2 ˝ X j˛jDm

A˛ .x; ; /

˛ ! C1 kk C kkp1

as kk ! 1I

4. A set \ ˚  y 2 Wpm .˝/ W Bˇ .x; y; Dy; : : : ; D m1 y/ 2 Lq .˝/ ; jˇ jm1

where Wpm .˝/ is a Sobolev space, it is not empty (see, for example, [Li69] where the cases are considered when Bˇ .x; / have a gradual growth as k k ! 1 and D.A/ is everywhere dense in X ).

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1 Preliminary Results

Thus, for y 2 D.A/ and  2 Wpm .˝/ there exists X Z a.y; / D A˛ .x; y; : : : ; D m y/D ˛ dxC j˛jm ˝

Z

X

C

Bˇ .x; y; : : : ; D m1 y/D ˇ dx:

jˇ jm1 ˝ ı

ı

Let Wpm .˝/  X  Wpm .˝/ where Wpm .˝/ is a completeness of infinitely differentiated finite functions of C01 .˝/ in Wpm .˝/. The form  ! a.y; / is linear and continuous on X and, therefore, a.y; / D hA.y/; iX , A.y/ 2 X  and on smooth functions the operator A W D.A/  X ! X  is determined by the identity X .1/j˛j D ˛ A˛ .x; y; : : : ; D m y/ A.y/ j˛jDm

X

C

.1/jˇ j D ˇ Bˇ .x; y; : : : ; D m1 y/:

jˇ jm1

When the conditions 1–4 are fulfilled A is the operator of variational calculus on D.A/. Assume for all y 2 D.A/ and ; ! 2 Wpm .˝/ X Z a1 .y; ; !/ D A˛ .x; y; : : : ; D m1 y; D m /D ˛ !dx j˛jm ˝

C

X

Z Aˇ .x; y; : : : ; D m y/D ˇ !dx;

jˇ jm1 ˝

a2 .y; !/ D

X

Z

Bˇ .x; y; : : : ; D m1 y/D ˇ !dx;

jˇ jm1 ˝

and the form ! ! a1 .y; ; !/ C a2 .y; !/ D a.y; ; !/ is continuous on X , thus A1 .y; /; !iX C hA2 .y/; !iX ; a.y; ; !/ D hb A.y; /; !iX D hb A.y; / D b A1 .y; / C A2 .y/ and b A.y; y/ D A.y/; where b A.y; / 2 X  ; b b A1 .y; y/ D A1 .y/. Because of the conditions 1–3 the operator A1 W X ! X  is the operator of variational calculus [Li69]. Let us now prove that for the operator A2 W D.A/  X ! X  the condition ( ) is true then we shall obtain the required result. Checking the condition “1 ”. Let D.A/ 3 yn ! y 2 D.A/ weakly in X then yn ! y strongly in Wpm1 .˝/, since the embedding Wpm .˝/  Wpm1 .˝/ is compact. Therefore, Bˇ .x; yn ; : : : ; D m1 yn / ! Bˇ .x; y, : : : ; D m1 y/ strongly in Lq .˝/. The last statement results from the theorem of M.A. Krasnoselsky. It means a2 .yn ; / ! a2 .y; / 8 2 X , therefore, A2 .yn / ! A2 .y/ weakly in X .

1.2 Classes of Multivalued Maps

89

Checking of the condition “2 ”. Let D.A/ 3 yn ! y 2 D.A/ weakly in X . Because of the above Bˇ .x; yn ; : : : ; D m1 yn / ! Bˇ .x; y; : : : ; D m1 y/ weakly in Lq .˝/, thus X Z Bˇ .x; yn ; : : : ; D m1 yn /D ˇ yn dx hA2 .yn /; yn iX D !

X

Z

jˇ jm1 ˝

Bˇ .x; y; : : : ; D m1 y/D ˇ ydx D hA2 .y/; yiX ;

jˇ jm1 ˝

since yn ! y strongly in Wpm1 .˝/. The space S.X I X  /. We will denote the family of all strict multivalued maps from X to X  by S.X I X  /. Let us introduce the following operations on the partially ordered by inclusion set S.X I X  /. I. For two arbitrary maps A; B 2 S.X I X  / there exists a single map C 2 S.X I X  / which is the sum of A and B (C D A C B), defined by the rule C.y/ D A.y/ C B.y/ D fa C b j a 2 A.y/; b 2 B.y/g; such that (a) (b) (c) (d)

A C B D B C A (commutative property); A C .B C C / D .A C B/ C C (associative property); There exists an element 0 2 S.X I X  / such that A C 0 D A 8A 2 S.X I X  /; For any A 2 S.X I X  / there exists an element .A/ 2 S.X I X  / such that A C .A/  0 and .A/ D A.

II. For any real number ˛ and for any element A 2 S.X I X  / an element ˛A 2 S.X I X  / (a distribution ˛ on A), defined by the rule .˛A/.y/ D ˛.A.y// D f˛a j a 2 A.y/g; such that (a) (b) (c) (d)

˛.ˇA/ D .˛ˇ/A; 1  A D A 8A 2 S.X I X  /; .˛ C ˇ/A  ˛A C ˇA; ˛.A C B/ D ˛A C ˛B.

Let us remark that the space S.X I X  / is not a linear space with respect to operations defined just now. Remark 1.39. Lemma 1.11 means that the family of all 0 -pseudomonotone multivalued maps which satisfy Property (˘ ) forms “a convex cone” in the space S.X I X  /. Definition 1.38. The multivalued map A W X  X  satisfies the uniform property ./C./ if for any bounded set D in X and for any c > 0 there exists c1 > 0 such that 1 c1 N kA.v/kC./  ŒA.v/; vC./ C kvkX 8v 2 D n f0g: c c

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1 Preliminary Results

Lemma 1.12. Let A W X1  X1 , B W X2  X2 be +-coercive maps which satisfies the uniform property ./C./ . Then the map C WD A C B W X  X  is uniformly +(–)-coercive. Proof. We prove this statement by contradiction. Let fxn gn1  X with xn ¤ 0N and kxn kX D kxn kX1 C kxn kX2 ! C1 and n ! C1. Taking into account that ŒC.xn /; xn C./  cC kC.xn /kC./ < C1; kxn kX n1 sup

(1.52)

where cC D minfcA ; cB g, cA ; cB > 0 are constants as in the conditions of uniform +(–)-coercivity of A and B respectively. Let A .r/ WD B .r/ WD

inf

ŒA.v/; vC./  cA kA.v/kC./ ; kvkX1

inf

ŒB.w/; wC./  cB kB.w/kC./ ; kwkX2

kvkX1 Dr

kwkX2 Dr

r > 0;

we remark that A .r/ ! C1; B .r/ ! C1 as r ! C1. In the case when kxn kX1 ! C1 as n ! C1 and kxn kX2  c 8n  1 we obtain ŒA.xn /; xn C./  cA kA.xn /kC./ kxn kX kxn kX1  A .kxn kX1 / ! C1 as n ! C1 kxn kX and moreover ŒB.xn /; xn C./  cB kB.w/kC./ kxn kX2  c1 !0 kxn kX kxn kX

as C n ! 1;

where c1 2 R is a constant from the uniform condition ./C./ with D D fy 2 X2 j kykX2  cg;

c D cB :

Therefore, ŒA.xn /; xn C./  cA kA.xn /kC./ ŒC.xn /; xn C./  cC kC.xn /kC./  kxn kX kxn kX ŒB.xn /; xn C./  cB kB.xn /kC./ C ! C1 as n ! C1; kxn kX and this is a contradiction to (1.52).

1.2 Classes of Multivalued Maps

91

If kxn kX1  c 8n  1 and kxn kX2 ! C1 as n ! C1 argumentation will be the same. When kxn kX1 ! C1 and kxn kX2 ! C1 as n ! C1 we obtain the contradiction ŒC.xn /; xn C./  cC kC.xn /kC./ kxn kX n1

C1 > sup

kxn kX1 kxn kX2  A .kxn kX1 / C B .kxn kX2 / kxn kX1 C kxn kX2 kxn kX1 C kxn kX2 ˚   min A .kxn kX1 /; B .kxn kX2 / ! C1: The Lemma is proved. The following Lemma is Weierstrass generalized Theorem [Va73].  Lemma 1.13. Let X be S a Banach space, K  X be a weakly star closed set and  L W X ! R D R fC1g be a weakly star lower semicontinuous functional. And either the set K is bounded or

lim

kvkX  !1

L.v/ D C1:

Then the functional L is bounded below on K, reaches on K its minimal value m and the set E D fv 2 K W L.v/ D mg is weakly star compact in X  . Proof. The proof is similar to the one of Theorem 9.3 from [Va73]. Proposition 1.75. Let A W X  X  be a bounded-valued -pseudomonotone operator. If yn ! y weakly in X and the inequality (1.11) holds true then, up to a 

subsequences fym g; fdm g, for each v 2 X there exists .v/ 2 co A.y/ for which lim hdm ; ym  viX  h .v/; y  viX :

(1.53)

m!1



Proof. Let yn ! y weakly in X , dn 2 co A.yn / and let (1.11) hold true. Then up to a subsequences,   lim hdm ; ym  viX  coA.y/; y  v ; 8 v 2 X:

m!1





The set co A.y/ is weakly star closed and bounded. Also the functional X  3 w 7! hw; y  viX ;

8 v 2 X;

(1.54)

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1 Preliminary Results

is weakly star lower semicontinuous. Then due to Lemma 1.13 there exists .v/ 2 

co A.y/ such that ŒA.y/; y  v D h .v/; y  viX : From the inequality (1.54) we obtain (1.53). Definition 1.39. A multivalued map A W X  X  satisfies Property .M / if from fyn gn0  X , dn 2 A.yn /; 8n  1, and yn * y0 2 X;

dn * d0 2 X  ; lim hdn ; yn iX  hd0 ; y0 iX ; n!1

the inclusion d0 2 A.y0 / follows. The next propositions concern 0 -pseudomonotone demiclosed maps disturbed by 0 -pseudomonotone maps. Such results allows us (by the help of corresponding theorems about properties of the resolving operator) to consider new classes of problems. So, we can state the existence of solutions and continuous (in some sense) dependence of solutions on functional parameters of the given problem. Proposition 1.76. Let A W X  X  be a 0 -pseudomonotone operator and let the map B W X  X  satisfy the following properties: 

1. The map co B W X  X  is compact, namely the images of bounded sets in X are precompact in X  ; 

2. The graph co B is closed in Xw  X  (namely with respect to the weak topology in X and the strong one in X  ). Then the map C D A C B is 0 -pseudomonotone. 

Proof. Let yn ! y weakly in X dn 2 co C.yn /, dn ! d weakly star in X  and lim hdn ; yn  yiX  0:

n!1







Since the operator B W X  X  is bounded then co C D co A C co B. Therefore 0



dn D dn0 C dn00 , dn0 2 co A.yn /, dn0 2 co B.yn /. In virtue of boundness of B we obtain that dn00 ! d 00 weakly star in X  namely dn0 ! d 0 D d  d 00 weakly star in X  . From the inequality (1.11) up to a subsequence fym g  fyn g we find ˝ ˝ ˛ ˛ 0  lim hdn ; yn  yiX  lim dn0 ; yn  y X C lim dn00 ; yn  y X n!1

n!1

˝

0  lim dm ; ym  y m!1

˛ X

n!1

˝

00 C lim dm ; ym  y m!1

˛ X

:



(1.55)

Since co B is a compact and its graph is closed in Xw  X  , we can assume that 

00 dm ! d 00 strongly in X  and moreover d 00 2 co B.y/. Then

1.2 Classes of Multivalued Maps

93

˝ 0 ˛ lim dm ; ym  y X  0:

m!1

Again up to a subsequence, since A is 0 -pseudomonotone we obtain ˝ 0 ˛ ; ym  v X  ŒA.y/; y  v ; 8v 2 X; lim dm

m!1

and then ˝ 0 ˝ 00 ˛ ˛ ; ym  v X C lim dm ; ym  v X lim hdm ; ym  viX D lim dm m!1 m!1 m!1   ˝ ˛  co A.y/; y  v C d 00 ; y  v X    co C.y/; y  v  ; 8v 2 X: The Proposition is proved. Proposition 1.77. Let A W X  X  be a 0 -pseudomonotone operator and let the 

embedding of X in some Banach space Y is compact and dense, and let co B W Y  

Y  be a locally bounded map such that the graph co B is closed in Y  Yw (with respect to the weak topology in Y and the weak star one in Y  ). Then C D A C B is a 0 -pseudomonotone map. 

Proof. Let (1.11) be valid. The operator co B is locally bounded namely for any y 2 X there exist N > 0 and " > 0 such that 

k co B./kC  N; if k  ykX  ": 

It is clear that every locally bounded operator is boundedvalued. Therefore, 





co C.y/ D co A.y/ C co B.y/ and dn D dn0 C dn00 , dn0 2 co A.yn /, dn00 2 co B.yn /. Due to compactness of the embedding X  Y we have that yn ! y strongly in Y 

and due to local boundness of co B a sequence fdn00 g is bounded in Y  (and in X  ) 00 00 g  fdn00 g such that dm ! d 00 weakly and hence the existence of a subsequence fdm     star in Y follows. The embedding operator I W Y ! X is continuous, hence I  00 remains continuous in weak star topologies [ReSi80]. Therefore dm ! d 00 weakly  0 00 0 00 star in X ; namely dm D dm  dm ! d D d  d weakly star in X  . Hence 00 ; ym  yiX ! 0: hdm 0 Then from (1.55) we obtain lim hdm ; ym  viX  0 whence up to a subsequence m!1

lim

mk !1

˝

˛   0 dm ; ymk  v X  co A.y; y  v/  ; 8 v 2 X: k

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1 Preliminary Results 



Further since the operator co B is closed in Y  Yw , we have that d 00 2 co B.y/ and lim

mk !1

˝ ˝ 0 ˝ 00 ˛ ˛ ˛ dmk ; ymk  v X D lim dm ; ymk  v X C lim dm ; ymk  v X k k mk !1

h

mk !1

i

   co A.y/; y  v C co B.y/; y  v   h  i D co C.y/; y  v ; 8 v 2 X: 



The Proposition is proved. Proposition 1.78. Let A W X  X  be an upper hemicontinuous operator with respect to the strong topology in X and the weak star topology in X  . Then A is radial continuous. Proof. It is well-known that A is u.h.c. [AuEk84], namely from xn ! x strongly in X it follows that lim ŒA.xn /; vC  ŒA.x/; vC ; 8 v 2 X:

n!1

We remark that u.h.c. operator is r.u.s.c. Hence its radial continuity easily follows. The Proposition is proved. Now let X be reflexive, Y be either reflexive or separable normalized space, Y  is an adjoint space, U is some nonempty set from Y  . Definition 1.40. The multivalued map A W X U  X  is called -quasimonotone, if for any fyn ; un gn1  X  U and y0 2 X , u0 2 U from yn ! y0 weakly in X , un ! u0 weakly star in Y  as n ! C1 and lim hdn ; yn  y0 iX  0;

n!1

where dn 2 coA.yn ; un /, 8n  1, the existence of subsequences fynk ; unk gk1 from fyn ; un gn1 such that lim hdnk ; ynk  wiX  ŒA.y0 ; u0 /; y0  w

8w 2 X

k!1

follow. Definition 1.41. The multivalued map A W X  U  X  satisfies Property SNk , if from yn ! y0 weakly in X , U 3 un ! u0 2 U weakly star in Y  , dn ! d weakly in X  (dn 2 coA.yn ; un / 8n  1) and hdn ; yn  y0 iX ! 0 it follows that d 2 coA.y0 ; u0 /.

as n ! 1

(1.56)

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95

Remark 1.40. If A W X U  X  satisfies Property SNk , then .A/ W X U  X  satisfies the same condition. The next proposition properly orders classes of maps of Sk and -quasimonotone types. Proposition 1.79. -quasimonotone multivalued map satisfies Property SNk . Proof. We consider U 3 un ! u0 2 U weakly star in Y  , yn ! y0 weakly in X , dn ! d weakly in X  (dn 2 coA.yn ; un /; 8n  1) and let (1.56) is true. Then lim hdn ; yn i D hd; y0 iX . Let us prove, that d 2 coA.y0 ; u0 /. In virtue of n!1

quasimonotony of A and (1.56), up to a subsequence fym ; um gm1 from fyn ; un gn1 it follows that lim hdm ; ym  wiX  ŒA.y0 ; u0 /; y0  w

8w 2 X:

m!1

So 8w 2 X ŒA.y0 ; u0 /; y0  w  lim hdm ; ym iX  lim hdm ; wiX D hd; y0  wiX : m!1

m!1

Due to Proposition 1.38 we obtain d 2 coA.y0 ; u0 /. The Proposition is proved. Lemma 1.14. Let A W X  U  X  and B W X  U  X  are -quasimonotone. Then C WD A C B W X  U  X  is -quasimonotone too. The proof is similar to the proof of Proposition 1.11. Definition 1.42. The mapping A W X  U  X  is demiclosed, if from yn ! y0 strongly in X , U 3 un ! u0 2 U weakly star in Y  , dn ! d weakly in X  (dn 2 coA.yn ; un / 8n  1) it follows that d 2 coA.y0 ; u0 /. Proposition 1.80. The mapping, that satisfies Property Sk is demiclosed. For any mapping A W X  X  we consider A W X  U  X  , A.y; u/ D A.y/, y 2 X, u 2 U . Definition 1.43. The multivalued map A W X  X  satisfies the property (Sk ), if the corresponding map A W X  U  X  satisfies Property (Sk ).

1.3 Subdifferentials in Infinite-Dimensional Spaces Subdifferential maps play an important role in the nonsmooth analysis and the optimization theory [Ps80, AuEk84, DeVa81], in nonlinear boundary value problems for partial differential equations, the theory of control of the distributed systems

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1 Preliminary Results

[ZgMe99, Li69], as well as the theory of differential games and mathematical economy [AuFr90, Ch97]. For basic properties of such maps we refer the reader to [AuEk84, DeVa81, IoTi79]. Here we will generalize basic properties of subdifferentials and local subdifferentials known for Banach spaces to the case of Frechet spaces. Let X be a Banach space, X  be its topologically dual, hf; xiX be pairing of the elements f 2 X  , x 2 X , U be a convex subset in X , F W X 7! R D R [ fC1g be a functional domF D fx 2 X jF .x/ ¤ C1g: The set @F .x0 I U / D fp 2 X  j hp; x  x0 iX  F .x/  F .x0 / 8x 2 U g refers to a local subdifferential of a functional F in a point x0 2 U . Some properties of locally subdifferential maps @F .I U / W U  X  in Banach spaces are investigated in [ZgMeNo04], and in finite-dimensional spaces were studied in the work [DeVa81] (in [DeVa81] local subdifferentials refer to conditional). Here it is shown, that, generally speaking, locally subdifferential maps are not bounded (Fig. 1.17). In our book the basic properties of subdifferential and locally subdifferential maps are generalized on Frechet spaces. Some of the outcomes, obtained here, are new even for the case of Banach spaces. Let X be a Frechet space, X  its topologically dual (adjoint) space. For x 2 X and f 2 X  , as usual, the symbol hf; xi stands for the bilinear pairing between X and X  . Assume that X is endowed with the topology generated by a family of seminorms f i g1 i D1 separating points of X . Recall that the topology is Hausdorff and metrizable by the metric d.x; y/ D

1 X i D1

2i

i .x  y/ : 1 C i .x  y/

(1.57)

Observe that d.x C h; y C h/ D d.x; y/, d.˛x; ˛y/ < j˛jd.x; y/ if j˛j > 1 and d.˛x; ˛y/  j˛jd.x; y/ if j˛j  1.

Fig. 1.17 Locally subdifferentiable nonconvex functional

1.3 Subdifferentials in Infinite-Dimensional Spaces

97

Let Y be a locally convex linear space and T W Y ! X be a linear continuous map. Recall that the adjoint (dual) transformation T  W X  ! Y  is given by a formula hx  ; T yi D hT  x  ; yi for y 2 Y , x  2 X  . For the existence and uniqueness of such transformation, see [Ru73]. Throughout the section, F stands for the functional F W X ! R [ fC1g and the symbol dom F denotes the set fx 2 X j F .x/ < C1g. Given a functional F and a convex body U such that intU  domF , a local subdifferential of F at the point x0 2 U \ domF is, by definition, the set @F .x0 I U / D f 2 X  j h; x  x0 iX  F .x/  F .x0 / for all x 2 U g Observe that @F .x0 I U1 /  @F .x0 I U2 /, if U1  U2 . In particular, @F .x0 I X / D @F .x0 /  @F .x0 I U /. The last set is called the subdifferential of F at the point x0 . Definition 1.44. The functional F W X 7! R refers to lower semicontinuous (l.s.c.) at the point x0 2 X , if for any sequence fxn gn1  dom F : xn ! x0 in X as n ! C1 lim F .xn /  F .x0 /: (1.58) n!C1

Definition 1.45. The functional F W X 7! R refers to lower semicontinuous on a set A  X , if 8x0 2 A F – l.s.c. at x0 . Definition 1.46. The functional F W X 7! R refers to upper semicontinuous (u.s.c.) at the point x0 2 domF , if for any sequence fxn gn1  dom F : xn ! x0 in X as n ! C1 lim F .xn /  F .x0 /: (1.59) n!C1

Definition 1.47. The functional F W X 7! R refers to upper semicontinuous on a set A  domF , if 8x0 2 A F – u.s.c. at x0 . Proposition 1.81. The functional F W X 7! R is l.s.c. on X if and only if 8 2 R the set (1.60) E D fx 2 dom F j F .x/  g is closed. Definition 1.48. The functional F W X 7! R refers to a convex on a convex set A  X , if 8x; y 2 A;

8˛ 2 .0; 1/

F .˛x C .1  ˛/y/  ˛F .x/ C .1  ˛/F .y/:

Remark 1.41. Further we assume that for each a 2 R and  > 0 (Fig. 1.18)   .C1/ D .C1/   D a C .C1/ D .C1/ C a D C1: Proposition 1.82. [AuEk84, p.191] Let X be a Banach space. Then the norm k  kX in X is subdifferentiable functional and

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1 Preliminary Results

Fig. 1.18 The local subdifferential map for F .x/ D j jxj  1j as x 2 U D RC

@k  kX .x/ D f p 2 X  j hp; xiX D kxkX ; kpkX  D 1 g 8x 2 X: In the case, when ˛ > 1 @

1

  k˛X .x/ D k  k˛1 @k  kX .x/ ˛1 g D f p 2 X  j hp; xiX D kxk˛X ; kpkX  D kxkX

˛k

8x 2 X:

Let ' W X ! R be a functional with dom' ¤ ;. Definition 1.49. The functional '  W X  ! R, defined by the rule   '  .p/ D sup hp; xiX  '.x/

8p 2 X  ;

x2X

is called adjoint functional to '. Similarly, if W X  ! R is a functional with dom defined by the rule 

 .x/ D sup hp; xiX  p2X 





W X ! R,

8x 2 X;

. The second adjoint function '  to ' is defined

is called adjoint functional to via formula '  D .'  / .

W X ! R be l.s.c., positively homogeneous convex

Proposition 1.83. Let function X . Then

ŒK; xC D where

.p/

¤ ;, then

.x/

K D fp 2 X  j hp; xiX 

Moreover, @ .x/ D K;

8x 2 X; .x/

8x 2 X g ¤ ;:

N if x D 0:

Remark 1.42. In virtue of [Ru73, Theorem 3.2, p.68] the set K is nonempty. Proof. From positive homogeneity for 

 .p/ D

we have that

N if p 2 K; 0; C1; if p 62 K

8p 2 X  :

(1.61)

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99

Fig. 1.19 The local subdifferential map for F .x/ D j jxj  1j as x 2 U D RC

Respectively 

 .x/ D sup hp; xiX 



p2X 

 .p/ D sup hp; xiX D ŒK; xC

8x 2 X:

p2K

Due to [AuEk84, Theorem 2, p.200] from convexity and from l.s.c. of on X it follows that  D . The formula (1.61) is the direct corollary of the definition N D 0 (Fig. 1.19). for K and .0/ The Proposition is proved. Definition 1.50. The functional F W X 7! R refers to weakly lower semicontinuous (w.l.s.c.) at the point x0 2 X , if for any net fx˛ g˛2I  dom F such that x˛ ! x0 in .X I X  / (1.62) lim F .x˛ /  F .x0 /; ˛

where lim F .x˛ / D inffb 2 R [ f˙1g j ˛

j b  is a cluster point for net fx˛ g˛2I in .X I X  /g: Definition 1.51. The functional F W X 7! R refers to w.l.s.c. on a set A  X , if 8x0 2 A F is w.l.s.c. at x0 . Proposition 1.84. The functional F W X 7! R is w.l.s.c. on X if and only if 8 2 R the set E , defined in (1.60), is closed in the .X I X  /-topology of the space X . Corollary 1.10. Let X be a Frechet space. Then every convex l.s.c. functional F W X ! R on X is w.l.s.c. on X . Proof. Let F W X ! R satisfies the latter conditions. Then due to Proposition 1.84 for an arbitrary fixed  2 R the set E , defined in (1.60) is closed in X . In virtue of the convexity for F , the given set is convex, so, due to [Ru73, p.81], it is weakly convex. Thus, from Proposition 1.84 w.l.s.c. for functional F on X follows. Theorem 1.8. Let a functional F W X ! R [ fC1g be given. Assume that there are a convex body U and a point x0 2 intU such that @F .x0 I U / ¤ ;. Then the functional F is weakly lower semicontinuous at x0 . Moreover, if @F .x0 I U / ¤ ; for every x 2 U , then F is convex on U .

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1 Preliminary Results

Proof. Let fx˛ g be a net converging to fx0 g and W  U be a neighborhood of fx0 g. Obviously there exists ˛0 such that x˛ 2 W for ˛ ˛0 . Let x  2 @F .x0 ; U /. For ˛ ˛0 , there is hx  ; x˛  x0 i  F .x˛ /  F .x0 /. Passing with x˛ to x0 , we deduce that lim F .x˛ /  F .x0 /. ˛

Now suppose that @F .x0 I U / ¤ ; for an arbitrary x0 2 U . Fix x0 2 U . For x  2 @F .x0 I U / and x1 ; x2 2 U , there is F .x1 /  F .x0 /  hx  ; x1  x0 i; and F .x2 /  F .x0 /  hx  ; x2  x0 i for all x1 ; x2 2 U . Let t 2 Œ0; 1. Adding the first inequality multiplied by t to the second one multiplied by 1  t, we obtain tF .x1 / C .1  t/F .x2 /  F .x0 / C hx  ; tx1 C .1  t/x2  x0 i: Since U is a convex set we can take x0 D tx1 C .1  t/x2 . The Theorem is proved. Theorem 1.9. Let X be a Frechet space, Y a locally convex linear space, F W X ! R [ fC1g and T W Y ! X a linear continuous map admitting an adjoint map T  . Let U  X be a convex body and V D T 1 .U /. Then for every T v 2 intU , where v 2 V , there is @.F ı T /.vI V / D T  .@F .T vI U // Proof. Let x 2 @F .T vI U /: Obviously, hx  ; x  T vi  F .x/  F .T .v// for every x 2 U: Taking x D T .y/ with y 2 V , we can rewrite the last inequality in the form hx  ; T y  T vi  F .T .y//  F .T .v// for every y 2 V or

hT  x  ; y  vi  .F ı T /.y/  .F ı T /.v/

for every y 2 V;

which means that T  x  2 @.F ıT /.vI V /: Thus @.F ıT /.vI V /T  .@F .T .v/I V //. To prove the inverse inclusion, take y  2 @.F ı T /.vI V /: Clearly hy  v; y  i  .F ı T /.y/  .F ı T /.v/

for every y 2 V:

Taking x  2 X  such that y  D T  x  , we can rewrite the last inequality in the form hx  ; T y  T vi  F .T .y//  F .T .v// for every y 2 V or

hx  ; x  T vi  F .x/  F .T .v// for every x 2 U;

which means that y  D T  x  2 T  .@F .T .vI V /// and this completes the proof.

1.3 Subdifferentials in Infinite-Dimensional Spaces

101

Theorem 1.10. Let U be a convex body in X , F W X 7! R [ fC1g be a convex functional on U and a lower semicontinuous functional on intU (intU  domF). Then for every x0 2 intU and every h 2 X , the quantity F .x0 C th/  F .x0 / t !0C t

DC F .x0 I h/ D lim

(1.63)

is finite and the following statements hold true: (i) There exists a counterbalanced (cf. [Ru73]) convex absorbing neighborhood of zero  (x0 C   intU ) such that for every h 2  F .x0 /  F .x0  h/  DC F .x0 I h/  F .x0 C h/  F .x0 /I

(1.64)

(ii) The functional intU  X 3 .xI h/ 7! DC F .xI h/ is upper semicontinuous; (iii) The functional DC F .x0 I / W X 7! R is positively homogeneous and semiadditive for every x0 2 intU ; (iv) There exist a neighborhood O.h0 / and a constant c1 > 0 such that for every x0 2 intU and every h0 2 X , jDC F .x0 I h/  DC F .x0 I h0 /j  c1 d.h; h0 /

for every h 2 O.h0 /:

Proof. First we introduce some auxiliary statements. Lemma 1.15. The functional F is locally upper bounded on intU , that is for every x0 2 intU there exist positive constants r and c such that F .x/  c; for each x 2 Br .x0 /, where Br .x0 / D fx 2 X j d.x; x0 / < r g. Proof. For arbitrary x0 2 intU there exists "1 > 0 such that B2"1 .x0 /  intU  domF , hence B"1 .x0 /  B2"1 .x0 /  domF . Since F is lower semicontinuous, than for each n D 1; 2; : : : the set An D fx 2 B"1 .x0 / jF .x/  ng is closed in X and

C1 [

An D B"1 .x0 /  dom F:

nD1

Since the metric space .B"1 .x0 /; d / is complete, due to the Baire Category Theorem, there exists n0 2 N such that intAn0 ¤ ; in B"1 .x0 /. We now prove that intAn0 ¤ ; in X . Since intAn0 ¤ ; in B"1 .x0 /, we conclude that there exist x1 2 intAn0 and "2 > 0 such that the following equality holds true: An0  fx 2 B"1 .x0 / j d.x; x1 / < "2 g D B"1 .x0 / \ B"2 .x1 / ¤ ;:

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1 Preliminary Results

Thus the following two cases are possible: (1) B"1 .x0 / \ B"2 .x1 / ¤ ;; (2) B"1 .x0 / \ B"2 .x1 / D ;; @B"1 .x0 / \ B"2 .x1 / ¤ ;. In the first case, the set B"1 .x0 / \ B"2 .x1 / is open in topology ; therefore, there exist x2 2 X and "3 > 0 such that B"3 .x2 /  B"1 .x0 / \ B"2 .x1 /  An0 . Thus, for each x 2 B"3 .x2 /, there is F .x/  n0 . Hence x2 2 intAn0 in X . In the second case, for an arbitrary x 2 @B"1 .x0 / \ B"2 .x1 /, there exists fxn gn1  B"1 .x0 / such that xn ! x as n ! C1. Since x 2 B"2 .x1 /, then there exists N such that for each n  N , xn 2 B"2 .x1 /. Therefore, xn 2 B"1 .x0 / \ B"2 .x1 / ¤ ;, and we may proceed further as in the first case. Thus intAn0 ¤ ; in X . Now we show that the functional F is upper bounded in some neighborhood "1 =d.x2 ;x0 / 0 x2 , where  D 1C" . Therefore, of x0 . Let x2 ¤ x0 , y D x2 C x1 1 =d.x2 ;x0 / "1 .x0  x2 /; d.y; x0 / D d d.x2 ; x0 / "1 d.x0 ; x2 / D "1 ; < d.x0 ; x2 /

y D x0 C



"1 .x0  x2 /; 0 d.x0 ; x2 /



that is y 2 B"1 .x0 /  domF . For an arbitrary x 2 B"3 .x0 /, we consider z D .x C x2  x0 /= D .x  .1  /y/=. Since 0 <  < 1, we conclude that  1  x  x  x  x0 "3 0 d.z; x2 / D d x2 C ; x2 D d ; 0 < d.x; x0 / < D "3 ;     hence z 2 B"3 .x2 /, and F .z/  n0 . Due to convexity of F , there is F .x/ D F .z C .1  /y/  F .z/ C .1  /F .y/  n0 C .1  /F .y/: From this we conclude that F is upper bounded in the neighborhood Br .x0 / with r D "3 and c D n0 C .1  /F .y/. Lemma 1.16. The functional F is locally Lipschitzean on intU , i.e., for every x0 2 intU there exist r1 > 0 and c1 > 0 such that jF .x/  F .y/j  c1 d.x; y/

for all x; y 2 Br1 .x0 /:

Proof. The local upper boundness of the functional F on intU follows from Lemma 1.15. Therefore, for every x0 2 intU there exist r > 0 and c > 0 such that F .x/  c for every x 2 Br .x0 /. d.x;x0 / , we put For an arbitrary x 2 Br .x0 / .x ¤ x0 / and t D rCd.x;x 0/ yD

1t x0 C .t  1/x D x0 C .x0  x/; t t

1.3 Subdifferentials in Infinite-Dimensional Spaces

103

where t 2 .0; 1/. Then

1t .x0  x/; 0 t

r r .x0  x/; 0 < d.x; x0 / D r; Dd d.x; x0 / d.x; x0 /

d.y; x0 / D d

i.e., F .y/  c. Due to convexity of F , F .x0 / D F .ty C .1  t/x/  tF .y/ C .1  t/F .x/  tc C .1  t/F .x/; or .1  t/F .x0 /  t.c  F .x0 // C .1  t/F .x/. Hence F .x0 /  F .x/  Now let z D

x.1/x0 

t .c  F .x0 // .c  F .x0 // D d.x; x0 /: 1t r

D x0 C

d.z; x0 / D d

xx0 , 

where D

x  x

0



d.x;x0 / r

(1.65)

2 .0; 1/. Then

 1 ; 0 < d.x; x0 / D r;

i.e., F .z/  c, and since F is convex, we obtain F .x/ D F . z C .1  /x0 /  F .z/ C .1  /F .x0 /  c C .1  /F .x0 / or

c  F .x0 / d.x; x0 /: r Relations (1.65) and (1.66) imply the following estimate F .x/  F .x0 /  .c  F .x0 // D

jF .x/  F .x0 /j 

c  F .x0 / d.x; x0 /: r

(1.66)

(1.67)

Now we show that the Lipschitz condition holds true for F on B"1 .x0 / with "1 D r=3. Hence in view of (1.67), for all x1 ; x2 2 B3"1 .x0 / F .x1 /  c, F .x2 /  c. If x1 2 B"1 .x0 /, then B2"1 .x1 /  Br .x0 /, that is x1 2 intU . Therefore, from (1.67) we obtain 8x 2 B2"1 .x1 /

jF .x/  F .x1 /j 

c  F .x1 / d.x; x1 /: 2"1

(1.68)

In particular, inequality (1.68) is valid for an arbitrary element of B"1 .x0 /. Further, due to (1.67) c  F .x1 /  .c  F .x0 // C jF .x1 /  F .x0 /j c  F .x0 / d.x1 ; x0 / < 2.c  F .x0 //:  .c  F .x0 // C r

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1 Preliminary Results

From the last relation, using (1.68) we finally obtain jF .x2 /  F .x1 /j 

c  F .x0 / d.x2 ; x1 / "1

for all x1 ; x2 2 B"1 .x0 /;

i.e., c1 D cF"1.x0 / , r1 D "1 . The Lemma is proved. Now we continue to prove Theorem. Let x0 2 intU and Br .x0 / D x0 C Br .0/. Then due to Lemmas 1.15 and 1.16 the upper boundness and the Lipschitz condition for F on Br .x0 / follow. We recall that, unlike in the case of a Banach space, Br .0/ is not absolutely convex, but at the same time there exists a convex absorbing counterbalanced set  D .x0 / in a basis of topology , such that   Br .0/. Then F .x/  c; for every x 2 x0 C ; jF .x1 /  F .x2 /j  c1 d.x1 ; x2 /

for all x1 ; x2 2 x0 C :

(1.69)

For each u 2 X there exists t D t.u/ > 0 such that t 1 u 2  (if u 2  then t D 1). So for each 2 .0; t 1  the element u 2 , as t  1 . Further, due to convexity of F , for every 1 ; 2 2 R such that 0 < 1  2  t 1 , there follows:





1

1 C .x0 C 2 u/  F .x0 / F .x0 C 1 u/  F .x0 / D F x0 1 

2

2

1

1  1 F .x0 / C F .x0 C 2 u/  F .x0 /

2

2

1 D .F .x0 C 2 u/  F .x0 //:

2 .x0 / monotonely decreases as ! 0C. Hence the function 7! F .x0 Cu/F  For each u 2 , the quantity DC F .x0 I u/ is finite. In fact, ˛u 2  for every ˛ such that j˛j  1, therefore

DC F .x0 I u/ D inf

>0

F .x0 C u/  F .x0 /

 F .x0 C u/  F .x0 / < C1 as x0 C u 2 Br .x0 /  intU  dom F: On the other hand, for every 2 .0; 1/ x0 D u 2 , and moreover, for each 2 .0; 1/, 1 < F .x0 /  F .x0  u/ 

1 1C .x0

C u/ C

 1C .x0

 u/, i.e.,

F .x0 C u/  F .x0 / :

Thus, for each u 2 , 1 < F .x0 /  F .x0  u/  DC F .x0 I u/  F .x0 C u/  F .x0 / < C1;

1.3 Subdifferentials in Infinite-Dimensional Spaces

105

i.e., DC F .x0 I u/ 2 R for every u 2 . The validity of (1.64) follows from these facts. From (1.63) we immediately obtain DC F .x0 I ˛u/ D ˛DC F .x0 I u/

for all ˛ > 0 and u 2 X;

(1.70)

and since the set  is absorbing, then for each u 2 X there is ˛ > 0 such that ˛u 2 . Then from (1.70) we obtain DC F .x0 I u/ 2 R

for all x0 2 intU and u 2 X:

Now taking t > 0, we consider the function intU   3 .x; h/ 7! Ft .xI h/ D

F .x C th/  F .x/ : t

(1.71)

Lemma 1.17. For each pair .x0 I h0 / 2 intU  X there exists l > 0 such that for each t 2 .0; l/ the function Ft . I / is continuous at the point .x0 I h0 /. Proof. Let h0 2 X be arbitrary, x0 2 intU (the set  D .x0 / is defined above), then there exists t0 > 0 such that h0 2 t0 . For t 2 .0; l/, putting l D min. 2t10 ; 1/, we consider the function Ft .xI h/ in a neighborhood of the point .x0 ; h0 / W jFt .xI h/  Ft .x0 I h0 /j D

1 jŒF .x0 C th0 C .x  x0 / C t.h  h0 // t (1.72) F .x0 C th0 / C ŒF .x0 /  F .x/j:

If we take x 2 x0 C 14 , h 2 h0 C 14 , then 1 x0 C th0 2 x0 C   x0 C ; 2

t.h  h0 / 2

1 ; 4

1 ; x0 C th0 C .x  x0 / C t.h  h0 / 2 x0 C : 2 From (1.72) using (1.69), we derive .x  x0 / C t.h  h0 / 2

jFt .xI h/  Ft .x0 I h0 /j 

c1 .d.x C th; x0 C th0 / C d.x; x0 // ! 0 t

as x ! x0 , h ! h0 . The Lemma is proved. Lemma 1.17 implies the upper semicontinuity of the map intU  X 3 .xI h/ 7! DC F .xI h/ D inf Ft .xI h/ D inf Ft .xI h/; t >0

t 2.0;l/

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1 Preliminary Results

since it is a “pointwise infimum” of continuous functions. The positive homogeneity of DC F .x0 I / is obvious. Now we show that this map is semiadditive. Indeed, for all v1 ; v2 2 X F .x0 C t.v1 C v2 //  F .x0 / t v1 x0 Ct v2 2F . x0 Ct C /  2F .x0 / 2 2 D lim t !0C t F .x0 C tv1 /  F .x0 / F .x0 C tv2 /F .x0 / C lim  lim t !0C t !0C t t D DC F .x0 I v1 / C DC F .x0 I v2 /:

DC F .x0 I v1 C v2 / D inf

t >0

In order to complete the proof it suffices to show that the map DC F .x0 I / satisfies (iv). From semiadditivity it follows that jDC F .x0 I h/  DC F .x0 I h0 /j  maxfDC F .x0 I h  h0 /; DC F .x0 I h0  h/g  c1 d.h; h0 / for any h 2 h0 C 14 . The Theorem is proved. Definition 1.52. We call a set B  X  bounded in the .X  I X / topology (*-bounded), if sup jhy; xiX j < C1 for each x 2 X . y2B

It is obvious that each bounded set in X  is *-bounded. Definition 1.53. A multivalued map A W X  X  is called: (a) *-bounded, if for any bounded set B in X the image A.B/ is *-bounded in X  ; (b) *-upper semicontinuous, if for any set B open in the .X  ; X / topology the set A1 M .B/ D fx 2 X j A.x/  Bg is open in X ; (c) upper hemicontinuous, if the function X 3 x 7! ŒA.x/; yC D sup hd; yiX d 2A.x/

is upper semicontinuous for each y 2 X . Let us note that (c) follows from (b). Theorem 1.11. Let U be a convex body and intU  domF , where F W X ! R is a convex functional on U and a semicontinuous function on intU . Then (i) @F .xI U / is a nonempty convex compact set for every x 2 intU in the .X  I X / topology; (ii) @F .I U / W U  X  is a monotone map (on U );

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107

(iii) The map intU 3 x 7! @'.xI U /  X  is *-upper semicontinuous (on intU ) and Œ@'.x0 I U /; hC D DC '.x0 I h/

for all h 2 X and x0 2 intU:

(1.73)

Proof. First we prove condition (ii). Let x1 ; x2 2 U and i 2 @F .xi I U /, i D 1; 2. Then F .x2 /  F .x1 /  h1 ; x2  x1 iX ; F .x1 /  F .x2 /  h2 ; x1  x2 iX : Adding the first inequality to the second, we obtain h1  2 ; x1  x2 iX  0; or Œ@F .x1 I U /; x1  x2   Œ@F .x1 I U /; x1  x2 C

for all x1 ; x2 2 U:

The last relation proves the monotonicity on U . Convexity and weak star closure are obvious. Let us prove nonemptiness. Let us set an arbitrary x; h 2 intU and consider the real convex function '.t/ D F .x C t.h  x// defined on Œ0; 1. So there exist '.t/; '.tC/ such that '.t/  '.tC/ D lim

t !C0

'.t/  '.0/ D DC '.xI x  h/; t

or D '.xI x  h/  DC '.xI x  h/; where D '.xI v/ D DC '.xI v/. From Theorem 1.10 it follows that '.˛/  '.0/  '.1/  '.0/ 8˛ 2 .0; 1/ ˛ or DC F .xI h  x/  F .h/  F .x/

for all x; h 2 intU:

(1.74)

Lemma 1.18. For arbitrary x 2 intU there exists .x/ 2 X  such that D F .xI h/  h.x/; hiX  DC F .xI h/

for every h 2 X:

Proof. Let us fix h0 2 X and consider the one-dimensional subspace X0 D f˛h0 j ˛ 2 Rg. Let us choose an element  2 X  satisfying the following condition h; ˛h0 iX D DC F .x; ˛h0 /;

˛0

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(we remark that since x is an interior point of U , then due to Theorem 1.10 for every h 2 X there exists DC F .xI h/). It is possible to choose  in such a way, since X 3 h 7! DC F .xI h/ is a positively homogeneous functional. Further, taking into account the semiadditivity of X 3 h 7! DC F .xI h/, we obtain 0 D DC F .xI h  h/  DC F .xI h/ C DC F .xI h/ or  DC F .xI h/  DC F .xI h/:

(1.75)

Then for ˛ < 0, from (1.75), the following relation follows: h; ˛h0 iX D ˛DC F .xI h0 / D j˛jDC F .xI h0 /  j˛jDC F .xI h0 / D DC F .xI j˛jh0 / D DC F .xI ˛h0 /: Since h; viX  DC F .xI v/ for each v 2 X0 and X 3 h 7! DC F .xI h/ is a continuous positively homogeneous semiadditive functional, then according to the Hahn– Banach Theorem there exists 2 X  such that h ; hiX  DC F .xI h/ for each h 2 X and h ; h0 iX D h; h0 iX : Hence we obtain h ; hiX  DC F .xI h/ and h ; hiX D h ; hiX  DC F .xI h/ D D F .xI h/

for every h 2 X:

The last relation proves the required inequality. The Lemma is proved. Lemma 1.18 and inequality (1.74) guarantee the existence of .x/ 2 X  such that h.x/; h  xiX  DC F .xI h  x/  F .h/  F .x/

for every h 2 U;

i.e., .x/ 2 @F .x; U /, and hereby the nonemptiness of @F .x; U / is proved. Lemma 1.19. For every x0 2 intU , the following inequality holds true: @'.x0 I U / D fp 2 X  j hp; hiX  DC '.x0 I h/ for every h 2 X g: Proof. Let p 2 @F .x0 I U /. Then there exists an open convex set V containing zero such that x0 C V  intU and hp; hiX  F .x0 C h/  F .x0 / Hence, hp; hiX 

F .x0 C th/  F .x0 / t

for every h 2 V:

for every t 2 .0; 1/:

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109

Due to Theorem 1.10, hp; hiX  inf

t >0

F .x0 C th/  F .x0 / D DC F .x0 I h/ t

for every h 2 V:

Since the set V is absorbing and functions X 3 h 7! DC F .xI h/;

X 3 h 7! hp; hiX

are positively homogeneous, then hp; hiX  DC F .x0 I h/

for every h 2 X:

On the other hand, let for every h 2 X the relation hp; hiX  DC F .x0 I h/ hold true. Due to Theorem 1.10, there follows the existence of a counterbalanced convex absorbing neighborhood of zero  (x0 C   intU ) such that DC F .x0 I v/  F .x0 C v/  F .x0 /

for every v 2 :

Let us fix an arbitrary h 2 U \domF: Then there is ˛ 2 .0; 1/ such that ˛.hx0 / 2 . Therefore, ˛  hp; hx0 iX D hp; ˛ .h  x0 /iX  DC F .x0 I ˛.h  x0 //  F .x0 C ˛.hx0 //  F .x0 /  ˛F .h/ C .1  ˛/F .x0 /  F .x0 / D ˛.F .h/F .x0 //: Hence we obtain that hp; hx0 iX  F .h/F .x0 / for each h 2 U \domF , and for this reason hp; h  x0 iX  F .h/  F .x0 / for each h 2 U . Hence p 2 @F .x0 I U /. The Lemma is proved. By Lemma 1.19 it immediately follows that Œ@F .x0 I U /; hC  DC F .x0 I h/

for every h 2 X;

that is, due to Lemma 1.19, fp 2 X  j hp; h  x0 iX  Œ@F .x0 I U /; h  x0 C for every h 2 X g  fp 2 X  j hp; hiX  DC F .x0 I h  x0 / for every h 2 X g D @F .x0 I U /: On the other hand, every element p 2 @F .x0 I U / satisfies the condition hp; hiX  Œ@F .x0 I U /; hC 8h 2 X , which proves the inverse inclusion. Therefore, equality (1.73) holds.

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Further, due to (1.73) and Theorem 1.10, @F .I U / is upper hemicontinuous on intU . Moreover, the boundedness of @F .x0 I U / follows from the estimate Œ@F .x0 I U /; hC D DC F .x0 ; h/  c1 d.h; 0/ for every h 2 ; where  is absorbing. So, by virtue of the Banach–Alaoglu Theorem (cf. [Ru73]), @F .x0 I U / is a compact set in the .X  ; X / topology. Under these conditions, upper hemicontinuity of the map @F .I U / and the Castaing Theorem (cf. [AuEk84]) imply *-upper semicontinuity of @F .I U / on intU: This completes the proof of Theorem 1.11. Lemma 1.20. Let F W X ! R be a convex l.s.c. functional. Then for each x0 2 intdom F D X we have @F .x0 / D fp 2 X  j hp; uiX  DC F .x0 I u/ 8u 2 X g: From Theorem 1.11 the next one directly follows. Theorem 1.12. Let F W X 7! R be a convex on U lower semicontinuous on intU functional. Then for each x0 2 intdom ' the set @'.x0 / is nonempty convex compact in .X  I X /-topology, the map intdom ' 3 x 7! @'.x/  X  is *-u.s.c. and the following equality takes place Œ@'.x0 /; uC D DC '.x0 I u/

8u 2 X:

(1.76)

Corollary 1.11. Let '  W X  ! R be a adjoint functional to '. Then @'  W X   X is a closed map in X   intdom ' concerning .X  ; X /-topology on X  . Proof. Proving Theorem 1.12 we showed that the map @' is upper hemicontinuous on intdom ' and due to [Chi97, Proposition 3.5] it has the closed graph in X   intdom ' in the .X  ; X / topology on X  . Therefore the required statement follows from the proportion . 2 @'.x//

”

.x 2 @'  .//:

The Corollary is proved. Theorem 1.13. Let F1 ; F2 W X ! R and U D U1 \ U2 , where intU ¤ ;, U1 ; U2 are convex sets and @F1 .x1 I U1 / ¤ ;; @F2 .x2 I U2 / ¤ ;

for all x1 2 U1 ; x2 2 U2 :

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111

Then @F .xI U / ¤ ; for every x 2 U , where F D F1 C F2 , and @F .xI U / D @F1 .xI U / C @F2 .xI U /

for every x 2 intU:

Proof. Suppose that x 2 U . It is clear that @F .xI U /  @F1 .xI U / C @F2 .xI U /  @F1 .xI U1 / C @F2 .xI U2 / ¤ ;: In order to complete the proof, it is necessary to show that for every x 2 intU and for every h 2 X the following equality is fulfilled: DC F .xI h/ D DC F1 .xI h/ C DC F2 .xI h/:

(1.77)

Indeed, since functions F; F1 ; F2 satisfy assumptions of Theorem 1.8, then all conditions of Theorem 1.11 hold true for them as well. Thus, due to equality (1.73) and [ReSi80, Proposition 1] Œ@F .xI U /; hC D DC F .xI h/ D DC F1 .xI h/ C DC F2 .xI h/ D Œ@F1 .xI U /; hC C Œ@F2 .xI U /; hC D Œ@F1 .xI U / C @F2 .xI U /; hC for all x 2 intU and h 2 X: Hence @F .xI U / D @F1 .xI U / C @F2 .xI U /

for every x 2 intU:

Now we prove (1.77). Due to Theorem 1.8, the statement of Theorem 1.10 for functions F; F1 ; F2 holds true. Consequently, for all x 2 intU and h 2 X , we obtain F .x C th/  F .x/ t !0C t F1 .x C th/  F1 .x/ C F2 .x C th/  F2 .x/ D lim t !0C t F1 .x C th/  F1 .x/ F2 .x C th/  F2 .x/ C lim D lim t !0C t !0C t t D DC F1 .xI h/ C DC F2 .xI h/:

DC F .xI h/ D lim

The Theorem is proved. Definition 1.54. Suppose that U is a convex body. The functional F W X 7! R [ fC1g (intU  domF ) is said to be upper bounded on intU if for every bounded set B  intU the image F .B/ is upper bounded in R. The following result is new even in the case of X being a Banach space. Theorem 1.14. Let F W X 7! R be a convex lower semicontinuous functional. Then the following statements are equivalent:

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1 Preliminary Results

(a) F is an upper bounded functional on X ; (b) A multivalued map @F ./ D @F .I X / is *-bounded on X . Proof. The following statements are true. Lemma 1.21. If B is a bounded set in X and C is a *-bounded set in X  , then the quantity sup sup jhp; xiX j is finite. x2B p2C

Proof. Let .x/ D sup jhp; xiX j. *-boundedness of C implies that the given funcp2C

tional is well defined on X . We remark that .x/ D .x/ for x 2 X . Moreover, is convex positively homogeneous and lower semicontinuous as the supremum of convex positively homogeneous continuous functionals. Hence, due to Lemma 1.16, is continuous on X , i.e., is a continuous seminorm on X . By [ReSi80, Theorem V.23], the boundedness of B in X implies that sup sup jhp; xiX jD sup .x/
x2B

The Lemma is proved. Definition 1.55. Let X be a separable locally convex topological space, U  X be an unbounded convex body. Then the functional F W U 7! R [ fC1g is called coercive on U if F .x/ ! C1 as .x/ ! C1, x 2 U , where an arbitrary continuous seminorm on X . Lemma 1.22. Let B  X be a nonempty set satisfying one of the two conditions: .i / B is bounded, .i i / F is coercive on B. Then inf F .x/ > 1. x2B

Proof. For some integer n, we consider the following set: An D fx 2 B j F .x/  ng ¤ ;: The boundedness of An follows from the boundedness of B or coercivity of F . Indeed, if the set An is unbounded, then there exists a continuous seminorm and a sequence fxn gn1  B such that .xn / ! C1. Thus we obtain F .xn / ! C1, and this fact contradicts the construction of An . Therefore, taking into account Theorem 1.11 and Lemma 1.21 with C D fpg, p 2 @F .0/, we deduce that inf F .x/  F .0/  sup jhp; xij > 1. This completes the proof. x2B

x2B

Let us continue the proof of Theorem 1.14. Let the set B be bounded in X . First we assume that the multivalued map @F ./ is *-bounded on X . Then, by definition of a subdifferential, F .x0 /  F .x/  hpx ; x0  xiX

for all x 2 B and px 2 @F .x/:

Whence for all x 2 B and px 2 @F .x/, we obtain

1.3 Subdifferentials in Infinite-Dimensional Spaces

113

F .x/  F .x0 / C hpx ; x  x0 iX  jF .x0 /j C

sup

jhp; x  x0 iX j

p2@F .B/

 jF .x0 /j C

sup

sup

x2x0 CB p2@F .B/

jhp; xiX j:

Lemma 1.21 and the fact that x0 C B is the bounded set in X yield sup

sup

x2x0 CB p2@F .B/

jhp; xiX j < C1:

Moreover, let the functional F be upper bounded. Then, due to Theorem 1.11, for every u 2 X there is sup

jhp; uiX j D sup

sup hp; uiX

x2B p2@F .x/

p2@F .B/

D sup Œ@F .x/; uC D sup DC F .xI u/: x2B

x2B

Further, from Theorem 1.10 we infer that sup DC F .xI u/  sup .F .x C u/  F .x//  sup F .x/  inf F .x/ DW I:

x2B

x2B

x2BCu

x2B

Since B, B C u are bounded sets in X , then (due to Lemma 1.22 and the definition of an upper bounded functional) the quantity I is finite. Consequently, sup hp; uiX < C1 for every u 2 X . Hence, the set @F .B/ is *-bounded. p2@F .BIU /

Theorem 1.14 is proved. Remark 1.43. For an arbitrary multivalued map A W Y  X  X  , coA and coA stand for multivalued maps defined as follows: coA.y/ WD co.A.y//, coA.y/ WD co.A.y// for every y 2 Y . Remark 1.44. Lemma 1.22 holds true if X is reflexive, but F W X ! R [ fC1g is weakly lower semicontinuous. Corollary 1.12. Let '1 ; '2 W X 7! R be lower semicontinuous convex functionals upper bounded on X . Then @'1 C @'2 W X  X  is a *-bounded *-upper semicontinuous map with compact values in the .X  ; X / topology. Proof. The map G D @'1 C @'2 is upper hemicontinuous, since it is the sum of upper hemicontinuous maps. Also, @'i D co@'i .i D 1; 2/. Now we prove that coG D G. As coG D G, i.e., coG  @'1 C @'2 D G, it remains to prove the inverse inclusion. Let u 2 coG.y/, then there exists a net fu˛ g 2 G.y/ such that u˛ ! u in X  , and u˛ D u0˛ C u00˛ , where u0˛ 2 @'1 .y/; u00˛ 2 @'2 .y/. Since @'1 .y/, @'2 .y/ are compact sets in .X  ; X /-topology, we deduce that u D u0 Cu00 ; u0 2 @'1 .y/, u00 2 @'2 .y/, i.e., coG.y/  G.y/. Thus, G satisfies all conditions of the Castaing Theorem, whence *-upper semicontinuity of the map @'1 C @'2 follows. The *-boundedness of the map @'1 C @'2 follows from a similar statement for @'1 and @'2 .

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1 Preliminary Results

For an arbitrary bounded set B, images @'1 .B/ and @'2 .B/ are *-bounded in X  . Then jhg; xiX j D

sup g2@'1 .B/C@'2 .B/



sup g1 2@'1 .B/

jhg1 ; xiX j C

sup

sup

g1 2@'1 .B/ g2 2@'2 .B/

sup g2 2@'2 .B/

jhg1 C g2 ; xiX j

jhg2 ; xiX j < C1

for every x 2 X;

i.e., @'1 C @'2 is a *-bounded set in X  . The Corollary is proved. Theorem 1.15. Let X be a reflexive Frechet space, ' W X ! R [ fC1g weakly lower semicontinuous functional, B  dom' a closed convex set. Moreover, suppose that one of the following conditions holds: (a) Set B is bounded in X ; (b) The functional ' is coercive on B. Then functional ' is lower bounded on B and reaches its exact lower bound d , and the set K D fx 2 B j '.x/ D d g is weakly compact in X . Proof. Due to Lemma 1.22 and Remark 1.44, the functional ' is lower bounded. Therefore, there exists a net fx˛ g˛  B such that lim '.x˛ / D d D inf '.x/ < C1: ˛

x2B

The set fx˛ g˛ is bounded in X due to either the boundedness B or coercivity of '. Hence, in virtue of the Banach–Alaoglu Theorem, there exists a subnet (which we also denote by fx˛ g˛ ) such that x˛ ! x0 in .X I X  /-topology of the space X , and x0 2 B, because the set B is closed in .X I X  /-topology. Hence, due to the lower semicontinuity of the functional ' in .X I X  /-topology, we obtain '.x0 /  lim '.x˛ / D lim '.x˛ / D d; ˛

˛

i.e., x0 2 K. Finally, let fx˛ g˛  K be an arbitrary net. The set K is bounded by the construction. Consequently, we may assume that x˛ ! x0 in .X I X  /-topology. So, '.x0 /  lim '.x˛ / D d , whence x0 2 K. ˛

The Theorem is proved. Now we consider a functional ' W X 7! R D R [ f1; C1g under the condition int U  dom ', where dom ' D fx 2 X j j'.x/j < C1g:

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115

Let us remark that the global Lipschitz condition for a functional ' on X does not guarantee the locally Lipschitz condition for a real function R 3 t ! .t/ D '.x C ty/ for some x; y 2 X . Let us illustrate this on the following example. Example 1.4. Let X D C.R/ be the Frechet space with the metric [Ru73]: d.x; y/ D

X

max jx.s/  y.s/j

2

n

n1

s2Œn;n

1 C max jx.s/  y.s/j

;

x; y 2 X:

s2Œn;n

N The global Lipschitz condition for the For any x 2 X we set '.x/ WD d.x; 0/. functional ' on X follows from the estimation N  d.y; 0/j N  d.x; y/ jd.x; 0/ Suppose

N x.t/ D 0;

y.t/ D 4t;

t 2R

8x; y 2 X: .x; y 2 X /:

We will show that the real function  is not Lipschitzean in any neighborhood of zero, i.e., there exists the numerical sequence ftn gn1 such that tn ! 0 as n ! 1 and j.tn /  .0/j D C1: lim n!1 jtn j Indeed, if we set tn D 4n , n  1 we obtain: max 4s  4n X .tn / j.tn /  .0/j s2Œm;m n m D D4 2 jtn j tn 1 C max 4s  4n m1 s2Œm;m

D

X

2m

m1



n1 X

m

n X

4 2m  1 C 4mn mD1 1 C 4mn

2m D 2n  1 ! C1; n ! 1:

mD0

The upper and the lower Clarke derivatives for a main local Lipschitzean functional ' defined on Banach space X are introduced as follows [Cl90]: 1 .'.v C ˛h/  '.v//; ˛ 1 # 'C l .x; h/ D .'.v C ˛h/  '.v//: lim v!x; ˛&0C ˛ "

'C l .x; h/ D

lim

v!x; ˛&0C

(1.78) (1.79)

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1 Preliminary Results

Here x 2 dom ', h 2 X . We remark that 'C" l .x; h/  'C# l .x; h/. This definition is correct due to the local Lipschitz condition of ' and to normability of the space X . It turns out that in Frechet space even the global Lipschitz condition for the functional ' does not guarantee existence of upper and lower Clarke derivatives on the whole space X . Let us illustrate this statement using the following example. Example 1.5. Let us take X , d.; /, ', x, y, , ftn gn1 from example 1.4. Then, 1 .'.v C ˛y/  '.v// v!x; ˛&0C ˛ 1 .tn /  lim .2n  1/D C 1: .'.x C ˛y/  '.x//  lim  lim n!1 tn n!1 ˛&0C ˛

'C"l .x; y/ D

lim

Similarly, replacing z D y we obtain: 'C# l .x; z/ D

1 1 .'.v  ˛y/  '.v// D lim .'.z/  '.z C ˛y// ˛ ˛ v!x; ˛&0C z!x; ˛&0C

D

lim

1 " .'.z C ˛y/  '.z// D 'C l .x; y/ D 1: z!x; ˛&0C ˛ lim

Thus, it is necessary to reinforce the local Lipschitz condition (on the interior of the effective definitional set) for definition of upper and lower Clarke derivatives in the sense of (1.78) and in the sense of (1.79) respectively for some functional defined in a Frechet space such that (a) The given condition will coincide with the local Lipschitz condition for any functional defined on a Banach space; (b) Main convex lower semicontinuous functional defined in the Frechet space satisfies the given condition on the interior of the effective definitional set; (c) The existence of upper and lower Clarke derivatives on the interior of the effective definitional set of the functional will follow from the given condition. Definition 1.56. The functional ' W X ! R is called M -local Lipschitzean on intdom ' if for any x0 2 intdom ' there exists convex counterbalanced absorbing neighborhood of zero O [Ru73] (x0 C O  dom ') and such constant C > 0, that j'.x C t.y  x//  '.x/j  C  t

8x; y 2 x0 C O

8t 2 .0; 1/:

(1.80)

Theorem 1.16. For main M -local Lipschitzean functionals defined on the Frechet space the conditions a–c are satisfied. Proof. At first let us ascertain, that M -local Lipschitz condition is stronger than local Lipschitz condition. Let ' W X ! R be a main M -local Lipschitzean on intdom ' functional (X is the Frechet space). Then for any x0 2 intdom ' there exists convex counterbalanced

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117

absorbing neighborhood of zero O (x0 C O  dom ') and a constant C > 0 such that the inequality (1.80) is valid. Since the metric d is coordinated with the topology of the space X , then there exists ı > 0 such that B4ı .x0 / D fx 2 X j d.x; x0 / < 4ıg  x0 C O: For any x; y 2 Bı .x0 / (x ¤ y) let us set zDxC

2ı .y  x/; d.y; x/

tD

d.y; x/ 2 .0; 1/: 2ı

Remark that d.z; x0 /  d.x; x0 / C d.z; x/  ı C d

2ıx 2ıy ; : d.y; x/ d.y; x/

Since d.x; y/  d.x; x0 / C d.y; x0 / < 2ı we see that d

2ıx 2ıy ; d.y; x/ d.y; x/



2ı d.y; x/ D 2ı: d.y; x/

So d.z; x0 /  3ı < 4ı. Thus, j'.y/  '.x/j D j'.x C tz/  '.x/j  C  t D

C d.y; x/: 2ı

This means that the functional ' is Lipschitzean on Bı .x0 /. (a) It is enough to show that in the case of a Banach space X , the main local Lipschitzean on intdom ' functional ' is M -local Lipschitzean on intdom '. Let x0 2 intdom ' and let constants c; ı > 0 be such that j'.x/  '.y/j  ckx  ykX

8x; y 2 Br .x0 / D fz 2 X j kz  x0 kX < ıg:

Absolute convexity of neighborhood of zero Br .0/ D Br .x0 /  x0 in X implies that for any x; y 2 Br .x0 /, t 2 .0; 1/ j'.x C t.y  x//  '.x/j  ctkx  ykX  2cr  t: The last relation proves the statement a). (b) Let ' W X ! R [ fC1g be a main convex lower semicontinuous functional. From Theorem 1.16 the local Lipschitz condition for ' on intdom ' follows. Let us show that ' satisfies the stronger condition, namely M -local Lipschitz condition on intdom '. From Theorem 1.15 it follows that the functional ' is locally bounded from above on intdom ', i.e., 8x0 2 intdom' there exist positive constants r and c such that '.x/  c 8x 2 Br .x0 /. Since the metric d is coordinated with the topology of the

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1 Preliminary Results

space X there exists an absolutely convex neighborhood of zero O such that 8x 2 x0 C 3O

'.x/  c:

Let x; y 2 x0 C O, t 2 .0I 1 be arbitrary fixed. In virtue of convexity of ' we have: '.x C t.y  x//  '.x/  t.'.y/  '.x//  t.c  '.x//:

(1.81)

On the other hand, let

D

t 2 .0; 1/; 1Ct

z D x C t.y  x/ 2 x0 C O;

D

x C .  1/z :

Remark that in virtue of absolute convexity of O we obtain:  DxC

1

.x  z/ D x C .x  y/ 2 x0 C 3O:

So './  c. Thus '.x/ D '.  C .1  /z/  './ C .1  /'.z/  c C .1  /'.z/: Hence

.c  '.x//: 1

The last inequality together with (1.81) means that '.x/  '.z/ 

j'.x C t.y  x//  '.x/j  t.c  '.x//: Specifically, by replacing in the last relation y D x, x D x0 , t D 1, we obtain: c  '.x/  c  '.x0 / C j'.x0 C x  x0 /  '.x0 /j  2.c  '.x0 // and j'.x C t.y  x//  '.x/j  2.c  '.x0 //  t for all x; y 2 x0 C O, t 2 .0I 1. (c) Let us show that from M -local Lipschitz condition on intdom ' of the main functional ' existence of the upper Clarke derivative for arbitrary x 2 intdom ' and h 2 X in the sense of (1.78) follows. Indeed, suppose x0 2 intdom ' is an arbitrary fixed, O (x0 C O  dom ')is an absolutely convex neighborhood of zero and C > 0 is a constant such that the inequality (1.80) is valid. Since O is absorbing then for an arbitrary h 2 X there exists t0 D t0 .h/ > 0 such that t0 h 2 12 O. Then due to (1.80), 1 8v 2 x0 C O; 8 2 .0; t0 / 2

ˇ ˇ ˇ '.v C h/  '.v/ ˇ C ˇ : ˇ ˇ t ˇ

0

(1.82)

1.3 Subdifferentials in Infinite-Dimensional Spaces

119

On the other hand, since '.v C th/  '.v/ '.v C th/  '.v/  inf sup v!x0 ; ˛&0C t ı>0 t 2.0;ı/; v2B .x0 / t ı lim

and since the metric d is coordinated with the topology of the space X there exists ı > 0 such that 1 ı < t0 ; Bı .x0 /  x0 C O 2 and due to (1.82) we have "

'C l .x0 ; h/ 

C '.v C th/  '.v/  < C1: t t0 t 2.0;ı/; v2Bı .x0 / sup

Similarly, for arbitrary x0 2 intdom ' and h 2 X 'C# l .x0 ; h/  

C > 1: t0 .h/

So, 8x0 2 intdom '; 8h 2 X

 1 < 'C# l .x0 ; h/  'C" l .x0 ; h/ < C1:

This means, that lower and upper Clarke derivatives exist on intdom '. The Theorem is proved (Fig. 1.20). Proposition 1.85. Let ' W X ! R ba a main M -locally Lipschitzean functional. Then

The locally Lipschitz condition The M-locally Lipschitz condition The convexity with lower semicontinuity

Fig. 1.20 Some properties for main functional in Frechet space

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1 Preliminary Results

'C" l .x0 ; h/ D .'/"C l .x0 ; h/;

'C# l .x0 ; h/ D 'C" l .x0 ; h/

(1.83)

for all x0 2 intdom ', h 2 X . For an arbitrary x0 2 intdom ' the functional "

X 3 h ! 'C l .x0 ; h/ is sublinear and continuous, and the functional X 3 h ! 'C# l .x0 ; h/ is superlinear and continuous. Proof. Let x0 2 intdom ', h 2 X . Then, by definition, 'C" l .x0 ; h/ D

'.x  ˛h/  '.x/ : x!x0 ; ˛&0C ˛ lim

Let us set x  ˛h D z. Then x D z C ˛h, hence '.z/  '.z C ˛h/ ˛ .'/.z C ˛h/  .'/.z/ " D .'/C l .x0 ; h/: lim D z!x0 ; ˛&0C ˛

'C" l .x0 ; h/ D

lim

z!x0 ; ˛&0C

Similarly, 'C# l .x0 ; h/ D

'.x C ˛h/  '.x/ ˛ x!x0 ; ˛&0C lim

D

lim

x!x0 ; ˛&0C

.'/.x C ˛h/  .'/.x/ ˛

D .'/"C l .x0 ; h/ D 'C" l .x0 ; h/: Let us show the sublinearity of "

X 3 h ! 'C l .x0 ; h/: From the equality "

'.x C ˛h/  '.x/ x!x0 ; ˛&0C ˛ '.x C ˛h/  '.x/ D   'C" l .x0 ; h/ lim  D x!x0 ; ˛&0C ˛

'C l .x0 ; h/ D

lim

1.3 Subdifferentials in Infinite-Dimensional Spaces

121

it is follows that this functional is positively homogeneous. Let us show that it is subadditive. Using the subadditive property of the upper limit, we have '.x C ˛h1 C ˛h2 /  '.x/ ˛ x!x0 ; ˛&0C '.x C ˛h1 C ˛h2 /  '.x C ˛h1 / lim  x!x0 ; ˛&0C ˛ '.xC˛h1 /'.x/ D 'C" l .x0 ; h1 /C'C" l .x0 ; h2 /: lim  x!x0 ;˛&0C ˛

"

'C l .x0 ; h1 C h2 / D

lim

The superlinearity of

X 3 h ! 'C# l .x0 ; h/

is obvious now. Let us prove continuity of the functional. Suppose O (x0 C O  dom ') is absolutely convex neighborhood of zero O and C > 0 is a constant such that the inequality (1.80) holds true. Remark that for arbitrary h 2 X and  2 .0; 1= .h// such that h 2 O=2, where .h/ D infft > 0 j x 2 tO=2g is the Minkovsky functional (in virtue of absolute convexity and since O=2 is open we have that the functional X 3 h ! .h/ is a continuous seminorm [Ru73]). Let us consider two cases: (i) .h/ > 0. From (1.80) it follows that ˇ ˇ ˇ '.v C th/  '.v/ ˇ C ˇ : ˇ ˇ ˇ t 

8v 2 x0 C O=2; 8t 2 .0; 1/; 8 2 .0; 1= .h//

1 If we put in the last inequality  D 2 .h/ (in virtue of (1.78) and (1.79)) we finally obtain: (1.84) j'C" l .x0 ; h/j  2C .h/; j'C# l .x0 ; h/j  2C .h/:

(ii) .h/ D 0. From (1.80) due to the definition of it follows that 8v 2 x0 C O=2; 8t 2 .0; 1/; 8 > 0

ˇ ˇ ˇ '.v C th/  '.v/ ˇ C ˇ : ˇ ˇ ˇ t 

In virtue of (1.78) and (1.79) we obtain "

j'C l .x0 ; h/j 

C ; 

#

j'C l .x0 ; h/j 

C 

8 > 0:

(1.85)

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1 Preliminary Results

Thus from (1.84) and (1.85) we have "

j'C l .x0 ; h/j  2C .h/;

#

j'C l .x0 ; h/j  2C .h/

8h 2 X:

(1.86)

Remark that here C and ./ depend on x0 2 intdom ' and do not depend on h 2 X . From (1.86) it follows that upper and lower Clarke derivatives as direction functions are continuous in zero. From sublinearity of X 3 h ! 'C" l .x0 ; h/ and from superlinearity of X 3 h ! 'C# l .x0 ; h/ continuity of given maps on the whole space X follows. The Proposition is proved. Theorem 1.17. The functional ' W X ! R is M -locally Liptschitz on intdom ' if and only if for each x0 2 intdom ' there exists a continuous seminorm on X and a constant C > 0 such that j'.u1 /  '.u2 /j  C .u1  u2 / 8ui 2 X W .ui  x0 / < 1; i D 1; 2:

(1.87)

Proof. Let the functional ' W X ! R be M -locally Lipschitzean on intdom '. Then for fixed x0 2 intdom ' there exists an absolutely convex neighborhood of zero O (x0 C O  dom '), there exists a constant C > 0 such that (1.80) holds true. We denote a Minkovsky functional for a set O=6 at the point x by .x/. We remark that ./ is a continuous seminorm on X and the set fx j .x/  1g is contained in O=4. Then for arbitrary u; v 2 X : .u  x0 / < 1; .v  x0 / < 1 and we have three different cases: (i) 0 < .v  u/ < 1. Then hD

vu 2 O=4; .v  u/

v D u C .v  u/h 2 x0 C O

and, due to (1.80), j'.v/  '.u/j D j'.u C .v  u/h/  '.u/j  C .v  u/: (ii) .v  u/  1. Then v  u 2 O=2 and there exists t0 > 1 such that for each

2 .1; t0 / h D .v  u/ 2 O=2; v D u C h = 2 x0 C O: Hence, j'.v/  '.u/j D j'.u C h = /  '.u/j 

C :

1.3 Subdifferentials in Infinite-Dimensional Spaces

123

So, when & 1C taking into account that .v  u/  1 we obtain the required inequality (1.87). (iii) .v  u/ D 0. Due to the definition of the Minkovsky functional we have that 8 > 0 h D

1 .v  u/ 2 O=4; v D u C h 2 x0 C O:

Hence, j'.v/  '.u/j D j'.u C h /  '.u/j  C : So, when & 0C, taking into account .v  u/ D 0, we obtain the inequality (1.87). From (i)–(iii) validity of (1.87) follows. Now let the condition (1.87) with x0 2 intdom ', with continuous on X seminorm and a constant C > 0 be valid. Let us consider a set fh 2 X j .h/ < 1=2g in the capacity of absolutely convex neighborhood of zero O. Then for each x; y 2 x0 C O j'.x C t.y  x//  '.x/j  C .t.y  x//  C t: The Theorem is proved. Theorem 1.18. Let the main functional ' W X ! R be M -locally Lipschitzean " on intdom '. Then for each h 2 X the upper Clarke derivative 'C l .; h/ is upper semicontinuous on intdom '. Proof. Let the functional ' W X ! R be M -locally Lipschitzean on intdom '. Let us fix h 2 X and x0 2 intdom '. Proposition 1.86. For each x0 2 intdom ' and h 2 X "

1 .'.x C ˛h/  '.x//; ı&0C d.xIx0 /<ı ˛&0C ˛

(1.88)

1 .'.x C ˛h/  '.x//: ˛

(1.89)

'C l .x0 ; h/ D lim

sup

lim

'C# l .x0 ; h/ D lim

inf

lim

ı&0C d.xIx0 /<ı ˛&0C

Proof. Now we prove the validity of the inequality (1.88). Let us set 1 .'.x C ˛h/  '.x//; x!x0 ; ˛&0C ˛ 1 A2 D inf sup lim .'.x C ˛h/  '.x//: ı&0C d.xIx0 /<ı ˛&0C ˛ A1 D 'C" l .x0 ; h/ D

lim

We remark that from M -locally Lipschitz condition for ' on intdom ' it follows that A2 2 R. After fixing " > 0 let us consider the sequence fxn gn1  intdom '

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1 Preliminary Results

such that d.xn I x/ ! 0 as n ! 1 and 1 .'.xn C ˛h/  '.xn // > A2  ": ˛&0C ˛ lim

For each n  1 there exists number ˛n 2 .0I 1=n/ such that 1 .'.xn C ˛n h/  '.xn // > A2  ": ˛n The given inequality shows that A1  A2  ": Since " is an arbitrary positive number we can see that A1  A2 . In virtue M -locally Lipschitz condition for ' there exist C > 0 and a continuous seminorm on X such that j'.u1 /  '.u2 /j  C .u1  u2 / 8ui 2 X W .ui  x0 / < 1; i D 1; 2: Let us consider N D fx 2 X j .x/ D 0g and a factor space M WD X=N with a factor norm k.x/kM D inf .x  z/; z2N

where  W X ! M is factor map [Ru73, p. 39-41]. Let us set ..x// D '.x/ 8x 2 X: Then, for each vi 2 M : kvi  .x0 /kM < 1, i D 1; 2 j .v1 / 

.v2 /j  C kv1  v2 kM :

We remark that 1 . .x C ˛.h//  .x//; x!.x0 /; ˛&0C ˛ 1 A2 D inf sup lim . .x C ˛.h//  ı&0C kx.x0 /kM <ı ˛&0C ˛

A1 D

lim

.x//:

Further, following by the proof of [DemVas81, Proposition 2.1.3] we obtain that A2  A1 .

1.4 Minimax Inequalities in Finite-Dimensional Spaces

125

The Proposition is proved. The second inequality (1.89) can be proved similarly. The further proof is similar to the proof of [DemVas81, Proposition 2.1.1]. The Theorem is proved. Together with well-defined on X M -locally Lipschitz function ' on int dom' we will consider the so called generalized Clarke gradient at the point x 2 intdom ' defined in the following way: "

@C l '.x/ D fp 2 X  j hp; v  xiX  'C l .x; v  x/

8v 2 X g;

x 2 X:

Corollary 1.13. Let the main functional ' W X ! R be M -locally Lipschitzean on intdom '. Then the generalized Clarke gradient @C l ' is upper *-semicontinuous on intdom '. The proof follows from Theorem 1.18, from Castaing Theorem [AuEk84] and from Theorem 1.12.

1.4 Minimax Inequalities in Finite-Dimensional Spaces The role of the classical acute angle Lemma [ZgMe99] in demonstration of solvability for nonlinear operator equations with monotone coercive maps in finitedimensional space is well-known. In the second section the minimax inequalities are investigated so that in the next section by means of this apparatus multivalued analogues of “acute angle Lemma” are proved. Suppose X is a Hausdorff topological space, Y is a topological vector space, N is a convex subset in Y , f W X  N ! R is some function. The multivalued map D W N ! 2X (2X is a family of all subsets of the space X ) is called strict if DomD D fy 2 N jD .y/ ¤ ; g D N: Let Bc .N I X / be a family of all strict upper semicontinuous multivalued maps [ZgMe99] with compact values (images). For each D 2 Bc .N I X / we set f # .D/ D sup inf f .d.y/; y/; f } .D/ D inf sup f .d.y/; y/; y2N d 2D

d 2D y2N

where the notation d 2 D means that d is a selector of the multivalued map D (namely d .y/ 2 D .y/ 8y 2 N ). It is clear that the following inequality holds true: f # .D/  f } .D/ 8D 2 Bc .N I X / :

126

1 Preliminary Results

Theorem 1.19. Let the following conditions be fulfilled: .1/ There can be found y0 2 N such that 8 2 R the set fx 2 X jf .x; y0 /   g is relatively compact in X ; .2/ 8y 2 N the function X 3 x 7! f .x; y/ is lower semicontinuous and 8x 2 X the function N 3 y 7! f .x; y/ is concave. Then there exists xN 2 X such that sup f .x; N y/  f # .D/  f } .D/ ; 8D 2 Bc .N I X / :

(1.90)

y2N

Proof. Let = .N / be a family of all finite subsets from N . In virtue of [AuEk84, Theorem 6.2.6] there can be found an element xN 2 X , such that N y/ D sup f .x; y2N

inf sup f .x; y/ :

sup

K2=.N / x2X y2K

(1.91)

We set K D fy1 ; : : : ; yn g and ( n SC

D 2

n RC

ˇ n ) ˇX ˇ i D 1 : ˇ ˇ i D1

Since N is the convex set we see that coK  N , moreover D .coK/  X 8D 2 Bc .N I X /. In this case n X

inf max f .x; yi / D inf sup

x2X i D1;:::;n

x2X 2S n C



inf

i f .x; yi / 

i D1

sup

x2D.coK/ 2S n

C

 infn

!

n X

! i f .x; yi / 

i D1

0 0 0 1 11 n n X X sup @ i f @ d i @ j yj A ; yi AA

2SC 2S n C

i D1

(1.92)

j D1

where d W N ! X .i D 1; : : : ; n/ is some fixed selector family of the multivalued map D. Let us consider the function F .; y/ D

n X i D1

0

0

i f @ d i @

n X j 1

1

1

j yj A ; yi A ;

(1.93)

1.4 Minimax Inequalities in Finite-Dimensional Spaces

127

where selectors di 2 D satisfy the condition f .di .y/ ; yi / D inf f .d.y/; yi /; 8y 2 N:

(1.94)

d 2D

Such selectors exist due to lower continuity of the function f by the first variable while the second one is fixed and due to compactness of values of the map D. Proposition 1.87. A function F defined by the relations (1.93), (1.94) is lower semibounded by  while  is fixed. Proof. At first we prove that for each i D 1; : : : ; n the function 0

0

n 3  7! Fi ./ D i f @di @ SC

n X

1

1

j yj A ; yi A

(1.95)

j D1

is lower semicontinuous. The following Proposition has an independent value. Lemma 1.23. Let X , Y be Hausdorff topological spaces and G W X ! 2Y be a compact-valued map. Then the following properties are equivalent: .a/ The map G is upper semicontinuous at the point x0 2 X . .b/ For an arbitrary net fx˛ g convergent to x0 and ˛ 2 G .x˛ / there can be isolated a subnet f g of the net f˛ g such that  ! 0 in Y and 0 2 G .x0 /. Proof. Let us prove the implication a ) b. Let the map G be upper semicontinuous at the point x0 , we consider an arbitrary x˛ ! x0 in X and ˛ 2 G .x˛ /. As it is known [ZgMe99, Proposition 1.4.8] the map G is upper semicontinuous at the point x0 if and only if from the convergence x˛ ! x0 it follows that the set G .x0 / attracts the net of the sets fG .x˛ /g (see Proposition 1.8). If here G does not satisfy the property b) then there can be found at least one net ˛ 2 G .x˛ / ; which does not have a subnet convergent in Y to any element from G .x0 / : This means that for an arbitrary v 2 G .x0 / there exists a neighborhood  ./ ; which seldom meet the net f˛ g ; namely there can be found T s D s . .// ; such that f˛ g  .v/ D ;; 8˛  s: Due to compactness of G .x0 / the family f ./ jv 2 G .x0 / g forms a covering from which there can be isolated a finite subcovering T Sm f .k / jk D 1; : : : ; m g : Moreover if ˛  max s . .vk // then f˛ g kD1  .vk / D ;; and this fact kD1;:::y;m

contradicts to the attraction of fG .x˛ /g to the set G .x0 / : Therefore an arbitrary net ˛ 2 G .x˛ / has a subnet f g convergent in the space Y to the element 0 : Let us prove that 0 2 G .x0 / : If 0 … G .x0 / then due to compactness of the image G .x0 / in Y there exist disjoint neighborhoods  .G .x0 // and  .0 /. Indeed for each  2 G .x0 / there exist disjoint neighborhoods  ./ and  .0 /,

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1 Preliminary Results

and the family of f ./ j 2 G .x0 / g covers G .x0 / : Let f .i / ji D 1; : : : ; l g be S a finite subcovering then it is sufficient to set  .G .x0 // D li D1  .i / ;  .0 / D Tl i D1 i .0 / : So, the net f˛ g is frequent in  .0 / and does not leave  .G .x0 // starting with some index ˛0 and this fact is contradictory. The implication b ) a is proved in [ZgMe99, see Proposition 1.4.11] (see Proposition 1.11). ˛ 0 n We will continue the proof of Proposition P 1.87. Let  !  in SC and let us consider ˛ 2 D .y ˛ / ; where y ˛ D njD1 ˛j yj ; ˛ D di .y ˛ / : Then y ˛ ! Pn 0 y0 D 1.23 there can be found subnet f g j D1 j yj and in virtue of Lemma   such that  ! 0 in X , moreover 0 2 D y 0 . And since the function f is lower semicontinuous by the first variable then

0

0

i f .0 ; yi /  lim i f @di @ v

n X

1

1

vj yj A ; yi A ;

j D1

    here Fi 0  i f .0 ; yi /. We prove that lim Fi .˛ /  Fi 0 . Let us denote ˛   b D lim Fi .˛ /. Assume the contrary, namely b < Fi 0 . We consider the net   ˚ ˇ ˛ Pn ˇ  f˛ g such that b D lim Fi ˇ , and respectively y ˇ D  j D1 j yj , ˇ   ˇ 2 D y ˇ . n 0o ˚  Due to the condition b of Lemma 1.15 we can isolate subnets y ˇ ; ˇ 0 of nets ˚ ˇ ˚    0 y , ˇ respectively, such that y ˇ ! y 0 in N , ˇ 0 !  0 in X and  0 2 D y 0 . Here    0    0 b D lim Fi; ˇ D lim Fi; ˇ D lim i 0 f di00 y ˇ ; yi ˇ ˇ0 ˇ0     0    0  i f  ; yi  i f di y ; yi D Fi 0  0 where di0 y ˇ D ˇ 0 : The obtained contradiction proves lower semicontinuity of functions Fi defined n 3  7! F .; / by the equality (1.95) 8i D 1; : : : ; n. Then the function SC is also lower semicontinuous as a finite sum of lower semicontinuous functions. Proposition 1.87 is proved. n n For each  2 SC the function SC 3  7! F .; / is affine therefore due to the classical Ky Fan inequality for finite-dimensional spaces

inf

sup F .; /  sup F .; / :

n n 2SC 2SC

n 2SC

(1.96)

On the other hand the function y 7! f .x; y/ is concave. Hence for each selector d 2 D we have

1.4 Minimax Inequalities in Finite-Dimensional Spaces

F .; / D

n X

0

0

i f @ d i @

i D1

0 0

 f @d @

n X

1

i yj A ; yi A 

j D1 n X

1

1

j yj A ;

j D1

n X

1

129 n X

i f

j D1

di

n X

i yj

! ; yi

i D1

i yi A :

i D1

Since the last relation is valid 8d 2 D we see that 1 1 0 0 n n X X j yj A ; i yi A  f # .D/ F .; /  inf f @d @ d 2D

!

j D1

n 8 2 SC :

i D1

From here and from (1.96), (1.92) we find inf sup f .x; y/  f # .D/ :

x2X y2K

(1.97)

Since the inequality (1.97) is true 8K 2 = .N /, in view of (1.91) we obtain (1.90). The Theorem is proved. n ! N For each K D fy1 ; : : : ; yN g 2 = .N / let us consider a map ˇk W SC n P i yi . The finite topology on N is a final topology defined by the rule ˇk ./ D i D1

[ReSi80] with respect to a family of maps fˇk jK 2 = .N /g.   The set N with the finite topology we denote by Nf and Bc Nf I X is a space of upper semicontinuous maps from Nf in X with compact images.   Proposition 1.88. For validity of the inclusion D 2 Bc Nf I X it is necessary andsufficient  that for an arbitrary K 2 = .N / the composition Dk D D ı ˇk 2 n Bc SC IX The proof follows from the general properties of multivalued maps [ZgMe99]. Analyzing the proof of Theorem 1.19 we come to the following Proposition. Theorem 1.20. Let the conditions 1, 2 of Theorem 1.19 be fulfilled. Then there exists x 2 X such that sup f .x; y/  f # .D/  f } .D/

  8D 2 Bc Nf I X

y2N

Zeroes of multivalued maps. Let Y and Y  be a dual pair of topological vector spaces. Suppose P  Y  is a closed convex cone, P   Y is its negative polar cone, namely ˇ n o ˇ P  D y 2 Y ˇhp; yiY  0 8p 2 P ;

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1 Preliminary Results

where h; iY W Y   Y ! R is the canonical duality, K is a compact space, Cv .Y / is a family of all closed convex subsets of Y . 

Theorem 1.21. Let F W K ! 2Y be a strict multivalued map and the following conditions be satisfied: (1) The map F is P -upper hemicontinuous, namely 8y 2 P  the real function K 3 x 7! ŒF .x/ ; yC D sup hg .x/ ; yiY g2F

is upper semicontinuous; (2) The set F .x/ C P is nonempty convex and closed weakly star; (3) There exists D 2 Bc .P  I K/ such that sup ŒF .d .y// ; yC  0 8y 2 P  :

d 2D

Then there can be found xN 2 K such that 0 2 F .x/ N CP

(1.98)

and the set Z .P / D fx 2 K j0 2 F .x/ C P g is nonempty and closed. Proof. Let us introduce on K  P  the function f .x; y/ D ŒF .x/ ; yC which is lower semicontinuous by x and concave by y. In virtue of Theorem 1.19 there can be found an element xN 2 K for which sup f .x; N y/  f # .D/

y2P 

8D 2 Bc .P  I K/ :

Therefore (see Condition (3)), inf f .d.y/; y/  0, so f # .D/  0 and we d 2D

obtain ŒF .x/ N ; yC  0 8y 2 P  . We remark that ( ŒP; yC D

0; y 2 P  ; C1; y … P  ;

and using the properties of support functions we obtain 0  ŒF .x/ N ; yC C ŒP; yC D ŒF .x/ N C P; yC whence (see Condition (2)) 0 2 F .x/ N C P:

8y 2 Y;

1.4 Minimax Inequalities in Finite-Dimensional Spaces

131

Corollary 1.14. Let the conditions 1, 2 of Theorem 1.22 be fulfilled and sup ŒF .x/ ; yC  0

8y 2 P 

x2K

The set Z .P / is nonempty and closed. Proof. It is sufficient to consider a “constant” multivalued map D W P   K .D .y/ D K 8y 2 P  /. Using Lemma 1.23 it is easy to see that D 2 Bc .P  I K/. Definition 1.57. The point xN 2 K is called P -critical point of the map F if it satisfies the relation (1.98). Theorem 1.22. Suppose that the conditions 1, 2 of Theorem 1.21 are satisfied and all P -critical points in K of the map F are isolated. Then the number of such points is finite. Proof. We are going to prove the Theorem by contradiction. Let us assume that the family of P -critical points in K is infinite. Then in virtue of compactness of K from this family there can be isolated a convergent subnet x˛ ! x0 in K where 0 2 F .x˛ / C P . Hence under the condition 1 we find 0  limŒF .x˛ / C P; yC D limŒF .x˛ / ; yC  ŒF .x0 / ; yC ˛

8y 2 P  ;

˛

and therefore, 0  ŒF .x0 / C P; yC D ŒF .x0 / ; yC C ŒP; yC

8y 2 Y

or (see Condition (2) of Theorem 1.21) 0 2 F .x0 / C P; namely x0 is a P -critical point of the map F , moreover it is not isolated and this fact contradicts to the conditions of the Theorem. Theorem 1.23. Let conditions 1, 2 of Theorem 1.21 be fulfilled and 8" > 0 there can be found D" 2 Bc .P  I K/ such that sup ŒF .d" .y// ; yC  " 8y 2 P  :

(1.99)

d" 2D"

Then there exists at least one element xN 2 K such that 0 2 F .x/ N C P and the set Z .P / is closed. Proof. As in the case of Theorem 1.21 the following inequality must be proved: N y/  sup sup f .x;

y2P 

inf f .d" .y/ ; y/ D f # .D" /

y2P  d" 2D"

8" > 0;

132

1 Preliminary Results

where f .x; y/ D ŒF .x/ ; yC . Hence sup f .x; N y/  inf f # .D" / : ">0

y2P 

From here and from the condition (1.99) we obtain the estimation inf inf f .d" .y/ ; y/  inf " D 0

">0 d" 2D"

">0

8y 2 P  :

Further the proof must be completed in the same way as in Theorem 1.21. The Theorem is proved. Let us consider a special case when Y is a reflexive Banach space; Yw is a space Y provided with the weak topology; K is closed convex bounded set in Y (and therefore in Yw too.) Then (multivalued, generally speaking) projection map K W Y ! 2K from Y on K is defined by the formula ˇ  ˇ K .y/ D w 2 K ˇˇ kw  ykY D inf kv  ykY : v2K 

It is clear that DomK D Y and K .y/ D y 8y 2 K namely K is a multivalued retraction. Lemma 1.24. The multivalued map K W Y  K has closed and convex values and it is upper semicontinuous from Y onto K  Yw . Proof. Closeness and convexity of K .y/ 8y 2 Y are obvious. To prove upper semicontinuity we use Lemma 1.23 keeping in mind that K is a compact subset in Yw and that the map K is compact-valued. Suppose yn ! y0 in Y . Let us consider a sequence n 2 K .yn /. Without any restrictions we can assume that n ! 0 weakly in Y (if it is not weakly convergent then in virtue the Banach– Alaoglu Theorem we should pass to a subsequence) and here 0 2 K. Then 8w 2 K kn  yn kY D inf kv  yn kY  kw  yn kY ; v2K

passing here to the limit and taking into account weak semicontinuity of the norm in a Banach space we find k0  y0 kY  lim kn  yn kY  lim kw  yn kY D kw  y0 k 8w 2 K n!1

n!1

or k0  y0 kY  inf kw  y0 kY ; w2K

where 0 2 K, namely 0 2 K .y0 /. Now it remains to use Lemma 1.23. The Lemma is proved.

1.4 Minimax Inequalities in Finite-Dimensional Spaces

133

Remark 1.45. If K is a compact set in Y then K W Y ! 2K is compact-valued and upper semicontinuous from Y onto K  Y . Theorem 1.24. Suppose Y , Y  are Hausdorff locally convex spaces, P  Y is a convex set, K is a compact set, F W K  Y  and the following conditions hold true: (1) A multivalued map F W K  Y  has nonempty closed convex values and F .x/ is an equicontinuous set in Y  [Sc71] 8x 2 K; (2) 8y 2 P a functional K 3 x ! ŒF .x/; yC is upper semicontinuous; (3) 9 D 2 Bc .P  I K/ such that sup ŒF .d.y//; yC  0;

8y 2 P:

d 2D

Then there can be found xN 2 K such that 0 2 F .x/ N C P ; where P  D fw 2 Y  j hw; yiY  0 8y 2 P g. Proof. The proof is based on the following Proposition which is a generalization of the Weierstrass Theorem on the case of locally bounded sets. Lemma 1.25. Suppose W is locally convex space, W  is its topologically adjoint, b D R [ f1g E  W  is a set closed in the topology .W  I W /, L W E ! R is its upper semicontinuous main functional in the topology .W  I W /. Besides, let either the set E be equicontinuous in W  , or the following analogue of coercivity hold true: for an arbitrary set U  W  , which is not equicontinuous and   R there exists w 2 U such that L.w /  . Then the functional L is upper bounded on E and reaches on E its upper boundary l, and here the set fw 2 EjL.w/ D lg is compact in the topology .W  I W /. The Theorem is proved. The following Theorem in a certain sense is adjoint with Theorem 1.24. Theorem 1.25. Suppose Y is a Hausdorff reflexive locally convex set, K is a compact set, P is a convex set in Y  and the following properties take place: (1) F W K ! Cv .Y /, F .x/ is a bounded set in Y 8x 2 K; (2) 8y 2 P a function K 3 x ! ŒF .x/; yC is upper semicontinuous; (3) Let for some D 2 Bc .P I K/ sup ŒF .d.y//; yC  0 8y 2 P:

d 2D

Then the inclusion 0 2 F .x/ C P  has a solution xN 2 K.

134

1 Preliminary Results

In the proof of Theorem 1.25, similarly to the one of Theorem 1.25 the following case of the generalized Weierstrass Theorem is used. Lemma 1.26. Let E be weakly closed set in the reflexive locally convex space W , b D R [ fC1g be weakly lower semicontinuous and let let a functional L W E ! R one of the following conditions hold true: (a) E is a bounded set in W ; (b) The functional L is coercive on E, namely 8  2 R 9 L.Y /  .

Y 2 E such that

Then the functional L which is bounded below on E reaches on E its lower boundary l and the set fy 2 EjL.y/ D lg is weakly compact in W . Corollary 1.15. Let K be compact subset in Y (or Yw ) such that K has multivalued retraction D 2 Bc .Y I K/ : Let also F W K ! Cv .Y  / be upper hemicontinuous map and (1.100) sup ŒF .d .y// ; yC  0 8y 2 Y: d 2D

Then there exists xN 2 K such that 0 2 F .x/. N Proof. It is sufficient in Theorem 1.21 to set P D f0g. The following Proposition is a multivalued analogue of acute angle Lemma. Corollary 1.16. Suppose Y is finite-dimensional space, F W BN r ! Cv .Y / are strictly u.s.c. maps where BN r D fy 2 Y j kykY  rg : If here ŒF .y/; yC  0

8y 2 Y W

kykY D r;

(1.101)

then there exists xN 2 B r for which 0N 2 F .x/. N ˇ n o ˇ Proof. Let us set ır D y 2 Y ˇ kyk D r . We consider a multivalued map D W Y ! 2ır defined by the following relation D .y/ D

8 ry < kyk ; y ¤ 0; : N  ; y D 0;

where   ır is a family of all cluster points of the sequence n D yn ! 0.   Lemma 1.27. The map D belongs to the class Bc Y I BN r .

ryn kyn k

where

1.4 Minimax Inequalities in Finite-Dimensional Spaces

135

Proof. It is clear that the map D is closed-valued and continuous at all points y ¤ 0. Let us prove that for y D 0 this map is upper semicontinuous. Since ır is a compact set then D.0/ is a compact subset and again we use Lemma 1.23. Let yn ! 0 then ryn 2 ır and there exists a subsequence m ! 0 , here we have 0 2  n D ky nk whence upper semicontinuity follows. For y ¤ 0 in virtue of (1.101) we have 



 ry r ry ry F ;y ; D F  0; kyk kyk kyk kyk C C and for y D 0 sup ŒF .l/; 0C D 0 and hence the condition (1.99) holds true and l2 N

therefore there exists xN 2 B r such that 0 2 F .x/. N Remark 1.46. In [AuEk84, Corollary 6.4.3] the analogical Proposition is proved under the condition stronger than (1.101), namely, ŒF .y/; y  0

8y 2 Y W

kykY D r:

In the space Rn let us consider a simplex ˇ n ) ˇX ˇ n SC D x 2 RnC ˇ xi D 1 : ˇ (

i D1

n ! Cv .Rn / be a strict upper hemicontinuous map and Corollary 1.17. Let F W SC n : ŒF .y/; yC  0 8y 2 SC

(1.102)

n Then there exists x 2 SC such that

0  F .x/ N  RnC :

(1.103)

n . We define a Proof. Let us set Y D Y  D Rn , P D RnC , P  D RnC , K D SC n SC n multivalued map D W RC ! 2 by the formula

( D .y/ D

Pn y

; y ¤ 0; i D1 yi N  ; y D 0;

n where the set   SC is selected by the same rule as in the proof ofCorollary 1.16. n Similarly to Lemma 1.27 it can be proved that D 2 Bc RnC I SC , and it is easy to verify that from (1.102) the condition (3) of Theorem 1.21 follows and hence (1.103) is valid.

136

1 Preliminary Results

Remark 1.47. The Proposition of Corollary 1.17 is contained in [AuEk84, Theorem n 6.4.1], however the singlevalued map C W RnC ! SC which is used in the proof of the above mentioned Theorem is not continuous in the natural topology. Remark 1.48. The compactness condition K can ben weakened ˇ if we replaceoit by ˇ  the following one: 9y0 2 P , for which K0;u D x 2 K ˇŒF .x/ ; yC  0 is a compact. Remark 1.49. Suppose K  Y is a compact convex subset, F W K ! Cv .Y  / is a strict upper hemicontinuous map, p D f0g. In this case, as it was remarked in [AuEk84, p.331], as a natural example of the map D we can take a subdifferential @K .p/ of the support function  .K; p/ D ŒK; pC of the set K. The function Y 3 p 7!  .K; p/ is convex, continuous, completely defined therefore @K W Y  ! 2K is strict upper semicontinuous map with closed convex images. So, @K 2 Bc .KI Y  /. But since @K does not have, generally speaking, a continuous selector then in [AuEk84, p.331, the formula (17)] an alternative condition of normality is introduced. Our constructions allow to make these condition natural (1.100) to prove solvability by use of Corollary 1.15.

References [AuEk84] Aubin J-P, Ekeland I (1984) Applied nonlinear analysis. Wiley-Interscience, New York [AuFr90] Aubin J-P, Frankowska H (1990) Set-valued analysis. Birkhauser, Boston [Br72] Brezis H (1972) Problemes unilateraux. J Math Pures Appl 51:377–406 [BrHe72] Browder FE, Hess P (1972) Nonlinear mapping of monotone type in Banach spaces. J Func Anal. doi:10.1016/0022-1236(72)90070-5 [Ch97] Chikrii AA (1997) Conflict-controlled processes. Kluver, Boston [Cl90] Clarke FH (1990) Optimization and nonsmooth analysis. SIAM, Philadelphia [DeVa81] Dem’yanov VF, Vasiliev LV (1981) Non-differentiated optimization [in Russian]. Nauka, Moscow [Du65] Dubinskii YuA (1965) Weak convergence in nonlinear elliptic and parabolic equations. Mat Sb 67:609–642 [GaGrZa74] Gajewski H, Gr¨oger K, Zacharias K (1974) Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Akademie, Berlin [HuPa97] Hu S, Papageorgiou NS (1997) Handbook of multivalued analysis, vol I: Theory. Kluwer, Dordrecht [HuPa00] Hu S, Papageorgiou NS (2000) Handbook of multivalued analysis. vol II: Applications. Kluwer, Dordrecht [IoTi79] Ioffe AD, Tihomirov VM (1979) Theory of extremal problems. North Holland, Amsterdam [IvMe88] Ivanenko VI, Mel’nik VS (1988) Variational methods in control problems for systems with distributed parameters [in Russian]. Naukova dumka, Kiev [Li69] Lions JL (1969) Quelques methodes de resolution des problemes aux limites non lineaires. Dunod Gauthier-Villars, Paris [Me84] Mel’nik VS (1984) Method of monotone operators in theory of optimal systems with restrictions. Docl AN USSR Ser A 7:64–67 [Me97] Mel’nik VS (1997) Critical points of some classes of multivalued mappings. Cybern Syst Anal. doi:10.1007/BF02665895

References

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[MeSo97] Mel’nik VS, Solonoukha OV (1997) Stationary variational inequalities with multivalued operators. Cybern Syst Anal. doi:10.1007/BF02733070 [Pa87] Pachpatte BG (1987) Existense theorems for hammerstein-type integral equation. An sti Univ Iasi Sec Ia 33:197–201 [Pa85] Panagiotopoulos PD (1985) Inequality problems in mechanics and applications: convex and nonconvex energy functions. Birkh¨auser, Boston [PeKaZa08] Perestyuk MO, Kasyanov PO, Zadoyanchuk NV (2008) On solvability for the second order evolution inclusions with the volterra type operators. Miskolc Math Notes 9:119–135 [Ps80] Pshenichniy BN (1980) Convex analysis and extremal problems [in Russian]. Nauka, Moscow [ReSi80] Reed M, Simon B (1980) Methods of mathematical physics. vol. 1: Functional analysis. Academic, New York [Ru73] Rudin W (1973) Functional analysis. McGraw-Hill, New York [Sc71] Schaefer H (1971) Topological vector spaces. Springer, New York [Sh97] Showalter R (1997) Monotone operators in banach space and nonlinear partial differential equations. American Mathematical Society, Providence, RI [Sk94] Skrypnik IV (1994) Methods for analysis of nonlinear elliptic boundary value problems. American Mathematical Society, Providence, RI [Tr78] Triebel H (1978) Interpolation theory, function spaces, differential operators. NorthHolland, Amsterdam [Va73] Vainberg MM (1973) Variation methods and method of monotone operators. Wiley, New York [VaMe99] Vakulenko AN, Mel’nik VS (1999) Solvability and properties of solutions of one class of operator inclusions in Banach spaces. Naukovi visti NTUU “KPI” 3: 105–112 [ZaKa07] Zadoyanchuk NV, Kasyanov PO (2007) Faedo–Galerkin method for nonlinear second-order evolution equations with Volterra operators. Nonlinear Oscillations. doi:10.1007/s11072-007-0016-y [ZgMe99] Zgurovsky MZ, Mel’nik VS (1999) Nonlinear analysis and control of infinitedimentional systems [in Russian]. Naukova dumka, Kiev [ZgMe04] Zgurovsky MZ, Mel’nik VS (2004) Nonlinear analysis and control of physical processes and fields. Springer, Berlin [ZgMeNo04] Zgurovsky MZ, Mel’nik VS, Novikov AN (2004) Applied methods of analysis and control by nonlinear processes and fields [in Russian]. Naukova Dumka, Kiev

Chapter 2

Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Abstract In this chapter we propose the general approach for studying mathematical models of stationary geophysical processes and fields of different nature. In the first section we study functional-topological properties of the resolving operator for operator inclusions with multi-valued weakly coercive maps of pseudo-monotone type. In the second section we investigate properties of solutions for parameterized operator problems with non-smooth, probably, discontinuous or multi-valued interaction functions. Further, we investigate variational inequalities in locally convex spaces. In Sect. 2.4 we develop the multi-valued penalty method for the investigation of stationary unilateral problems of non-linearized viscoelasticity theory. Further, we consider a general approach to the investigation of contact problems which can be described by equations of Hammerstein type, study operators equations with non-coercive maps, develop the method of simulated control with corresponding applications which demonstrate some advantages of the presented mathematical toolbar. The exposition is accompanied by classes of real mathematical models with different non-linear relationships between determinative parameters.

2.1 Strong Solutions for Parameterized Operator Inclusions Suppose X is a reflexive Banach space, Y  is a normalized space, U  Y  is a  nonempty subset, A W X  U ! 2X is a multivalued map, f 2 X  , u 2 U are some fixed elements, Kf;u D fy 2 X j f 2 A .y; u/g : In this section we will investigate some properties of the set Kf;u . We denote a family of all nonempty closed convex subsets of the space X  by Cv .X  /, Br D fy 2 X j kykX  rg. Theorem 2.1. Let for any u 2 U A.; u/ W X ! Cv .X  / be a bounded pseudomonotone map and let the coercivity condition be satisfied: kyk1 X ŒA .y; u/ ; yC ! C1 as kykX ! 1: M.Z. Zgurovsky et al., Evolution Inclusions and Variation Inequalities for Earth Data Processing I, Advances in Mechanics and Mathematics 24, c Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-13837-9 2, 

(2.1) 139

140

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Then for any f 2 X  there can be found r > 0 such that the set Kf;u nonempty and weakly compact.

T

Br is

Proof. Let us fix u 2 U . Without loss of generality we denote A.y/ D A.y; u/ 8y 2 X . Let F .X / be a family of all finite-dimensional subspaces of the space X . For F 2 F .X / by IF W F ! X we denote the canonical embedding and by  IF W X  ! F  we denote its adjoint operator. We set AF D IF AIF W F ! 2F . Lemma 2.1. For each F 2 F .X /, AF D coAF and it is an upper semicontinuous map. Proof. For an arbitrary x 2 F the set AF .x/ is convex and closed in F  . The convexity is obvious, let us prove that it is closed. Suppose zn 2 AF .x/ and zn ! z0 in F . Let us consider gn 2 A .x/ ; zn D w  IF gn . The sequence fgn g is bounded in X  and we can assume that gn ! g0 in X  , moreover g0 2 A .x/, since A D coA. Then however zn ! z0 2 AF .x/, namely AF .x/ 2 Cv .F  / 8x 2 F . Let us prove upper semicontinuity. The set AF .x/ is bounded, and hence it is compact in F  . Let us assume that at the point x0 2 F the map AF is not upper semicontinuous. Then there can be found " > 0 such that in every ball   1 B 1 .x0 / D x 2 F j kx  x0 kF < n n a point xn can be chosen such that AF .xn / 6 B" .AF .x0 // D fz … F  j dist .z; AF .x0 // < "g : Let us consider sequences fxn g ; fzn g where zn 2 AF .xn / nB" .AF .x0 // ;

xn 2 B 1 .x0 / : n

The sequence xn converges to x0 in F , and the sequence zn is bounded due to boundness of the map AF . Without loss of generality we assume that zn ! z0 in F  . In virtue of Propositions 1.48 and 1.49 the map AF is demiclosed whence closeness of AF follows. So z0 2 AF .x0 /, and this fact contradicts to zn … B" .AF .x0 //. The Lemma is proved. Let us consider the function  W RC ! R,  .r/ D

inf

kykX Dr

kyk1 X ŒA .y/ ; yC :

It is clear that lim  .t/ D C1. Then t !1

ŒA .y/  f; yC  . .kykX /  kf kX  / kykX ;

2.1 Strong Solutions for Parameterized Operator Inclusions

141

and there can Tbe found r > 0 such that for any y 2 @Br . For F 2 F .X / we set B r;F D B r F and obtain the similar estimation for the map AF , namely for any x 2 @B r;F ŒAF .x/  fF ; xC  0 where fF D IF f . For the map AF Corollary 1.16 is applicable, namely for any F 2 F .X / there exists f such that (2.2) fF 2 AF .yF / ; here yF 2 B r;F . In virtue of Proposition 1.39 the inclusion (2.2) is equivalent to the following inequality: (2.3) ŒAF .yF / ; xC  hfF ; xiF 8x 2 F: For an arbitrary F0 2 F .X / we set GF0 D

\ ˚ ˇ yF 2 B r;F ˇ yF satisfies (2.3)g: F F0

The set GF0 ¤ ; and it is contained in the closed ball B r . Moreover S for an arbitrary family F1 ; : : : ; Fn 2 F .X / and F 2 F .X / such that F  niD1 Fi we have T w w ; ¤ GF  niD1 GFi , namely the system of sets G F is centered where G F is a weak closureTof the set GF in X . Then in virtue of reflexivity of the space X there w exists y0 2 F 2F .X/ G F . We prove that y0 2 Kf;u . Let us fix an arbitrary w 2 X and choose F0 2 F .X / under the condition y0 ; w 2 F0 . Then there can be found a sequence fyn gn1  GF0 such that yn * y0 in X and also dn0 2 AFn .yn /, here dn0 D fn where yn 2 Fn \ B r ;

F0  Fn 2 F .X /;

fn D IFn f:

Therefore, lim hdn ; yn  y0 iX D lim hf; yn  y0 iX D 0;

n!1

dn0

where D subsequences

IFn dn , fdm g 

n!1

and since the map A is -pseudomonotone then up to a fdn g, fym g  fyn g for which

lim hdm ; ym  wiX  ŒA.y0 /; y0  w

8w 2 X;

m!1

whence we get ŒA.y0 /; y0 w  hf; y0 wiX or ŒA.y0 /; y0 wC  hf; y0 wiX 8w 2 X that is equivalent to the inclusion A.y/ 3 f . Therefore nonemptyness of Kf;u is proved. Weak compactness Kf;u can be easily verified. The Theorem is proved. Remark 2.1. It is clear that Theorem 2.1 remains valid for both pseudomonotone and s -pseudomonotone maps. Let for u 2 U K0;u D fy 2 X jA.y; u/ 3 0g be a set of critical points of the map A.; u/.

142

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Theorem 2.2. Suppose A.; u/ W B r ! Cv .X  / is a bounded demiclosed map for some u 2 U . Let Condition ˛.Br / for A.; u/ be satisfied and let ŒA.y; u/; yC  0 8y 2 @Br :

(2.4)

Then the set K0;u \ B r is nonempty and weakly compact. Proof. Let us fix u 2 U . To simplify the proof let us suppose that the space X is separable and A.y/ D A.y; u/ 8y 2 X . Suppose fhi g1 i D1 is its arbitrary complete system and each n elements h1 :::::hn are linearly independent. We denote a linear span of the elements h1 :::::hn by Xn , Jn W Xn ! X is an embedding operator, Jn W X  ! Xn is adjoint operator. For each n we define a finite-dimensional  multivalued map An W Xn ! 2Xn by the rule An .y/ D

Jn A.Jn y/

D

[ d 2A

(

n X hd.y/; hi iX hi

) ; y 2 B n;r D Br

\

Xn :

i D1

To prove the Theorem we need the following Lemma. Lemma 2.2. The map An is convex-valued and upper semicontinuous on B n;r . Proof. For an arbitrary y 2 B n;r the set An .y/ is convex and closed in Xn . Convexity immediately follows from linearity of the operator Jn . Let us prove that it is closed. Suppose kn 2 An .y/ is an arbitrary sequence such that kn !  n in Xn . We consider its inverse image k by the map Jn , namely k 2 A.Jn y/;

kn D Jn k :

A sequence fk g is bounded in X  and due to Banach–Alaoglu Theorem there can be isolated a weakly convergent subsequence m ! , and here  2 A.Jn y/ since A.Jn y/ 2 Cv .X  /. Therefore, n D Jn .m / !  n D Jn ./: m

Hence,  n 2 An .y/ and closure is proved. Let us prove that for every n a map An W Xn ! Cv .Xn / is upper semicompact, namely for an arbitrary sequence yk 2 Bn;r convergent to yk 2 B n;r there can be found a subsequence fym g and dm 2 An .ym / such that dm ! d in Xn and d 2 An .y/. So let fyk g be an arbitrary convergent sequence in B n;r and dk 2 An .yk /. From boundness of the map A boundness of An for each n follows, hence from the sequence fdk g there can be isolated a convergent subsequence dm ! d in Xn . Since from demicloseness of A and closeness of An it follows that d 2 An .y/ namely the map An is upper semicompact. To complete the proof of Lemma 2.2 we use the following Proposition. Proposition 2.1. Let X and Y be Hausdorff topological spaces. Each upper semicompact map F W X ! 2Y is upper semicontinuous.

2.1 Strong Solutions for Parameterized Operator Inclusions

143

Proof. Assume the contrary. Let x0 2 X be a point at which F is not upper semicontinuous. Then there can be found a neighborhood V of the set F .x0 / such that in any neighborhood of the point U there exist xU 2 U such that F .xU / 6 V . Let us consider nets fxU 2 U jU 2 ˝x0 g; fU 2 F .xU /g; where ˝x0 is an ordered by inclusion filter of neighborhoods of the point x0 . At the same time the net fxU g converges to the point x0 and due to upper semicompactness of the map F there can be isolated a subnet fu g convergent in Y to the point 0 and at the same time 0 2 F .x0 /. This contradicts to the fact that U 2 V 8U 2 ˝x0 . The Proposition is proved. The Lemma is proved. For the map An the inequality (2.4) is valid, namely ŒAn .y/; yC  0

8y 2 @Bn;r :

Hence by Lemma 2.2 and by Corollary 1.16 the existence of an element yn 2 Bn;r such that (2.5) 0 2 An .yn / follows. The sequence fyn g is bounded in X and we may assume that yn ! y0 2 Br weakly in X (if the sequence is not weakly convergent we pass to a subsequence). Since a family of maps fJn g separates points we see that in virtue of (2.5) there exists d 2 A for which d.yn / ! 0 weakly in X  . Let wn 2 X be such that wn ! y0 strongly in X . Then in virtue of Proposition 1.38 ŒA.yn /; yn  y  ŒA.yn /; yn  wn  C ŒA.yn /; wn  y0 C

(2.6)

The first summand in the right side of the inequality (2.6) is nonpositive since (2.5) holds. Let us obtain an estimate of the second summand: ŒA.yn /; wn  y 0 C  kA.yn /kC kwn  y0 kX  l kwn  y0 kX  Hence from inequality (2.6) we have lim ŒA.yn /; yn  y0   0;

n!1

and since the map A has Property ˛.B r / we have that yn ! y0 strongly in X and at the same time d.yn / ! 0 weakly in X  . ButT the operator A is demiclosed therefore 0 2 A.y0 /. So we proved that the set K0;u B r is Tnonempty. Now it remains to prove the weak compactness only. Let fyn g  K0;u Br , then there can be isolated a subsequence fym g such that ym ! y0 weakly in X and 0 2 A.ym /. Therefore, ŒA.ym /; ym  y0   0;

144

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

and hence we conclude (in virtue of Condition ˛) that ym ! y0 strongly in X . From demicloseness of A we obtain 0 2 A.y0 /. The Theorem is proved. A critical point y0 2 K0;u of a map A is called an isolated point if there exists a closed ball B r .y0 / D fy 2 X j ky  y0 kX  rg; that does not contain any other isolated point of the map A but y0 . Let D be an arbitrary bounded open subset in X . 

Theorem 2.3. Suppose for some fixed u 2 U A.; u/ W D  X ! 2X is demiclosed map for which Condition ˛.D/ is satisfied. Let it have isolated critical points only in D. Then the number of such points is finite. Proof. Assume the contrary, namely that a set of critical points in D is infinite and let us choose a sequence fy k g from it. We assume that yk ! y0 weakly in X . Since 0 2 A.yk ; u/ we have ŒA.yk ; u/; yk  y0   0 and in virtue of Condition ˛.D/ yk ! y0 strongly in X  . Moreover there exists a selector d./ 2 A.; u/ such that d.yk / ! 0 weakly in X  . Hence we conclude that y0 2 D and 0 2 A.y0 ; u/. The obtained critical point occurs not isolated and this fact contradicts to the conditions of the Theorem. The Theorem is proved. Theorem 2.4. Let A W X  U  X  satisfies Property SNk , U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in X  . Then \ [

w

K.um ; fm /  K.u; f /;

(2.7)

n1 mn w

where U is the weak closure of U  X in X . Proof. Let U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in X  . Let T S w K.um ; fm / is nonempty (otherwise us check (2.7). We assume that the set n1 mn T S w the inclusion (2.7) holds true). So, if y0 2 K.um ; fm / then there is such n1 mn

sequence fnk gk1 that y0 is a weak cluster point in X of fynk gk1 : 8k  1 ynk 2 K.unk ; fnk /. We remark that hfnk ; ynk  y0 iX ! 0 as k ! 1. Thus, due to Property SNk for A, f 2 A.y0 ; u/, i.e., y0 2 K.u; f /. The weak closureness of K.u; f / follows from (2.7). The Theorem is proved.

2.2 Parameterized Operator Inequalities

145

2.2 Parameterized Operator Inequalities Suppose X is reflexive Banach space, X  is its topologically adjoint, Y  is adjoint space to some either separable normalized or reflexive Banach space, U  Y  is  nonempty subset. For multivalued map A W X  U ! 2X in addition to ŒA.y; u/; wC D sup hd .y; u/ ; wiX ;

8y; w 2 X;

d 2A

and ŒA.y; u/; w D inf hd .y; u/ ; wiX ; d 2A





let us consider associated maps coA W X  U ! 2X , coA W X  U ! 2X defined by the following relations coA .y; u/ D co .A .y; u//, coA .y; u/ D co .A .y; u//. 

Proposition 2.2. Let a map A W X  U ! 2X be bounded-valued. Then d 2 coA .y; u/ if and only if ŒA.y; u/; wC  hd; wiX 8w 2 Z, where Z is some dense subset in X . Proof. For an arbitrary v 2 X 9 fvn g  Z that vn ! v in X , and ŒA.y; u/; vn C  hd; vn iX :

(2.8)

Due to boundness of A.y; u/, for each n D 1; 2; : : : 9d 2 coA .y; u/ such that ŒA.y; u/; vn C D hd; yn iX . Without loss of generality we may assume (otherwise we pass to a subsequence) that dn ! d0 .v/ D d0 weakly in X  and d0 2 coA .y; u/. Then in virtue of (2.8) for v 2 X we have hd0 ; viX  hd; viX whence ŒA.y/; vC  hd0 .v/ ; viX  hd; viX 8v 2 X and in virtue of Proposition 1.39 d 2 coA .y/. The inverse statement is obvious.  N We remind that the map A W X ! 2X is called -pseudomonotone (pseudomonotone, respectively), if from weak convergence yn ! y in X and from the inequality lim hdn ; yn  yiX  0;

n!1

where dn 2 coA .yn / (dn 2 A .yn / respectively) there can be isolated the following subsequences fym g  fyn g, fdm g  fdn g such that lim hdm ; ym  wiX  ŒA .y/ ; y  w

8w 2 X:

m!1

Remark 2.2. The following implication is valid: “A is -pseudomonotone” ) “A N is -pseudomonotone”. N The important class of -pseudomonotone maps is parameterized multivalued maps which occur in the control theory, namely the following Proposition holds.

146

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Proposition 2.3. Suppose A W U  X ! X  , U is bounded weakly star closed subset in the space Y  . Let the operator A be quasimonotone [ZgMe04], namely let from U 3 un ! u weakly star in U , yn ! y weakly in X and lim hA .un ; yn / ; yn  yiX  0;

n!1

the existence of subsequences fum g  fun g, fym g  fyn g such that lim hA .um ; ym / ; ym  wiX  hA .u; y/ ; y  wiX

8w 2 X

n!1

ˇ n o ˇ N follow. Then the map A .y/ D A .u; y/ ˇ u 2 U is -pseudomonotone. Proof. Let yn ! y weakly in X and lim hdn ; yn  yiX  0 where dn 2 A .yn / n!1

8n  1. Then 9un 2 U such that dn D A .un ; yn /. Due to boundness of the set U we may assume that un ! u weakly star in U , and due to quasimonotony hA .un ; yn / ; yn  yiX ! 0 and lim hA .um ; ym / ; ym  wiX  hA .u; y/ ; y  wiX

8w 2 X;

n!1

where it is sufficient to set dm D A .um ; ym /. Remark 2.3. It is easy to give an example of the parameterized map: A D coA. In N this case -pseudomonotony coincides with -pseudomonotony. Suppose @ .X / D fF  X j F is a linear subspace, dim F < 1g, IF W F ! X is an embedding operator, IF W X  ! F  is its adjoint operator. For each F 2  @ .X / let us consider the map AF W F  U ! 2F defined by the commutative property of the following diagram: A

X " IF

!

F

!

AF



2X # IF 2F



namely AF D IF A. 

We remind that the map A W X ! 2X is called finite-dimensionally bounded if  8F 2 @ .X / the map AF W F ! 2F is bounded. Every bounded map is finitedimensionally bounded but the inverse Proposition does not hold. Together with a  map A W X  U ! 2X and elements f 2 X  , u 2 U let us consider an operator inclusion A .y; u/ 3 f: (2.9) Definition 2.1. An element y 2 X is called a weak solution of the inclusion (2.9) if ŒA .y; u/ ; wC  hf; wiX

8w 2 X:

(2.10)

2.2 Parameterized Operator Inequalities

147 

Theorem 2.5. Let for any u 2 U A.; u/ W X ! 2X be a strict -pseudomonotone finite-dimensionally bounded map and for any f 2 X  there exists r > 0 such that ŒA .y; u/  f; yC  0

8y 2 @Br  X:

Then 8f 2 X  , u 2 U there exists a weak solution of the operator inclusion (2.9). Proof. Let us fix u 2 U . To simplify the proof without loss of generality we denote  A.y/ D A.y; u/ 8y 2 X . For each F 2 @ .X / we set AF D IF coA W F ! 2F . 

Remark that AF D IF coA .x/ 2 Cv .F  /. The last relation can be obtained from the following Lemma. Lemma 2.3. The following equality is valid coIF A.x/ D IF coA.x/

8x 2 F:

Proof. First of all let us remark that coIF A D IF coA. Indeed, 8x 2 F the embedding IF A .x/  IF coA .x/ takes place, therefore, coIF A .x/  IF coA .x/ since IF coA .x/ is a convex set. Let us prove the inverse embedding. Let  2 IF coA .x/, hence there can be found hP2 coA .x/ such that  D IF h. In itsPturn 9 fi gniD1 n n and fdi gniD1 i D1 i di . But Pnthat i  0, PinD1 i D 1, di 2 A.x/ and h D then  D i D1 i IF di D i D1 i hi , hi 2 IF A .x/, that is  2 coIF A .x/ and hence coIF A .x/  IF coA .x/. Therefore, coIF A.x/  IF coA.x/ 8x 2 F and coIF A D IF coA. On the other hand coIF A.x/  IF coA.x/ 8x 2 F therefore coIF A.x/  IF coA.x/, and since for continuous maps the embedding takes place IF coA.x/  IF coA.x/ we have that IF coA.x/  IF coA.x/ and this fact proves the equality coIF A D IF coA. 

Lemma 2.4. The map AF D IF coA W F ! 2F is upper semicontinuous. Proof. The proof is similar to the one of Lemma 2.1. T For each F 2 @ .X / we set Br;F D B r F . Then ŒAF .x/  fF ; xC D ŒA .x/  f; xC  0

8x 2 @B r;F ;

where fF D IF f . So for the map AF ./  fF Corollary 1.16 is applicable whence it follows that 8F 2 @ .X / 9yF 2 Br;F such that AF .yF / 3 fF :

(2.11)

In virtue of Proposition 1.39 the inclusion (2.11) is equivalent to the inequality ŒAF .yF / ; wC  hfF ; wiF

8w 2 F

(2.12)

148

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

For an arbitrary F0 2 @ .X / we set GF0 D

[

fyF 2 Br;F j yF satisfies (2.12)g:

F F0

It is clear that the set GF0 is nonempty and that it is contained in the ball B r . Besides for an arbitrary finite familyTF1 ; : : : ; Fn 2 @ .X / and F 2 @ .X / such that S n n i D1 Fi  F we have ; ¤ GF  i D1 GFi . w w Therefore the system of sets fG F g is centered, where G F is a weak closure of the set GF in X . In Theorem B r is a compact in the weak Tvirtue of Banach–Alaoglu w topology. Hence, F [email protected]/ G F ¤ ;. T w We consider y0 2 F [email protected]/ G F . Let us fix an arbitrary w 2 X and choose F0 2 @ .X / under the condition y0 ; w 2 F0 . Then there can be found a sequence to y0 in the space X and also dn0 2 AFn .yn / fyn g 2 GF0 weakly convergent T 0 .dn D fn /, where yn 2 B r Fn , F0  Fn 2 @ .X /, fn D IFn f . In this case ˝ ˛ lim hdn ; yn  y0 iX D lim dn0 ; yn  y0 Fn D lim hfn ; yn  y0 iFn

n!1

n!1

n!1

D lim hf; yn  y0 iX D 0; n!1

where dn0 D IFn dn , dn 2 coA .yn /. Since the map A is -pseudomonotone then at least for some subsequences fym g  fyn g, fdm g  fdn g we have lim hdm ; ym  wiX  ŒA .y0 / ; y0  w

8w 2 X;

m!1

whence ŒA .y0 / ; y0  w  hf; y0  wiX or ŒA .y0 / ; vC  hf; viX 8v 2 X and the Theorem is proved. Remark 2.4. Analyzing the proof of Theorem 2.5, there should be remarked that all its conclusions are valid for a weaker boundness condition, namely for the case when the map A is finite-dimensionally locally bounded: 8F 2 @ .X / and yF 2 F 9N > 0 and " > 0 such that 

kAF .F /kC D sup kdF .F /kF   N dF 2AF

8 kF  yF kF  ": 

Theorem 2.6. Suppose for any u 2 U A.; u/ W X ! 2X is a strict -pseudomonotone map and 8F 2 @ .X / an operator AF .; u/ W F ! Cv .F  / is upper semicontinuous and the condition (2.12) holds true. Then Theorem 2.5 takes place. Suppose V , W are reflexive Banach spaces, A W V  U  V  , B W W  U  W are multivalued maps, the space X D V \ W is dense in V and in W , then the 

2.2 Parameterized Operator Inequalities

149

dual space X  D V  C W  . For f 2 X  and v 2 X we set hf; viX D hf1 ; viV C hf2 ; viW ; where f D f1 C f2 , f1 2 V  , f2 2 W  . For a fixed f 2 X  , u 2 U let us investigate the operator inclusion A.y; u/ C B.y; u/ 3 f:

(2.13)

We denote by F .X / a filter of all finite dimensional subspaces in X . Let for F 2 F .X / IF W F ! X be the embedding operator, let IF W X  ! F  be the adjoint operator. For each F 2 F .X / we consider the map AF W F  U  F  generated by the map A D A C B W X  U  X by the rule AF D IF A. Theorem 2.7. Let A W V  U  V  , B W W  U  W  be -quasimonotone and for any u 2 U A.; u/, B.; u/ be finite-dimensionally locally bounded maps and let both coercive conditions be satisfied ŒA.y; u/; yC  A .kykV / kykV

(2.14)

ŒB.y; u/; yC  B .kykW / kykW

(2.15)

where the functions A ./ W RC ! R, B ./ W RC ! R are bounded below on any segment, A .s/ ! C1, B .s/ ! C1 as s ! 1 and one of the operators is bounded-valued. Then for each f 2 X  , u 2 U the operator inclusion (2.13) has a weak solution, namely 9y 2 X such that ŒA.y; u/; vC C ŒB.y; u/; vC  hf; viX

8v 2 X:

(2.16)

Theorem 2.8. A W V  U  V  , B W W  U  W  be -quasimonotone and for any u 2 U A.; u/, B.; u/ satisfy (2.14), (2.15) and one of them is bounded-valued and the operator A .; u/ D A.; u/ C B.; u/ W X  X  has a property (˘ ) and is finite-dimensionally locally bounded. Then for each f 2 X  , u 2 U there can be found r > 0 such that Ku;f \ BN r is nonempty and weakly compact where BN r is a closed ball in X with the radius r, Ku;f D fy 2 X jy satisfies (2.16)g: Moreover, if U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in X  . Then \ [ n1 mn

w

Kum ;fm  Ku;f :

(2.17)

150

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Proof. Let us prove Theorem 2.7. Let us fix an arbitrary u 2 U . We consider the operator A D .A C B/.; u/ W X  X  ;

A .y/ D A.y; u/ C B.y; u/;

y 2 X:

Due to Lemmas 1.1, 1.9 and 1.11 to solve problem (2.16) is remains to find y 2 X such that ŒA .y/; vC  hf; viX ; where A W X  X  is finite-dimensionally locally bounded, -pseudomonotone map for which the coercivity condition kyk1 X ŒA .y/; yC ! C1 as kykX ! 1

(2.18)

is satisfied. Let us denote ˚ Kf D y 2 X j ŒA .y/; vC  hf; viX ;

 8v 2 X :

Suppose BN r is a closed ball in X with the radius r. Proposition 2.4. Suppose A W X  X  is finite-dimensionally locally bounded pseudomonotone map and (2.18) is valid then for each f 2 X  there can be found r > 0 such that the set Kf \ BN r is nonempty and weakly compact. Proof. For each F 2 F .X / we set ANF D IF coA W F  F  and remark that ANF .x/ 2 Cv .F  / 8x 2 F . The last relation follows from Lemma 2.3. Let us consider the function  W RC ! R, defined by the relation .r/ D

inf

kykX Dr

kyk1 X ŒA .y/; yC :

In virtue of condition (2.18) it would be easy to make sure that .r/ ! 1 as r ! 1, and ŒA .y/  f; yC  . .kykX /  kf kX  / kykX : Hence there can be found r > 0 such that ŒA .y/  f; yC  08y 2 @Br :

(2.19)

Similarly to the proof of Lemma 2.1 there can be showed that 8F 2 F .X / ANF W F  F  is upper semicontinuous map.

2.2 Parameterized Operator Inequalities

151

We set BN r;F D F \ BN r , then 8x 2 @Br;F the estimation 

ANF .x/  fF ; x

 C

 0;

(2.20)

where fF D IF f takes place. Indeed, using (2.19)   ANF .x/  fF ; x C D D

sup dF 2ANF .x/

sup d 2coAN.x/

hdF  fF ; xiF

hd  f; xiX D ŒA.x/  f; xC  0:

Here dF D IF d , d 2 coA .x/. Hence in virtue of the properties of the upper and the lower forms and of finitedimensional local boundness A we conclude that ANF W F  F  is an upper semicontinuous map. Hence all the conditions of multivalued analogue of acute angle Lemma (Corollary 1.16) hold true, by this Lemma the existence of yF 2 BNr;F for which ANF .yF / 3 fF follows and this fact is equivalent to ŒA .yF /; xC  hf; xiX

8x 2 F

(2.21)

Further with some technical modifications the proof of the Proposition can be completed in the same way as the correspondent part of the one of Theorem 2.1. For each F0 2 F .X / we define GF0 D

[ ˚ yF 2 B r;F jŒAF .yF /; x C  hfF ; xiF

8x 2 F



F F0

Obviously GF0 ¤ ; and it is contained in B r which is a weakly compact set due to Banach–Alaoglu Theorem. Let us consider a family of nonempty subsets w

fGF jF 2 F .X /g and fG F jF 2 F .X /g; w

where G F is a weak closure of the set GF in X . S For any finite family F1 ; : : : ; Fn 2 F .X / and F 2 F .X / such that Fi  F i D1

we have GF 

n \

GFi ¤ ;:

i D1

Hence the family fGF jF 2 F .X /g and especially w

fG F jF 2 F .X /g

152

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

are centered families in the weak compact B r , therefore [ReSi80] \

w

G F ¤ ;:

F 2F .X/

T

Let us fix an arbitrary v 2 X . We consider y0 2

w

F 2F .X/

G F and choose F0 2 F .X /

such that y0 ; v 2 F0 . Then there can be found a sequence yn 2 Fn  F0 (yn 2 B r ), Fn 2 F .X / weakly convergent to y0 in X and also dn0 2 AFn .yn /, dn0 D fn D IFn f . In this case lim hdn ; yn  y0 iX D lim hdn0 ; yn  y0 iFn

n!1

n!1

D lim hfn ; yn  y0 iF D lim hf; yn  y0 iX D 0; n!1

n!1

where dn 2 coA .yn /, dn0 D IFn dn , in virtue of IFn coA .yn / D coIFn A .yn / D coAFn .yn /: Since the operator A W X  X  is -pseudomonotone there can be found subsequences fym g, fdm g such that lim hdm ; ym  viX  ŒA .y0 /; y0  v

8v 2 X:

(2.22)

m!1

On the other hand for v 2 X and 8m 0 hdm ; ym  viX D hdm ; ym  viFm D hfm ; ym  viFm D hf; ym  viX ;

that is lim hdm ; ym  viX D hf; y0  viX : m!1

From here and from (2.22) we obtain hf; y0  viX D lim hdm ; ym  viX  ŒA .y0 /; y0  v

8v 2 X;

m!1

that is equivalent to ŒA .y0 /; wC  hf; wiX

8w 2 X;

that is Kf \ Br ¤ ;. Now let fyn g  Kf \ Br be an arbitrary sequence. It is bounded and we may assume that yn ! y0 weakly in X (up to a subsequence), at the same time ŒA .yn /; wC  hf; wiX

8w 2 X;

2.2 Parameterized Operator Inequalities

153

that is equivalent to the inclusion coA .yn / 3 f . Hence for each n  1 there can be found dn 2 coA .yn / such that dn D f . Hence we immediately conclude that lim hdn ; yn  y0 iX  0

n!1

and due to -pseudomonotony of the operator A 9fym g  fyn g, fdm g  fdn g such that hf; y0  viX D lim hdm ; ym  viX  ŒA .y0 /; y0  v

8v 2 X;

m!1

that is ŒA .y0 /; wC  hf; wiX 8w 2 X . So, y0 2 Kf \ Br . Since ŒA .y/; vC D ŒA.y; u/; vC C ŒB.y; u/; vC the Theorem is proved. Remark 2.5. Theorem 2.8 can be proved similarly. The proof for (2.17) is similar to the proof of (2.7). Theorem 2.9. Let A W X  U  X  satisfies property SNk , U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in X  . Then \ [

w

K.um ; fm /  K.u; f /;

(2.23)

n1 mn w

where U is the weak closure of the set U  X in X . Moreover, if there are u 2 U , f 2 X  , r > 0 such that ŒA.y; u/  f; yC  0

8y 2 X W kykX D r;

(2.24)

and X 3 y ! A.y; u/ is finite-dimensionally locally bounded and satisfies Property (˘ ), then K.u; f / is nonempty and weakly closed set in X . The proof is similar to the previous one. Remark 2.6. The sufficient condition for (2.24) is C-coercivity condition for A.; u/ W X  X . Now we consider the bounded multivalued map A W X  X  , that satisfies Property (Sk ), it is C-coercive, but it is not -coercive, it is not -pseudomonotone and A is not -pseudomonotone too. Example 2.1. Let n  2, ˝  Rn be a bounded domain, XR D H01 .˝/ is a real Sobolev space [GaGrZa74], X  H 1 .˝/, .u; v/L2 D u.x/v.x/dx, u; v 2 ˝ R L2 .˝/; ..u; v// D ru  rvdx is the inner product in H01 .˝/, u; v 2 H01 .˝/; a  b ˝ p is the inner product of vectors a; b 2 Rn ; kukH 1 .˝/ D ..u; u//, u 2 X . 0

154

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

For any y 2 X let us set A.y/ D fdiv˛ryj˛ 2 Œ1; 1g: We remark that A W X  X  , coA.y/ D A.y/ D A.y/ 8y 2 X . Let us check that A satisfies the property Sk . Let yn ! y weakly in X , dn ! d weakly in X  (dn D ˛n 4yn 2 A.yn /, ˛n 2 Œ1; 1, n  1) and hdn ; yn yi ! 0 as n ! 1. Then 8n  1 hdn ; yn  yi D ˛n h4yn ; yn  yi D ˛n kyn  yk2X  ˛n h4y; yn  yi: Due to yn ! y weakly in X and j˛n j  1 8n  1, then ˛n kyn  yk2X ! 0 as n ! 1. Hence there is a subsequence fym ; ˛m gm  fyn ; ˛n gn1 such that either ˛m ! 0 or kym  ykH 1 .˝/ ! 0. At that case when ˛m ! 0 (taking into 0

account that 4 W H01 .˝/ ! H 1 .˝/ is a bounded operator) we have that dm D ˛m 4ym ! 0N 2 A.y/. On another hand when ym ! y strongly in H01 .˝/ (taking into account that j˛m j  1 8m) we have that up to a subsequence fyk ; ˛k g  fym ; ˛m g ˛k 4yk ! ˛4y strongly in X for some ˛ 2 Œ1; 1. So, d D ˛4y 2 A.y/. C-coercivity for A follows from ŒA.y/; yC D sup ˛h4y; yi D kyk2X

8y 2 X:

j˛j1

The boundedness for A follows from the estimation kA.y/kC  kykX 8y 2 X . A is not -coercive, because ŒA.y/; y D kyk2X 8y 2 X . Now let us show that A is not -pseudomonotone. We consider an arbitrary orthonormal vector system fyn gn1 in infinite-dimensional Hilbert space X D H01 .˝/. It is well-known, that yn ! 0N weakly in X . Let dn D 4yn 2 A.yn / N D kyn k2 D 1 8n  1. So, lim hdn ; yn  0i N < 8n  1. Then hdn ; yn  0i X n!1 N D 1 < 0 D ŒA.0/; N 0N  0 N  for any subsequence 0, but limhdm ; ym  0i m

fym ; dm gm  fyn ; dn gn . In virtue of A.y/ D A.y/ 8y 2 X we obtain that A is not -pseudomonotone too. Example 2.2. Let us consider examples related to parameterized operator inclusions and multivariational inequalities and illustrate substantial dependence of Property SNk on a choice of topology in a space of parameters. Firstly we will build a bounded mappings that satisfies Property SNk , but it is not -quasimonotone and A is not -quasimonotone too. Assume that n  2, ˝  Rn is a bounded domain with a rather smooth boundary  , X D H01 .˝/ is a real Sobolev space [GaGrZa74], X  H 1 .˝/, Z .u; v/L2 .˝/ D

u.x/v.x/dx; ˝

u; v 2 L2 .˝/I

2.2 Parameterized Operator Inequalities

155

is the scalar product in W W  D L2 .˝/, Z ru  rvdx;

..u; v// D

u; v 2 H01 .˝/

˝

is the scalar product in H01 .˝/; a  b is the scalar product of vectors a; b 2 Rn ; p kukH 1 .˝/ D ..u; u//, u 2 X . Suppose 1 ; 2 are the given functions with L1 .˝/ 0 such that 1 .x/  2 .x/ a.e. in ˝; (2.25) where ˛ > 0. Let us set    U D U D uij .x/ 1i;j n

ˇ  ˇ uj i D uij 2 L1 .˝/ 8i; j D 1; n; ˇ ˇ 1 .x/  uij .x/  2 .x/a.e. in ˝; 8i; j D 1; n :

U forms a nonempty set of uniformly bounded, symmetric square matrices. Also let Y D ŒL1 .˝/nn . Then Y  D ŒL1 .˝/nn . Consider the set V D fU D Œu1 ; u2 ; : : : ; un  2 ŒL1 .˝/nn W divui D 0 8i D 1; 2; : : : ; ng; where the value of the operator div on a vector u 2 ŒL2 .˝/n (see for detail [KapKasKoh08, p. 1628]) is defined as the element of the space H 1 .˝/ such that Z hdivu; iH 1 .˝/ D 

.u; r/Rn dx; 8 2 H01 .˝/:

0

˝

T We say that a functional parameter U is admissible if U 2 V U , where the set U is defined above. The set of all admissible parameters is denoted by U. Note that U is sequentially compact in the weakly star topology of the space ŒL1 .˝/nn (see for detail [KapKasKoh08, p. 1628]). Let us consider the operator A.y; U / D div.U .x/ry/ D 

n X i;j D1

@ @xi

uij .x/

@y ; @xj

regarded as a mapping A W X  U ! X  . Now we prove that the parameterized multi-valued mappings satisfies Property SNk (see for detail [KapKasKoh08, p. 1629]) by the lemma on compensated compactness [KapKasKoh08]. Lemma 2.5. The multivalued mappings A W X  U ! X  satisfies Property SNk . Proof. Let Uk D Œu1k ; u2k ; : : : ; unk  ! U0 D Œu10 ; u20 ; : : : ; un0  *-weakly in Y  , yk ! y0 weakly in X , dk ! d0 weakly in X  and lim hdk ; yk  y0 iX D 0;

k!C1

156

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

where dk D A.yk ; Uk /, Uk 2 U 8k  1. Let us set pk D ui k , p0 D ui 0 , vk D ryk , v0 D ry0 . Since divpk D 0 and rotvk D 0 8k  1 (see for detail [KapKasKoh08, p. 1626, 1629]), ryk ! ry0 weakly in ŒL2 .˝/n and Uk ! U0 *-weakly in ŒL1 .˝/nn and weakly in ŒL2 .˝/nn , we see that by the lemma on compensated compactness, Z lim

k!1 ˝

D

.Uk ryk ; /Rn dx D

i D1

n Z X i D1

˝

Z

n X

.ui 0 ; ry0 /Rn i dx D

lim

k!1 ˝

.ui k ; ryk /Rn i dx

Z ˝

.U0 ry0 ; /Rn dx;

8 2 ŒC01 .˝/n :

Since the set C01 .˝/ is dense in L2 .˝/, we see that Uk ryk ! U0 ry0 weakly in ŒL2 .˝/n . Hence, dk D div.Uk ryk / ! div.U0 ry0 / D d0 weakly in X  : So, d0 D A.y0 ; U0 /, that was to be shown. The Lemma is proved. Taking into account (2.25), neither A no A is -quasimonotone in the general case (see for detail [KapKasKoh08, p. 1624-1625]). Moreover, if we set A .y/ D fA.y; U /jU 2 Ug; y 2 X; we obtain a bounded multi-valued mapping which satisfies Property Sk , but it is not -pseudomonotone mapping and A is not -pseudomonotone mapping too. A sufficient condition for C-coercivity A is for example such condition: 9˛ > 0 W for a.e. x 2 ˝

2 .x/  ˛; 1 .x/ D 0:

(2.26)

Lemma 2.6. Under condition (2.26), a multivalued mapping A is C-coercive mapping. Proof. Indeed, if we set U0 .x/ D diag.˛; ˛; : : : ; ˛/ for a.e. x 2 ˝ (note that U0 2 U), then ŒA .y/; yC  hA.y; U0 /; yiX D ˛kyk2X The Lemma is proved.

8y 2 X:

2.2 Parameterized Operator Inequalities

157

Remark 2.7. Under condition (2.26), 8y 2 X

ŒA .y/; y  0:

Hence, A , moreover, is not -coercive mapping. Let us consider a convex s.s.c. functional W R ! R such that .u/ 2 L1 .˝/, for any u 2 W . Let us denote a subdifferential of by ˚ W R  R. It is well known [Ba76, p.61], that ' W W ! R, defined by Z ' .u/ D

.u .x// dx; ˝

is convex, lower demicontinuous functional on W . Moreover, w 2 @' .u/ if and only if w .x/ 2 ˚ .u .x// almosteverywhere in ˝ and w 2 W .  For fixed f 2 X  and U D uij .x/ 1i;j n 2 U let us consider such problem: n X



i;j D1

@ @xi



@y uij .x/ @xj

C ˚.y.x// 3 f .x/ for a.e. x 2 ˝;

y.x/ D 0 a.e. on :

(2.27)

(2.28)

Multiplying (2.27) by smooth finite functions in the domain ˝ and “integrating by parts” (see for detail [Li69, GaGrZa74]) we obtain such problem: hA.y; U /; w  yiX C '.w/  '.y/  hf; w  yiX

8w 2 X:

(2.29)

Solutions of problem (2.29) are called the generalized solutions of problem (2.27)– (2.28). Let us denote the set of generalized solutions of problem (2.27)–(2.28) by K.f; U /  X . Hence, in consequence of Theorem 2.9 and upper given assumptions the next corollary is fulfilled. Corollary 2.1. Let Un ! U *-weakly in Y  , fn ! f strongly in X  , Un 2 U 8n  1. Then w \ [ K.Um ; fm /  K.U ; f /; n1 mn w

where Q is a weak closure of the set Q  X in X . Moreover, there exists such U 2 U that for an arbitrary f 2 X  the set K.U ; f / is nonempty and weakly closed in set X . Remark 2.8. For a fixed f 2 X  let us consider the set of “admissible pairs” G.f / of problem (2.29), i.e., G.f / D f.y; U /jU 2 U; y 2 K.U ; f /g  X  Y  :

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2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

In Corollary 2.1, in particular, it is stated the set of “admissible pairs” G.f / of problem (2.29) is nonempty and closed in the topology of weak convergence in X and weakly star topology of the space Y  .

2.3 Variation Inequalities in Banach and Frechet Spaces Suppose V and W are reflexive Banach spaces, the space X D V \ W is dense in V and in W then the dual space X  D V  C W  . For f 2 X  and v 2 X we set hf; viX D hf1 ; viV C hf2 ; viW ; where f D f1 C f2 , f1 2 V  , f2 2 W  . Let Y  is a normalized space, U  Y  is  a nonempty subset, A W V  U ! 2V is a strict multivalued map, ' W W ! R D R [ fC1g is a convex function. Let us consider the inequality ŒA .y; u/ ; v  yC C ' .v/  ' .y/  hf; v  yiV

8v 2 X;

(2.30)

where f 2 V  , u 2 U . 

Theorem 2.10. Suppose A W V  U ! 2V is -quasimonotone and for any u 2 U A.; u/ finite-dimensionally locally bounded map and the following coercivity condition is satisfied: 9y0 2 X such that kyk1 V ŒA .y; u/ ; y  y0 C ! C1 as kykV ! 1: Besides suppose ' W W ! R is a convex lower semicontinuous function, dom ' D fv 2 W j ' .v/ ¤ C1g D W and lim

kykW !1

kyk1 W ' .y/ D C1:

(2.31)

Then for any f 2 V  , u 2 U the variation inequality (2.30) has at least one solution. Moreover, if U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in V  . Then \ [

w

Kum ;fm  Ku;f ;

(2.32)

n1 mn

where Ku;f D fy 2 X jy satisfies (2.30)g: Proof. Let us fix u 2 U . Without loss of generality we denote A.y/ D A.y; u/ 8y 2 X . Under the conditions of the Theorem the function ' is subdifferentiable and the subdifferential @' .y/ is a nonempty closed bounded convex subset in W   8y 2 W and the map @' W W ! 2W is upper semicontinuous.

2.3 Variation Inequalities in Banach and Frechet Spaces

159

Together with the inequality (2.30) let us consider the operator inclusion coA .y/ C @' .y/ 3 f:

(2.33)

We set A D coA C @'. In virtue of the obvious equality 2V

 CW 



D 2V C 2W





we have A W X D V \ W ! 2X , here the map A is strict. Let us continue the proof of the Theorem. As it is known (Lemma 1.7,  Corollary 1.29), @' W W ! 2W is the -pseudomonotone bounded-valued map,  therefore in virtue of Lemma 1.11 the map A D coA C @' W X ! 2X is also -pseudomonotone. Further due to property (2.31) and to the inequality ' .y0 / C Œ@' .y/ ; y  y0 C  ' .y/ we obtain

kyk1 W Œ@' .y/ ; y  y0 C ! C1 as kykW ! 1:

Then in virtue of Lemma 1.9 the operator A is coercive.  The operator @' W W ! 2W is locally bounded (Lemma 1.3). Hence it is finite-dimensionally locally bounded. Therefore the map A D coA ./ C @' ./ is finite-dimensionally locally bounded too. In this case all conditions of Theorem 2.5 (we keep in mind Remark 2.4) are satisfied. Due to this Theorem the solvability of the inclusion (2.33) which is equivalent to the variation inequality (2.30) (see Remark 2.11) follows. The proof of (2.32) is similar to the proof of (2.7). The Theorem is proved. Now suppose A W V  U  V ;

B WW U W

are multivalued maps. Let us investigate the variation inequality with multivalued maps of the following type ŒA.y; u/; vyC CŒB.y; u/; vyC C'.v/'.y/  hf; vyiX

8v 2 X; (2.34)

where ' W X ! R is a convex functional, f 2 X  . Theorem 2.11. Suppose operators satisfy the conditions of Theorem 2.7 (or 2.8), ' W X ! R is a convex lower semicontinuous functional. Then for any f 2 X  , u 2 U the variation inequality (2.34) has at least one solution. Moreover, if U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in V  . Then \ [ n1 mn

w

Kum ;fm  Ku;f ;

(2.35)

160

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

where Ku;f D fy 2 X jy satisfies (2.34)g: Proof. Let us fix u 2 U . Without loss of generality we denote A.y/ D A.y; u/, B.y/ D B.y; u/ 8y 2 X . Under the conditions of the Theorem the functional ' generates strict subdifferential map @' W X  X  which is upper semicontinuous monotone and has closed convex bounded values in X  . Therefore @' W X  X  is -pseudomonotone finite-dimensionally locally bounded map. Together with (2.34) let us consider the operator inclusion coA.y/ C coB.y/ C @'.y/ 3 f:

(2.36)

The inclusion (2.36) is equivalent to the variation inequality (2.20). To complete the proof it remains to remark that the operator A D coA C coB C @' satisfies all conditions of Theorem 2.7 (or 2.8) whence solvability of (2.36) follows. The proof of (2.35) is similar to the proof of (2.7). Now suppose X is a Frechet space, that is locally convex linear topological space and at the same time it is a complete metric space [Ru73], X  is its locally adjoint, h; iX W X   X ! R is a paring. Let a functional ' have the following appearance: '.y/ D '1 .y/ C '2 .y/  hf; yiX

8y 2 U;

(2.37)

where U is nonempty convex set, f 2 X  . Suppose a functional '1 W X ! 7 R is 7 R is convex lower semicontinuous on X (intdom '1 ¤ ;), a functional '2 W X ! convex on U and dom '1  dom '2 . Theorem 2.12. Let all listed above conditions hold. Then the following properties are equivalent: 1/ x0 2 intdom '1 \ U;

'.x0 / D inf '.x/I x2U

2/ x0 2 intdom '1 \ U; Œ@'1 .x0 I U /; x  x0 C C '2 .x/  '2 .x0 /  hf; x  x0 iX 8x 2 U:

(2.38)

Proof. Let us consider the implication 1 ) 2. Let the point x0 2 intdom '1 \ U satisfy 1. Then for any x 2 U and t 2 Œ0; 1 we have

2.3 Variation Inequalities in Banach and Frechet Spaces

161

'.x0 / D '1 .x0 / C '2 .x0 /  hf; x0 iX  '1 .x0 C t.x  x0 // C '2 .x0 C t.x  x0 //  hf; x0 C t.x  x0 /iX  '1 .x0 C t.x  x0 // C t'2 .x/ C .1  t/'2 .x0 /  thf; x  x0 iX  hf; x0 iX : Hence '1 .x0 C t.x  x0 //  '1 .x0 / C '2 .x/  '2 .x0 /  hf; x  x0 iX t or after passing to the limit as t ! 0C we have DC '1 .x0 I x  x0 / C '2 .x/  '2 .x0 /  hf; x  x0 iX : The last relation holds true due to Theorem 1.10 with ' D '1 and U D dom '1 which is a convex body. We remark also that for ' D '1 on X all conditions of Theorem 1.12 hold. Therefore in virtue of relation (1.76) we come to the following inequality: Œ@'1 .x0 I U /; x  x0 C C '2 .x/  '2 .x0 /  Œ@'1 .x0 I X /; x  x0 C C '2 .x/  '2 .x0 / D DC '1 .x0 I x  x0 / C '2 .x/  '2 .x0 /  hf; x  x0 iX 8x 2 U; since @'1 .x0 I U /  @'1 .x0 I X / D @'1 .x0 /: Now on the contrary, let the inequality (2.38) hold true. Using the definition of the local subdifferential @'1 .x0 I U / we have that for any x 2 U '1 .x/  '1 .x0 / C '2 .x/  '2 .x0 /  Œ@'1 .x0 I U /; x  x0 C C '2 .x/  '2 .x0 /  hf; x  x0 iX ; hence '.x/  '.x0 /, that is equivalent to 1. The Theorem is proved. Remark 2.9. In the literature the inequality (2.38) is called the variation inequality with multivalued maps. In Banach spaces such maps are being actively studied. Theorem 2.13. Suppose X is a reflexive Frechet space, functionals '1 ; '2 W X ! R are convex, lower semicontinuous on X , U  X is nonempty convex closed set, and a functional ' is of (2.37) type, and also either the functional ' is coercive on U or the set U is bounded. Then the variation inequality (2.38) has at least one solution x0 2 U . Moreover the set K D fy 2 U jy is a solution of (2.38)g is weakly compact in X .

162

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Proof. In our case dom ' D dom '1 D dom '2 D X and we occur (due to Lemma 1.22) under the conditions of Theorem 1.15 with ' D ' and B D U , whence it follows that the problem '.x/ ! inf;

x2U

has a solution x0 2 U and moreover the set of such solutions is weakly compact in X . To complete the proof it remains to apply Theorem 2.12. The Theorem is proved.

2.4 The Penalty Method for Multivariation Inequalities Suppose X is a reflexive Banach space, Y  is a normalized space, U  Y  is a  nonempty subset, A W X  U ! 2X is a multivalued map, f 2 X  , u 2 U are some fixed elements, K is a nonempty closed convex set in the space X , ' W X ! R D R[fC1g is a convex function with dom ' ¤ ;. Further the following objects will be investigated: ŒA.y; u/;   yC  hf;   yiX ŒA.y; u/;   yC C './  '.y/  hf;   yiX

8 2 KI

(2.39)

8 2 K \ dom ':

(2.40)



Theorem 2.14. Let A W X  U ! X be a singlevalued and quasimonotone, ' W X ! R be a convex lower semicontinuous functional with dom ' ¤ ;, U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in X  . Then \ [

w

K.um ; fm /  K.u; f /;

(2.41)

n1 mn w

where U is the weak closure of U  X in X , K.u; f / D fy 2 K \ dom ' j y satisfies (2.40)g: Proof. Let U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in X  . Let us T S w check (2.41). We assume that the set K.um ; fm / is nonempty (otherwise n1 mn T S w the inclusion (2.41) holds true). So, if y0 2 K.um ; fm / then there is such n1 mn

sequence fnk gk1 that y0 2 dom ' \ K is a weak cluster point in X of fynk gk1 : 8k  1 ynk 2 K.unk ; fnk /. We remark that hfnk ; ynk  y0 iX ! 0 as k ! 1 and lim hA.ynk ; unk /; ynk  y0 iX  lim hf;   y0 iX C '.y0 /  lim '.ynk /  0;

k!1

k!1

k!1

2.4 The Penalty Method for Multivariation Inequalities

163

because ' is weakly lower semicontinuous [ZgMe04]. Thus, due to the quasimonotony for A, hA.ynk ; unk /; ynk  y0 iX ! 0 as k ! 1 and lim hA.ynk ; unk /; w  y0 iX  hA.y0 ; u/; w  y0 iX

k!1

8w 2 X:

So, 8w 2 dom' \ K hf; w  y0 iX D lim hfnk ; w  ynk iX  lim hA.ynk ; unk /; w  y0 iX k!1

k!1

C '.w/  lim '.ynk /  hA.y0 ; u/; w  y0 iX C '.w/  '.y0 /; k!1

i.e., y0 2 K.u; f /. The weak closureness of K.u; f / follows from (2.41). The Theorem is proved. Following [Fr82] let us consider the case when X D V \ W where V and W are reflexive Banach spaces, 

A W V ! 2V ; f 2 V  C W  ; kkX D kkV C kkW : Let ˇ W X ! W  be a penalty operator corresponding to the set K, namely K D fy 2 V \ W jˇ.y/ D 0g. We will consider that the penalty operator ˇ W X ! W  is bounded, demicontinuous and monotone [Li69]. As it is known for closed convex K in reflexive spaces the penalty operator always exists. In [Li69] for the existence strict convexity of norms in X and X  is required. However due to the E. Asplund Theorem [Di75] in a reflexive Banach space there can be chosen a norm equivalent to the initial one, such that this norm is strictly convex together with its adjoint. Theorem 2.15. Suppose K ¤ ; is a closed convex set in the space X D V \ W , A W V  V  is a bounded pseudomonotone (in the sense of Definition 1.28) map and there can be found y0 2 K such that for some fixed "0 > 0 kyk1 X



1 ŒA.y/; y  y0  C hˇ.y/; y  y0 iX "0

 ! C1

(2.42)

as kykX ! 1. Then 8f 2 V  C W  9 fy" g  X , where 1 .coA/.y" / C ˇ.y" / 3 f; "

" 2 .0; "0 /

(2.43)

w

y" ! y in X as " ! 0, and weak cluster point y 2 K satisfies the inequality (2.39). Proof. Let the penalty operator ˇ W X ! W  be bounded, demicontinuous and monotone [Li69]. For each " > 0 let us consider the multivalued map

164

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

1  A" D coA C ˇ W X ! 2X : " Its boundness immediately follows from boundness of A and ˇ. 

Lemma 2.7. For each " > 0 the operator A" W X ! 2X is pseudomonotone. w

Proof. Let yn ! y in X and  lim ŒA" .yn /; yn  y D lim

n!1

n!1

1 ŒA.yn /; yn  y C hˇ.yn /; yn  yiW "

  0:

Here using the facts that 8y 2 X

g D g1 C g2 2 V  C W  D X  ;

hg; yiX D hg1 ; yiV C hg2 ; yiW :

 ˚ and up to a subsequence ynk  fyn g we obtain one of two relations: lim

nk !1



A.ynk /; ynk  y

 

 0;

lim

nk !1

˝

ˇ.ynk /; ynk  y

˛ W

0

(2.44)

If the first relation in (2.44) takes place then (again we pass to a subsequence if necessary) due to pseudomonotony of A we have lim

nk !1

  A.ynk /; ynk  w   ŒA.y/; y  w

8w 2 V

(2.45)

And this inequality implies that   A.ynk /; ynk  y  ! 0: But then lim

nk !1

˝ ˛ ˇ.ynk /; ynk  y W  0;

and since the operator ˇ is pseudomonotone (Proposition 1.62) we have lim

nk !1

˝ ˛ ˇ.ynk /; ynk  w W  hˇ.y/; y  wiW

8w 2 W:

(2.46)

From inequalities (2.45), (2.46) which are valid for w 2 V \ W we find lim

nk !1



A" .ynk /; ynk 

 

 ˝  1 ˛ lim ˇ.ynk /; ynk  W  lim A.ynk /; ynk   n !1 " k nk !1  ŒA" .y/; y  

8 2 X:

In case when the second relation in (2.44) holds true argumentation is similar. The Lemma is proved.

2.4 The Penalty Method for Multivariation Inequalities

165

Let us continue the proof of the Theorem. Since the operator A"0 is -coercive (condition (2.42)) and 8" 2 .0; "0 /, y 2 X 1 ŒA.y/; y  y0  C hˇ.y/; y  y0 iW " 1 D ŒA.y/; y  y0  C hˇ.y/  ˇ.y0 /; y  y0 iW " 1  ŒA.y/; y  y0  C hˇ.y/  ˇ.y0 /; y  y0 iW "0 1 D ŒA.y/; y  y0  C hˇ.y/; y  y0 iW ; "0 then 8" 2 .0; "0 / the operator A" is -coercive and the operator inclusion (2.43) has at least one solution y" 2 V \ W . Now let us direct " ! 0. We prove that the sequence fy" g is bounded in X . Indeed due to condition (2.42)   1 ŒA.y /; y  y  C /; y  y hˇ.y i ky" k1 " " 0  " " 0 W X "   1 1  ky" kX ŒA.y" /; y"  y0  C hˇ.y" /; y"  y0 iW "0 as ky" kX ! 1, " ! 0, since the operator ˇ is monotone. Therefore ky" kX  C where the constant C does not depend on " > 0. Since the map A is bounded then in virtue of Proposition 1.38



kA.y" /kC D co A.y" /  l: C

From inclusion (2.43) and have jhˇ.y" /; y" iW j  " jhf; y" iX C ŒA.y" /; y" C j    " kf kX  ky" kX C kA.y" /kC ky" kV  "k: Therefore, hˇ.y" /; y" iW ! 0

as " ! 0

and similarly 8w 2 X

hˇ.y" /; viW ! 0:

Due to Banach–Alaoglu Theorem from the sequence fy" g there can be isolated w weakly convergent subsequence (we will keep the initial notation for it) yn ! y in X . For an arbitrary v 2 X hˇ.y" /  ˇ.v/; y"  viW  0;

166

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

therefore hˇ.v/; y  viW  0: We set v D y  tw, where t > 0, 8w 2 X then hˇ.y  tw/; twiW D t hˇ.y  tw/; wiW  0: Dividing the last relation by t, passing to the limit as t ! 0C and taking into account demicontinuity of the operator ˇ we have hˇ.y/; wiW  0

8w 2 X;

therefore ˇ.y/ D 0 and y 2 K. For an arbitrary  2 K we obtain ŒA.y" /  f;   y" C 

1 hˇ./  ˇ.y" /;   y" iW  0; "

or ŒA.y" /;   y" C  hf;   y" iX . In virtue of Proposition 1.38 the last inequality is equivalent to the following one: ŒA.y" /; y"    hf; y"  iX

8 2 K

(2.47)

Since y 2 K from the last inequality we have that lim ŒA.y" /; y"  y  0:

"!0



Due to pseudomonotony of the map A W V ! 2V (up to a subsequence) we obtain: lim ŒA.y" /; y"    ŒA.y/; y  

"!0

8! 2 K;

and together with the inequality (2.47) it implies hf; y  iX  ŒA.y/; y  

8 2 K:

Taking into account Proposition 1.38 we obtain ŒA.y/;   yC  hf;   yiX

8 2 K:

This fact proves the Theorem. 

Theorem 2.16. Suppose A W X ! 2X is bounded demiclosed map such that Condition ˛.X / and the coercivity condition (2.42) are satisfied. Then all statements of Theorem 2.15 hold true, moreover y" ! y in X as " ! 0.

2.4 The Penalty Method for Multivariation Inequalities

167

Proof. Due to the results of the first chapter for each " > 0 the operator 1 A" D coA C ˇ " has Property ˛.X / and it is demiclosed as the sum of demiclosed and demicontinuous maps. Therefore for each " > 0 solvability of operator inclusion (2.43) follows from the results obtained above. Similarly to the proof of Theorem 2.15 it is established that w

ky" kX  C;

y" ! y in X;

y 2 K;

and also lim ŒA.y" /; y"  y  0:

"!0

But then (up to a subsequence) we may assume that y" ! y in X . Due to boundness of the map A 8" > 0 there exists d" 2 .coA/.y" / such that hd" ; y"  iX D ŒA.y" /; y"   ; w

where kd" kX   k. We have d" ! d in X  up to a subsequence and since the operator A is demiclosed we see that d 2 .coA/.y/ and from (2.47) we obtain ŒA.y/; y    lim hd" ; y"  iX D lim ŒA.y" /; y"    hf; y  iX "!0

8 2 K;

"!0

that is equivalent to ŒA.y/;   yC  hf;   yiX

8 2 K:

The Theorem is proved. The following Proposition is a generalization of Theorem 2.16. Theorem 2.17. Suppose X D V \ W , V and W are reflexive Banach spaces, K  is closed and convex in X , A W V ! 2V is a bounded demiclosed operator which satisfies Condition ˛.V / .˛2 .V // and 9y0 2 K such that 8" > 0   1 y  y ŒA.y/; y  y ! C1 as kykX ! 1:  C hˇ.y/; kyk1 i 0  0 W X " (2.48) Then 8" > 0 and f 2 X  D V  C W  the operator inclusion 1 A" .y/ D coA.y/ C ˇ.y/ 3 f "

(2.49)

168

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

is solvable and from the sequence of its solutions fy" g there can be isolated a subsequence fy g such that w

y ! y

in X;

y ! y

in

V

as ! 0

and the element y 2 K satisfies the inequality ŒA.y/;   yC  hf;   yiX Proof. Let B W X ! 2W conditions:



8 2 K:

(2.50)

be a multivalued map which satisfies the following

(a) B is a bounded radial semicontinuous operator with s.b.v.; w (b) If yn ! y in X and ŒB.y/; yn  y ! 0; then there can be found a subsequence fynk g  fyn g such that lim

nk !1

  B.ynk /; y  ynk   0:

Let us consider an operator inclusion of the following type coA.y/ C coB.y/ 3 f;

(2.51)

where f 2 X  . 

Lemma 2.8. Suppose A W V ! 2V is a bounded demiclosed map such that con dition ˛.V / .˛2 .V // is satisfied, a map B W X ! 2W satisfies conditions a, b and ˚  ŒA.y/; yC C ŒB.y/; yC ! C1 as kykX ! 1: kyk1 X Then for each f 2 X  the operator inclusion (2.51) is solvable. 

Proof. We set A D coA C coB and prove that the map A W X ! 2X is w pseudomonotone. Let X 3 yn ! y in X and lim ŒA .yn /; yn  y D lim fŒA.yn /; yn  y C ŒB.yn /; yn  y g  0 n!1 (2.52) Isolating a subsequence, from (2.52) we obtain one of two relations: n!1

lim ŒA.yn /; yn  y  0;

n!1

lim ŒB.yn /; yn  y  0:

n!1

(2.53)

In the first case due to boundness of the map A and to Condition ˛.V / (˛2 .V / respectively) there can be found dn 2 coA.yn / for which ŒA.yn /; yn  y D hdn ; yn  yiV ! 0 as n ! 1

2.4 The Penalty Method for Multivariation Inequalities

169

(we are speaking about a subsequence of the sequence fyn g). Therefore we come to the second relation from (2.53) and together with semiboundness of variation of the map B it ensures ŒB.yn /; yn  y ! 0. Hence from the condition b we obtain lim ŒB.yn /; y  yn   0:

n!1

For arbitrary h 2 W \ V and > 0 we set w. / D y C .h  y/: 0

ŒB.yn /; yn  w. /  ŒB.w. //; yn  w. /C  C.RI kyn  w. /kX /; and since (see Proposition 1.38) ŒB.yn /; yn  w. /  ŒB.yn /; yn  yC C ŒB.yn /; yn  h ; taking into account the previous inequality we find

ŒB.yn /; y  h  ŒB.yn /; y  yn  C ŒB.yn /; yn  w. / 0

 ŒB.yn /; y  yn  C ŒB.w. //; yn  w. /C  C.RI kyn  w. /kX / or after passing to the limit as n ! 1: 0

lim ŒB.yn /; y  h  ŒB.w. //; y  hC  C.RI ky  hkX /: n!1

Dividing the obtained inequality by > 0, passing to the limit as ! 0C and taking into account r.s.c. of the operator B and properties of the function C we have lim ŒB.yn /; y  h  lim ŒB.w. //; y  hC  ŒB.y/; y  h :

n!1

!0C

Since ŒB.yn /; yn  h  ŒB.yn /; yn  y C ŒB.yn /; y  h and ŒB.yn /; yn  y ! 0 we obtain that lim ŒB.yn /; yn  h  lim ŒB.yn /; y  h  ŒB.y/; y  h 8h 2 X: n!1 (2.54)

n!1

Further 8h 2 X and n D 1; 2; : : : 9'n 2 coA.yn / such that ŒA.yn /; yn  h D h'n ; yn  hiV ; where without loss of generality we may assume that 'n ! ' in V  and yn ! y  in V . Then due to demicloseness of the map A W V ! 2V , ' 2 coA.y/,

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2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

lim ŒA.yn /; yn  h D lim h'n ; yn  yiV n!1

n!1

D h'; y  hiV  ŒA.y/; y  h

8h 2 X:

From here and from relation (2.54) we finally obtain lim ŒA .yn /; yn  h  lim ŒA.yn /; yn  h C lim ŒB.yn /; yn  h

n!1

n!1

n!1

 ŒA.y/; y  h C ŒB.y/; y  h D ŒA .y/; y  h

8h 2 X;

and pseudomonotony of the map A is established. Therefore solvability of inclusion (2.51) is a direct corollary of results from previous sections. The Lemma is proved. Remark that the penalty operator ˇ W X ! W  satisfies all conditions of Lemma 2.8, hence 8" > 0 the inclusion (2.49) has at least one solution y" . From coercivity condition (2.48) and from the inequality ŒA.y" /; y"  y0  C

1 hˇ.y" /; y"  y0 iW D ŒA" .y" /; y"  y0  "

 hf; y"  y0 iX  kf kX  ky"  y0 kX

(2.55)

we obtain the estimation ky" kX  l. Indeed fixing some "0 > 0 for 0 < "  "0 due to monotony of ˇ we have 

A"0 .y" /; y"  y0

 

D ŒA.y" /; y"  y0  C

1 hˇ.y" /  ˇ.y0 /; y"  y0 iW "0

 ŒA" .y" /; y"  y0  ; but

  ky" k1 X A"0 .y" /; y"  y0  ! C1

(2.56)

as

ky" kX ! 1;

hence the required estimation follows from (2.55) and (2.56). Therefore without w any restrictions we may assume that y" ! y in X and by the same way as in Theorem 2.15 we prove the inclusion y 2 K. Besides 8" > 0 ŒA.y" /; y"  h  hf; y"  hiX 8h 2 K; whence lim ŒA.y" /; y"  y  0

"!0

and in virtue of Property ˛.V / y" ! y in V (for the corresponding subsequence).  Analyzing the proof of Lemma 2.8 it is easy to see that the map A W V ! 2V is

2.4 The Penalty Method for Multivariation Inequalities

171

pseudomonotone, hence lim ŒA.y" /; y"  h  ŒA.y/; y  h

8h 2 K;

"!0

or ŒA.y/; y  h  hf; y  hiX

8h 2 K;

that is equivalent to the inequality (2.50). 

Theorem 2.18. Suppose A W V ! 2V is a bounded -pseudomonotone map and the coercivity condition (2.48) holds true. Then for each " > 0 the inclusion (2.49) is solvable 8f 2 X  and from the sequence of solutions fy" g there can be isolated w a subsequence fy g such that y ! y in X as ! 0 where y 2 K and it satisfies (2.50). Proof. Similarly to the proof of Theorem 2.17 let us consider the map A D coA C  coB, where the operator B W W ! 2W satisfies the condition (a). 

Proposition 2.5. Suppose A W V ! 2V is a bounded -pseudomonotone map, the  operator B W W ! 2W satisfies the condition a and the coercivity condition form Lemma 2.8 holds true. Then 8f 2 X  the operator inclusion (2.51) has at least one solution. Proof. It is clear that the map A D coA C coB W X ! 2X



is bounded as a vector sum of bounded maps. Let us prove its -pseudomonotony. w Let y ! y in X , dn 2 coA .yn / and lim hdn ; yn  yiX  0;

n!1

where dn 2 V  C W  . Similarly to Lemma 1.1 it can be proved that coA D A . Therefore dn D dn0 C dn00 where dn0 2 coA.yn /, dn00 2 coB.yn / and then up to a subsequence one of the relations ˝ ˛ lim dn0 ; yn  y V  0;

n!1

˝ ˛ lim dn00 ; yn  y W  0

n!1

(2.57)

holds true. Let the second relation in (2.57) take place. Then from the condition of s.b.v. for the operator B we find ˝

dn00 ; yn  y

˛ W

!0

and for arbitrary > 0 and h 2 W ˝ 00 ˛ 0 dn ; yn  . / W  ŒB.yn /; yn  . /  ŒB.. //; yn  . /C  C.RI kyn  . /kX /;

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2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

where . / D y C .h  y/. Hence ˝ ˛ 0 1 lim dn00 ; y  h W  ŒB.. //; y  hC  C.RI ky  hkX /

n!1

and taking into account properties of the function C and the condition a we have ˝ ˝ ˛ ˛ lim dn00 ; yn  h W D lim dn00 ; y  h W

n!1

n!1

 lim ŒB.. //; y  hC  ŒB.y/; y  h 8h 2 W;

(2.58)

!0C

namely the operator B is -pseudomonotone. Further since ˝

dn00 ; yn  y

˛ W

! 0;

the first relation in (2.57) takes place and due to -pseudomonotony of the map A ˝ ˛ lim dn0 ; yn  h V  ŒA.y/; y  h 8h 2 V;

n!1

and together with (2.58) it proves -pseudomonotony of the map A . Let F .X / be a family of all finite-dimensional subspaces of the space X . For F 2 F .X / by IF W F ! X we denote the canonical embedding operator (kIF yF kX D kyF kF 8yF 2 F ), and IF W X  ! F  is its adjoint operator. Suppose AF D  IF BIF W F ! 2F , where Cv .F  / is a family of all nonempty closed convex subsets of the space F  . Lemma 2.9. For each F 2 F .X / AF W F ! Cv .F  / and it is the upper semicontinuous map. Proof. For an arbitrary yF 2 F the set AF .yF / is convex and closed in F  . Convexity is a direct corollary of linearity of the operator IF . Let us prove closeness. Let n 2 AF .yF / be an arbitrary sequence such that n ! 0 in F  and we consider its inverse image n of the map IF , that is n D IF . n /, n 2 AF .IF yF /. The sequence f n g is bounded in X  and without loss of generality we may assume that w

n ! 0 in X  where 0 2 A .IF yF / since A D co A . Therefore n D IF n ! IF 0 D 0 and 0 2 AF .yF /; that is AF .yF / 2 Cv .F  /. Let us prove upper semicontinuity of the map AF . The set AF .yF / is bounded and hence it is compact in F  . We assume that at the point yF0 2 F the map AF is not upper semicontinuous. Then there can be found " > 0

2.4 The Penalty Method for Multivariation Inequalities

173

for which in every ball

˚  B1=n .yF0 / D yF 2 F j yF  yF0 F < 1=n there can be found yFn such that ˚  AF .yFn / 6 B" .AF .yF0 // D F 2 F  jdist.F ; AF .yF0 // < " : Let us consider sequences fyFn g and fFn g where Fn 2 AF .yFn /nB" .AF .yF0 //;

yFn 2 B1=n .yF0 /:

The sequence fyFn g converges to yF0 in F and the sequence fFn g is bounded in F  due to boundness of the maps A and IF therefore we may assume that Fn ! F0 in F  . From Propositions 1.48, 1.49 demicloseness of A and hence closeness of the map AF follow. Therefore F0 2 AF .yF0 / and this fact contradicts to Fn … B" .AF .yF0 //. So for each F 2 F .X / the map AF is upper semicontinuous and has closed convex values. Lemma 2.9 is proved. Let us consider a function  W RC ! R, defined by the formula .r/ D

ŒA .y/; yC : kykX

inf

kykX Dr

The function  has the following property: lim .r/ D C1:

r!1

Indeed let B r D fy 2 X j kykX  rg;

@Br D fy 2 X j kykX D rg:

From boundness of A it follows that 8" > 0 9y" 2 @Br such that .r/ > ky" k1 X ŒA .y" /; y" C  "; where " does not depend on r. Hence due to the coercivity condition from Lemma 2.8 we obtain the required property of the function  . Therefore for a fixed f 2 X  ŒA .y/  f; yC  ..kykX /  kf kX  / kykX such that ŒA .y/  f; yC  0

8y 2 @Br

(2.59)

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2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

For F 2 F .X / we set B r;F D B r \ F . Then from (2.59) we obtain the similar estimation for AF that is ŒAF .yF /  fF ; yF C  0 8yF 2 @Br;F ; therefore Corollary 1.16 can be applied here, namely 8F 2 F .X / there exists yF 2 F such that AF .yF / 3 fF ; fF D IF f; (2.60) where yF 2 B r;F . Inclusion (2.60) is equivalent to the following inequality (Proposition 1.39): ŒAF .yF /; F C  hfF ; F i

8F 2 F:

(2.61)

Hence there exists a sequence where yF 2 B r and yF satisfies (2.61). For an arbitrary F0 2 F .X / we set GF0 D

[ ˚

 yF 2 B r;F j yF satisfies (2.61) :

F F0

The set GF0 is nonempty and it is contained in the ball B r . Moreover for an arbin S Fi the trary finite family F1 ; : : : ; Fn 2 F .X / and F 2 F .X / such that F  i D1

following relation takes place n [

GFi  GF ¤ ;;

i D1 w

w

namely the system of sets fG F g is centered where G F is a closure of the set GF in the weak topology of the space X . Then from reflexivity of the space X the existence of y0 such that [ w y0 2 GF (2.62) F 2F .X/

follows. Let us prove that y0 is a solution of inclusion (2.51). Let us fix an arbitrary w 2 X and choose F0 2 F .X / under the condition y0 ; w 2 F0 . In virtue of (2.62) w there exists a sequence fyn g  GF0 such that yn ! y0 in X and also n 2 An .yn / for which n D fn where Fn 2 F .X /, yn 2 Fn \ B r .F0  Fn /, fn D IFn f , An D IFn A . Then obviously hn ; yn  y0 iFn D hdn ; yn  y0 iX D hfn ; yn  y0 iFn D hf; yn  y0 iX ; where dn 2 A .yn /. Therefore lim hdn ; yn  y0 iX D lim hf; yn  y0 iX D 0;

n!1

n!1

2.4 The Penalty Method for Multivariation Inequalities

175

and since the map A is -pseudomonotone there can be found subsequences fynk g  fyn g and fdnk g  fdn g such that lim

nk !1

˝ ˛ dnk ; ynk  w X  ŒA.y0 /; y0  w 8w 2 X;

whence we obtain the inequality hf; yn  wiX  ŒA .y0 /; y0  w : It is equivalent to the inclusion A .y0 / 3 f in virtue of Proposition 1.39 and this fact proves Proposition 2.5. Let us continue the proof of the Theorem. From the last Proposition it follows that for each " > 0 the inclusion (2.49) has at least one solution y" and similarly to the prove of Theorem 2.17 we establish the boundness in X of the sequence fy" g. w We assume that y" ! y in X . Repeating the corresponding part of the proof of Theorem 2.15 we make sure that y 2 K. Let us consider a sequence d" 2 coA.y" / such that 1 d" C ˇ.y" / D f: " Then for an arbitrary w 2 K hd" ;   y" iV D

1 hˇ./  ˇ.y" /;   y" iW C hf;   y" iX  hf;   y" iX ; "

and here due to the boundness of the map A W V ! 2V w d" ! d in V  . But from the previous inequality we have



we may assume that

lim hd" ; y"  yiV  0:

"!0

Since the operator A W V ! 2V necessary, we have



is -pseudomonotone, up to a subsequence if

lim hd" ; y"  wiV  ŒA.y/; y  w

8w 2 K:

"!0

Hence from the inequality lim hd" ; y"  wiV  hf; y  wiX

"!0

8w 2 K

we obtain the required relation hf; y  wiX  ŒA.y/; y  w which is equivalent to inequality (2.50).

8w 2 K;

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2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

When analyzing the proof of Proposition 2.5 it is clear that the following result is valid. 

Corollary 2.2. Suppose A W X  U ! 2X is a -quasimonotone map and for any u 2 U A.; u/ is bounded and C-coercive map. Then 8f 2 X  ; u 2 U the set Kf;u D fy 2 X jA .y; u/ 3 f g is nonempty and 8r > 0 Kf;u \ B r is weakly compact. Moreover if U 3 un ! u 2 U weakly star in Y  , fn ! f strongly in X  , then w \ [ K.um ; fm /  K.u; f /: n1 mn

Remark 2.10. Theorems 2.17, 2.18 and also Theorems 2.15, 2.16 remain valid if coercivity condition (2.48) is replaced by a weaker condition: there can be found "0 > 0, y0 2 K such that kyk1 X



1 ŒA.y/; y  y0 C C hˇ.y/; y  y0 iW "

 ! C1

(2.63)

as kykX ! 1. Proof. Let (2.63) be true then for an arbitrary fixed f 2 X  there exists r0 D r0 .f I "0 / such that   A"0 .y/; y  y0 C  hf; y  y0 iX 8y 2 @Br0 : Similarly to the proof of Proposition 2.5 we make sure that there exists a function 0 W RC ! R such that   A"0 .y/  f; y  y0 C  0 .kykX / kykX ; where 0 .s/ ! C1 as s ! 1: Then 8" < "0   ŒA" .y/; y  y0 C  A"0 .y/; y  y0 C ; therefore   ŒA" .y/  f; y  y0 C  A"0 .y/  f; y  y0 C  08y 2 @Br0 : Hence 8" < "0 the inclusion (2.49) has at least one solution in the ball B r0 which radius does not depend on ". Now let us study a variation inequality of the (2.40) type: ŒA.y/;   yC C './  '.y/  hf;   yiX

8 2 K \ dom '

2.4 The Penalty Method for Multivariation Inequalities

177

where K is a closed convex set in X , ' W X ! R D R [ fC1g is a main convex lower semicontinuous function, dom ' D fx 2 X j'.x/ ¤ C1g: Together with the inequality (2.40) let us consider 8 2 dom ' ŒA.y/;   yC C

1 hˇ.y/;   yiW C './  '.y/  hf;   yiX : "

(2.64)



Theorem 2.19. Suppose A W V ! 2V is a bounded -pseudomonotone map, ' W X ! R is a main convex lower semicontinuous function, there exist "0 > 0 and y0 2 K \ dom ' such that   1 ŒA.y/; y  y  C C '.y/ ! C1 y  y kyk1 hˇ.y/; i 0  0 W X "0

(2.65)

as kykX ! 1. Then for each f 2 X  and for each " > 0 there exists at least one solution y" 2 dom ' of inequality (2.64) and the family of such solutions is weakly compact in X . If besides spen dom ' D X then from the sequence fy" g there can be isolated w a subsequence fy g such that y ! y 2 K \ dom ' in X and it satisfies (2.40). Proof. Similarly to the proof of Theorem 2.18 let us consider the operator A D  coA C coB where the map B W W ! 2W satisfies the condition (a) and also the variation inequality connected with it ŒA.y/;   yC C './  '.y/  hf;   yiX

8 2 dom ':

(2.66)

Proposition 2.6. Let 9y0 2 dom ' and kyk1 X fŒA .y/; y  y0  C '.y/g ! C1

as

kykX ! 1:

(2.67)

Then 8f 2 X  inequality (2.66) is solvable and the family of solutions is weakly compact in dom '. Proof. Using Mosko method [Mo67] we define XO D X  R;

KO D epi' D f.yI / 2 X  Rj'.y/  g;

AO.y/ O D .A .y/I 0/ 2 XO  D X   R;

yO D .yI ˛/ 2 XO :

Similarly to the proof of Proposition 2.5 -pseudomonotony of the map A is established and hence the map AO is -pseudomonotone too. The set KO is closed and convex, similarly to [MeSo97, Theorem 3] the equivalency of (2.66) with

178

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

the inequality

h i D E AO.y/; O O  yO  fO; O  yO C

8O 2 KO

X

(2.68)

where fO D .f I 1/ is proved. For each R > 0 let us consider the set O ky  y0 kX C j˛  '.y0 /j  Rg; KO R D f.yI ˛/ D yO 2 Kj bounded in XO and for a fixed R > 0 we study the inequality (2.68) for O 2 KO R . Its solvability follows from the next statement which is significant enough itself. 

Lemma 2.10. Suppose X is a reflexive Banach space, A W X ! 2X is a bounded -pseudomonotone map and K is a closed convex bounded subset in X . Then for each f 2 X  there can be found one y 2 K such that ŒA .y/;   yC  hf;   yiX

8 2 K:

(2.69)

And here the solution family of inequality (2.69) is weakly compact in K. Proof. Let F .X / be a family of all finite-dimensional subspaces of the space X . For each F 2 F .X / similarly to the proof of Theorem 2.18 we set AF D IF A IF , KF D K \ F and consider a finite-dimensional variation inequality ˛ ˝ ŒAF .yF /; F  yF C  IF f; F  yF

8F 2 KF

(2.70)

Let v1 ; v2 ; : : : ; vk be a basis in F and let us provide F with a structure of a Hilbert space with an inner product .; /F . Namely, we identify spaces F F  . In this case k X AF .yF / D hA .yF /; vi iX vi i D1

and (2.70) is equivalent to the inequality .yF ; F  yF /F  ŒyF C fF  coAF .yF /; F  yF 

8F 2 KF ;

where fF D IF f and coAF .yF / D IF coA .yF /. Due to Lemma 2.9 the map coAF W F ! Cv .F / is upper semicontinuous therefore 8" > 0 there exists "-approximation [LaOp71], namely there can be found a continuous singlevalued map d" W F ! F for which  .graphd" ; graphcoAF / D

sup

.; graphcoAF / < ":

2graphd"

Here graphd" , graphcoAF are the graphs of the maps d" and coAF respectively therefore is the natural metric on F  F .

2.4 The Penalty Method for Multivariation Inequalities

179

For a fixed " > 0 let us consider an "-approximation d" and its associated variation inequality .yF ; F  yF /F  .yF C fF  d" .yF /; F  yF /F

8F 2 KF

(2.71)

Let KF W F ! KF be a projection operator on the set KF . As it is known 8yF 2 F an element KF .yF / can be completely characterized by the condition .KF .yF /  yF ; F  KF .yF //F  08F 2 KF and hence the inequality (2.71) is equivalent to the relation yF D KF .yF C fF  d" .yF //: Therefore due to the fixed point Theorem 8" > 0 there exists yF ;" 2 KF which satisfies (2.71). Let us consider the sequence fyF ;" g">0 which is bounded due to boundness of the set KF . Further from boundness of the map AF and from the fact that d" .KF /  coAF .KF / boundness of the sequence fd" .yF ;" /g">0 in F follows. Therefore we may assume that yF ;" ! yF ;0 in F and d" .yF ;" / ! d0 in F where d0 2 coAF .yF ;0 / [LaOp71]. Passing to the limit as " ! 0 in the inequality (2.71) where instead of yF we substitute yF ;" we get .yF ;0 ; F  yF ;0 /F  .yF ;0 C fF  d0 ; F  yF ;0 /F

8F 2 KF ;

and since d0 2 coAF .yF ;0 / we see that ŒAF .yF ;0 /; F  yF ;0 C  .d0 ; F  yF ;0 /F  .fF ; F  yF ;0 /F

8F 2 KF :

Hence solvability of the inequality (2.70) for each F 2 F .X / is established. Moreover 9l > 0 such that kyF kX  l 8F 2 F .X /. For some F0 2 F .X / we set GF0 D

[ ˚

 yF 2 KF j ŒAF .yF /  f F ; F  yF C  0 8F 2 KF :

F F0

Obviously GF0 ¤ ; and the system fGN Fw g where fGN Fw g is a weak closure of the set GF is centered. Therefore \ w GF ; 9y0 2 F 2F .X/

and y0 2 K. Let us prove that y0 is a solution of the inequality (2.69). For any w 2 X we choose F0 2 F .X / from the condition y0 ; w 2 KF0 . Then there can w be found a sequence fyn g  GF0 such that yn ! y0 and also dn 2 coA .yn / for which

180

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

lim hdn ; yn  y0 i D lim ŒA .yn /; yn  y0  n!1 ˝ ˛  lim fF0 ; yn  y0 F D lim hf; yn  y0 iX D 0:

n!1

n!1

n!1

Therefore due to -pseudomonotony of the map A we obtain (for corresponding subsequences) hf; y0  wiX D lim

˝

nk !1

˝ ˛ ˛ f; ynk  w X  lim dnk ; ynk  w X nk !1

 ŒA .y0 /; y0  w

8w 2 K;

namely y0 satisfies the inclusion (2.69). The weak compactness property of solution family is a direct corollary of -pseudomonotony and boundness of K. Lemma 2.10 is proved. So in virtue of Lemma 2.10 there exists yOR 2 KO R such that h

AO.yOR /; O  yOR

i C

E D  fO; O  yOR

X

8O 2 KO R

(2.72)

Since yO0 D .y0 ; '.y0 // 2 KO R , substituting in (2.72) O D yO0 we have ŒA .yR /; yR  y0  C R  hf; yR  y0 iX C '.y0 /;

(2.73)

and since '.yR /  R ; y0 2 dom ' we get ŒA .yR /; yR  y0  C '.yR /  hf; yR  y0 iX C '.y0 /  kf kX  kyR  y0 kX C C1  C.1 C kyR kX /: Hence in virtue of condition (2.67) we obtain the estimation kyR kX  l and in virtue of (2.73) we have R  hf; yR  y0 iX C '.y0 / C ŒA .yR /; y0  yR C  .kf kX  C kA .yR /kC / kyR  y0 kX C '.y0 /  k In the same time setting C" .r/ D

inf

kykX Dr

ŒA .y/; y  y0  C '.y/ ; kykX

due to the coercivity condition (2.67) and due to boundness of the map A we find R  '.yR /  C" .kyR kX / kyR kX  ŒA .yR /; yR  y0 

 .C" .kA .yR /kX /  kA .yR /k / kyR kX  kA .yR /k ky0 kX  CO kyR kX ;

2.4 The Penalty Method for Multivariation Inequalities

181

whence kyR kX C jR j  C1 ; where the constant C1 does not depend on R. Therefore kyR  y0 kX C jR  '.y0 /j  C2 and for R > C2 an element yOR D .yR ; R / is a solution of inequality (2.68). Indeed O for small enough t 2 Œ0; 1 for an arbitrary O 2 K, Ot D t O C .1  t/yOR 2 KO R and due to inequality (2.72) we obtain i E h D O AO.yOR /; O  yOR  fO; O  yOR 8O 2 K: C

XO

Since inequality (2.68) is equivalent to (2.66), we have solvability of inequality (2.66) established. Weak compactness of the solution family can be easily proved. Proposition 2.6 is proved. Since the operator ˇ W X ! W  satisfies all requirements of the map B W X ! 2 then solvability of inequality (2.64) for each " > 0 is established. Here the family fy" g">0 is bounded in X due to coercivity condition (2.65) and due to the estimation W

1 hˇ.y" /; y"  y0 iW C '.y" / "0 1  ŒA.y" /; y"  y0  C hˇ.y" /  ˇ.y0 /; y"  y0 iW C '.y" / "  hf; y"  y0 iX C '.y0 /80 < " < "0 :

ŒA.y" /; y"  y0  C

w

Therefore we may assume that y" ! y0 in X . From the previous inequality we obtain '.y" /  '.y0 / C .kA .y" /kC C kf kX  / ky"  y0 kX  k0 : Namely y 2 dom ' and since 8v 2 dom ' hˇ.y" /; y"  viW  ".kf kX  C kA .y" /kC / ky"  y0 kX C "'.v/  "'.y0 /; we have lim hˇ.y" /; y"  viW  0:

"!0

Due to monotony of ˇ it follows that: 0  lim hˇ.y" /; y"  viW  lim hˇ.v/; y"  viW D hˇ.v/; y  viW 8w 2 dom ': "!0

"!0

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2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

For an arbitrary w 2 intdom ' there exists " > 0 such that y C tw 2 dom '

8jtj < ":

Substituting in the last inequality v D y C tw.t > 0/ we obtain hˇ.y C tw/; wiW  0 8w 2 intdom '; whence as t ! 0C we find hˇ.y/; wiW  0: Similarly analyzing the case t < 0 we come to the relation hˇ.y/; wiW  0 namely hˇ.y/; wiW D 0

8w 2 intdom ';

but then hˇ.y/; wiW D 0

8w 2 dom ';

and therefore for w 2 spen dom '. Since spen dom ' is dense in X then ˇ.y/ D 0 that is y 2 K. Further for an arbitrary w 2 K \ dom ' ŒA.y" /  f; w  y"  C '.w/  '.y" /  ŒA.y" /; y"  y0  1 C hˇ.w/  ˇ.y" /; w  y" iW  0; " whence lim ŒA.y" /; y"  y D lim hd" ; y"  yiV  0;

"!0

"!0

where d" 2 coA.y" / and due to -pseudomonotony '.w/ C hf; y  wiX  lim .hd" ; y"  wiV C '.y" // "!0

 ŒA.y/; y  w C '.y/8w 2 K \ dom ': The Theorem is proved. Again we consider the inequality of (2.40) type: ŒA.y/;   yC C './  '.y/ > hf;   yiX 8  2 K \ dom ' 

(2.74)

for the case when dom ' D X . As before AW V ! 2V , X D V \ W , K is a closed convex subset in X , 'W X ! R is a convex lower semicontinuous function, f 2 X  . Let us cite for the map A sufficient conditions such that the inequality (2.74) was strictly solvable (had strict solutions).

2.4 The Penalty Method for Multivariation Inequalities

183

Together with the inequality (2.74) we consider the operator inclusion connected with it 1 (2.75) coA.y/ C @'.y/ C ˇ.y/ 3 f; " 

where @'W X ! 2X is a subdifferential of the function '. Proposition 2.7. Each solution y 2 Dom.@'/ of the inclusion (2.75) satisfies the variation inequality 1 ŒA.y/;   yC C hˇ.y/;   yiW C './  '.y/  hf;   yiX 8  2 dom ' " (2.76) Proof. Let y 2 Dom.@'/  dom ' satisfies the inclusion (2.75) then there can be found d.y/ 2 coA.y/ and h.y/ 2 @'.y/ such that 1 d.y/ C h.y/ C ˇ.y/ D f: " However hh.y/;   yiX 6 './  '.y/

8  2 dom ';

therefore 1 ŒA.y/;   yC C hˇ.y/;   yiW C './  '.y/ > hd.y/;   yiV " 1 Chh.y/;   yiX C hˇ.y/;   yiW > hf;   yiX 8  2 dom ': " The Proposition is proved. Proposition 2.8. Each solution y 2 intdom ' of the inequality (2.76) is a solution of the inclusion (2.75) if dom ' D X . Proof. For a main convex lower semicontinuous function ' its subdifferential @' W intdom '  X ! 2X



is an upper semicontinuous map and @'./ is a nonempty closed convex bounded subset of X  8  2 intdom '. Since '.y/  './ > hd./; y  i 8 y 2 intdom '; where d./ 2 @'./ we have '.y/  './ > Œ@'./; y  C :

184

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Then taking into account the inequality (2.76) we find 1 ŒA.y/;   yC C hˇ.y/;   yiW > hf;   yiX C Œ@'./; y  C : " The set intdom ' is convex therefore  D y C t.w  y/ 2 intdom ' 8 w 2 intdom '; 8 t 2 Œ0; 1; hence we obtain the inequality 1 ŒA.y/; w  yC C hˇ.y/; w  yiW  hf; w  yiX C Œ@'.y  t .y  w//; y  wC : "

Since each upper semicontinuous map is radial semicontinuous, passing in the last inequality to the limit as t ! 0C we obtain 1 ŒA.y/; w  yC C hˇ.y/; w  yiW > hf; w  yiX C Œ@'.y/; y  w " whence 1 ŒA.y/; w  yC C Œ@'.y/; w  yC C hˇ.y/; w  yiW " > hf; w  yiX 8 w 2 intdom ':

(2.77)

Let us show that the relation (2.77) holds true 8 w 2 X . Let wn 2 intdom ' and wn ! w in X . Then 1 ŒA.y/; wn  yC C Œ@'.y/; wn  yC C hˇ.y/; wn  yiW > hf; wn  yiX : (2.78) " Remark that lim ŒA.y/ C @'.y/; wn  yC 6 ŒA.y/ C @'.y/; w  yC :

n!1

Indeed, 9 dn 2 coA.y/ C @'.y/ such that hdn ; wn  yiX D ŒA.y/ C @'.y/; wn  yC and without loss of generality we assume that dn * d . Then passing to the limit in (2.78) we obtain 1 ŒA.y/; w  yC C Œ@'.y/; w  yC C hˇ.y/; w  yiW > hf; w  yiX 8 w 2 X; "

2.4 The Penalty Method for Multivariation Inequalities

185

whence

1 coA.y/ C @'.y/ C ˇ.y/ 3 f: " The Proposition is proved.

Remark 2.11. If dom ' D X then due to Propositions 2.7, 2.8 it follows that the solution family of the operator inclusion (2.75) coincide with the solution family of the variation inequality (2.76). 

Theorem 2.20. Suppose AW V ! 2V is a bounded -pseudomonotone map, 'W X ! R is a convex lower semicontinuous function, dom ' D X and there can be found "0 > 0; y0 2 K such that   1 ŒA.y/; y  y ! C1 kyk1  C hˇ.y/; y  y i 0 C 0 W X "0

(2.79)

as kykX ! 1. Then 8 " 6 "0 the inequality (2.76) is solvable and from the sequence of its solutions fy" g there can be isolated a subsequence fy g such that y * y in X , y 2 K and it satisfies the inequality (2.74). Proof. For " 6 "0 let us consider the operator inclusion (2.75) which is equivalent to the variation inequality (2.76) due to Remark 2.11. We set A" .y/ D coA.y/ C B" .y/; where

1 B" .y/ D @'.y/ C ˇ.y/: " 

For each " > 0 the operator B" W X ! 2W is bounded, monotone and radial semicontinuous, namely it fulfills the condition (a) of the proof of Theorem 2.17. There fore the map A" W X ! 2X is bounded and -pseudomonotone (Proposition 2.6). Lemma 2.11. For each " 6 "0 the operator inclusion A" .y/ 3 f

(2.80)

is solvable. Proof. Let y0 2 K be an element from the condition (2.79). Let us consider the map A"; y0 .y/ D A" .y Cy0 / which is, obviously, bounded and -pseudomonotone. Moreover for an arbitrary " 6 "0 we obtain y /; b y  y0 C ŒA"; y0 .y/; yC D ŒA" .y C y0 /; yC D ŒA" .b 1 y /; b y  y0 iW C Œ@'.b D ŒA.b y /; b y  y0 C C hˇ.b y /; b y  y0 C "

186

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

1 hˇ.b y /; b y  y0 iW C Œ@'.y0 /; b y  y0  "0 1 > ŒA.b y /; b y y0 C C hˇ.b y /; b y  y0 iW  k@'.y0 /k kb y y0 kX ; "0 > ŒA.b y /; b y  y0 C C

where b y D y C y0 . Hence due to coercivity condition (2.79) we have kyk1 X ŒA"; y0 .y/; yC ! C1 as kykX ! 1; 

namely the operator A"; y0 W X ! 2X satisfies all requirements of Corollary 2.2, whence validity of the Lemma follows. Let y" be a solution of inclusion (2.75) for " 6 "0 . Similarly to Remark 2.10 we conclude that ŒA" .y/  f; yC > 0 8 y 2 @Br0 ; where r0 does not depend on " 6 "0 . Therefore for each " 6 "0 the inclusion (2.75) has a solution in the ball Br0 , namely ky" kX 6 r0 and without any restrictions we may assume that y" * y in X . The map A D coA C @'W X ! 2X



is bounded, -pseudomonotone and coA D A . Further similarly to the proof of Theorem 2.18 we prove that y 2 K and the following inequality is valid hf; y  wiX > ŒA.y/; y  w C Œ@'.y/; y  w 8 w 2 K; which due to the relation './  '.y/ > Œ@'.y/;   yC is equivalent to inequality (2.74). Definition 2.2. An element y 2 K \ dom ' is called a strict solution (s-solution) of the variation inequality (2.74) if there can be found a selector d of the multivalued map coA such that hd.y/; w  yiX C '.w/  '.y/ > hf; w  yiX 8 w 2 K \ dom ': 

Theorem 2.21. Suppose AW V ! 2V is a bounded s -pseudomonotone map, dom ' D X and the coercivity condition (2.79) is valid. Then for each " 6 "0 the variation inequality (2.76) has an s-solution. Sequence of solutions fy" g contains a subsequence fy g such that y * y in X; y 2 K and it is an s-solution of the inequality (2.74). Proof. Since s -pseudomonotony of the map A implies its -pseudomonotony then for each " 6 "0 the inclusion (2.75) is solvable. Namely there exists y" 2 X and

2.4 The Penalty Method for Multivariation Inequalities

187

selectors l.y" / 2 coA.y" /; for which

h.y" / 2 @'.y" /;

l.y" / C h.y" / C "1 ˇ.y" / D f;

(2.81)

therefore y" is an s-solution of inequality (2.76) where kykX 6 r0 (see the proof of Theorem 2.20). Let d" D l.y" / C h.y" / and y" * y in X . Then in virtue of (2.81) hd" ; y"  wiX 6 hf; y"  wiX

8 w 2 K;

and since y 2 K then lim hd" ; y"  yiX 6 0:

"!0

Let us establish s -pseudomonotony for the map A D coA C @'. It is sufficient to prove s -pseudomonotony of the map @' (Proposition 1.55). This property follows from Lemma 1.7. Hence, up to a subsequence if necessary we conclude that hf; y  wiX > lim hd.y" /; y"  wiX > hd.y/; y  wiX 8 w 2 K; "!0

where d.y/ 2 A .y/ or hd.y/; w  yiX > hf; w  yiX 8 w 2 K: In the same time d.y/ D l.y/ C h.y/ where l.y/ 2 coA.y/; h.y/ 2 @'.y/ therefore hl.y/; w  yiV C '.w/  '.y/ > hl.y/; w  yiV C hh.y/; w  yiX > hf; w  yiX 8 w 2 X; and this fact proves the Theorem. Example 2.3. Under conditions of Example 2.2 we additionally suppose that

0,

K D fy 2 W jy.x/  0 for a.e.x 2 ˝g is a nonempty closed convex set in the space W , U0 D fU 2 U j for a.e. x 2 ˝; 8 2 Rn . ; U .x/ /Rn  0g:

(2.82)

Note that U0 is sequentially compact set in the weakly star topology of the space ŒL1 .˝/nn (see [KapKasKoh08, p. 1628]). Lemma 2.12. The multivalued mappings A W X  U0  X  satisfies the condition of 0 -quasimonotonicity.

188

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Proof. Let Uk D Œu1k ; u2k ; : : : ; unk  ! U0 D Œu10 ; u20 ; : : : ; un0  *-weakly in Y  , yk ! y0 weakly in X , dk ! d0 weakly in X  and lim hdn ; yn  yiX  0;

n!C1

where dk D A.yk ; Uk /, Uk 2 U0 8k  1. Let us set pk D ui k , p0 D ui 0 , vk D ryk , v0 D ry0 . Since divpk D 0 i rotvk D 0 8k  1 (see for details [KapKasKoh08, p. 1626, 1629]), ryk ! ry0 weakly in ŒL2 .˝/n and Uk ! U0 *-weakly in ŒL1 .˝/nn and weakly in ŒL2 .˝/nn , by the lemma on compensated compactness, Z lim

k!1 ˝

D

.Uk ryk ; /Rn dx D

i D1

n Z X i D1

˝

Z

n X

lim

k!1 ˝

.ui k ; ryk /Rn i dx

Z

.ui 0 ; ry0 /Rn i dx D

˝

.U0 ry0 ; /Rn dx;

8 2 ŒC01 .˝/n :

Since the set C01 .˝/ is dense set in L2 .˝/, we see that Uk ryk ! U0 ry0 weakly in ŒL2 .˝/n . Therefore, dk D div.Uk ryk / ! div.U0 ry0 / D d0 weakly in X  :

(2.83)

Hence, d0 D A.y0 ; U0 /. Let us show that lim hA.yk ; Uk /; yk  y0 iX  0:

(2.84)

k!C1

Indeed, from (2.82) and symmetry of matrices Un we have that 8k  1 hA.yk ; Uk /; yk  y0 iX  hA.y0 ; Uk /; yk  y0 iX D hA.yk ; Uk /; y0 iX  hA.y0 ; Uk /; y0 iX : In consequence of Uk D Œukij 1i;j n ! U0 D Œu0ij 1i;j n *-weakly in ŒL1 .˝/nn ,

@y0 @xj

2 L2 .˝/, j D 1; n and (2.83), we have that: X Z

hA.y0 ; Uk /; y0 iX D !

X Z 1i;j n ˝

1i;j n ˝

u0ij .x/

ukij .x/

@y0 @y0 dx @xj @xi

@y0 @y0 dx D hA.y0 ; U0 /; y0 iX k ! C1 @xj @xi

2.4 The Penalty Method for Multivariation Inequalities

189

and hA.yk ; Uk /; y0 iX ! hd0 ; y0 iX D hA.y0 ; U0 /; y0 iX k ! C1: This implies (2.84). From (2.83) and (2.84) we have that 8w 2 X

lim hA.yk ; Uk /; yk  wiX  lim hA.yk ; Uk /; y0  wiX k!C1

k!C1

D hA.y0 ; U0 /; y0  wiX : The lemma is proved.   For fixed f 2 X  and U D uij .x/ 1i;j n 2 U0 let us consider such problem: 

n X i;j D1

@ @xi



@y uij .x/ @xj

 f .x/ for a.e. x 2 ˝;

y.x/  0 for a.e. x 2 ˝; 1

n X @y @ @ uij .x/ C f .x/A y.x/ D 0 for a.e. x 2 ˝; @xi @xj 0

(2.85)

(2.86) (2.87)

i;j D1

y.x/ D 0 a.e. on :

(2.88)

Multiplying (2.27) by rather regular non-negative finite functions in the domain ˝ and “integrating by parts” (see for details [Li69, GaGrZa74]) we obtain such problem:  hA.y; U /; w  yiX  hf; w  yiX 8w 2 K; (2.89) y 2 K: Solutions of problem (2.89) are called the generalized solutions of problem (2.85)– (2.88). Let us denote the set of generalized solutions of problem (2.85)–(2.88) by K.f; U /  X . Hence, in consequence of Corollary 2.2 and upper given assumptions the next corollary is fulfilled. Corollary 2.3. Let Un ! U *-weakly in Y  , fn ! f weakly in X  , Un 2 U0 8n  1. Then w \ [ K.Um ; fm /  K.U ; f /; n1 mn w

where Q is weak closure of the set Q  X in X . Moreover, there exists such U 2 U0 that for an arbitrary f 2 X  the set K.U ; f / is nonempty and weakly closed in X , 9 fy"n gn1  X such that takes place the next inclusion coA.y"n ; u/ C

1 coˇ.y"n / 3 f "n

8n  1;

190

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces w

"n ! 0C, y"n ! y in X as n ! 1, and weakly limit point y satisfies (2.89), ˇ W W ! Cv .W  / is the penalty operator. Remark 2.12. For a fixed f 2 X  let us consider the set of “admissible pairs” G.f / of problem (2.89), i.e., G.f / D f.y; U /jU 2 U0 ; y 2 K.U ; f /g  X  Y  : In Corollary 2.3, in particular, it is stated that the set of “admissible pairs” G.f / of problem (2.89) is nonempty and closed in the topology of weak convergence in X and the weakly star topology of the space Y  . Remark 2.13. Solenoidal controls in coefficients of the main part of the heat conductivity equation, may be for the first time, were introduced in [KapKasKoh08]. In the present book this idea is realized for the case of problems with degeneration. Particularly, it is shown that to prove closedness of the set of admissible solutions for the given control problem in the Cartesian product of natural topologies we do not need uniform ellipticity of the main part of the corresponding differential operator.

2.5 Nonlinear Operators Equations of the Hammerstein Type. System of Operators Equations In this section we consider the problems of existence of solutions for operators equations of the Hammerstein type and study the properties of totality of solutions under some conditions of operators. Let X; U be Banach spaces. Let us consider the equation x C BF .u; x/ D f;

x 2 X;

u 2 U;

where F W U  X ! X  is nonlinear map, B W X  ! X is linear operator, f 2 X . Let us suggest, that the space X has Property ./: (1) X is reflexive; (2) There exist monotone hemicontinuous operator A, that operates from adjoint space X  into X and that A0 D 0, A is continuous in 0 of space X  and hAz; ziX  C kzk˛X  , where C > 0; ˛ > 1; z 2 X  . As the examples of such spaces we can consider reflexive Banach spaces X , for which the norm in X  is differentiable by Frechet. In such spaces the operator A has the form Az D kzk˛X  grad kzkX  ;

˛ > 0;

z ¤ 0;

z 2 X I

A0 D 0:

Let us give the typical examples of systems of functional equations of different type.

2.5 Nonlinear Operators Equations of the Hammerstein Type. System of Operators Equations 191

Example 2.4. Let us consider the next system of functional equations, that consists of nonlinear integral equation as well as nonlinear partial differential equation with boundary conditions Z x .!/ C

K .!; w/ h .w; z .w/ ; x .w// d w D g .!/

(2.90)

˝



N X @z @z @ D f .!/ ; gj !; x .!/ ; ;:::;  @!j @!1 @!N

(2.91)

zj@˝ D 0;

(2.92)

j D1

where ˝ is bounded domain in RN ; @˝ be its boundary. Let ˝ be domain in RN and let h D h.xI / be the function, that is defined for almost all x 2 ˝ and all  2 Rm . It is said, that the function h has the Caratheodory property (h 2 CAR ), if: (1) For all  2 Rm the function ˝ 3 x ! h.x; / is measurable; (2) For almost all x 2 ˝ the function Rm 2  ! h.x; / is continuous. The meaning of Caratheodory property we will consider in the next Theorem. Theorem 2.22. Let h 2 CAR and ui .x/, i D 1; : : : ; m, are measurable functions on ˝. Then the combined function g.x/ D h.x; u1 .x/; : : : ; um .x// is measurable on ˝. Let h D h.xI / be the function, that is defined for x 2 ˝ and  2 Rm , and let h 2 CAR. The operator H , that is defined on various well-ordered sets of m measurable functions ui .x/ i D 1; : : : ; m, by the formula H.u1 ; : : : ; um /.x/ D h.x; u1 .x/; : : : ; um .x//; x 2 ˝ is called the Nemitsky operator. Theorem 2.22 guarantees, that the Nemitsky operator moves up the set of measurable functions into the measurable function. The next Theorem is the dominant in the theory of boundary problems. Theorem 2.23. Let p1 ; p2 ; : : : ; pm and r are real numbers, Pi  1.i D 1; : : : ; m/, V  1 and h D h.xI / .x 2 ˝;  2 Rm / such, that h 2 CAR. For arbitrary ui 2 LPi .˝/ .i D 1; : : : ; m/ the relation H.u1 ; : : : ; um / 2 LV .˝/ takes place if and only if the next condition is true: for a.e. x 2 ˝ jh.xI 1 ; : : : ; m /j  g.x/ C c

m X i D1

where g 2 LV .˝/; c > 0.

ji jPi jV ;

192

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

At that the Nemitsky operator H W LP1 .˝/  : : :  LPi .˝/ ! LV .˝/ is continuous. Let the next properties are true: (1) The function K .!; w/ is continuous by both variables in the domain ˝  ˝  R N  RN ; @z (2) Functions x; z and their partial derivatives @! ; j D 1; : : : ; N , belong to the j space of integrable functions of degree that equals to p, i.e., x; z;

@z 2 Lp .˝/ ; @!j

1 1 C D 1I p q

p  2;

(3) The function h .!; / 2 CAR, that corresponds to the map h .!; z .!/; x .!// and satisfies the condition jh .!; /j  a .!/ C c1

2 X

ji jp1 ;

(2.93)

i D1

where a .!/ 2 Lq .˝/ ;

 D .1 ; 2 / 2 R2 ;

c1 > 0I  @z (4) Functions gi .!; / 2 CAR, which correspond to maps gi !; x .!/ ; @! ; : : :, 1  @z @!N and satisfy the conditions N C1 X ˇ ˇ ˇgj .!; /ˇ  bj .!/ C cj j i jp1 ;

(2.94)

i D1

where

D . 1 ; : : : ; N C1 / 2 RN C1 ;

bj .!/ 2 Lq .˝/ ;

cj > 0I

(5) The functional f can be represented in the form X

f D

D ˛ v˛ ;

v˛ 2 Lq .˝/ ;

j˛j2

where D ˛ is the differential operator of the form D˛ D

@j˛j ˛N @x1˛1 ; : : : ; @xN

;

˛ D .˛1 ; : : : ; ˛N / ;

j˛j D ˛1 C : : : C ˛N :

2.5 Nonlinear Operators Equations of the Hammerstein Type. System of Operators Equations 193

The derivative D ˛ u of distribution u on ˝ is represented by the formula .D ˛ u/ .'/ D .1/j˛j u .D ˛ '/ ; D .˝/ D f' 2 C01 .˝/ W p .'/ D

X

' 2 D .˝/ ; sup j ˛ .!/ D ˛ ' .!/jg;

˛ !2˝

where f ˛ g is some family of functions, which are defined and continuous on the set ˝.y These conditions allow us to represent the system (2.90), (2.91) with boundary conditions (2.92) in the form of system of operators equations. Indeed, denoting F .z; x/ .!/ D h .!; z .!/ ; x .!// and taking into account, that W01;p .˝/  Lp .˝/ ; in view of (2.93) we will obtain the nonlinear Nemitsky operator 1;p

F W W0

.˝/  Lp .˝/  Lp .˝/  Lp .˝/ ! Lq .˝/ :

The integral operator let us denote as Z .By/.!/ D K .!; w/ y .w/ d w ˝

and, taking into account, that the function K .!; w/ is continuous, we can suggest, that this function K .!; w/ 2 Lp .˝/ by the variable of integration !. Therefore, the linear operator B W Lq .˝/ ! Lp .˝/. Then (2.90) we can write in the form of operators equation x C BF .z; x/ D g: This equation in the literature is called the not uniform equation of the Hammerstein type. Let us now pass on to the operator representation of (2.91) with boundary conditions (2.92). Let us denote the linear operator L W z ! gradz D , where the vector function D

@z @z ;:::; @!1 @!N



and linear operator Gj .x; / D Gj .x; Lz/ D gj

@z @z !; x .!/ ; ;:::; @!1 @!N

D 'j .!/:

Here ' D .'1 ; : : : ; 'N / D .G1 ; : : : ; GN / D G: Taking into account boundary conditions (2.92) we suggest, that z 2 W01;p .˝/.

194

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Then the operator 1;p

L W W0

.˝/ ! Lp .˝/  : : :  Lp .˝/ D LN p .˝/; ƒ‚ … „ N

and in view of property (4) operators C1 Gj W Lp .˝/  : : :  Lp .˝/ D LN .˝/ ! Lq .˝/ ; p ƒ‚ … „

j D 1; : : : ; N:

N C1

Denoting the vector-function ' D .'1 ; : : : ; 'N / 2 LN q .˝/ and according to 4), 'i 2 W 1;q .˝/ ; j D 1; : : : ; N . Let us denote the divergence of vector-function ' with inverse sign in the form L ' D 

N X @'j ; @!j

j D1

and the space

h

i W01;p .˝/ D W 1;q .˝/

is the range of values of this operator. Therefore, we can write in the operator form (2.91) with boundary conditions (2.92) G .x; z/ D L G .x; Lz/ D f; where L W W01;p .˝/ ! LN p .˝/ ;

 N C1 G W LN .˝/ ! W 1;q .˝/ ; p

h i  N L W W 1;q .˝/ ! W01;p .˝/ D W 1;q .˝/ : Let us note, that the set W 1;q .˝/ we can identify with the set ( 

f W f 2 D .˝/ ;

) f .'/ sup < C1 ; '2D.˝/ k'kW 1;p .˝/

where D  .˝/ is the set of linear continuous functionals on D .˝/. Let us note, that the space W01;p .˝/ is the closing of the set C01 .˝/ in 1;p .˝/ with regard to norm W

2.5 Nonlinear Operators Equations of the Hammerstein Type. System of Operators Equations 195

0 B kzkW 1;p .˝/ D @

Z

˝

0 @

X

11=p

1p=2 2 jD ˛ zj A

C d! A

:

j˛j1

For f 2 D  .˝/ and ' 2 D .˝/ we write f .'/ D hf; 'i. This function h:; :i, defined on D  .˝/  D .˝/, is called the scalar product of D  .˝/ and D .˝/. Let us now write the system of nonlinear functional equations (2.90), (2.91) with boundary conditions (2.92) in the form of system of operators equations of the Hammerstein type of the next form x C BF .z; x/ D g

(2.95)

G .x; z/ D f:

(2.96)

Example 2.5. Let us consider the next system of functional equations Z K .!; w/ h .w; z .w/ ; x .w// d w D g .!/

x .!/ C

(2.97)

˝

@z @2 z @2 z @z D f .!/ ;:::; ; ; : : : ; G !I x .!/ ; 2 @!1 @!N @!12 @!N

(2.98)

with boundary condition zj@˝ D 0:

(2.99)

Let: (1) The function K .!; w/ 2 C .˝  ˝/ and x; z;

@z @2 z ; 2 Lp .˝/ ; @!j @!j2

j D 1; : : : ; N;

p  2I

(2) The function h .!; / 2 CAR, that corresponds to the map h .!; z .!/ ; x .!// and satisfies the condition jh .!; /j  a .!/ C c1

2 X

ji jp1 ;

i D1

where a .!/ 2 Lq .˝/ ;

 D .1 ; 2 / 2 R2 ;

c1 > 0I

(3) Function G .!; / 2 CAR, which corresponds to the map

@z @z @2 z @2 z G !; x .!/ ; ;:::; ; ; : : : ; 2 @!1 @!N @!12 @!N

196

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

and satisfies the condition jG .!; /j  b .!/ C c

lC1 X

j i jp1 ;

i D1

where b .!/ 2 Lq .˝/ ;

D . 1 ; : : : ; lC1 / 2 RlC1 ;

c > 0I

(4) The functional f can be represented in the form f D

X

D ˛ v˛ ;

v˛ 2 Lq .˝/ :

(2.100)

j˛j2

T Taking into account boundary conditions (2.99), we suggest z 2 W01;p .˝/ Wp2 .˝/. Let us write the first equation of this system (similarly to Example 2.4) in operator form, taking into account 1;p

F W W0

.˝/  Lp .˝/  Lp .˝/  Lp .˝/ ! Lq .˝/ :

Let us now consider the operator form of (2.98) with boundary conditions (2.99). Denoting the left part of (2.98) as the nonlinear operator

@z @2 z @2 z @z ;:::; ; ; : : : ; G .x; z/ .!/ D G !; x .!/ ; 2 @!1 @!N @!12 @!N

;

we will obtain the Nemitsky operator  lC1 G W Lp .˝/ ! Lq .˝/ : The functional f 2 Lq .˝/ can be represented in the form of (2.100), therefore, h i f 2 W 1;q .˝/ D W01;p .˝/ . So, we can write the system (2.97), (2.98) with boundary conditions (2.99) in the operator form x C BF .z; x/ D g; (2.101) G .x; z/ D f

(2.102)

Example 2.6. Let us consider the system of nonlinear functional equations, that consists of nonlinear integro-differential equation and the second order partial differential quasilinear equation Z x .!/ C ˝



@z @z K .!; w/ h w; z .w/ ; ;:::; ; x .w/ d w D g .!/ @!1 @!N

(2.103)

2.5 Nonlinear Operators Equations of the Hammerstein Type. System of Operators Equations 197

N X @z @z @2 z D f .!/ ; C G !I x .!/ ; z .!/ ; ;:::; @!1 @!N @!j2 j D1

(2.104)

with boundary conditions zj@˝ D 0;

j D 1; : : : ; N:

(2.105)

Let: (1) The function K .!; w/ 2 C .˝  ˝/ and x; z;

@z @2 z ; 2 Lp .˝/ ; @!j @!j2

j D 1; : : : ; N;

p  2I

(2) The function h .!; / 2 CAR, that corresponds to the map

@z @z h !; z .!/ ; ;:::; ; x .!/ @!1 @!N



and satisfies the condition jh .!; /j  a .!/ C c1

N C2 X

ji jp1 ;

i D1

where a .!/ 2 Lq .˝/ ;

 D .1 ; N C2 / 2 RN C2 ;

c1 > 0I

(3) The function G .!; / 2 CAR, that corresponds to the map

@z @z ;:::; G !; x .!/ ; z .!/ ; @!1 @!N and satisfies the condition jG .!; /j  b .!/ C c

N C2 X

j i jp1 ;

i D1

where b .!/ 2 Lq .˝/ ;

D . 1 ; : : : ; N C2 / 2 RN C2 ;

(4) The functional f can be represented in the form f D

X j˛j2

D ˛ v˛ ;

v˛ 2 Lq .˝/ :

c > 0I

198

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Taking into account boundary conditions (2.105), we suggest 1;p

z 2 W0

.˝/  Lp .˝/ :

Here, denoting maps Z By.!/ D

K .!; w/ y .w/ d w; ˝

y 2 Lq .˝/ ;



@z @z F .z; x/ .!/ D h !; z .!/ ; ;:::; ; x .!/ ; @!1 @!N

@z @z G .x; z/ .!/ D G !; x .!/ ; z .!/ ; ; ;:::; @!1 @!N Lz D

N X @2 z ; @!j2 j D1

we will obtain operators, which operates 1;p

F W W0

.˝/  Lp .˝/  Lp .˝/  Lp .˝/ ! Lq .˝/ ;

G W Lp .˝/ 

1;p W0

B W Lq .˝/ ! Lp .˝/ ;

.˝/  Lp .˝/  Lp .˝/ ! Lq .˝/ ; L W W 2;p .˝/ !Lq .˝/ ;

and we will obtain the system of operators equations x C BF .z; x/ D g; Lz C G .x; z/ D f: We considered systems of functional equations, which consist of the second order partial differential equations. Analogically, we can consider equations and systems of arbitrary even order. Let us consider some well-known definitions. Let EX and EY are normalized spaces. Definition 2.3. The map G W D.G/ ! EY , where D.G/  EX is called strenuously continuous, if it transforms any weakly converging sequence into converging sequence. Definition 2.4. The map G W D.G/  EX ! EY is called hemicontinuous in the point x0 2 D .G/, if for every vector x, such, that x0 Ctx 2 D .G/ as 0  t  ˛ the sequence G .x0 C tn x/ converges to G .x0 / weakly, i.e., G .x0 C tn x/ ! G .x0 / weakly in EY as tn ! 0C. Definition 2.5. The operator T from normalized space E into E  (where E  is adjoint space to E) is called semimonotone, if there exists such strenuously continuous map C W E ! E  , that the G D T C C W E ! E  is monotone.

2.5 Nonlinear Operators Equations of the Hammerstein Type. System of Operators Equations 199

Let us consider now the Theorems of existence of solutions of not uniform equation of the Hammerstein type and study the properties of totality of solutions. We will suggest, that Banach space X has Property ./ I U is bounded set in U , and the space U is adjoint with some Banach space W . Theorem 2.24. Let the operator B W X  ! X be linear continuous positive and has the right backward; F .u; :/ W X ! X  8u 2 U is radially continuous operator with semibounded variation, that satisfies the condition ˛ ˝ F .u; x/  B 1 g; x X  0 if

kxkX >  > 0;

 D const.

Then the set Ku D fx 2 X W x C BF .u; x/ D gg is nonempty and weakly compact, when u 2 U , g 2 X are fixed. Proof. The proof of this Theorem methodically is not different from the proof of Theorem in the case of uniform equation [Va73]. therefore, let us consider some fundamental places of proof. Let us consider the sequence of operators ˚n ! D A

  ! C B  ! C BF u; B  ! ; n

! 2 X  .n D 1; 2; : : :/ ;

(2.106)

which operates from X  into X . Here the operator A W X  ! X is monotone hemicontinuous, continuous in 0 of the space X  such, that A0 D 0, and so hA!; !iX D k!k˛C1 X  ; ˛ > 0; ˛ D const. In view of property ./ of the space X , there exists the operator A with upper mentioned properties. Let us note, that from hemicontinuity of operator A it follows the radial continuity, and from relation hA!; !iX k!k˛C1 X D D k!k˛X  ! 1 k!kX  k!kX 

as

k!kX  ! 1

its coercivity follows. Taking into account the properties of operators A; B; F we draw the conclusion, that the operator ˚n is coercive radially continuous with semibounded variation. Then, in view of [Va73, Corollary 4.1], the set of solutions of equation ˚n ! D g is nonempty for every n, g 2 X and u 2 U . This solution we will denote as !n . Then from (2.106) we have A

  !n C B  !n C BF u; B  !n D g: n

(2.107)

200

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

From here, taking into account, that operator B is linear and has the right backward E D !  ˛ ˝  n A ; !n C hB!n ; !n iX C F u; B  !n  B 1 g; B  !n X D 0: n X

(2.108)

In the equality (2.108) the first and the second terms are positive, when !n ¤ 0, as !n satisfies the equality (2.107), then the third term is negative, whence, in view of conditions of Theorem kxn kX D kB  !n kX   for all n. Also we have k!n kX   1 D const, since the linear continuous positive operator B  W X  ! X can’t transform nonbounded set into bounded set. From the equality (2.107) we will obtain E D !  ˛ ˝  n F u; B  !n ; B  !n X D  A ; !n  hB!n ; !n iX C hg; !n iX : n X From here, taking into account, that hA!; !iX D k!k˛C1 X  we have ˝   ˛ 1 F u; B  !n ; B  !n X D  ˛ k!n k˛C1 X   hB!n ; !n iX C hg; !n iX n

(2.109)

and we draw the conclusion, that jhF .u; B  !n / ; B  !n iX j  c1 D const, since all terms in the right part of equality (2.109) are bounded in view of boundness of set f!n g  X  and linear continuity of operator B. Then, in view of Remark 1 [LaSoUr68] we have

 

F u; B  !n   c2 D const as X

kB  !n kX  :

Let us note here, that c2 does not depend on u, as in the equality (2.109) the right part does not depend on u. Taking into account the form of operator A, and that operator A is continuous in 0 of the space X  and A0 D 0, we have lim A

n!1

!n D0 n

(2.110)

Sequences fB  !n g D fxn g  X and fF .u; B  !n /g  X  are bounded and in view of reflexivity of spaces X and X  we can choose a subsequences, which weakly converge respectively to x0 and to y0 , i.e., B  !m ! x0 weakly in X FB  !m ! y0 weakly in X  . Then, passing to the limit in the equality (2.107), in view of (2.110), we will obtain x0 C By0 D g; (2.111) since B is linear and continuous operator. Further, using that fact, that operator F is operator with semibounded variation (i.e., F .u; / W X ! X  8u 2 U is operator with semibounded variation) we can prove, that in the equality (2.111): y0 D F .u; x0 / :

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

201

And we can finish the proof of the first part of the Theorem, i.e., Ku ¤ ; is nonempty. Let us pass to the proof of the second part of the Theorem. Let us prove, that the set Ku is weakly compact: from the arbitrary sequence fxn g  Ku we can choose a subsequence fxm g  fxn g such, that xm ! x0 weakly in X , at that x0 2 Ku . From the first part of the proof of the Theorem we know, that kxn kX  , kF .u; xn /kX   C1 , xn 2 X , when u 2 U is fixed. Therefore, in view of reflexivity of spaces 9 fxm g  fxn g, xm ! x0 weakly in X and F .u; xm / ! y0 weakly in X  . So, in the equation xn C BF .u; xn / D g passing to the weak limit and taking into account the continuity of the operator B, we will obtain x C By0 D g: Using that fact, that the operator F is radially continuous with semibounded variation, we can establish y0 D F .u; x0 / and, thus, x0 C BF .u; x0 / D g: The Theorem is proved. In the case of semicontinuous operator F the next analogical Theorem is true. Theorem 2.25. Let the next conditions are true: (1) Banach space X has Property ./. (2) Operator B is linear continuous positive from X  in X . (3) Hemicontinuous bounded and semicontinuous operator F .u; :/ W X ! X  as every u 2 U satisfy inequality ˝ ˛ F .u; x/  B 1 g; x  0;

if kxkX >  > 0

. D const/ :

Then the set Ku;g D fx 2 X W x C BF .u; x/ D gg

8g 2 X

is nonempty and weakly compact.

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization Let the state of the object, that we consider, is described by nonlinear operator equation F x D f;

202

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

where F W X ! X  is nonlinear operator, X is Banach space, X  is its adjoint, x 2 X; f 2 X  . If the operator F satisfies the next conditions: (1) F satisfies one kind of continuity (radial continuity, hemicontinuity, demicontinuity); (2) F is monotone, semimonotone or with semibounded variation; (3) F is coercive or coercive in the modified form, i.e., hF x; xi  0;

if

kxkX > ;

(2.112)

then the set K1 D fx 2 X W F x D 0g is nonempty [Va73, GlLiTr81, LaSoUr68]. For example, let us consider the next statements. Theorem 2.26. [GlLiTr81]. Let the operator F W X ! X  is radially continuous monotone coercive. Then the set of solutions of equation F x D f;

x 2 X;

(2.113)

at fixed f 2 X  is nonempty, weakly closed and convex. From the proof of these statements it follows, that the condition (2.112) will guarantee kxkX   8x 2 K1 .y For uniform equations the condition (2.112) is modified and has the form: (2.114) hF x  f; xi  0; if kxkX > : For the system (2.113) the condition (2.114) may be unfulfilled. As a result of this there emerges the next problem: find such operator M W X ! X , that the equation FM x D f has solutions f 2 X  , at that the set K2 D fx 2 X W FM x D f g is bounded, i.e., kxkX   8x 2 K2 . Let the reflexive Banach space X is compactly and densely embedded in respective Hilbert space H , i.e., X  H  X  , X  is adjoint space to X I nonlinear operator F W H ! H is Lipschitz. At that 8x1 ; x2 2 X we will obtain jhF x1  F x2 ; x1  x2 iX j D j.F x1  F x2 ; x1  x2 /j  kF x1  F x2 kH kx1  x2 kH  C1 kx1  x2 kH kx1  x2 kH  C1 C2 kx1  x2 k2X D C0 kx1  x2 k2X : From here C1 kx1  x2 k2H  hF x1  F x2 ; x1  x2 iX ;

8x1 ; x2 2 X:

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

203

0

Here the norm kkH D kkX is compact in comparison with the norm kkX , i.e., if xn ! x0 weakly in X , then xn ! x0 strongly in H , since the space X is compactly embedded in H . Thus, we obtained the next statement. Proposition 2.9. Let reflexive Banach space X is compactly and densely embedded in Hilbert space H , and nonlinear operator F W H ! H is Lipschitzean. Then the narrowing of operator A on space X is the operator with semibounded variation 0 with the norm kkX D kkH , that is compact with regard to the norm kkX .y Theorem 2.27. Let Banach space X be compactly embedded in Hilbert space H and satisfies Property ./, F W H ! H be nonlinear Lipschitzean operator and its narrowing on the space X satisfies the next condition: 9 > 0;

hF x  f; xiX > 0;

at that

if

kxkX D :

(2.115)

Then there exists such operator M W H ! H , that the equation FM x D f;

x 2 X;

(2.116)

has the solution in the ball with radius , at that the totality of solutions K2 D fx 2 X W FM x D f g is weakly compact, and also K2  K1 D fx 2 X W F x D f g : Proof. Let us consider the continuous operator M W H ! H , that reflects elements x 2 H such, that kxkH > , on the sphere with radius , and elements x 2 H such, that kxkH  , leaves without changes. For example, ( Mx D

 kxkH

x; if kxkH > ; x; if kxkH  

This operator is monotone and Lipschitzean. Indeed, let X be Hilbert space. Then (1) If x1 ; x2 2 X such, that kx1 kH  ;

kx2 kH  , then

.M x1  M x2 ; x1  x2 / D .x1  x2 ; x1  x2 /  0I (2) If x1 ; x2 2 X such, that kx1 kH  ;

kx2 kH > , then

.M x1  M x2 ; x1  x2 / D x1 

 x2 ; x1  x2 kx2 kH



204

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

  .x1 ; x2 / C D .x1 ; x1 /  1 C .x2 ; x2 / kx2 kH kx2 kH

   kx1 kH  kx2 kH C  kx1 k2H  1 C kx2 k2H kx2 kH kx2 kH D kx1 k2H  kx1 kH  kx2 kH   kx1 kH C  kx2 kH D .kx1 kH  kx2 kH /  .kx1 kH  /  0; since both factors are not positive; (3) If x1 ; x2 2 X such, that kx1 kH > ;

kx2 kH  , then

 .M x1  M x2 ; x1  x2 / D x1  x2 ; x1  x2 kx1 kH

  .x1 ; x1 /  C 1 .x1 ; x2 / C .x2 ; x2 / D kx1 kH kx1 kH

 C 1  kx1 kH kx2 kH C kx2 k2H   kx1 kH  kx1 kH D .kx1 kH  kx2 kH / .  kx2 kH /  0;

since both factors are not negative; (4) If x1 ; x2 2 X such, that kx1 kH > ;

kx2 kH > , then

  .M x1  M x2 ; x1  x2 / D x1  x2 ; x1  x2 kx1 kH kx2 kH

     .x1 ; x2 / C .x1 ; x1 /  C .x2 ; x2 / D kx2 kH kx1 kH kx1 kH kx2 k H

  kx1 kH kx2 kH C kx2 kH C  kx1 kH  kx1 kH kx2 k H D  kx1 kH   kx2 kH   kx1 kH C  kx2 kH D 0: Now, if Banach space X is “embeddable”, i.e., there exists Hilbert space H such, that the next conditions hold: (1) X is embedded in H , and it is dense in H , and H is embedded in X  , and it is dense in X  , at that the topologies of spaces X and H are coordinated; (2) 8x 2 X the next equality is true .y; x/ D hz; xi, where y 2 H and z 2 X  , then z D y. Then for arbitrary x1 ; x2 from the subspace X of the space H (i.e., 8x1 ; x2 2 X  H ) we can define the bilinear form hM x1  M x2 ; x1  x2 i as the scalar product in H . At that, the proof of the monotony of operator M in X is analogical with points (1)–(4). Now let us pass to proof that fact, that operator M is Lipschits. Indeed, let the space X be Hilbert and Banach embeddable.

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

Then (1) If x1 ; x2 2 X such, that kx1 kH  ;

kx2 kH  , then

kM x1  M x2 k D kx1  x2 kH I (2) If x1 ; x2 2 X such, that kx1 kH  ;

kx2 kH > , then

kx1  x2 k2H  kM x1  M x2 k2H D .x1  x2 ; x1  x2 /  x1 

  x2 ; x1  x2 kx2 kH kx2 k H  .x1 ; x2 / D .x1 ; x1 /  2 .x1 ; x2 / C .x2 ; x2 /  .x1 ; x1 / C 2 kx2 kH

 2 2 .x1 ; x2 /  2 .x ; x / D  2  2  kx2 kH 2 2 2 kx k kx2 kH 2 H

 kx2 k2H  2 kx1 kH  kx2 kH C 2 kx1 kH  2 D .kx2 kH  / .kx2 kH C   2 kx1 kH /  0; since both factors are not negative, and, therefore, kM x1  M x2 kH  kx1  x2 kH I (3) If x1 ; x2 2 X such, that kx1 kH > ;

kx2 kH  , then

kx1  x2 k2H  kM x1  M x2 k2H D .x1  x2 ; x1  x2 /

  x1  x2 ; x1  x2  kx1 kH kx1 kH D .x1 ; x1 /  2 .x1 ; x2 / C .x2 ; x2 /  .x2 ; x2 /  2 C2 .x1 ; x2 /  .x1 ; x1 / kx1 kH kx1 k2H

 .x1 ; x2 /  2 D kx1 k2H  2  2 kx1 kH  kx1 k2H  2 kx1 kH  kx2 kH C 2 kx2 kH  2 D .kx1 kH  / .kx1 kH C   2 kx2 kH /  0; since both factors are not negative, and, therefore, kM x1  M x2 kH  kx1  x2 kH I

205

206

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

(4) If x1 ; x2 2 X such, that kx1 kH > ;

kx2 kH > , then

kx1  x2 k2H  kM x1  M x2 k2H D .x1  x2 ; x1  x2 /

     x1  x2 ; x1  x2 kx1 kH kx2 kH kx1 kH kx2 k H D .x1 ; x1 /  2 .x1 ; x2 / C .x2 ; x2 / 

2 kx1 k2H

.x1 ; x1 /

2 2 .x1 ; x2 /  .x2 ; x2 / kx1 kH  kx2 kH kx2 k2H

 .x1 ; x2 / D kx1 k2H C kx2 k2H  22  2  2 kx1 kH  kx2 kH

  kx1 k2H C kx2 k2H  22  2  2 kx1 kH kx2 kH kx1 kH  kx2 kH C2

D kx1 k2H C kx2 k2H  22  2 kx1 kH kx2 kH C 22 D .kx1 kH  kx2 kH /2  0

and, therefore, kM x1  M x2 kH  kx1  x2 kH : Thus, we proved, that operator M is Lipschitzean. Therefore, the narrowing of operator ˚ D FM W H ! H on the space X is the operator with semibounded variation. Further, 8x 2 X such, that kxkX >  let us note, that kxkX D kxkH , taking into account (2.115) and equality kM xkX D , we have: 

  0 < hFM x  f; M xiX D FM x  f; x kxkX X  D hFM x  f; xiX < hFM x  f; xiX D h˚x  f; xiX : kxkX Thus, the operator ˚ W X ! X  is Lipschitzean (from here it follows its continuity), with semibounded variation and satisfies the condition h˚x  f; xiX > 0;

if

kxkX  :

(2.117)

Under the condition of the Theorem in view of property ./ in reflexive space X there exists such operator A W X ! X  , that is monotone, hemicontinuous, continuous in 0 of space X , and A0 D 0, at that hAx; xiX  C kxk˛X ;

when constans C > 0; ˛ > 1:

(2.118)

Then we can set Ax D kxk˛X grad kxkX ;

whence

: hAx; xi D kxk˛C1 X

(2.119)

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

207

Let us consider the sequence of operators Tn x D A

x C ˚x: n

Here operators A . / for every n are coercive, and operator ˚ satisfies the inequality n (2.117). Then ˝ x ˛ ˝ ˛ An;x n A xn ; xn hTn x; xi h˚x; xi h˚x; xi D C D C kxkX kxkX kxkX kxkX kxkX ˛C1 kxkX   n ˛C1 x x kxk˛X C f; ! C1;  n C D n kxkX kxkX kxkX since the first component converges to infinity as kxkX ! 1 for every fixed n, and the second component is bounded in view of the fact, that f is linear continuous x belong to the unit ball of the space X . From here functional, and elements kxk X we draw the conclusion, that operators Tn are coercive. From the hemicontinuity of and that fact, that operator ˚ is Lipschitzean it follows the radial continuity of A . / n operator Tn .y Besides that, operators A . / n are monotone, and operator ˚ is operator with semibounded variation. Then for every n and 8f 2 X  equations Tn x D A

xn C ˚xn D f n

(2.120)

have solutions. Let us denote these solutions by xn . From the equality

as for every xn ¤ 0

E D x n A ; xn C h˚Fxn ; xn i D hf; xn i : n ˛ ˝ xn A n ; xn > 0, we obtain

(2.121)

h˚xn  f; xn i < 0: From here, taking into account the condition (2.117), we find kxn k  . Further, from (2.121), taking into account (2.119), we have E D x n h˚xn ; xn i D hf; xn i  A ; xn n

x ˛C1 kxn k˛C1

n D hf; xn i  n C hf; xn i  : n X n˛ Since, in the last expression both components are bounded, then in view of Remark 1 [LaSoUr68], we draw the conclusion, that k˚xn kX  C3 D const as kxn kX  . Now from the expression

208

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

kxn k˛C1 X  d D const n˛ it follows, that d kxn k˛C1 X  ! 0 as n ! 1; n˛C1 n

i.e.,

xn ! 0: n

Taking into account, that the operator A is continuous in 0 of the space X and A0 D 0, we have xn lim A ! 0: (2.122) n!0 n Sequences fxn g  X and f˚xn g  X  are bounded and in view of reflexivity of spaces X and X  from them we can choose such subsequences, that fxn g  fxm g ! x0 weakly in X , f˚xn g  f˚xm g ! '0 weakly in X  . Passing to the limit in (2.121), in view of (2.122) from (2.120) we will obtain '0 D f: Let us show, that '0 D ˚x0 . Since, operator ˚ is operator with semibounded variation, then for all x 2 X such, that kxk  , we have   0 h˚x  xm ; x  xm i  C I kx  xm kX D c0 kx  xm k2H : In view of (2.121), this inequality will have the form E D x m h˚x; x  xm i  h˚Fxm ; xi  A ; xm C hf; xm i m   0  C I kx  xm kX D c0 kx  xm k2H : From here, passing to the limit, we will obtain   0 h˚x; x  x0 i  h'0 ; xi C hf; x0 i  C I kx  x0 kX D c0 kx  x0 k2H ; ˝ ˛ since A xmm ; xm ! 0, and the norm kkH is compact with regard to the norm kkX . From the last inequality, taking into account '0 D f , we have   0 h˚x  f; x  x0 i  C I kx  x0 kX D c0 kx  x0 k2H : From here, in view of [LaSoUr68, Theorem 1.3], we will obtain ˚x0 D f or FM x0 D f , and from inequality (2.117) we draw the conclusion, that kx0 kX < .y Let us show now, that the totality of solutions K2 is weakly compact. Since the operator ˚ is the operator with semibounded variation, then for all x 2 X , such, that kxkX   and xm 2 K2 , we have

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization



0

209



h˚x  ˚xm ; x  xm i  C I kx  xm kX D c0 kx  xm k2H : passing to the limit as m ! C1 at ˚xm D FM xm D f , we will obtain   0 h˚x  f; x  x0 i  C I kx  x0 kX D c0 kx  x0 k2H : Therefore, in view of [LaSoUr68, Theorem 1.3], we have ˚x0 D f or FM x0 D f , and also kx0 kX  . Then from the definition of operator M FM x0 D F x0 D f: Thus, we proved, that the totality of solutions of equation (2.116) is bounded and weakly compact, and also K2  K1 . The Theorem is proved. Let the operator M W X ! X satisfies upper mentioned conditions. Definition 2.6. The operator F W X ! X  we will call M -monotone, if superimposed operator ˚ D FM W X ! X  is monotone, i.e., 8x1 ; x2 2 X the next inequality takes place h˚x1  ˚x2 ; x1  x2 i D hFM x1  FM x2 ; x1  x2 i  0: Definition 2.7. The operator F W X ! X  we will call M -semimonotone, if there exists strenuously continuous operator C W X ! X  such, that the operator ˚ D .FM C CM / W X ! X  is monotone. Theorem 2.28. Let Banach space X has Property ./, operator F W X ! X  is radially continuous M -monotone, 9 > 0, for that hF x  f; xiX > 0;

if

kxkX D  D const.

Then the equation FM x D f;

x 2 X;

has the solution in the ball with radius , at that the totality of solutions K2 D fx 2 X W FX D f g is weakly compact, and also K2  K1 D fx 2 X W F x D f g : Theorem 2.29. Let Banach space X has Property ./, operator F W X ! X  is hemicontinuous M -semicontinuous, 9 > 0, for that hF x  f; xiX > 0;

if

kxkX D  D const:

210

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Then the equation FM x D f;

x 2 X;

has the solution in the ball with radius , at that the totality of solutions K2 D fx 2 X W FX D f g is weakly compact, and also K2  K1 D fx 2 X W F x D f g : Remark 2.14. These Theorems we can prove in the same way as we proved Theorem 2.27, since monotone operator is the particular case of the operator with semibounded variation. Let us consider the methods of normalization for noncoercive nonlinear operator equations of Hammerstein type. Let X be reflexive Banach space, that compactly and densely embedded in Hilbert space H , X  H  X  , where X  is adjoint space to X , at that X has Property ./. Let us consider the nonlinear operator equation of Hammerstein type x C BF .x/ D g;

g 2 X:

(2.123)

Here F W X ! X  is nonlinear operator, B W X  ! X is linear continuous positive operator. Let us note, that in work [Va73] Theorems of existence of solutions for uniform equation of Hammerstein type have been proved by the method of monotone operators x C BF .x/ D 0; in Banach spaces, which have Property ./, under the conditions, when the operator B is linear continuous positive, and the operator F is hemicontinuous bounded monotone (or semibounded), that satisfies the condition (of coercivity in modified form) hF x; xi > 0; if kxkX   D const: In the given paragraph we will investigate nonuniform equation (2.123) under weaker conditions. We suggest, that the norm in X is differentiable by Frechet. Theorem 2.30. Let B W H ! H be continuous positive linear operator, nonlinear operator F W H ! H be Lipschitzean, i.e., 8x1 ; x2 2 X takes place the next kF x1  F x2 kH  r kx1  x2 kH ;

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

211

where r D const, and its narrowing on the space X satisfies the next condition: 9

for that

˝

F x  B 1 g; x

˛ X

0

8x 2 X

if

kxkX D :

(2.124)

Then there exists such regularizer M W H ! H , that the equation x C BF .M x/ D g;

g 2 X;

has the solution in the ball with radius , at that the totality of solutions Kg2 D fx 2 X W x C BF .M x/ D gg is weakly compact and Kg2  Kg1 D fx 2 X W x C BF .x/ D gg : Proof. Let define the operator M by the rule ( Mx D

 kxkH

x; if kxkH > ; x; if kxkH  :

(2.125)

We have already known from the previous, that this operator is monotone and Lipschitzean. Besides that, the operator ˚ D FM W H ! H , that is the superposition of Lipschitzean operators F and M , also is Lipschitzean. Then, it is not heavy to draw the conclusion, that the narrowing of operator ˚ on the space X is the operator with semibounded variation. Further, 8x 2 X such, that kxkX   takes place kM xkX D . Here we mean the narrowing of operator M on X . ( Mx D

 kxkH

x; if kxkH > ; x; if kxkH  :

Then, taking into account (2.124) and (2.125) we have ˝ ˛ ˝ ˛ 0 < FM x  B 1 g; M x X  ˚x  B 1 g; x X :

(2.126)

Thus, we will obtain the equation x C B˚ .x/ D g;

g 2 X:

(2.127)

Let us note, that from linearity and positivity of operator B it follows its reversibility. In view of property ./ in the space X  there exists the operator A W X  ! X , that is monotone, hemicontinuous in 0 of the space X  , and A0 D 0, and also hA!; !iX  D k!k˛C1 X ;

˛ > 0:

(2.128)

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2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

For (1.56) let us consider the sequence of operators ˚n ! D A

! C B  ! C B˚B  !; n

8! 2 X  :

(2.129)

Taking into account, that the operator A is monotone, hemicontinuous coercive, and the condition (2.128) holds, the operator ˚ W X ! X  is radially continuous with semibounded variation, the operator B W X  ! X is positive continuous, we draw the conclusion, that operators ˚n are coercive radially continuous with semibounded variation. Then the set of solutions ˚n ! D A

! C B  ! C B˚B  ! D g; n

! 2 X ;

(2.130)

is nonempty for every n. From here E D ! n A ; !n C hB  !n ; !n iX C hB˚B  !n ; !n iX D hg; !n iX ; n X

(2.131)

or, taking into account, that the operator B has the right inverse, and in view of linearity, we find E D ! ˛ ˝ n A ; !n C hB!n ; !n iX C ˚B  !n  B 1 g; B!n X D 0: n X In this equality the first and the second components are positive for !n ¤ 0, and, then, since !n satisfies the equality (2.131),the third component is negative, whence, in view of conditions of Theorem kxn kX D kB  !n kX   for any n. Besides that, k!n kX   1 D const, since the linear continuous positive operator B  W X  !X can’t transform unbounded set into bounded set. From the equality (2.131) we will obtain E D ! n h˚B  !n ; B  !n iX D  A ; !n  hB!n ; !n iX C hg; !n iX : n X From here, taking into account, that hA!; !iX D k!k˛C1 X  , we have h˚B  !n ; B  !n iX D 

1 k!n k˛C1 X   hB!n ; !n iX C hg; !n iX ; n˛

and we draw the conclusion, that j h˚B  !n ; B  !n i jX  c1 D const, since all the components in the right part of the last equality are bounded in view of the boundness of the set f!n g  X  and continuity of the operator B. Then, in view of Remark 1 [LaSoUr68] we have k˚B  !n kX   c2 D const

as

kB  !n kX  :

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

213

Now from the expression E D ! 1 n A ; !n D ˛ k!n k˛C1 X   d D const; n n X we obtain d 1 ! 0 as n ! 1; k!n k˛C1 X  n˛ n

i.e.,

k!n kX  ! 0: n

Taking into account, that the operator A is continuous in 0 of the space X  , and also, that A0 D 0, we have !n lim A D0 (2.132) n!1 n Now, in view of reflexivity of space X , we can choose the subsequences: fB  !n g  fB  !m g D fxm g ! x0 weakly in X and f˚xm g ! y0 weakly in X  . Passing to the limit in the equality (2.131) in view of (2.132), we will obtain x0 C By0 D g;

(2.133)

since B is linear continuous operator. Now let us prove, that in the equality (2.133) y0 D ˚x0 . Using that fact, that the operator ˚ W X ! X  is the operator with semibounded variation, for 8x 2 X such, that kxkX  , we can write   h˚x  ˚B  !m ; x  B  !m iX  C I kx  B  !m kX ; or ˝

 ˛ ˝ ˛ ˛ ˝  ˚x; x  B  !m  ˚B  !m ; x X  ˚B  !m ; B  !m X  C I x  B  !m X (2.134)

Taking into account, that B  !m D A

!m  BFB  !m C g; m

and substituting this expression in (2.134), we will obtain D !m E C h˚B  !m ; giX h˚x; x  B  !m iX  h˚B  !m ; xiX  ˚B  !m ; A m X    h˚B  !m ; B˚B  !m iX  C I kx  B  !m kX :

214

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Here, as m ! 1: (1) lim h˚B  !m ; B˚B  !m iX  hy0 ; By0 iX , i.e., the functional !! h!; B!i m!1 is weakly lower semicontinuous, since the operator B is linear bounded and positive (see Example 8.6 [Va73]); (2) h˚x; x  B  !m iX ! h˚x; x  x0 iX , since B  !m ! x0 weakly in X I (3) h˚B  !m ; xiX ! hy0 ; xiX and h˚B  !m ; giX ! hy0 ; giX , since ˚B  !m ! in X˛ I ˝y0 weakly  (4) ˚B !n ; A !mm X ! 0, since !mm ! 0 and A0 D 0I (5) C .I kx  B  !m kX / ! C .I kx  x0 kX /, since this function C is continuous by the second argument. Thus, passing to the limit in the last inequality, we will obtain h˚x; x  x0 iX  hy0 ; xiX C hy0 ; giX  hy0 ; By0 iX  C .I kx  x0 kX / ; or h˚x; x  x0 iX  hy0 ; x C By0  giX  C .I kx  x0 kX / : From By0  g D x0 , or h˚x  y0 ; x  x0 iX  C .I kx  x0 kX / : Then in view of Theorem 2.1 [Va73], we have ˚x0 D y0 : Now the equality (2.133) has the form x0 C B˚x0 D g: And, thereby, the first part of the Theorem is proved, i.e., the set Kg2 is nonempty. Now let us pass to the second part of the Theorem. Let us prove, that the set Kg2 is weakly compact, i.e., from the arbitrary sequence fxn g  Kg2 we can choose a subsequence fxm g  fxn g such, that xm ! x0 weakly in X , at that x0 2 Kg2 . From the first part of proof we know, that kxn kX  ;

kFxn kX   C1 D const;

8xn 2 Kg2 :

Therefore, in view of the reflexivity of spaces: fxn g  fxm g ! x0 weakly in X and f˚xn g  f˚xm g ! y0 weakly in X  : Thus, in the equation xn C B˚xn D g

(2.135)

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

215

passing to the limit, and, taking into account the continuity of the operator B, we will obtain x0 C By0 D g: Further, acting in the same way as in the first part of proof of the given Theorem and using the property of the operator ˚, we establish y0 D ˚x0 and, thereby, x0 C B˚x0 D g: Indeed, 8x 2 X such, that kxkX   and xm 2 Kg2 , we have h˚x  ˚xm ; x  xm iX  C .I kx  xm kX / : From here, taking into account (2.135) we will obtain h˚x; x  xm iX  h˚xm ; xiX C h˚xm ; giX  h˚xm ; B˚xm iX  C .I kx  xm kX / :

Here, passing to the limit, we find h˚x  y0 ; x  x0 iX  C .I kx  x0 kX / : Then, in view of Theorem 2.1 [LaSoUr68], we have, that y0 D ˚x0 , therefore, takes place. The Theorem is proved. In conclusion of this section let us consider solvability questions of nonlinear operator systems, containing equations of Gammerstein type. Let X; Y; Z be Banach spaces, Z  be a space adjoint to Z. Definition 2.8. An operator A W X  Y  Z ! Z  we call the operator with uniformly semibounded variation (u.s.b.v.), if for arbitrary bounded sets X0  X; Y0  Y and arbitrary z1 ; z2 2 Z such that kzi kZ  R, i D 1; 2, the following inequality is satisfied hA .x; y; z1 /  A .x; y; z2 / ; z1  z2 i   inf Cx;y .RI kz1  z2 kZ / ; x2X0 y2Y0

where for every .xI y/ 2 X0  Y0 the function Cx;y .:; :/ W
216

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

x C BF .u; z; x/ D g; G .v; x; z/ D f;

g 2 X;

f 2 Z;

u 2 U; v 2 V:

(2.136) (2.137)

Here F W U  Z  X ! X  and G W V  X  Z ! Z  are nonlinear operators, B W X  ! X is linear continuous positive operator. Let us investigate existence questions and solutions totality properties for systems of nonlinear operator equations (2.136), (2.137) for fixed u 2 U and v 2 V . Theorem 2.31. Let the operator B W X  ! X be linear continuous positive; the operator F .u; z; :/ W X ! X  8 .uI z/ 2 U  Z be radial continuous with semibounded variation, and let the operator satisfy the following condition ˝ ˛ F .u; z; x/  B 1 g; x  0 8 .uI z/ 2 U  Z if kxkX >   0;  D const;

(2.138)

and also let it be strenuously continuous along the second argument, i.e., the operator F .u; :; x/ W Z ! X  8 .uI z/ 2 U  Z transforms any sequence zn ! z0 weakly in Z into the sequence fF .u; zn ; x/g ! F .u; z0 ; x/ strongly in X  ; the operator G .v; x; :/ W Z ! Z  be radial continuous with u.s.b.v., and let the operator satisfy the condition hG .v; x; z/  f; zi  0

8 .vI x/ 2 V  X; if kzkZ > ı  0; ı D const; (2.139)

and let it be strenuously continuous along the second argument. Then the set 1 Ku;v D f.xI z/ 2 X  Z W

x C BF .u; z; x/ D g;

G .v; x; z/ D f g

is nonempty and weakly compact for each .uI v/ 2 U  V . Proof. Due to Property ./ in the spaces X  and Z there exist operators A1 W X  ! X and A2 W Z ! Z  , which are monotone hemicontinuous, continuous at zeroes of the spaces X  and Z respectively, A1 0 D 0; A2 0 D 0, and also ˛C1 hA1 !; !i D k!k˛C1 X  and hA2 z; zi D kzkZ  ;

˛>0

(2.140)

For the system (2.136), (2.137) let us consider under fixed .uI v/ 2 U V sequences of operators ˚1n W X  ! X; ˚2n W z ! Z  , defined by equations ˚1n ! D A1

  ! C B  ! C BF u; z; B  ! ; n

˚2n z D A2

z C G .v; x; z/ ; nC1

z 2 Z;

! 2 X ;

.uI z/ 2 U  Z; (2.141)

.vI x/ 2 V  X;

n D 1; 2; : : : I (2.142)

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

217

Similarly with written above, the operators ˚1n , ˚2n are coercive radial continuous with u.s.b.v. Then, due to [LaSoUr68, Corollary 4.1] solution sets of the equations ˚1n ! D g; (2.143) ˚2n z D f;

(2.144)

under fixed parameters are nonempty for each n. When n D 1 in (2.143) substituting z D z1 , where kzkZ  ı, the corresponding solution we denote by !1 and, therefore, B  !1 D x1 . In (2.144), substituting x D x1 , the corresponding solution we denote by z D z2 . Further in (2.143), assuming z D z2 , we denote the corresponding solution by !2 and B  !2 D x2 , in (2.144), assuming x D x2 , we denote the corresponding solution by z3 and so on, iterating this procedure, from (2.143), (2.144) we have A1

  !n C B  !n C BF u; zn ; B  !n D g; n

u 2 U;

(2.145)

and

znC1 C G .v; xn ; znC1 / D f; nC1 Hence, from (2.145) we have A2

D A1

v 2 V:

(2.146)

E  ˛ ˝  !n ; !n C hB  !n ; !n i C BF u; zn ; B  !n ; !n D hg; !n i ; n

or taking into account that the operator B has its right inverse, we find D A1

E  ˛ ˝  !n ; !n C hB!n ; !n i C F u; zn ; B  !n  B 1 g; B!n D 0: n

(2.147)

In this equality the first and the second summands are positive when !n ¤ 0, and then, since !n satisfies the equality (2.147), therefore, the third summand is negative, whence due to the conditions of the Theorem kxn kX D kB  !n kX   for each n. Besides, as it was mentioned in previous investigations k!n kX   1 D const, since linear continuous positive operator B  W X  ! X cannot transform unbounded sets into bounded ones. Similarly, from (2.146) we have  A2 or

 znC1 ; znC1 C hG .v; xn ; znC1 / ; znC1 i D hf; znC1 i ; nC1 

 znC1 ; znC1 C hG .v; xn ; znC1 /  f; znC1 i D 0: A2 nC1

(2.148)

(2.149)

Here the first summand for zn ¤ 0 is positive, then the second summand is negative and, therefore, from the condition (2.139) we conclude that kznC1 kZ  ı.

218

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

From the equality (2.146) we obtain E D !  ˛ ˝  n F u; zn ; B  !n ; B  !n D  A1 ; !n  hB!n ; !n i C hg; !n i : n Hence, keeping in mind that hA!; !i D k!k˛C1 X  , we have ˝   ˛ 1 F u; zn ; B  !n ; B  !n D  ˛ k!n k˛C1  hB!n ; !n i C hg; !n i ; n

(2.150)

and then hF .u; zn ; B  !n / ; B  !n i  c1 D const, since all summands in the right part of the last equality are bounded due to boundness of the set f!n g  X  and to the linear continuity of the operator B. Then in virtue of Remark 1 from [LaSoUr68] we have

 

F u; zn ; B  !n   c2 D const; as kB  !n k  : X X Here let us remark that c2 does not depend on u and zn , since in the equality (2.150) the right part does not depend on u and zn . Similarly form the equality (2.148) we conclude that kG .v; xn ; znC1 /kX   c3 D const; as kznC1 kZ  ı: Let us mention, that the constant c3 does not depend on v and xn as well. Now from the equation E D ! 1 n A1 ; !n D ˛ k!n k˛C1 X   d1 D const; n n   1 znC1 ; znC1 D  d2 D const; A1 kznC1 k˛C1 Z nC1 .n C 1/˛

or

we have d1 1 ! 0; k!n k˛C1 X  n˛ n

asn ! 1; i.e.,

k!n kX  !0 n

and d2 1 ˛C1 ! 0;  ˛ kznC1 kZ .n C 1/ nC1

asn ! 1; i.e.,

kznC1 kZ ! 1: nC1

Taking into account that the operators A1 and A2 are continuous at zero of the space X  and Z respectively, and also that A1 0 D 0 and A2 0 D 0, we have lim A1

n!1

!n znC1 D 0 and lim A2 D 0: n!1 n nC1

(2.151)

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

219

Now due to reflexivity of the spaces X and Z we conclude that there can be isolated subsequences fB  !n g  .B  !m / D fxm g ! x0 weakly in X , fF .u; zm ; xm /g ! y0 weakly in X  , fzn g  fzm g ! z0 weakly in Z; fG .v; xm ; zm /g ! 0 weakly in Z  . Then, passing to a weak limit, in equalities (2.145), (2.146) as m ! 1 in virtue of (2.151), we have x0 C By0 D g; (2.152) and 0

D f:

(2.153)

Now let us prove that in the equality (2.152) y0 D F .u; z0 ; x0 / and in the equality (2.153) 0 D G .v; x0 ; z0 /. Using the fact that the operator F .u; zm ; / W X ! X  with u.s.b.v., for each x 2 X such that kxk  , we can write ˝

 ˛    F .u; zm ; x/  F u; zm ; B  !n ; x  B  !m   inf Cu;zm I kx  B  !m kX ; u2U

zm 2Z0

or ˝ ˛ ˝   ˛ ˝   ˛ F .u; zm; x/ ; x  B  !m  F u; zm ; B  !m ; x  F u; zm ; B  !m ; B  !m Cu;zm .I kx  B!m kX / : (2.154)  inf .u;zm /2U Z0

Taking into account B  !m D A

  !m  BF u; z; B  !m C g; m

and substituting the expression in (2.154) we obtain  ˛ D   !m E ˝  hF .u; zm ; x/ ; x  B  !m i  F u; zm ; B  !m ; x  F u; zm ; B  !m ; A m  ˛ ˝    ˛ ˝  inf C F u; z; B  !m ; g  F u; zm ; B  !m ; B u; zm ; FB  !m 

.u;zm /2U Z0

Cu;zm .I kx  B!m kX / : In this inequality as m ! 1: (1) lim hF .u; zm ; B  !m / ; BF .u; zm ; B  !m /i  hy0 ; By0 i, i.e., the functional m!1

! ! .!; B!/ is weakly lower semicontinuous, since the operator B is linear bounded positive; (2) hF .u; zm ; x/ ; x  B  !m i ! hF .u; z0 ; x/ ; x  x0 i, since B  !m ! x0 weakly in X , and the operator F .u; :; x/ W Z ! X  is strenuously continuous; (3) hF .u; zm ; B  !m / ; xi ! hy0 ; xi and hF .u; zm ; B  !m / ; gi ! hy0 ; gi, since   F ˝ .u; zm ; B !m / ! !ym0 ˛weakly in X !;m (4) F .u; zm ; B !n / ; A m ! 0, since m ! 0 and A0 D 0;

220

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

(5) Cu;zm .I kx  B  !m kX / ! Cu;z0 .I kx  x0 kX /, since the function C is continuous 

inf

.u;zm /2U Z0

    Cu;zm I kx  B  !m kX   inf Cu;zo I kx  B  !m kX : u2U

Therefore, passing to the limit in the initial inequality, we obtain hF .u; z0 ; x/ ; x  x0 i  hy0 ; xi C hy0 ; gi  hy0 ; By0 i  inf Cu;z0 .I kx  x0 kX / ; u2U

or hF .u; z0 ; x/ ; x  x0 i  hy0 ; x C By0  gi   inf Cu;z0 .I kx  x0 kX / : u2U

From (2.152) By0  g D x0 , therefore hF .u; z0 ; x/  y0 ; x  x0 i   inf Cu;z0 .I kx  x0 kX / : u2U

Then we have F .u; z0 ; x0 / D y0 : Now the equality (2.152) takes the following form x0 C BF .u; z0 ; x0 / D g: Further let us show that the pair .x0 I z0 / satisfies the second equation of the system as well. Since the operator G .v; xm ; :/ W Z ! Z  is the operator with u.s.b.v., then for 8z 2 Z such that kzkZ  ı, the following relation takes place hG .v; xm ; z/i  hG .v; xm ; zmC1 / ; z  zmC1 i 

inf

.v;xm /2V X0

  Cv;xm ıI kz  zmC1 kZ

Hence, keeping in mind (2.146), i.e., G .v; xm ; zmC1 / D f  A2

zmC1 ; mC1

we have   zmC1 ; z  zmC1  hf; z  zmC1 i hG .v; xm z/ ; z  zmC1 i C A2 mC1 Cv;xm .ıI kz  zmC1 kZ / :  inf .v;xm /2V X0

Taking into account that the operator G is strenuously continuous along the second zmC1 argument and that A2 mC1 ! 0 as m ! 1, and also that the function Cv;xm is

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

221

continuous, from the last inequality we obtain hG .v; x0 ; z/ ; z  z0 i  hf; z  z0 i   inf Cv;x0 .ıI kz  z0 kZ / ; v2V

or hG .v; x0 ; z/  f; z  z0 i   inf Cv;x0 .ıI kz  z0 kZ / : v2V

Therefore, G .v; x0 ; z0 / D f: 1 is Thus the first part of the Theorem is proved, namely the fact that the set Ku;v nonempty. Now we begin to prove the second part of the Theorem. Let us prove that the 1 1 is weakly compact, i.e., from arbitrary sequences fxn I zn g  Ku;v there can set Ku;v be isolated subsequences fxm I zm g  fxn I zn g such that xm ! x0 weakly in X , 1 zm ! z0 weakly in Z, and moreover .x0 I z0 / 2 Ku;v . From the first part of the proof we know that

kxn kX  ; kzn kZ  ı;

kF .u; zn ; xn /kX   C1 D const; kG .v; xn ; zn /kZ   C2 D const;

u 2 U; v 2 V:

Therefore due to reflexivity of the spaces fxn g  fxm g ! x0 weakly in X , fF .u; zn ; xn /g  fF .u; zm ; xm /g ! y0 weakly in X  , fzn g  fzm g ! z0 weakly in Z and fG .v; xn ; zn /g  fG .v; xm ; zm /g ! 0 weakly in Z  . Thus in the equations xn C BG .u; zn ; xn / D g

(2.155)

G .v; xn ; zn / D f;

(2.156)

and passing to the limit and taking into account continuity of the operator B, we obtain x0 C By0 D g;

(2.157)

and 0

D f:

Further, acting similarly with the first part of the proof of the given Theorem there can be established that y0 D F .u; z0 ; x0 / and 0 D G .v; x0 ; z0 / and thus x0 C BG .u; z0 ; x0 / D g;

(2.158)

G .v; x0 ; z0 / D f:

(2.159)

222

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Indeed, for all x 2 X and z 2 Z such that kxkX   and kzkZ  ı, we have hF .u; zm ; x/  F .u; zm ; xm / ; x  xm i  

inf

.u;zm /2U Z0

Cu;zm .I kx  xm kX / ;

and hG .v; xm ; z/  G .v; xm ; zm / ; z  zm i 

inf

.v;xm /2V 2X0

Cv;xm .ıI kz  zm kZ / :

Hence, in view of (2.155), (2.156), we obtain: hF .u; zm ; x/ ; x  xm i  hF .u; zm ; xm / ; xi C hF .u; zm ; xm / ; gi  hF .u; zm ; xm / ; BF .u; zm ; xm /i 

inf

.u;zm /2U Z0

Cu;zm .I kx  xm kX / ;

and hG .v; xm ; z/ ; z  zm i  hf; z  zm i  

inf

.v;xm /2V 2X0

Cv;xm .ıI kz  zm kZ / :

Here, passing to the limit, similarly with the first part of the proof, we establish hF .u; z0 ; x/  y0 ; x  x0 i   inf Cu;z0 .I kx  x0 kX / ; u2U

and hG .v; x0 ; z/  f; z  z0 i   inf Cv;x0 .ıI kz  z0 kZ / : v2V

Then we have that y0 D F .u; z0 ; x0 / and G .v; x0 ; z0 / D f , therefore, (2.158) and (2.159) take place. The Theorem is proved. Corollary 2.4. Let the operator B W X  ! X be linear continuous positive; the operator F .u; z; :/ W X ! X  8 .uI z/ 2 U Z be radial continuous with u.s.b.v., and let the operator satisfy hF .u; z; x/ ; xi  0

8 .uI z/ 2 U  Z; if kxkX >  > 0;  D const;

and also let it be strenuously continuous along the second argument. The operator G .v; x; :/ W Z ! Z  be radial continuous with u.s.b.v. and satisfy the condition hG .v; x; z/ ; zi  0 8 .vI x/ 2 V  X; if kzkZ > ı > 0; and be strenuously continuous along the second argument.

ı D const;

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

223

Then the set 2 Ku;v D f.xI z/ 2 X  Z W x C BF .u; z; x/ D 0;

G .v; x; z/ D 0g

is nonempty and weakly compact. Corollary 2.5. Let all conditions of Theorem 2.31 be fulfilled, The operator L W Z ! Z  be linear continuous positive. Then the set 3 D f.xI z/ 2 X  Z W x C BF .u; z; x/ D g; Ku;v

Lz C G .v; x; z/ D f g

is nonempty and weakly compact. Corollary 2.6. Let for the operators B; F; G all conditions of Theorem 2.31 be fulfilled, except the condition (2.139). The operator L W Z ! Z  be linear continuous and the following condition holds true hLz C G .v; x; z/  f; zi  0

8 .vI x/ 2 V  X; if kzkZ > ı > 0:

3 is nonempty and weakly compact. Then the set Ku;v

Definition 2.9. We will call the operator T W X Y Z ! Z  radial semimonotone one (r.s.m.) if there exists strenuously continuous map C W Z ! Z  such that the map G D T .x; y; :/ C C from Z to Z  8 .x; y/ 2 X0  Y0 is monotone. Let us consider the system x C BF .u; z; x/ D g; G .v; x; z/ D f;

x; g 2 X; 

z 2 Z; f 2 Z ;

u 2 U;

(2.160)

v 2 V;

(2.161)

of nonlinear operator equations, where F W U Z X ! X  and G W V X Z ! Z  are r.s.m. operators, and B W X  ! X is linear continuous positive operator. Theorem 2.32. Let all following conditions be fulfilled: (1) Banach spaces X and Z  have Property ./; (2) F .u; z; :/ W X ! X  8 .uI z/ 2 U  Z is hemicontinuous bounded operator, satisfying the inequality ˝

˛ F .u; z; x/  B 1 g; x  0 8 .uI z/ 2 U  Z; if

kxkX >  > 0; (2.162)

and also it is strenuously continuous along the second argument; (3) The operator G .v; x; :/ W Z ! Z  is hemicontinuous 8 .vI x/ 2 V  X , satisfying the condition hG .v; x; z/  f; zi  0; if

kzkZ > ı > 0 .ı D const/ ;

and also it is strenuously continuous along the second argument.

(2.163)

224

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Then the set 1 KU;V D f.xI z/ 2 X  Z W x C BF .u; z; x/ D gI

G .v; x; z/ D f g

is nonempty and weakly compact. Proof. Let us remark that from linearity and positivity of the operator B its invertibility follows. Similarly with the previous section, in virtue of Property ./ in the spaces X  and Z there exist operators A1 W X  ! X and A2 W Z ! Z  respectively, which are monotone, hemicontinuous, continuous at zeroes of the spaces X  and Z respectively, and A1 0 D 0, A2 0 D 0, and also ˛C1 ; hA1 !; !iX  D k!k˛C1 X  and hA2 z; ziZ D kzkZ

˛ > 0:

(2.164)

For the system (2.160), (2.161) let us consider sequences of operators ˚1n ! D A1

  ! C B  ! C BF u; z; B  ! ; ! 2 X  ; .uI z/ 2 U  Z; n

(2.165)

z C G .v; x; z/ ; z 2 Z; .vI x/ 2 V  X; n D 1; 2; : : : ; (2.166) nC1 In virtue of our conditions we conclude that the operators ˚1n ; ˚2n are coercive hemicontinuous r.s.m. Then due to Theorem 18.4 [Va73] solution sets for the equations ˚2n z D A2

˚1n ! D A1

  ! C B  ! C BF u; z; B  ! D g; n

! 2 X  ; .uI z/ 2 U  Z

z C G .v; x; z/ D f; z 2 Z; .vI x/ 2 V  X; n D 1; 2; : : : I nC1 are nonempty for each n. For n D 1 in (2.165) substituting z D z1 , where kz1 kZ  ı, we denote the corresponding solution by !1 and, therefore, B  !1 D x1 . In (2.166) assuming x D x1 , we denote the corresponding solution by z2 . Further in (2.165), substituting z D z2 , we denote the corresponding solution by !2 and B  !2 D x2 , and in (2.166), assuming x D x2 , we denote the corresponding solution by z3 . Iterating the procedure, from (2.165) (2.166) we obtain ˚2n z D A2

A1

  !n C B  !n C BF u; zn ; B  !n D g; n

u 2 U;

and

(2.167)

znC1 C G .v; xn ; znC1 / D f; v 2 V: (2.168) nC1 These last equalities coincide with the equalities (2.145) and (2.146) respectively. Acting similarly to the previous section, we have: A2

kxn kX  ;

kF .u; zn ; xn /kX   C1 D const;

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

225

kzn kZ  ı; kG .v; xn ; zn /kZ   C2 D const; !n znC1 ! 0 and A2 ! 0 as n ! 1: A1 n nC1 Therefore in view of reflexivity of the spaces there can be isolated subsequences: fxn g  fxm g ! x0 weakly in X and

fF .u; zn ; xn /g  fF .u; zm ; xm /g ! y0 weakly in X  ; fzn g  fzm g ! z0 weakly inZ

and fG .v; xn ; zn /g  fG .v; xm ; zm /g !

0

weakly in Z  :

Thus, in (2.167) and (2.168), passing to the limit and keeping in mind continuity of the operator B, we obtain x0 C By0 D g; (2.169) and 0

D f:

Further, using the fact that the operators F and Z are hemicontinuous r.s.m. the following equations can be established: y0 D F .u; z0 ; x0 /

and

0

D G .v; x0 ; z0 / :

Thus we have x0 C BF .u; z0 ; x0 / D g;

(2.170)

G .v; x0 ; z0 / D f:

(2.171)

Since the operator F is r.s.m., there exists a strenuously continuous operator C1 W X ! X  such that the operator T1 D ŒF .u; zm ; :/ C C1  W X ! X  is monotone. Then 8x 2 X the following relation takes place hT1 x  T1 xm ; x  xm i  0: Therefore hT1 x; x  xm i  hT1 xm ; xi  hT1 xm ; xm i D hF .u; zm ; xm / ; xm i C hC1 xm ; xm i :

But in virtue of (2.167) xm D B  !m D g  A1

  !m  BF u; zm ; B  !m ; m

226

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

then D !m E hT1 x; x  xv i  hT1 xm ; xi  F .u; zm ; xm / ; A m   ˛ ˝   ˛ ˝  C F u; zm ; B  !m ; BF u; zm ; B  !m  F u; zm ; B  !m ; g D ˛  !m E ˝ C C1 xm ; A C C1 xm ; BF u; zm ; B  !m  hC1 xm ; gi : m Hence, as m ! 1 we draw the following conclusions: hT1 x; x  xm i D hF .u; zm ; x/ ; x  xm i C hC1 x; x  xm i ! hF .u; z0 ; x/ ; x  x0 i C hC1 x; x  x0 i

(1)

since xm ! x0 weakly in X and the operator F .u; :; x/ W Z ! X  is strenuously continuous; (2) F .u; zm ; xm / ! y0 weakly in X  and the operator C1 is strenuously continuous; (3) lim hF .u; zm ; B  !m / ; BF .u; zm ; B  !m /i  hy0 ; By0 i, since the functional n!1

(4) (5) (6) (7) (8)

!i is lower ˛ semicontinuous; ˝! ! hF !; BF F .u; zm ; B  !m / ; A !mm ! 0, since A !mm ! 0;  ˝hF .u; zm ;!Bm ˛!m / ; gi ! hy0 ; gi; C1 xm ; A m ! 0; hC1 xm ; BF .u; zm ; B  !m /i ! hC1 x0 ; By0 i, since the operator C1 is strenuously continuous; hC1 xm ; gi ! hC1 x0 ; gi. Therefore, from (2.150) as m ! 1 we have hT1 x; x  x0 i  hy0 ; xi C hC1 x0 ; xi  hy0 ; By0 i  hy0 ; gi C hC1 x0 ; By0 i  hC1 x0 ; gi :

From this inequality, taking into account (2.169), after some simplification we obtain hT1 x  .y0 C C1 x0 / ; x  x0 i  0; or in view of T1 D ŒF .u; z0 ; :/ C C1  we have hF .u; z0 ; x/ C C1 x  .y0 C C1 x0 / ; x  x0 i  0: Hence, due to Lemma 18.1 [Va73] F .u; z0 ; x0 / C C1 x0 D y0 C C1 x0 ; therefore, y0 D F .u; z0 ; x0 / and thus (2.170) takes place. Further, since the operator G is r.s.m., there exists a strenuously continuous operator C2 W Z ! Z  such that the operator T2 D ŒG .v; xm ; :/ C C2  W Z ! Z  is

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

227

monotone. Then 8z 2 Z the following relation takes place hT2 z  T2 zm ; z  zm i  0: So, we have hT2 z; z  zm i  hT2 zm ; zi  hT2 zm ; zm i D  hG .v; xm ; zm / ; zm i  hC2 zm ; zm i : But in virtue of G .v; xm ; zm / D f  A2

zm ; m

from the last inequality we obtain E D z m hT2 z; z  zm i  hT2 zm ; zi  A2 ; zm  hf; zm i  hC2 zm ; zm i : m

(2.172)

Hence, as m ! 1 we draw the following conclusions: (1) hT2 z; z  zm i D hG .v; xm ; z/ C C2 z; z  zm i ! hG .v; x0 ; z/ C C2 z; z  z0 i ; (2) (3) (4) (5)

since the operator G .v; :; z/ W X ! Z  is strenuously continuous; ; zi D ˛hG .v; xm ; zm / C C2 zm ; zi ! h 0 C C2 z0 ; zi; ˝hT2 zzm zmC1 mC1 A2 mC1 ; zm ! 0, since A2 mC1 ! 0; hf; zm i ! hf; z0 i; hC2 zm ; zm i ! hC2 z0 ; z0 i.

Then from the inequality (2.172) we find hG .v; x0 ; z/ C C2 z; z  z0 i  h or, taking into account that

0

0

C C2 z0 ; zi   hf; z0 i  hC2 z0 ; z0 i ;

D f and simplifying the obtained we have

hG .v; x0 ; z/ C C2 z  .

0

C C2 z0 / ; z  z0 i  0:

Hence, due to Lemma 18.1 [Va73] G .v; x0 ; z0 / C C2 z0 D

0

C C2 z0

or 0

D G .v; x0 ; z0 / ;

i.e., the equality (2.171) takes place. The first part of Theorem is proved. 1 Now we pass to the second part. It is necessary to prove that the set Ku;v is weakly 1 compact, namely, for arbitrary fxn ; zn g  Ku;v there can be isolated subsequences, such that: fxn g  fxm g ! x0 weakly in X and fzn g  fzm g ! z0 weakly in Z, 1 .x0 I z0 / 2 Ku;v .

228

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

1 For .xn I zn / 2 Ku;v we have:

kxn kX  ;

kF .u; zn ; xn /kX   r1 D const;

y kzn kZ  ı;

kG .v; xn ; zn /kZ   r2 D const:

Then in virtue of reflexivity of the spaces there can be isolated subsequences, such that: fxn g  fxm g ! x0 weakly in X and fF .u; zn ; xn /g  fF .u; zm ; xm /g ! y0 x weakly in X  I fzn g  fzm g ! z0 weakly in Z and fG .v; xn ; zn /g  fG .v; xm ; zm /g !

0

x weakly in Z  :

1 and Since the operators T1 D ŒF .u; zm ; :/ C C1  W X ! X  8 .u; zm / 2 Ku;v  1 8 .v; xm / 2 Ku;v are monotone, then 8x 2 T2 D ŒG .v; xm ; :/ C C2  W Z ! Z X and 8z 2 Z we have

hT1 x  T1 xm ; x  xm i  0 and hT2 z  T2 zm ; z  zm i  0: That is, hT1 x; x  xm i  hT1 xm ; xi  hT1 xm ; xm i D hF .u; zm ; xm / ; xm i C hC1 xm ; xm i : Hence in view of xm D BF .u; zm ; xm /  g we obtain hT1 x; x  xm i  hT1 xm ; xi  hF .u; zm ; xm / ; BF .u; zm ; xm /i  hC1 xm ; gi  hF .u; zm ; xm / ; gi C hC1 xm ; BF .u; zm ; xm /i and also, taking into account G .v; xm ; zm / D f , we have hT2 z; z  zm i  hT2 zm ; zi  hT2 zm ; zm i D  hG .v; xm ; zm / ; zm i  hC2 zm ; zm i D  hf; zm i  hC2 zm ; zm i : Further, from these last two inequalities we conclude that the weak limits x0 and z0 satisfy x0 C BF .u; z0 ; x0 / D g; and G .v; x0 ; z0 / D f: Thus, the Theorem is proved. Corollary 2.7. Let all conditions of Theorem 2.32 be fulfilled; the operator L W Z ! Z  be linear continuous positive. Then the set

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

 3 Ku;v D

.xI z/ 2 X  Z W

x C BF .u; z; x/ D gI Lz C G.v; x; z/ D f

229



is nonempty and weakly compact. Corollary 2.8. Let all conditions of Theorem 2.32, excepting the condition (2.163) be satisfied, the operator L W Z ! Z  be linear continuous and the condition 8 .vI x/ 2 V  X hLz C G .v; x; z/  f; zi  0

if kzkZ > ı  0;

ı D const

3 be fulfilled. Then the set Ku;v is nonempty and weakly compact.

In this section let us consider the problems of regularization for noncoercive systems of operator equations, containing equations of Hammerstein type. Let X; Z be reflexive Banach spaces, compactly and densely embedded into corresponding Hilbert spaces H1 and H2 (i.e., X  H1  X  and Z  H2 Z  ), X  and Z  be adjoint spaces, moreover, X and Z  have Property ./. Let us consider a system of nonlinear operator equations, containing an equation of Hammerstein type x C BF .z; x/ D g; g 2 X; (2.173) f 2 Z:

G .x; z/ D f; 

(2.174)





Here F W ZX ! X and G W X Z ! Z are nonlinear operators, B W X ! X is a linear continuous positive operator, K 1 D f.xI z/ 2 X  Zjx C BF .z; x/ D g; G .x; z/ D f g: Previously we investigated the system (2.173)–(2.174) for coercive (in modified form) operators. Here we are going to study existence questions and properties of totality of solutions for the system of nonlinear operator equations (2.173), (2.174) for noncoercive operators. Theorem 2.33. Let the linear operator B W H1 ! H1 be continuous positive, the nonlinear operator F .z; :/ W H1 ! H1 be uniformly Lipschitzean, i.e., 8z 2 Z and 8x1 ; x2 2 X the following inequality takes place kF .z; x1 /  F .z; x2 /kH1  r1 kx1  x2 kH1 ; where r1 D const that does not depend on the variable z and its narrowing on the space X satisfies the following condition: 9 > 0

for which

˝

˛ F .z; x/  B 1 g; x X > 0 kxkX D ;  D const;

8z 2 Z; if

(2.175)

besides, F be strenuously continuous along the first argument under the second one fixed, i.e., The operator F .:; z/ W Z ! X  8x 2 X transforms any sequence

230

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

zn ! z0 , weakly converging in Z, into the sequence F .zn ; x/ ! F .z0 ; x/ that strongly converges in X  ; the nonlinear operator G .x; :/ W H2 ! H2 be uniformly Lipschitzean, and on the space Z it satisfies the following condition: 9 > 0;

for which hG .x; z/  f; ziZ > 08x 2 X; if kzkZ D ı; ı D const;

(2.176)

besides, F be strenuously continuous along the second argument under the first one fixed. Then there exist such operators: M1 W H1 ! H1

and M2 W H2 ! H2 ;

the system x C BF .z; M1 x/ D g; G .x; M2 z/ D f;

g 2 X;

f 2Z

(2.177)



(2.178)

has solutions only in a full sphere with the radius  of the space X and in a full sphere with the radius  of the space Z, moreover, the totality of solutions K 2 D f.xI z/ 2 X  Z

x C BF .z; M1 x/ D g;

G .x; M2 z/ D f g

is weakly compact and K 2  K 1 . Proof. Similarly with the proof of the previous Theorem let us define operators ( M1 x D

 kxkH1

x; if kxkH1 > ;

x; if kxkH1  I

( M2 z D

ı kzkH2

z; if kzkH2 > ı;

z; if kzkH2  :

We remark, that the operators M1 ; M2 are continuous, monotone and Lipschitzean. The operators ˚F1 .z; / D F .z; M1 .// W H1 ! H1 and ˚2 .x; / D G .x; M2 .// W H2 ! H2 ; represent superpositions of uniformly Lipschitzean operators F .z; / W; H1 ! H1 ;

G .x; / W H2 ! H2

and of Lipschitzean operators: M1 W H1 ! H1 ;

M2 W H2 ! H2 ;

are uniformly Lipschitzean as well. Therefore the narrowing of the operator ˚1 .z; / on the space X and the narrowing of the operator ˚2 .x; / on the space Z are

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

231

operators with r.s.b.v. Further, 8x 2 X and 8z 2 Z such that kxkX   and kzkZ  ı, equalities kM1 xkX D ; kM2 xkZ D ı take place. Then in view of (2.175) and (2.176) we have D E ˝ ˛  0 < F .z; M1 x/  B 1 g; M1 x X D F .z; M1 x/  B 1 g; kxk x X ˝ ˛ ˝ ˛ X  F .z; M1 x/  B 1 g; x X  F .z; M1 x/  B 1 g; x X D kxk X ˝ ˛ D ˚1 .z; x/  B 1 g; x X :

(2.179)

and D E 0 < hG .x; M2 z/  f; M2 ziZ D G .x; M2 z/  f; kzkı z Z

Z

D kzkı hF .x; M2 z/  f; ziZ  hG .x; M2 z/ f; ziZ Z D h˚2 .x; z/  f; ziZ : (2.180) Therefore we obtained the system x C B˚1 .z; x/ D g;

g 2 X;

(2.181)

f 2 Z:

˚2 .x; z/ D f;

(2.182)

Let us remark that from linearity and positivity of the operator B its invertibility follows. In virtue of Property ./ in the spaces X  and Z there exist operators A1 W X  ! X and A2 W Z ! Z  respectively, which are monotone, hemicontinuous, continues at zeroes of the spaces X  and Z respectively, and A1 0 D 0, A2 0 D 0, and also hA1 !; !iX  D k!k˛C1 X

and

; hA2 z; ziZ D kzk˛C1 Z

˛ > 0:

(2.183)

For the system (2.181), (2.182) let us consider a sequence of operators ˚1n ! D A1

˚2n z D A2

  ! C B  ! C B˚1 z; B  ! ; n

z C ˚2 .x; z/ ; nC1

z 2 Z;

8! 2 X  ;

.x/ 2 V  X;

.z/ 2 Z;

(2.184)

n D 1; 2; : : : I (2.185)

Taking into account that the operators A1 , A2 are monotone hemicontinuous coercive, the fact that the conditions (2.179) and (2.180) are fulfilled, that the operators ˚1 .z; :/ W X ! X  8z 2 Z and ˚2 .x; :/ W Z ! Z  8x 2 X are radial continuous with uniformly semibounded variation, and that the operator B W X  ! X is positive continuous, we conclude that the operators ˚1n , ˚2n are coercive radial continuous with uniformly semibounded variation. Then due to [LaSoUr68, Corollary 4.1] each of the equations

232

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

˚1n ! D A1 ˚2n z D A2

  ! C B  ! C B˚1 z; B  ! D g; n

z C ˚2 .x; z/ D f nC1

z 2 Z;

! 2 X ;

8x 2 X;

8z 2 Z;

(2.186)

n D 1; 2; : : : I (2.187)

is solvable for any n. For n D 1 in (2.186), substituting z D z1 where kz1 k  ı the corresponding solution we denote by !1 , therefore, B  !1 D x1 . In (2.187), assuming x D x1 , the corresponding solution we denote by z D z2 . Further in (2.186), assuming z D z2 , we denote the corresponding solution by !2 and B  !2 D x2 , and in (2.187), substituting x D x2 , we denote the corresponding solution by z3 . Iterating the procedure we obtain A1

  !n C B  !n C B˚1 zn ; B  !n D g; n

(2.188)

and

znC1 C ˚2 .xn ; znC1 / D f: nC1 Hence, from (2.187) we have A2

D A1

(2.189)

E   ˛ ˝ !n ; !n C hB  !n ; !n iX C B˚1 zn ; B  !n ; !n X D hg; !n iX ; (2.190) n X

or, taking into account that the operator B has its right inverse D A1

E  ˛ ˝  !n ; !n C hB!n ; !n iX C ˚1 zn ; B  !n  B 1 g; B!n X D 0: (2.191) n X

In this equality the first and the second summands are positive for !n ¤ 0, and then, since !n satisfies the equality (2.191), the third summand is negative, whence, due to the conditions of the Theorem kxn kX D kB  !n kX <  for any n. Besides, as it was mentioned in the previous Theorems k!n kX  < 1 D const, since the linear continuous positive operator B  W X  ! X cannot transform and unbounded set into bounded one. Similarly, from (2.191) we have  A2

znC1 ; znC1 nC1

 C h˚2 .xn ; znC1 / ; znC1 iZ D hf; znC1 iZ ;

  znC1 ; znC1 C h˚2 .xn ; znC1 /  f; znC1 iZ D 0: A2 nC1 Z

or

(2.192)

Z

(2.193)

Here the first summand as zn ¤ 0 is positive. Then the second one is negative, and therefore, due to the condition (2.180) we conclude that kznC1 kZ  ı. From the equality (2.190) we obtain ˝

E D !   ˛ n ˚1 zn ; B  !n ; B  !n X D  A1 ; !n  hB  !n ; !n iX C hg; !n iX : n X

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

233

Hence, in view of the fact that hA1 !; !iX D k!k˛C1 X  , we find ˝

  ˛ 1  ˚1 zn ; B  !n ; B  !n X D  ˛ k!n k˛C1 X   hB !n ; !n iX C hg; !n iX ; (2.194) n

and conclude that h˚1 .zn ; B  !n / ; B  !n iX  c1 , since all summands in the right part of the last equality are bounded in virtue of boundness of the set f!n g  X  and continuity of the operator B. Then we have

 

˚1 zn ; B  !n   c2 X

when

kB  !n kX  :

(2.195)

kznC1 kZ  ı:

(2.196)

Similarly, from the equality (2.192) we conclude k˚2 .xn ; znC1 /kX   c3

when

Let us remark, that the constant c3 does not depend on xn . Now from the relation D ! E 1 n A1 ; !n D ˛ k!n k˛C1 X ; n n X 

and

A2 we have

znC1 ; znC1 nC1

 D Z

1  d2 ; kznC1 k˛C1 Z .n C 1/˛

d1 1 k!n kX  ! 0 as n ! 1; i.e., !0 k!n k˛C1 X  ˛ n n n

and d2 1 kznC1 kZ ˛C1 ! 0 as n ! 1; i.e., ! 1:  ˛ kznC1 kZ .n C 1/ nC1 nC1 Keeping in mind that the operators A1 and A2 are continuous at zeroes of the spaces X  and Z respectively, and also that A1 0 D 0 and A2 0 D 0, we find lim A1

n!1

!n D 0; n

lim A2

n!1

znC1 D 0: nC1

(2.197)

Now in virtue of reflexivity of the spaces X and Z there can be isolated subsequences fB  !n g  fB  !m g D fxm g ! x0 weakly in X , f˚1 .zm ; xm /g ! y0 weakly in X  , fzn g  fzm g ! z0 weakly in Z and f˚2 .xm ; zm /g ! 0 weakly in Z  . Passing to the limit in the equalities (2.192), (2.193), in virtue of (2.197), we obtain (2.198) x0 C By0 D g; and 0

Let us prove that y0 D ˚1 .z0 ; x0 / and

D f: 0

D ˚2 .x0 ; z0 /.

(2.199)

234

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

In view of the fact that ˚1 .zm ; :/ W X ! X  is the operator with r.s.b.v., for x 2 X such that kxkX  , we can write ˝

  ˛   ˚1 .zm ; x/  ˚1 zm ; B  !m ; x  B  !m X   inf Czm I kx  B  !m kX ; zm 2Z0

or h˚1 .zm ; x/ ; x  B  !m iX  h˚1 .zm ; B  !m / ; xiX h˚1 .zm ; B  !m / ; B  !m iX   inf Czm .I kx  B  !m kX / :

(2.200)

z2Z

Taking into account that B  !m D A

  !n  B˚1 zm ; B  !m C g; n

and substituting this expression in (2.200), we obtain  ˛ ˝  h˚1 .zm ; x/ ; x  B  !m iX  ˚1 zm ; B  !m ; x X D  !m E  ˛  ˝   ˚1 F zm ; B  !m ; A C ˚1 zm ; B  !m ; g X   m X ˛   ˝   ˚1 zm ; B  !m ; B zm ; ˚1 B  !m X  inf Czm I kx  B  !m kX : zm 2Z

In this inequality as m ! 1: (1) lim h˚1 .zm ; B  !m / ; B˚1 .zm ; B  !m /iX  hy0 ; By0 iX , i.e., the functional m!1

! ! h!; B!iX is weakly lower semicontinuous; (2) h˚1 .zm ; x/ ; x  B  !m iX ! h˚1 .z0 ; x/ ; x  x0 iX , since B  !m ! x0 weakly in X and in view of strenuous continuity of the operator ˚1 .:; x/ W Z ! X ; (3) h˚1 .zm ; B  !m / ; xiX ! hy0 ; xiX and h˚1 .zm ; B  !m / ; giX ! hy0 ; giX ,   ; ˛ y0 weakly in!X ˝since ˚1 .zm ; B !m!/m! m (4) ˚1 .zm ; B !m / ; A m X ! 0, since m ! 0 and A0 D 0; (5)  lim inf CZm .I kx  B  !m kX /  CZ0 .I kx  x0 kX /, since this m!1 zm 2Z

function is continuous along the second argument and      inf Czm I kx  B  !m kX  Cz0 I kx  B  !m kX : zm 2Z0

Therefore, passing to the limit in the last inequality, we obtain h˚1 .z0 ; x/ ; x  x0 iX  hy0 ; xiX C hy0 ; giX  hy0 ; By0 iX  Cz0 .I kx  x0 kX / ; or h˚1 .z0 ; x/ ; x  x0 iX  hy0 ; x C By0  giX  Cz0 .I kx  x0 kX / : (2.201)

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

235

From (2.198) By0  g D x0 and keeping this in mind h˚1 .z0 ; x/  y0 ; x  x0 iX  Cz0 .I kx  x0 kX / :

(2.202)

Then due to Theorem 2.1 [LaSoUr68] we have ˚1 .z0 ; x/ D y0 ; and the equality (2.198) takes the form: x0 C B˚1 .z0 ; x0 / D g: Further, let us show that the pair .x0 I z0 / satisfies the second equation of the system as well. Since the operator ˚2 is the operator with r.s.b.v., then for 8z 2 Z such that kzkZ  ı, the following inequality takes place h˚2 .x; zm /i  h˚2 .xm ; zmC1 / ; z  zmC1 iZ   inf Cxm .ıI kz  zmC1 kZ / : xm 2X0

Hence, in view of the equality ˚2 .xm ; zmC1 / D f  A2

zmC1 ; mC1

we find   zmC1 ; z  zmC1  hf; z  zmC1 iZ h˚2 .xm ; zmC1 / ; z  zmC1 iZ C A2 mC1 Z   inf Cxm .ıI kz  zmC1 kZ /: xm 2X0

z

mC1 Since the operator G is strenuously continuous along the second argument, A2 mC1 ! 0 as m ! 1, and also that the function Cxm is continuous along the second argument, from the last inequality we obtain

h˚2 .x0 ; z/ ; z  z0 iZ  hf; z  z0 iZ  Cxm .ıI kz  z0 kZ / ; or h˚2 .x0 ; z/  f; z  z0 iZ  Cxm .ıI kz  z0 kZ / : Therefore, due to Minty Lemma ˚2 .x0 ; z0 / D f: This proves the first part of the Theorem, i.e., the fact that the set K 2 is nonempty.

236

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Now we pass to the second part of the proof. Let us show that the set K 2 is weakly compact, i.e., from arbitrary sequences fxn I zn g  K 2 there can be isolated subsequences fxm g  fxn g fzm g  fzn g, such that xm ! z0 weakly in X , zm ! z0 , moreover .x0 I z0 / 2 K 2 . From the proof of the first part of the Theorem we know that kxn k  ; k˚1 .zn ; xn /kX   C1 8xn 2 X; kzn k  ı;

k˚2 .xn ; zn /kZ   C2

8zn 2 Z:

Therefore, in virtue of the reflexivity of the spaces X and Z we may assume that fxn g  fxm g ! x0 weakly in X , f˚1 .zn ; xn /g  f˚1 .zm ; xm /g ! y0 weakly in X  , fzn g  fzm g ! z0 weakly in Z and f˚2 .xn ; zn /g  f˚2 .xm ; zm /g ! 0 weakly in Z  . Therefore in the equations xm C BF1 .zm ; xm / D g

(2.203)

F2 .xm ; zm / D f;

(2.204)

we can pass to the limit as m ! 1, and whence x0 C By0 D g; and 0

D f:

Further, similarly with the proof of the first part of the given Theorem, we can establish that y0 D F1 .z0 ; x0 /, D ˚2 .x0 ; z0 / and thus x0 C B1 .z0 ; x0 / D g;

(2.205)

˚2 .x0 ; z0 / D f:

(2.206)

Indeed, for 8x 2 X and 8z 2 Z such that kxkX   and kzkZ  ı, we have h˚1 .zm ; x/  K1 .zm ; xm / ; x  xm iX   inf Czm .I kx  xm kX / ; zm 2K 2

and h˚2 .xm ; z/  ˚2 .xm ; zm / ; z  zm iZ   inf Cxm .ıI kz  zm kZ / : xm 2K 2

Hence, taking into account (2.203), (2.204), we obtain h˚1 .zm ; x/ ; x  xm i  h˚1 .zm ; xm / ; xiX C h˚1 .zm ; xm / ; giX  h˚1 .zm ; xm / ; B˚1 .zm ; xm /iX  inf Czm .I kx  xm kX / ; zm 2K 2

2.6 Nonlinear Non-coercive Operator Equations and Their Normalization

237

and h˚2 .xm ; z/ ; z  zm iZ  hf; z  zm i   inf Cxm .ıI kz  zm kZ / : xm 2K 2

Here, passing to the limit as m ! 1 we find h˚1 .z0 ; x/  y0 ; x  x0 iX   inf Czm .I kx  xm kX / ; zm 2K 2

and h˚2 .x0 ; z/  f; z  z0 iZ   inf Cxm .ıI kz  zm kZ / : xm 2K 2

Then due to Theorem 2.1 [LaSoUr68] we have y0 D ˚1 .z0 ; x0 / and ˚2 .x0 ; z0 / D f , therefore (2.205) and (2.206) hold true. The Theorem is proved. So the set K 2 D f.xI z/ 2 X  Z W D f.xI z/ 2 X  Z W

x C B˚1 .z; x/ D g; ˚2 .x; z/ D f g x C BF .z; M1 x/ D g; G .x; M2 z/ D f g

is nonempty, weakly compact and 8 .xI z/ 2 K 2 kxkX   and kzkZ  ı. Besides, keeping in mind the definitions of the operators M1 and M2 , we conclude that 8 .xI z/ 2 K 2 the following equalities hold true: x C BF .z; M1 x/ D x C BF .z; x/ D g; G .x; M2 z/ D G .x; z/ D f: Hence K 2  K 1 D f.xI z/ 2 X  Z

x C BF .z; x/ D g;

G .x; z/ D f g :

Definition 2.10. The operator F W Z  X ! X  we call M -uniformly monotone if the superpositional operator ˚ .z; :/ D F .z; M .// W X ! X  is uniformly monotone, i.e., 8x1 ; x2 2 X and 8z 2 Z the following inequality takes place h˚ .z; x1 /  ˚ .z; x2 / ; x1  x2 i D hF .z; M x1 /  F .z; M x2 ; x1  x2 / ; x1  x2 iX  0

238

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Definition 2.11. The operator F W X ! X  we call M -uniformly semimonotone, if there exists a strenuously continuous operator C W X ! X  such that the operator ˚ .z; / D ŒF .z; M .// C CM ./ W X ! X 

8z 2 Z

is monotone. The following statements hold true. Corollary 2.9. Let the operator B W X  ! X be linear continuous positive, the operator F .z; :/ W X ! X  8z 2 Z be radial continuous M -monotone and satisfy the condition: ˝

˛ F .z; x/  B 1 g; x > 0

8z 2 Z;

 D const; (2.207) and also let it be strenuously continuous along the first argument under the second one fixed, the operator G .x; :/ W Z ! Z  be radial continuous M -monotone and satisfy the condition hG .x; z/  f; zi > 0 8x 2 X;

if

kxkX D  > 0;

kzkZ D ı > 0;

ı D const;

(2.208)

and let it be strenuously continuous along the first argument under the second one fixed. Then the statement of Theorem 2.33 is valid. Corollary 2.10. Let the operator B W X  ! X be linear continuous positive, the operator F .z; :/ W X ! X  8z 2 Z be hemicontinuous M -semimonotone, let it satisfy the condition (2.207), and also let it be strenuously continuous along the first argument. Let the operator G .x; :/ W Z ! Z  be hemicontinuous M semimonotone, satisfying the condition (2.208) and let it be strenuously continuous along the first argument. Then the statement of Theorem 2.33 is valid. Corollary 2.11. Let the conditions of Theorem 2.33 (or of the corollaries 2.9, 2.10) be satisfied. Then the system of operator equations 

x C BF .z; x/ D g G .x; z/ D f

has a solution .x0 I z0 / 2 X  Z, moreover kx0 kX <  and kz0 kZ < ı.

2.7 Some Example Connected with Membranes Theory Let us consider some example connected with unilateral restrictions and friction in the flat and elastic membranes theory [DuLi76].

2.7 Some Example Connected with Membranes Theory

239

Example 2.7. Suppose ˝ is a bounded domain in n-dimensional Euclidean space Rn with the regular boundary @˝ [GaGrZa74], a function h W R ! R satisfies the following conditions: (1) For all s 2 R the function ˝ 3 x ! h .x; s/ is measurable; (2) For almost all x 2 ˝ functions g0 ; g1 2 L1 .˝/ and a number ˛  0 be such that for all s 2 R and almost all x 2 ˝ the following estimations are valid g0 .x/  h .x; s/  g1 .x/ C ˛ jsjp ;

(2.209)

where 1 < p < 1. In Banach space Lp .˝/ (the space of the functions summable together with their p degree) let us consider the integral functional Z ' .y/ D

h .x; y .x// dx:

(2.210)

˝

Lemma 2.13. Let the conditions (1)–(3) be valid. Then ' W Lp .˝/ ! R is a convex lower semicontinuous functional, dom' D Lp .˝/, and here ˚  @' .y/ D  2 Lp .˝/ j .x/ 2 @s h .x; y .x// a.e in ˝ ; where p 1 C q 1 D 1 and @s h .x; s/ is a subdifferential of the function h .x; s/ by the second variable. Proof. Convexity is obvious and lower semicontinuity follows from the lower estimation (2.209) and from Lebesgue–Fatou Theorem. From the upper estimation (2.209) we obtain that dom ' D Lp .˝/. It is well known [EkTe99, Proposition 9.5.1] that the adjoint functional looks like Z '  ./ D h .x;  .x// dx; where  2 Lp .˝/ ; h .x; s/ D sup fts  h.x; t/g : t 2R

˝

The relation  2 @' .y/ is equivalent to the equality ' .y/ C '  ./ D

Z y .x/  .x/ dx; ˝

or

Z



 h .x; y .x// C h .x;  .x//  y .x/  .x/ dx D 0

˝

Since the integration element is nonnegative h .x; y .x// C h .x;  .x// D y .x/  .x/ a.e.; that is equivalent to the inclusion  .x/ 2 @s h .x; y .x// a.e. in ˝.

240

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

Therefore the embedding is proved and  ˚ @' .y/   2 Lq .˝/ j .x/ 2 @s h .x; y .x// a.e D N If  2 N then, obviously, ' .y/ C '  ./ D h; yi whence  2 @' .y/. Suppose x D .x1 ; x2 ; : : : ; xn / 2 ˝; ˛PD .˛1 ; : : : ; ˛n / is a multiindex with integer nonnegative components ˛i ; j˛j D niD1 ˛i , D ˛ y .x/ D

@ @x1

˛1

:::

@ @xn

˛n

y .x/ ; D k y D fD ˛ yj j˛j D kg

We denote a number of different multiindexes ˛; ˇ whose length does note exceed m and m  1 respectively by M1 ; M2 . Let Wpm .˝/ be Sobolev spaces containing functions from Lp .˝/ such that their generalized partial derivatives up to o

m-th order inclusively also lie in Lp .˝/, let Wpm .˝/ be a subspace of the space Wpm .˝/ which is a closure of the space of smooth finite functions in Wpm .˝/. Consider the family A˛ .x; ; / .j˛j  m/ of real defined in ˝  RM2  RM1 functions such that the following conditions [Li69] are satisfied: (a) For almost all x 2 ˝ the function RM2  RM1 3 . I / ! A˛ .x; ; / is continuous and for all ;  the function ˝ 3 x ! A˛ .x; ; / is measurable; (b) There exists a function l 2 Lp .˝/ and a constant c > 0 such that   jA˛ .x; ; / j  C k kp1 C kkp1 C l .x/ for a.e. x 2 ˝I (c) For almost all x 2 ˝ and arbitrary bounded k k X j˛jDm

A˛ .x; ; / a

1 kk C kkp1

! C1 as kk ! 1I

(d) For almost all x 2 ˝ and for all     0  X   0 00 00 0 00 A˛ x; ;   A˛ x; ;  ˛  ˛ > 0 as  ¤  : j˛jDm

 ˚ We set ıy D y; Dy; : : : ; D m1 y then 8y; w 2 Wpm .˝/ the form X Z a .y; w/ D A˛ .x; ıy; D m y/ D ˛ wdx j˛j<m ˝

is defined. Suppose V is a closed vector space in Wpm .˝/ such that o

Wpm .˝/  V  Wpm .˝/ ; K D fw 2 V jw .x/  0 a.e. in˝g :

2.7 Some Example Connected with Membranes Theory

241 

Let us consider a multivalued map A W V ! 2V generated by the formula ˚  R hA .y/ ; wiV D a .y; w/ C ˝ wdxj 2 @' .y/ D hA1 .y/ ; wiV C hA2 .y/ ; wiV

(2.211)

Proposition 2.10. Let the conditions (1)–(3) and (a)–(d) be valid. Then the map  A D A1 C A2 W V ! 2V is s -pseudomonotone. 

Proof. Under the conditions (a)–(d) the operator A1 W V ! 2V generated by the formula hA1 .y/ ; wiV D a .y; w/ is pseudomonotone [Li69]. Further in virtue of Lemma 1.7 the subdifferential @' W Lp .˝/ ! 2Lq .˝/ is s -pseudomonotone bounded-valued map. It remains to use Proposition 1.56. Theorem 2.34. Suppose the conditions of Proposition 2.10 are valid, f 2 V  and kyk1 V a .y; y/ ! C1

as

kykV ! 1:

Then for every " > 0 the inclusion X j˛jm

1 .1/j˛j D ˛ A˛ .x; ıy" ; D m y" / C @s h .x; y" /  jy" jp2 y" 3 f "

where y  .x/ D



(2.212)

0; if y .x/  0 y .x/ ; if y .x/ < 0

has a solution y" 2 V and from the sequence fy" g there can be isolated a subw sequence fy g such that y ! y in V , y 2 K and it is an s-solution of the inequality ŒA .y/ ;   yC  hf;   yiV 8 2 K: o

Remark 2.15. In the case when V D Wpm .˝/ for inclusion (2.212) we obtain homogenous Dirichlet problem and when V D Wpm .˝/ we obtain a Neumann problem. 

Proof. The operator A W V ! 2V generated by (2.211) is bounded, s -pseudomonotone and A D coA. Besides 0N 2 K and due to monotony of the subdifferential and of the penalty operator ˇ .y/ D jy  jp2 y  we have

242

2 Operator Inclusions and Variation Inequalities in Infinite-Dimensional Spaces

kyk1 V



1 a .y; y/ C ŒA2 .y/ ; yC C D hˇ .y/ ; yiV "



 kyk1 V fa .y; y/ C ŒA2 .0/ ; y g

 kyk1 V fa .y; y/  kA2 .0/k kykV g ! C1

as kykV ! 1, namely coercivity condition (2.79) holds true. Therefore all conditions of Theorem 2.21 hold true.

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Index

Acute-angle Lemma, xxiv, 164 Adjoint functional, 98 Adjoint with the vector process, 23 Admissible pairs, xvi Affine process, 23 A multivalued map that satisfies property (˘ ), 92, 106 An example of the operator of variational calculus on D.A/, 87 Aumann integral, 19

Banach, xxiii Banach space, 18 Basis theory, xxiv Bochner integrals, 19 Bounded, 35, 106 *-bounded set, 106

Canonical duality, 29 Cartesian product, 15 Centered, 3 Coercive, 112 C-coercivity, xxii C./-coercive, 35 -coercivity, xxii Compactness Theorems, xxiii Condition (˛), 41 Condition ./C./ , 37 Condition .˘ /, xxii, 37 Continuous, 10 Converging sequence of sets, 4 Convex, xxiii, 97

d-closed, 35 Demiclosed, 48 Diagonal product of the mappings, 15

Difference approximations, xxiii Differential equations with discontinuous right-hand side, xix Differential-operator inclusions, xix, xxiii Directed, 2 Dubinskii method, xxiv

Effective set, 2 Elliptic regularization, xxiii Embedding and approximation Theorems, xxiv Evolution inequalities on cones, xviii Evolution variation inequalities, xxiv

Faedo–Galerkin method, xxiii, xxiv Finite-dimensionally locally bounded, 35 Frechet, xxiii Free boundary problems, xxiv Friction, 38 ˛.G/.˛ 0 .G//, 41 ˛0 .G/.˛00 .G//, 41 ˛1 .G/, 41 ˛2 .G/, 42 ˛3 .G/.˛30 .G//, 42 Generalized pseudomonotone, xxi Generalized pseudomonotone on X operator, 75 Generalized pseudomonotony, xxii Generalized sequence (or net), 2 Graph F , 8

Harmful impurities expansion in the atmosphere, xxiv Hausdorff metric, 5 Hemicontinuous (h.c.), 26, 198

245

246 Image, 2 Inverse mapping, 8

Lipschitzean, 10 Lipschitzean mappings, 10 Locally bounded, 35 Local subdifferential, 96, 97 Lower form, xxii Lower limit, 3 Lower semicontinuous, xxiii, 5, 26, 97 Lower semifinite topology, 4 (l.s.c.), 97

Map, 198 Mapping, 9, 10, 15 Maximal monotone, 40 Measurable, 18 Membranes theory, 238 Method of finite differences, xxiii Method of semigroups, xxiii Minimax inequalities, xxiv M -local Lipschitzean, 116 M -monotone, 209 Modification of Caratheodory conditions, 19 Monotone, 34, 40 m-semiflow, 28 M -semimonotone, 209 Multivalued mapping, 2, 5, 10 Multivalued Nemitsky operator, 19 Multivalued penalty method, xxiv M -uniformly monotone, 237 M -uniformly semimonotone, 238

n-dimensional, Navier–Stokes problem, xvii Negative polar cone, 129 Nemitsky operator, 19 N -monotony, 34 Noncoercive problems, xxiv Nonlinear integro-differential equation, 196 (with N-s.b.v), 36 Operator with .M I N /-semi-bounded variation, 54 Operators equations of the Hammerstein type, 190

Parameterized operator inclusions, xxiv, 154 P -critical point, 131 Penalty method, 162

Index Physical anomalies conditions, xi Projection map, 132 Proper, 2 Property ./, 84 Property .˘ /, xxii Property ./, 190 Property .M /, 92 Property SNk , 94 Property A , 48 Pseudomonotone, xx .F I G/ -pseudomonotone, 45 .F I G/-pseudomonotone, 45 .F I G/w -pseudomonotone, 46 s -pseudomonotone, 64 -pseudomonotone on X, 37 0 -pseudomonotone on X, 37 Pseudomonotony in classical sense, xxii

-quasimonotone, 94 Quantum effects, 38 Quasiopen, 10

Radial continuous, 36 Radial lower semicontinuous, 36 Radial semicontinuous, 36 Radial semimonotone one (r.s.m.), 223 Radial upper semicontinuous, 36 (r.c.), 36 (r.l.s.c.), 36 (r.s.c.), 36 (r.u.s.c.), 36

Satisfies condition . /, 18 (s.b.v.), 36 Semimonotone, 198 Semimonotone (s.m.), 36 Singular perturbations, xxiii Singular perturbations method, xxiv Small (complete) pre-image, 2 s-mutually bounded, 80 Space X, 190 s-radial continuous map, 65 Strenuously continuous, 198 Strict, 2 Strict solution (s-solution), 186 Subdifferential maps, xxiii Subdifferential maps in locally-convex spaces, xxiv Submonotone (sub-m.), 37 Superposition operator, 19 System of functional equations, 191

Index System of nonempty subsets, 2 System of subsets, 3 Systems of Hammerstein type, xxiv Systems of operator inclusions, xxiv

The functional, 97, 99, 112, 116 The map, 48, 198 The mapping, 6, 7, 10 The multivalued map, 94 The multivalued mapping, 10, 18 The operator, 40, 84, 85, 198, 209, 215, 223, 237, 238 The operator of variational calculus on D.A/, 85 The operator with N -semibounded variation on X, 36 The operator with N -subbounded variation (N -sub-b.v.), 37 The operator with semibounded variation on X, 36 The operator with subbounded variation (sub-b.v.), 37 The operator with V -semibounded variation on X (with V -s.b.v.), 36 Theorems about continuous embedding, xxiv The sequence of sets, 4 The space S.XI X  /, 89 The theory of semigroups, xxiii The uniform property ./C./ , 89 (u.h.c.), 36

247 Uniformly C./-coercive, 35 Uniformly semibounded variation (u.s.b.v.), 215 Unilateral processes, xxiv Upper and the lower Clarke derivatives, 115 Upper form, xxii Upper hemicontinuous, 26, 36 Upper limit, 3 Upper locally Lipschitzean, 10 Upper semicompact, 9 Upper semicontinuous (u.s.c.), 97 Upper semicontinuous, 7 !-upper semicontinuous, 6 *-upper semicontinuous, 106 Upper semifinite topology, 4 Upper support function, 18

Variation inequality, xix, xxiv, 186 Vector process, 23 Vietoris topology, 4 Viscosity, 38 V -monotony, 34 Weakly C./-coercive, 35 Weakly lower semicontinuous (w.l.s.c.), 99 Weak solution of the inclusion, 146 w0 -monotony, 34 w0 -pseudomonotone multivalued maps, xxiii w0 -pseudomonotone (0 -pseudomonotone on W ), xxi w0 -pseudomonotony, xxii