Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions

Series in Real Analysis – Vol. 11

GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS Not Absolutely Continuous Solutions

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SERIES IN REAL ANALYSIS

Published Vol. 1:

Lectures on the Theory of Integration R Henstock

Vol. 2:

Lanzhou Lectures on Henstock Integration Lee Peng Yee

Vol. 3:

The Theory of the Denjoy Integral & Some Applications V G Celidze & A G Dzvarseisvili translated by P S Bullen

Vol. 4:

Linear Functional Analysis W Orlicz

Vol. 5:

Generalized ODE S Schwabik

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Uniqueness & Nonuniqueness Criteria in ODE R P Agarwal & V Lakshmikantham

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Henstock–Kurzweil Integration: Its Relation to Topological Vector Spaces Jaroslav Kurzweil

Vol. 8:

Integration between the Lebesgue Integral and the Henstock–Kurzweil Integral: Its Relation to Local Convex Vector Spaces Jaroslav Kurzweil

Vol. 9:

Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock–Kurzweil, and McShane Douglas S Kurtz & Charles W Swartz

Vol. 10: Topics in Banach Space Integration Òtefan Schwabik & Ye Guoju Vol. 11: Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions Jaroslav Kurzweil Vol. 12: Henstock–Kurzweil Integration on Euclidean Spaces Tuo Yeong Lee Vol. 13: Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock–Kurzweil, and McShane (2nd Edition) Douglas S Kurtz & Charles W Swartz

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Series in Real Analysis – Vol. 11

GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS Not Absolutely Continuous Solutions

Jaroslav Kurzweil Academy of Sciences, Czech Republic

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Series in Real Analysis — Vol. 11 GENERALIZED ORDINARY DIFFERENTIAL EQUATIONS Not Absolutely Continuous Solutions Copyright © 2012 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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Preface

The Kurzweil-Henstock integral is a nonabsolutely convergent integral. The aim of this treatise is the exploitation of this property in some convergence problems in ordinary differential equations and in some situations where solutions of infinite variation can occur. This leads to generalized differential equations the theory of which is presented in Chapters 6–18, 22–24, 27. Since the motion of Kapitza’s pendulum was a significant motivation for the theory, the equation of motion of Kapitza’s pendulum is exposed in Chapter 2 and various estimates for equations of a similar type are obtained by elementary methods in Chapters 3 and 4. The concept of the Kurzweil-Henstock integral and of the generalized differential equation goes back to 1957. The preparation of this book started in 2004 after W. N. Everitt had encouraged me to build a solid theory of the generalized differential equations. The results in this direction were reported and discussed regularly in the Seminar on Differential Equations and Integration Theory in the Institute of Mathematics of the Academy of Sciences of the Czech Republic in Prague. I wish to thank the participants of the seminar for their contributions and comments, in particular to J. Jarn´ık, B. Maslowski, I. Vrkoˇc, M. Tvrd´ y ˇ and the late S. Schwabik. The last two of them in addition read and commented the manuscript. E. Ritterov´a was very helpful by typing several versions of the manuscript. The research connected with the preparation of this book was supported by the grant No. IAA100190702 of the Grant Agency of the Acad. Sci. of the Czech Republic and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503. Prague, May 2011

Jaroslav Kurzweil v

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Contents

Preface

v

1.

Introduction

1

2.

Kapitza’s pendulum and a related problem

9

3.

Elementary methods: averaging

11

4.

Elementary methods: internal resonance

15

5.

Strong Riemann-integration of functions of a pair of coupled variables

27

6.

7. 8.

9.

Generalized ordinary differential equations: Riemann-solutions (concepts)

Strong

Functions ψ1 , ψ2

37 43

Strong Riemann-solutions of generalized differential equations: a survey Approximate solutions: boundedness vii

47 51

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Generalized Ordinary Differential Equations

10. Approximate solutions: a Lipschitz condition

59

11. Approximate solutions: convergence

63

12. Solutions

69

13. Continuous dependence

77

14. Strong Kurzweil Henstock-integration of functions of a pair of coupled variables

83

15. Generalized differential equations: Henstock-solutions

97

Strong Kurzweil

16. Uniqueness

101

17. Differential equations in classical form

105

18. On a class of differential equations in classical form

111

19. Integration and Strong Integration

119

20. A class of Strong Kurzweil Henstock-integrable functions

127

21. Integration by parts

135

22. A variant of Gronwall inequality

145

23. Existence of solutions of a class of generalized ordinary differential equations

155

24. A convergence process as a source of discontinuities in the theory of differential equations

165

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Contents

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ix

25. A class of Strong Riemann-integrable functions

177

26. On equality of two integrals

185

27. A class of Generalized ordinary differential equations with a restricted right hand side

187

Appendix A.

Some elementary results

189

Appendix B.

Trifles from functional analysis

191

Bibliography

193

Symbols

195

Subject index

197

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Chapter 1

Introduction

A solution of a classical differential equation x˙ = f (x, t)

(1.1)

is a function u such that its derivative u˙ is at every τ equal to f (u(τ ), τ ), i.e. in a neighborhood of τ the linear function t → u(τ ) + f (u(τ ), τ ) (t − τ ) is a good approximation of u. Usually f and u are Rn -valued functions. Given u(a) = y the value u(T ) is approximately equal to y+

k ∑

f (u(τi ), τi ) (ti − ti−1 )

i=1

for T > a. Here a = t0 < t < · · · < tk = T, (1.2) τi ∈ [ti−1 , ti ], τi being called the tag of the interval [ti−1 , ti ] and the partition of [a, b ] into intervals [ti−1 , ti ] is sufficiently fine. Moreover, u is a solution of the Volterra integral equation ∫ T u(T ) = y + f (u(t), t) dt (1.3) a

and vice versa, every solution of (1.3) is a solution of (1.1) and fulfils u(a) = y. A generalized ordinary differential equation (GODE) d x = Dt F (x, τ, t) dt 1

(1.4)

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Generalized Ordinary Differential Equations

depends on a function F of three variables and its solution is a function u such that the function t → u(τ ) + F (u(τ ), τ, t) − F (u(τ ), τ, τ ) is a good approximation of u in a neighborhood of any τ. The value u(T ) is approximately equal to u(a) +

k ∑

[F (u(τi ), τi , ti ) − F (u(τi ), τi , ti−1 )].

(1.5)

i=1

In fact, the sum in (1.5) can be viewed as an approximation of an integral which is denoted by ∫ T Dt F (u(τ ), τ, t). (1.6) a

The quality of approximation depends on the interpretation of the concept that the partition of [a, T ] is fine. By definition, u is a solution of (1.4) if it fulfils ∫ T u(T ) = u(a) + Dt F (u(τ ), τ, t)

(1.7)

a

for T > a, which is a Volterra-type integral equation. The concept of a fine partition of [a, b ] admits various interpretations and two of them are crucial in this treatise. Let ξ > 0, [a, T ] ⊂ R. A set {([ti−1 , ti ], τi ) ; i = 1, 2, . . . , k} is a ξ-fine partition of [a, T ] if (1.2) holds, if ti − ti−1 ≤ ξ and τi ∈ [ti−1 , ti ] for i = 1, 2, . . . , k. Let δ be a positive function on [a, T ], i.e. δ : [a, T ] → R+ . A set {([ti−1 , ti ], τi ) ; i = 1, 2, . . . , k} is a δ-fine partition of [a, T ] if (1.2) holds and if τi − δ(τi ) ≤ ti−1 ≤ τi ≤ ti ≤ τi + δ(τi )

for i = 1, 2, . . . , k.

These two concepts of a fine partition of [a, T ] are a basis for two concepts of a solution of (1.4). Let us proceed to the concept of a solution u of (1.4) without making precise the integral in (1.6). Let X be a Banach space, ∥x∥ denoting the norm of x for x ∈ X. Assume that [a, b ] ⊂ R, u : [a, b ] → X and that there exists ξ0 > 0 such that F (u(τ ), τ, t) is defined for τ, t ∈ [a, b ],

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Introduction

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|t − τ | ≤ ξ0 . u is called an SR-solution (Strong Riemann solution) of (1.4) if for every ε > 0 there exists ξ > 0 such that k ∑

∥u(ti ) − u(ti−1 ) − F (u(τi ), τi , ti ) + F (u(τi ), τi , ti−1 )∥ ≤ ε

i=1

for every ξ-fine partition {([ti−1 , ti ], τi ); i = 1, 1, . . . , k} of [a, b ]. Assume that [a, b ]⊂ R, u: [a, b ]→ X and that there exists δ0 : [a, b ]→ R+ such that F (u(τ ), τ, t) is defined for τ, t ∈ [a, b ], |t − τ | ≤ δ0 (τ ). u is called an SKH-solution (Strong Kurzweil-Henstock solution) of (1.4) if for every ε > 0 there exists δ : [a, b ] → R+ such that k ∑

∥u(ti ) − u(ti−1 ) − F (u(τi ), τi , ti ) + F (u(τi ), τi , ti−1 )∥ ≤ ε

i=1

for every δ-fine partition {([ti−1 , ti ], τi ) ; i = 1, 2, . . . , k} of [a, b ]. This definition is correct since for every δ : [a, b ] → R+ there exists a δ-fine partition of [a, b ]. Let ξ0 > 0, 0 < ξ < ξ0 , and let δ(τ ) ≤ ξ for τ ∈ [a, b ]. Then every δ-fine partition of [a, b ] is a ξ-fine partition of [a, b ]. Therefore every SRsolution of (1.4) is an SKH-solution of (1.4) but not vice versa. The origin of the concept of a GODE goes back to the averaging principle by which the solutions of x˙ = h0 (x) +

k ∑

hj (x) cos(σj t/ε + ηj ),

(σj ̸= 0)

(1.8)

j=1

tend for ε → 0 to the solutions of x˙ = h0 (x)

(1.9)

which is called the averaged equation since 1 T →∞ T

∫

T

h0 (x) = lim

[ h0 (x) +

0

k ∑

] hj (x) cos(σj t/ε + ηj ) dt .

j=1

The right hand side of (1.8) does not converge pointwise for ε → 0 which gave an impulse to tying together the convergence of solutions with various types of convergence of the right hand sides of the differential equations.

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Generalized Ordinary Differential Equations

The book consists of five parts. In the first part (Chapters 2–4) Kapitza’s pendulum is briefly treated. Its equation is transformed to an equation of the form (1.8). Let λi : R → R, i = 1, 2, . . . , k, ∫ 1 λj (t + 1) = λj (t), λj (s) ds = 0, hj : Rn → Rn . 0

Let u : [a, b ] → Rn be a solution of x˙ = h0 (x) +

n ∑

hj (x) λj (t/ε),

(1.10)

j=1

and let w : [a, b ] → Rn be a solution of (1.9), u(a) = w(a). Then ∥u(t) − w(t)∥ ≤ κ ε for t ∈ [a, b ] for some κ > 0 and ε sufficiently small, i.e. w is an approximation of u of order ε. Moreover, an approximation w e of u of order ε2 can be defined by the formula k ∑ w(t) e = w(t) + hj (w(t)) Λj (t/ε) j=1

∫

1

where Λj : R →R is the primitive of λj such that that w(a) e = w(a) + ε

k ∑

Λj (s) ds = 0. (Observe 0

hj (w(a)) Λj (a/ε).)

j=1

The second part (Chapters 5–13) contains a theory of GODEs in the form d x = Dt G(x, τ, t) (1.11) dt where G fulfils conditions (8.2)–(8.7). In simplified form conditions (8.3), (8.4) may be written as ∥G(x, τ, t) − G(x, τ, s)∥ ≤ κ |t − s|α ,

(1.12)

∥G(x, τ, t) − G(x, τ, s) − G(y, τ, t) + G(y, τ, s)∥ ≤ κ ∥x − y∥ |t − s|β , (1.13) where κ > 0, 0 < α ≤ β, α + β > 1. Conditions (8.5)–(8.7) are of a similar type. A local existence theorem and a continuous dependence theorem are valid for the equation of the type (1.11). By the continuous dependence theorem there exists a function Ω : R → R such that Ω(η) → 0 for η → 0 and

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Introduction

5

the following situation takes place: Let G∗ fulfil (8.1)–(8.7), let v : [a, b ] → X be a solution of (1.11) and let v ∗ : [a, b ] → X be a solution of d x = Dt G∗ (x, τ, t), dt

(1.14)

v(a) = v ∗ (a). If G∥(x, τ, t) − G∗ (x, τ, t)∥ ≤ η (for (x, τ, t) from a suitable set) then ∥v(t) − v ∗ (t)∥ ≤ Ω(η) for t ∈ [a, b ]. A uniqueness result is postponed to Chapter 16 since it is proved for the concept of SKH-solutions which is wider than the concept of SR-solutions. A solution u: [a, b ] → X of (1.11) fulfils ∥u(t) − u(s)∥ ≤ κ1 |t − s|α , where κ1 > 0, t, s ∈ [a, b ], but need not have finite variation on any interval. By definition u is a solution of the corresponding integral equation of Volterra type. The main object of the third part (Chapters 14–18) is the concept of an SKH-solution of the GODE (1.11), no specific assumptions on G being imposed. In particular: (i) If a < c < b, u : [a, b ] → X and if u is an SKH-solution of (1.11) on [a, c] and on [c, b] then it is an SKH-solution on [a, b ] (which need not hold for SR-solutions in general). (ii) If u : [a, b ] → X is an[SKH-solution of (1.11) on [c, b ] for every c]such that a < c < b and if u(c) − u(a) − G(u(a), a, c) − G(u(a), a, a) → 0 for c → a, then u is an SKH-solution of (1.11) on [a, b ]. Let g depend on the variables x, t and u : [a, b ] → X. Then u is called an SKH-solution of the differential equation in the classical form x˙ = g(x, t)

(1.15)

if ∫

∫

T

u(T ) − u(S) = (SKH)

T

Dt [g(u(τ ), τ ) t] = (SKH) S

g(u(t), t) dt . S

Conditions on g, G are found such that u is a solution of (1.15) if and only if it is a solution of (1.11). The following result is an illustration of the above concepts: ( ) ( ) Let h : X × R \ {0} → X, H : X × R \ {0} → X, the main assumption

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Generalized Ordinary Differential Equations

6

being that h and H are bounded and continuous, ∂ H(x, t) = h(x, t) and 1 ≤ α < 1 + 12 β. ∂t Put

 −α −β   t h(x, t ) g(x, t) = 0   |t|−α h(x, −|t|−β )

if t > 0, if t = 0, if t < 0.

Then for every y ∈ X there exists a unique SKH-solution u : [a, b ] → X of (1.15), u(a) = y. u need not be absolutely continuous on any neighbourhood of 0. Let Φ : [a, b ] → R be nondecreasing, Φ(τ ) − limt→τ,t<τ Φ(t) < 1 for a < τ ≤ b. The fourth part (Chapters 19–24) contains a theory of the GODE d x = F (x, τ, t), dt

(1.16)

where F : X × [a, b ] 2 → X fulfils

∥F (x, τ, t) − F (x, τ, s)∥ ≤ (1 + ∥x∥) Φ(t) − Φ(s)

and three conditions of a similar type. For every y ∈ X there exists a unique SKH-solution u : [a, b ] → X of (1.16), u(a) = y. This solution is a function of bounded variation and depends continuously on y and F. Put F ◦ (x, T ) = (SKH)

∫

T

Dt F (x, τ, t). a

Then u is an SKH-solution of (1.16) if and only if it is an SKH-solution of d x = F ◦ (x, t). dt

(1.17)

In other words: a process which is described by equation (1.16) is described by equation (1.17) as well. Moreover, for a given F there may exist a sequence of functions Fi , i ∈ N, which are continuous and fulfil similar conditions as F and if ui : [a, b ] → X is a solution of d x = Dt Fi (x, τ, t), dt

ui (a) = y,

then ui is continuous, the sequence ui is bounded and ui (t) → u(t) almost everywhere. Observe that Chapter 19 contains only such results which

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Introduction

7

follow directly from the definition of the SKH-solution and that rather ∫ b general results on the existence of (SKH) Dt U (τ, t) and on integrationby-parts are presented in Chapters 20,21.

a

The main topic of the fifth part (Chapters 25–27) is the treatment of the GODEs d x = Dt G(x, τ, t) , dt d x = Dt G◦ (x, τ, t) , dt

(1.18) (1.19)

where G: X × [a, b ]2 → X fulfils conditions which are similar to the conditions imposed on G in the GODE (8.8) and ∫ t ◦ G (x, t) = (SR) Dt G(x, σ, s). (1.20) a

In the simplified form G fulfils ∥G(x, τ, t) − G(x, τ, s)∥ ≤ κ (1 + ∥x∥) |t − s|α

for x ∈ X, τ, t, s ∈ [a, b ] ,

∥G(x, τ, t) − G(x, τ, s) − G(z, τ, t) + G(z, τ, s)∥ ≤ κ ∥x − z∥ |t − s|β for x, z ∈ X, τ, t, s ∈ [a, b ] , where κ > 0, 0 < α ≤ β, α + β > 1, and three similar conditions. The integral in (1.20) exists and G◦ fulfils analogical conditions as G. Moreover, u is a solution of (1.18) if and only if it is a solution of (1.19). The main part of the book was written in the years 2007-2009. The results on the relation of the GODEs (1.15), (1.16) and on the relation of the GODEs (1.18), (1.19) were added to the text in the final phase of its preparation.

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Chapter 2

Kapitza’s pendulum and a related problem

In [Kapitza (1951a)], [Kapitza (1951b)] the author studied the equation ¨ = (g L−1 − L−1 (A ω) ω sin ω t) sin Θ Θ

(2.1)

which describes the motion of a pendulum the support of which is vibrating with a large frequency ω and a small amplitude A. Here g, L, ω, A ∈ R+ , L is the length of the pendulum and g is the gravitational constant. Putting ˙ − L−1 (Aω) cos ω t sin Θ = Φ Θ the equation (2.1) is transformed to       1 −1 2 sin Θ − 2 (L A ω) sin Θ cos Θ 

˙ = Φ + L−1 A ω sin Θ cos ω t , Θ ˙ = g L−1 Φ

− 12 (L−1 A ω)2 sin Θ cos Θ cos 2ω t −L−1 A ω Φ cos Θ cos ω t .

     

(2.2)

If ω is large and A is small (i.e. if A ω ≤const.) then the solutions of (2.2) are close to the solutions of the averaged equation } ˙ = Φ, Θ (2.3) Φ˙ = g L−1 sin Θ − 21 (L−1 A ω)2 sin Θ cos Θ . The relation of solutions of (2.2) and (2.3) can be studied as the relation of solutions of x˙ = f (x) + h(x, t, t/ε),

ε>0

(2.4)

and the solutions of x˙ = f (x) , 9

(2.5)

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Generalized Ordinary Differential Equations

10

the main assumption being the existence of a bounded function H(x, τ, t) such that ∂ H(x, τ, t) = h(x, τ, t) . ∂t This problem will be tackled by elementary methods in Chapter 3. In Chapter 4 an equation of a more special form than (2.4) is treated but more precise results are obtained. An insight into the problem of averaging is given by the equation x˙ = x ε−α cos(2π t/ε) + ε−α sin(2π t/ε)

(2.6)

where 0 < α ≤ 1/2. Let y ∈ R and denote by uα,ε : [−1, 1] → R the solution of (2.6) fulfilling x(0) = y. It is well known that uα,ε (t) = exp

( sin(2π t/ε) ) 2π εα−1

(2.7)

(

) ∫ t ( sin(2πτ /ε) ) sin(2πτ /ε) y + exp − dτ 2π εα−1 εα 0

and it can be deduced that ε1−2 α t ≤ const. ε1−α , uα,ε (t) − y + 4π

(2.8)

const. being independent of α, ε, t. Hence, for ε → 0,  y if α < 12 , uα,ε (t) → y − t if α = 1 . 2 4π In other words, uα,ε tends to a solution of x˙ = 0 if α < 21 and ε → 0 while uα,ε tends to a solution of x˙ = − 1/(4 π) if α = 21 and ε → 0. If α = 0 then (cf. (2.8)) ∥u1,ε (t) − y∥ ≤ const. ε . 2.1. Remark. In [Lojasiewicz (1955)] the author treated an equation the particular case of which is (2.1) and obtained the corresponding convergence results. J. Jarn´ık approached an analogous problem by means of the Kurzweil-Henstock integral and related ideas, see [Jarn´ık (1965)]. A different method was used in [Mitropolskij (1971)].

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Chapter 3

Elementary methods: averaging

The aim of this chapter is to estimate the difference of the solutions uε of x˙ = f (x) + h(x, t/ε)

(3.1)

and the solutions w of the “averaged” differential equation x˙ = f (x) ,

(3.2)

the main assumption being the existence of a bounded H such that ∂ H(x, t) = h(x, t) . ∂t

(3.3)

Moreover, it is assumed that there exist µ, ν ∈ R+ such that f and its differential Df are bounded by µ

(3.4)

h, H and some of their derivatives are bounded by ν .

(3.5)

and

Let [a, b ] ⊂ R. It will be proved in this chapter that there exist κ1 , κ2 ∈ R, depending only on µ and b − a, such that the inequality ∥uε (t) − w(t)∥ ≤ ε (ν κ1 + ν 2 κ2 ),

t ∈ [a, b ]

(3.6)

holds for the solution uε of (3.1) and the solution w of (3.2) such that uε (a) = w(a). (3.6) is valid if ν is large provided ε > 0 is sufficiently small (cf. (3.7). Let 0 ≤ α < 21 and let uα,ε be a solution of x˙ = f (x) + ε−α h(x, t/ε) 11

(3.7)

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Generalized Ordinary Differential Equations

12

where f fulfils (3.4) and h, H fulfil (3.5). Then ε−α h, ε−α H

}

and some of their derivatives are bounded by ε−α ν .

(3.8)

(3.6) implies that ∥uα,ε (t) − w(t)∥ ≤ ε1−α ν κ2 + ε1−2α ν 2 κ3 ,

t ∈ [a, b ]

(3.9)

if uα,ε (a) = w(a). 3.1. Notation. Let R denote the set of reals, R+ the set of positive reals, R+ 0 the set of nonnegative reals, N the set of positive integers, N0 the set of nonnegative integers. Let R, µ, ν, ε ∈ R+ , n ∈ N .

(3.10)

Let [a, b ] denote the segment in R with endpoints a, b for a, b ∈ R, a < b. The vector space Rn is supposed to be equipped with a norm, ∥x∥ denoting the norm of x for x ∈ Rn . If r ≥ 0 then B(r) = {x ∈ Rn ; ∥x∥ ≤ r} (i.e. B(r) is the ball in Rn with center at 0 and radius r). If h : B(r) → Rn then D h(x) is the differential of h at x. Analogously, if h : B(r) × [a, b ] → Rn then D1 h(x, t) is the differential of h with respect to x at (x, t) and D2 h(x, t) is the differential of h with respect to t at (x, t). D2 h(x) is the differential of the second order of h at x. Let f : B(R) → Rn ,

h : B(R) × R → Rn ,

H : B(R) × R → Rn .

(3.11)

Assume that f, D f are continuous ,

(3.12)

∥f (x)∥ ≤ µ, ∥D f (x)∥ ≤ µ for x ∈ B(R) ,

(3.13)

h, D1 h, D1 H, D2 H are continuous ,

(3.14)

D2 H(x, t) = h(x, t) for x ∈ B(R), t ∈ R

(3.15)

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13

and ∥h(x, t)∥, ∥D1 h(x, t)∥, ∥H(x, t)∥, ∥D1 H(x, t)∥ ≤ ν

}

for x ∈ B(R), t ∈ R .

(3.16)

3.2. Remark. Conditions (3.13)–(3.16) may be fulfilled if h(x, t) =

k ∑

hi (x) λi (t)

(3.17)

i=1

where hi : B(R) → Rn , λi : R → R are continuous, if there exist ωi ∈ R+ ∫ ωi such that λi (t + ωi ) = λi (t) for t ∈ R and λi (t) dt = 0, i = 1, 2, . . . , k, since the primitive Λi of λi is bounded and H(x, t) =

k ∑

0

hi (x) Λi (t) .

i=1

3.3. Theorem. Assume that y ∈ B(R), that uε : [a, b ] → B(R) is a solution of (3.1), u(a) = y and that w : [a, b ] → B(R) is a solution of (3.2), w(a) = y. Then } ∥uε (t) − w(t)∥ (3.18) ≤ ε [(2 + µ)(b − a) eµ (b − a) ν + (b − a) eµ (b − a) ν 2 ] . Proof.

Observe that (cf. (3.15)) ε

d H(uε (t), t/ε) = ε D1 H(uε (t), t/ε) u˙ ε (t) + h(uε (t), t/ε) . dt

Hence after substituting for u˙ ε (t) we obtain ∫

t

h(uε (s), s/ε) ds = ε A (t) a

where A (t) = D1 H(uε (t), t/ε) − D1 H(y, a/ε) ∫ t [ ] − D1 H(uε (s), s/ε) f (uε (s)) + h(uε (s), s/ε) ds . a

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Generalized Ordinary Differential Equations

Moreover, ∫

∫

t

uε (t) = y +

t

f (uε (s)) ds + ∫

a

∫

a

h(uε (s), s/ε) ds a

t

f (uε (s)) ds + ε A (t) ,

= y+ t

w(t) = y +

f (w(s)) ds . a

Hence

∫

t

uε (t) − w(t) =

[ ] f (uε (s)) − f (w(s)) ds + ε A (t) .

a

(3.13) and (3.16) imply that ∥f (uε (s)) − f (w(s))∥ ≤ µ ∥uε (s) − w(s)∥ , ∥A (t)∥ ≤ 2 ν + (b − a) ν (µ + ν) and

∫

t

∥uε (t) − w(t)∥ ≤

µ ∥uε (s) − w(s)∥ ds a

+ ε [(2 + (b − a) µ) ν + (b − a) ν 2 ] . Therefore (3.18) is correct by Lemma A.1. Moreover, (3.6) holds with κ1 = (2 + µ) (b − a) e µ(b − a) ,

κ2 = (b − a) e µ (b − a) .

3.4. Remark. Let 0 < r < R, y ∈ B(r), [ ] ε (2 + µ) (b − a) ν + (b − a) ν 2 e µ (b − a) ≤ R − r .



(3.19)

Let there exist a solution w : [a, b ] → B(r) of (3.2), w(a) = y. By classical results on ordinary differential equations uε exists on an interval [a, c], a < c ≤ b and can be continued to [a, b ] (cf. (3.18)) and (3.19)).

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Chapter 4

Elementary methods: internal resonance

The solution w of (3.2) can be viewed as an approximation of the solutions uε of (3.1). A more precise approximation w eε is constructed in this chapter but fairly stronger conditions are imposed on the right-hand side of (3.1). 4.1. Notation. Let n, k ∈ N; ε, R, µ, ν ∈ R+ . Let f : B(R) → Rn , hi : B(R) → Rn , λi : R → R for i = 1, 2, . . . , k. Assume that f and hi are differentiable, ∥f (x)∥ ≤ µ,

}

∥Df (x)∥ ≤ µ ,

(4.1)

∥D f (x) − D f (¯ x)∥ ≤ µ ∥x − x ¯∥ for x, x ¯ ∈ B(R) , ∥hi (x)∥ ≤ ν k −1 ,

∥D hi (x)∥ ≤ ν k −1 ,

∥D hi (x) − D hi (¯ x)∥ ≤ ν k −1 ∥x − x ¯∥ for x, x ¯ ∈ B(R), λi

is continuous ,

∫

i = 1, 2, . . . , k ,

t+1

|λi (t)| ≤ 1, λi (t+1) = λi (t) ,

λi (s) ds = 0 , t

for i = 1, 2, . . . , k .

      

(4.2)

        

(4.3)

By Lemma A.2 there exists Λi : R → R such that d Λi (t) = λi (t), Λi (t + 1) = Λi (t) , dt ∫ t+1 Λi (t) d τ = 0, |Λi (t)| ≤ 12 t

15

      

(4.4)

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for t ∈ R, i = 1, 2, . . . , k. Put ∫ 1 ξi,j = Λi (τ ) λj (τ ) dτ,

i, j = 1, 2, . . . , k .

(4.5)

0

Observe that |ξi,j | ≤

1 2

(4.6)

and that d Λi (t) λj (t) + Λj (t) λi (t) = (Λi (t) Λj (t)) , dt ∫ 1 (Λi (τ ) λj (τ ) + Λj (τ ) λi (τ )) dτ 0

= Λi (1) Λj (1) − Λi (0) Λj (0) = 0 . Hence ξi,j + ξj,i = 0,

i, j = 1, 2, . . . , k .

(4.7)

If p, q : B(R) → R and if D p (x), D q(x) exist for x ∈ B(R) then D p ◦ q(x) is the composition of the maps D p and q, ( ) D p ◦ q(x) = D p (x) q(x) (4.8) n

and [p, q](x) denotes the Lie bracket of the vector fields p, q, i.e. } [ p, q](x) = D p ◦ q(x) − Dq ◦ p (x) = D p (x) q(x) − D q(x) p (x),

x ∈ B(R) .

(4.9)

Let y ∈ B(R), [a, b ] ⊂ R, ε, ν ∈ R+ . Assume that 0 < ε ≤ 1,

(4.10)

that uε : [a, b ] → B(R) is a solution of x˙ = f (x) +

k ∑

hi (x) λi (t/ε),

uε (a) = y

(4.11)

i=1

and wε : [a, b ] → B(R − 14 ε ν) is a solution of ∑ x˙ = f (x) − ε [hi , hj ](x) ξi,j , i>j

wε (a) = y − ε

∑ i

hi (y) Λi (a/ε) .

      

(4.12)

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Elementary methods : internal resonance

Put w eε (t) = wε (t) + ε

∑

hi (wε (t)) Λi (t/ε),

jk

17

t ∈ [a, b ] .

(4.13)

i

Assume that w eε (t) ∈ B(R) for t ∈ [a, b ] . Observe (cf. (4.2), (4.4)) that ∥w eε (t) − wε (t)∥ ≤ ε 14 ν,

(4.14)

t ∈ [a, b ]

(4.15)

which implies that w eε (t) ∈ B(R) for t ∈ [a, b ]. Finally, assume that ε ν2 ≤ 1 .

(4.16)

The goal of this Chapter is to obtain an estimate for ∥uε (t) − w eε (t)∥,

t ∈ [a, b ] .

The crucial step is the inequality ∫ t ∥uε (t) − w eε (t)∥ ≤ C + B ∥uε (s) − w eε (s)∥ ds,

t ∈ [a, b ] ,

(4.17)

a

where C, B are positive functions of µ, ν, ε, t. (4.17) implies that ∫ ( t ) ∥uε (t) − w eε (t)∥ ≤ exp B(s) ds max{C(τ ) ; τ ∈ [a, t]} a

for t ∈ [a, b ] (cf. Lemma A.1). Assumption (4.16) is introduced in order to ∫ t keep B(s) ds bounded. It will also make many formulas simpler. a

4.2. Lemma. Let [σ, s] ⊂ [a, b ]. Then ∥wε (s) − wε (σ)∥ ≤ (µ + ε ν 2 ) (s − σ) . Proof. since

(4.18)

(4.18) is a consequence of (4.12), (4.1), (4.2), (4.6), (4.7), (4.16) ∑

[hi , hj ](x) ξi,j =

i>j

and ε

∑ i,j

∑

D hi ◦ hj (x) ξi,j

i,j

∥D hi ◦ hj (x)∥ |ξi,j | ≤ ε ν 2 ≤ 1 . 

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4.3. Lemma. Let ρ ∈ R+ , [σ, σ + ε] ⊂ [a, b ] and let p : B(R)→Rn fulfil ∥p (x)∥ ≤ ρ, ∥p (x) − p (¯ x)∥ ≤ ρ∥x − x ¯∥

for x, x ¯ ∈ B(R) .

(4.19)

for t ∈ R .

(4.20)

Let Ξ : R → R be continuous and fulfil ∫ |Ξ(t)| ≤ 1, Ξ(t + 1) = Ξ(t),

t+1

Ξ(s) ds = 0 t

Then

∫

σ+ε

p (wε (s)) Ξ(s/ε) ds ≤ ε2 ρ (µ + 1) .

(4.21)

σ

Proof.

∫

σ+ε

p (wε (s)) Ξ(s/ε) ds = J (σ) + K(σ) , σ

where ∫

σ+ε

J (σ) =

[p (wε (s)) − p (wε (σ))] Ξ(s/ε) ds , σ

∫

σ+ε

K(σ) =

p (wε (σ)) Ξ(s/ε) ds . σ

By (4.16), (4.18) ∫

σ+ε

∥J (σ)∥ ≤

ρ (µ + 1) (s − σ) ds ≤ ρ (µ + 1) ε2 σ

and by (4.21) K(σ) = 0, i.e. (4.20) holds.



4.4. Lemma. Let p fulfil (4.19) and let Ξ fulfil (4.20). Then

∫

t

p (wε (s)) Ξ(s/ε) ds ≤ ε ρ [(µ + 1) (b − a) + 1]

a

Proof.

for a < t ≤ b .

  

(4.22)

Let a < t ≤ b and let m ∈ N0 and σℓ ∈ R be defined by

m ε < t − a ≤ (m + 1) ε,

σℓ = a + ℓ ε for ℓ = 0, 1, . . . , m .

(4.23)

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Then ∫

t

p (wε (s)) Ξ(s/ε)ds a

=

m−1 ∑∫ σℓ +ε

∫ t p (wε (s)) Ξ(s/ε)ds+ p (wε (s)) Ξ(s/ε)ds .

ℓ=0 σℓ

By (4.21)

∫

σm

σℓ +ε

p (wε (s)) Ξ(s/ε) ds ≤ ε2 ρ (µ + 1) .

σℓ

Moreover (cf. (4.19), (4.20), (4.23)),

∫

t

p (wε (s) Ξ(s/ε) ds ≤ ε ρ

and m ≤

σm

b−a . ε 

Therefore (4.22) is correct. 4.5. Lemma. Let p : B(R) → Rn be differentiable and fulfil ∥p (x)∥ ≤ ρ, ∥Dp (x)∥ ≤ ρ, ∥Dp (x) − Dp (¯ x)∥ ≤ ρ

} (4.24)

for x, x ¯ ∈ B(R) . Then

∫ t

[p (w eε (s)) − p (wε (s))]

  

a

≤ ε ρ [(b − a + 1) ν + (b − a) µ ν + (b − a) ν ] . 2

Proof.

2

 

Let a < t ≤ b. Then ∫

t

[p (w eε (s) − p (wε (s)))] ds = L + N a

where ∫

t

L=

[p (w eε (s) − p (wε (s)) − Dp (wε (s))(w eε (s) − wε (s))] ds , a

∫ N =

t

Dp (wε (s)) (w eε (s) − wε (s)) ds . a

(4.25)

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By (4.14), (4.15) p (w eε (s)) − p (wε (s)) ∫ 1 ( ) ) = Dp wε (s) + λ (w eε (s) − wε (s)) (w eε (s) − wε (s) dλ 0

and p (w eε (s)) − p (wε (s)) − Dp (wε (s))) (w eε (s) − wε (s)) ∫ 1 [ ( ] Dp wε (s)+λ[w eε (s)−wε (s)])−Dp (wε (s) (w eε (s)−wε (s)) dλ . = 0

Hence ∥p (w eε (s)) − p (wε (s)) − Dp (wε (s)) (w eε (s) − wε (s))∥ ≤ ρ (ε ν)2 since (cf. (4.15)) ∥w eε (s) − wε (s)∥ ≤ εν and ∥L∥ ≤ ρ (b − a)(ε ν)2 .

(4.26)

By (4.13) N =

∫ k ∑ ε i=1

t

D p (wε (s)) hi (wε (s)) Λi (s/ε) ds .

(4.27)

a

The integrals in (4.27) can be estimated similarly as in Lemma 4.4 since (cf. (4.2), (4.4)) ∥D p (x) hi (x)∥ ≤ ρ ν k −1 , ∥D p (x)hi (x) − D p (¯ x) hi (¯ x)∥ ≤ 2 ρ ν k −1 ∥x − x ¯∥

for x, x ¯ ∈ B(R) ,

and |Λi (t)| ≤

1 2

for t ∈ R .

Hence (cf. (4.4)) ∥N ∥ ≤ k ε · ε 2 ρ ν k −1 [(µ + 1) (b − a) + 1] 12 ≤ ε2 ρ [(µ + 1) (b − a) + 1] ν . (4.25) is a consequence of (4.26) and (4.28).

} (4.28) 

4.6. Theorem. There are functions βj : (R+ )2 → R+ , j = 1, 2, 3 such that } ∥uε (t) − w eε (t)∥ (4.29) ≤ ε2 [β1 (b − a, µ) ν + β2 (b − a, µ) ν 2 + β3 (b − a, µ) ν 3 ]

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for t ∈ [a, b ]. Proof.

By (4.11) ∫

uε (t) = y + ∫

f (uε (s)) ds + a

k ∫ ∑ i=1

t

hi (uε (s)) λi (s/ε) ds , a

t

hi (uε (s)) λi (s/ε) ds = ε hi (uε (t))Λi (t/ε) − ε hi (y)Λi (a/ε) ∫ t − ε D hi (uε (s))u˙ ε (s)Λi (s/ε)ds ,

a

∫

t

a

∫

t

t

Dhi (uε (s)) u˙ ε (s) Λi (s/ε) ds = Dhi (uε (s)) f (uε (s)) Λi (s/ε) ds a ∫ ∑ t + D hi (uε (s)) hj (uε (s)) Λi (s/ε) λj (s/ε) ds .

a

a

j

Hence

∑ ∑  hi (uε (t))Λi (t/ε) + y − ε hi (y) Λi (a/ε)  uε (t) = ε    i  ∫ t i ∫  t  ∑  + f (uε (s))ds − ε D hi ◦ f (uε (s)) Λi (s/ε) ds a a  i   ∑∫ t    −ε D hi ◦ hj (uε (s)) Λi (s/ε) λj (s/ε) ds .  

(4.30)

a

i,j

By (4.12) wε (t) = y − ε

∑

∫ hi (y) Λi (a/ε) +

i,j

f (wε (s)) ds a

∑∫ i

−ε

t

t

D hi ◦ hj (wε (s)) ξi,j ds a

since (cf. (4.9), (4.7)) ∑ ∑ [hi , hj ](x) ξi,j = D hi ◦ hj (x) ξi,j , i>j

        

x ∈ B(R).

i,j

Subtracting (4.31) from (4.30) we obtain uε (t) − wε (t) ∫ ∑ =ε hi (uε (t)) Λi (t/ε) + i

t a

[ ] f (uε (s)) − f (wε (s)) ds

(4.31)

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−ε −ε +ε

∑∫

t

∑∫

t

i

a

i,j

a

∑∫ i,j

D hi ◦ f (uε (s)) Λi (s/ε) ds D hi ◦ hj (uε (s)) Λi (s/ε) λj (s/ε) ds t

D hi ◦ hj (wε (s)) ξi,j ds ,

a

which can be transformed (cf. also (4.13)) to ∫ t [ ] uε (t) − w eε (t) = f (uε (s)) − f (w eε (s)) ds a

∫

[

t

+

] f (w eε (s)) − f (wε (s)) ds

a

−ε

∑ ∫ t[ ] D hi ◦ f (uε (s)) − D hi ◦ f (w eε (s)) Λi (s/ε) ds a

i

−ε

∑∫ ∫

t[

] D hi ◦ f (w eε (s)) − D hi ◦ f (wε (s)) Λi (s/ε) ds

a

i

                                       

   −ε D hi ◦ f (wε (s)) Λi (s/ε) ds    a     ∫  ∑ t[  ]  −ε D hi ◦ hj (uε (s)) − D hi ◦ hj (w eε (s)) Λi (s/ε) λj (s/ε) ds      i,j a    ∫   ∑ t[ ]   −ε D hi ◦ hj (w eε (s)) − D hi ◦ hj (wε (s)) Λi (s/ε) λj (s/ε) ds      i,j a   ∫  t  ∑ [ ]    −ε D hi ◦ hj (wε (s)) Λi (s/ε) λj (s/ε) − ξi,j ds .   a t

(4.32)

i,j

Now the terms on the right-hand side of (4.32) will be estimated. By (4.1) ∫ t

∫ t [ ]

f (uε (s)) − f (w eε )) ds ≤ µ ∥uε (s) − w eε (s)∥ ds . (4.33)

a

0

By Lemma 4.5 with p = f, ρ = µ

∫ t[ ]

f (w eε (s) − f (wε (s))) ds

a

[ ] ≤ ε2 ((b − a + 1) µ + (b − a) µ2 ) ν + (b − a) µ ν 2 .

    

(4.34)

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Elementary methods : internal resonance

Now, (4.1), (4.2), (4.4) and (4.10) imply that

∑∫ t[ ]

D hi ◦ f (uε (s)) − D hi ◦ f (w eε (s)) Λi (s/ε) ds

ε a

i

∫

t

≤ ε2µν

∥uε (s) − w eε (s)∥ 21 ds

0

∫

                  

t

≤µ

23

∥uε (s) − w eε (s)∥ ds

(4.35)

0

since ε ν ≤ 1 by (4.16) and (4.10). Similarly (cf. (4.15), (4.16))

∑∫ t[

]

Dhi ◦ f (w eε (s)) − Dhi ◦ f (wε (s)) Λi (s/ε) ds

ε a

i

∫

t

≤ ε2µν a

∥w eε (s) − wε (s)∥ 12 ds

              

(4.36)

≤ ε (b − a) µ ν 2 . ∑ Lemma 4.4 with p = i D hi ◦ f, ρ = 2 µ ν, Ξ = Λi gives (cf. (4.4)) 

∑∫ t



 D hi ◦ f (wε (s)) Λi (s/ε) ds 

ε   a  i  1 2 (4.37) ≤ ε 2 µ ν [(µ + 1) (b − a) + 1] 2        2 2 ≤ ε [(b − a + 1) µ ν + (b − a) µ ν] . 2

Further, (cf. (4.1), (4.2), (4.3), (4.4), (4.10), (4.16))

∑∫ t[

 ]

 D hi ◦ hj (uε (s)) − D hi ◦ hj (w eε (s)) Λi (s/ε) λj (s/ε) ds 

ε   i,j

a

∫

∫

t

≤ εν 2

∥uε (s) − w eε (s)∥ ds ≤

2

a

t

∥uε (s) − w eε (s)∥ ds .

   

(4.38)

a

Similarly

∫ t∑[

 ]

 D hi ◦ hj ( w eε (t)) − D hi ◦ hj (wε (t)) Λi (s/ε) λj (s/ε) ds 

ε  a i,j

  

εν ≤ ε2 (b − a) ν 3 . ≤ ε (b − a) 2 ν 2 2

Finally, by Lemma 4.4 with p = D hi ◦ hj ,

ρ = 2 ν 2 k −2 ,

Ξ = Λi λj − ξi,j ,

(4.39)

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we get

∑∫

ε i,j

t

[ ]

D hi ◦ hj (wε (s)) Λi (s/ε) λj (s/ε) − ξi,j ds

a

≤ ε2 2 ν 2 [(µ + 1) (b − a) + 1] 12

              

≤ ε2 [(b − a + 1) + (b − a) µ] ν 2

(4.40)

since |ξi,j | ≤ 12 , ∫ t+1 |Ξ(t)| ≤ 1, Ξ(t + 1) = Ξ(t), Ξ(s) ds = 0

|λj (t)| ≤ 1,

|Λi (t)| ≤ 12 ,

for t ∈ R

t

and by Lemma A.2 there exists a primitive Θ of Ξ such that Θ(t + 1) = Θ(t) and |Θ(t)| ≤ 14 for t ∈ R. Relations (4.32)–(4.40) imply that ∫ ∥uε (t) − w eε (t)∥ ≤ B

t

∥uε (s) − w eε (s)∥ ds + C ,

(4.41)

a

where B = 2 µ + 1 (cf. (4.33), (4.35), (4.38)). Moreover, C = ε2 [γ1 (b − a, µ) ν + γ2 (b − a, µ) ν 2 + γ3 (b − a, µ)ν 3 ] , where (cf. (4.34), (4.37)) γ1 (b − a, µ) = 2 (b − a + 1) µ + (b − a)}, µ2 and (cf. (4.34), (4.36), (4.40)) γ2 (b − a, µ) = b − a + 1 + 4 (b − a) µ and (cf. (4.39)) γ3 = b − a . By Lemma A.2, (4.29) is true with ( ) βj (b − a, µ) = γj (b − a, µ) exp (2 µ + 1) (b − a) ,

j = 1, 2, 3 .



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25

4.7. Remark. The only bound on ν in Theorem 4.6 is ε ν 2 ≤ 1 (cf. (4.16)). ¯ i : B(R) → Rn be differentiable and Let 0 ≤ α ≤ 21 , ν¯ ∈ R+ , y ∈ B(R), let h fulfil ¯ ¯ ¯ ¯ x)∥ ≤ ν¯ ∥x − x ∥h(x)∥ ≤ ν¯, ∥D h(x)∥ ≤ ν¯, ∥D h(x) − D h(¯ ¯∥ , for x, x ¯ ∈ B(R). Theorem 4.6 can be applied to the equation ∑ ¯ x˙ = f (x) + ε−α h(x) λi (t/ε) .

(4.42)

i

Let uε : [a, b ] → B(R) be a solution of (4.42), uε (a) = y, let wε : [a, b ] → B(R) be a solution of ∑ ¯ i, h ¯ j ](x) , x˙ = f (x) + ε1−2α [h wε (a) = y − ε

1−α

∑

i>j

hi (y) Λi (a/ε) ,

i

and w eε (t) = wε (t) + ε1−α

∑

hi (wε (t)) Λi (t/ε),

t ∈ [a, b ] .

i

Theorem 4.6 implies that ∥uε (t) − w eε (t)∥ ≤ ε2 [β1 (b − a, µ) ε−α ν¯ + β2 (b − a, µ) ε−2 α ν¯2 + β3 (b − a, µ) ε−3 α ν¯3 ]

for t ∈ [a, b ]

and uε (t) − w eε (t) → 0 uniformly on [a, b ] if α < 23 . 4.8 . Remark. Consider equation (4.11). Let uε , wε , w eε have the same meaning as in Notation 4.1. Assume in addition to (4.1)–(4.3) that a = 0,

Λi (a) = 0,

ξi,j = 0 for i, j = 1, 2, . . . , k .

(4.43)

Then wε is a solution of x˙ = f (x),

wε (a) = y.

Hence it is independent of ε and we may write w instead of wε , ∑ w eε (t) = w(t) + ε hi (w(t)) Λi (t/ε) i

(4.44)

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and by Theorem 4.6 (cf. (4.1), (4.2), (4.16)) ∑ ∥uε (t) − w(t) − ε hi (w(t)) Λi (t/ε)∥

 

i

≤ ε2 [β1 (b − a, µ) ν + β2 (b − a, µ) ν 2 + β3 (b − a, µ) ν 3 ]



(4.45)

for ε ≤ ν −2 . Equation (2.2) which describes the motion of Kapitza’s pendulum can be written in the form x˙ = f (x) + h1 (x) cos(4 π t/ε) + h2 (x) cos(2 π t/ε) where x=

( ) Θ Φ

( ,

f (x) =

,

)

0

L−1 A (2 π/ε) sin Θ

−L−1 A (2 π/ε) Φ cos Θ

,

) ,

λ1 (t) = cos 4π t,

Λ1 (t) =

sin 4 π t , 4π

λ2 (t) = cos 2π t,

Λ2 (t) =

sin 2 π t , 2π

ξi,j = 0

ω = 2 π/ε ,

− 21 (L−1 A 2 π/ε)2 sin Θ cos Θ (

h2 (x) =

)

g L−1 sin Θ

( h1 (x) =

Φ

(4.46)

for i, j = 1, 2 .

There exist R, µ, ν ∈ R+ such that (4.1) and (4.2) hold. The estimate (4.45) is valid in the case of equation (4.46).

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Chapter 5

Strong Riemann integration of functions of a pair of coupled variables

5.1. Notation. X is a Banach space, ∥x∥ Dom U ⊂ R2 , U : Dom U → X.

being the norm of

x ∈ X,

Given σ ∈ R we denote by U (σ, ·) the X-valued function which is defined for all τ ∈ R such that (σ, τ ) ∈ Dom U with the value U (σ, τ ) for such τ. Similarly for U (·, t) for a given t ∈ R. If a, b ∈ R, a ≤ b, we put [a, b ] = {t ∈ R ; a ≤ t ≤ b}. If v : [a, b ] → X, then ∫

b

(R)

v(t) dt a

denotes the classical Riemann integral. 5.2. Definition. U is SR-integrable (Strongly Riemann integrable) on [a, b ] and u is an SR-primitive of U on [a, b ] if there exists ξ0 ∈ R+ such that (τ, t) ∈ Dom U for τ, t ∈ [a, b ],

τ − ξ0 ≤ t ≤ τ + ξ0 ,

for every ε > 0 there exists ξ > 0 k ∑

such that

∥u(ti ) − u(ti−1 ) − U (τi , ti ) + U (τi , ti−1 )∥ ≤ ε

i=1

for every set A = (t0 , τ1 , t1 , τ2 , t2 , . . ., τk , tk ) fulfilling a = t0 ≤ τ1 ≤ t1 ≤ τ2 ≤ . . . ≤ τk ≤ tk = b , ti − ti−1 ≤ ξ,

i = 1, 2, . . . , k .

(5.1)                       

(5.2)

Briefly, U is called SR-integrable and u is called a primitive of U, (τ, t) is called a pair of coupled variables. 27

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5.3. Lemma. Assume that U is SR-integrable on [a, b ], u is its primitive, ε > 0. Let ξ > 0 correspond to ε by Definition 5.2, [S, T ] ⊂ [a, b ]. Then m ∑ ∥u(si ) − u(si−1 ) − U (σi , si ) + U (σi , si−1 )∥ ≤ ε (5.3) i=1

for every sequence A = (s0 , σ1 , s1 , . . . , σm , sm ) fulfilling S = s0 ≤ σ1 ≤ s1 ≤ · · · ≤ σm ≤ sm = T , si − si−1 ≤ ξ Proof.

}

for i = 1, 2, . . . , m .

(5.4)

Let the sequence A fulfil (5.4). There exists a sequence B = (r0 , ρ1 , r1 , . . . , ρn , rn )

such that a = r0 ≤ ρ1 ≤ r1 ≤ · · · ≤ ρn ≤ rn = S , ri − ri−1 ≤ ξ for i = 1, 2, . . . , r and a sequence C = (ℓ0 , λ1 , ℓ1 , . . . , λp , ℓp ) such that T ≤ ℓ0 ≤ λ1 ≤ ℓ − 1 ≤ . . . λp ≤ ℓp = b , ℓi − ℓi−1 ≤ ξ

for i = 1, 2, . . . , p .

Put ti = ri

for i = 0, 1, . . . , n ,

τi = ρi

for i = 1, 2, . . . , n ,

ti = si−n ,

τi = σi−n

ti = ℓi−n−m ,

τi = λi−n−m

for i = n + 1, n + 2, . . . , n + m , for i = n + m + 1, n + m + 2, . . . , k ,

where k = n + m + p. Then a = t0 ≤ τ1 ≤ t1 ≤ τ2 ≤ · · · ≤ τk ≤ tk = b , ti − ti−1 ≤ ξ, i = 1, 2, . . . , k . Therefore by (5.2) m ∑

∥u(sj ) − µ(sj−1 ) − U (σj , sj ) + U (σj , sj−1 )∥

j=1

≤

k ∑

∥u(ti ) − u(ti−1 ) − U (τi , ti ) + U (τi , ti−1 )∥ ≤ ε

i=1

and (5.3) is correct.



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SR-integration of functions of a pair of coupled variables

29

5.4 . Corollary. Let U be SR-integrable on [a, b ], u being its primitive, [S, T ] ⊂ [a, b ]. Then U is SR-integrable on [S, T ] and u is its primitive. 5.5. Lemma. Let U be SR-integrable and let u be a primitive of U on [a, b ]. (i) If y ∈ X and v(t) = u(t) + y for t ∈ [a, b ] then v is a primitive of U on [a, b ], (ii) if v is another primitive of U on [a, b ] then u(T ) − v(T ) = u(a) − v(a)

for T ∈ [a, b ] .

Proof. (i) is a direct consequence of Definition 5.2. In order to prove (ii) assume that v is a primitive of U on [a, b ] and that a < T ≤ b. By Corollary 5.4 U is SR-integrable on [a, T ], u and v being its primitives. Therefore there exists a sequence A = (t0 , τ1 , t1 , . . . , τk , tk ) such that a = t0 ≤ τ1 ≤ t1 ≤ · · · ≤ τk ≤ tk = T , k ∑ ∥u(ti ) − u(ti−1 ) − U (τi , ti ) + U (τi , ti−1 )∥ ≤ ε , i=1 k ∑

∥v(ti ) − v(ti−1 ) − U (τi , ti ) + U (τi , ti+1 )∥ ≤ ε .

i=1

Hence k ∑

∥u(ti ) − u(ti−1 ) − v(ti ) + v(ti−1 )∥ ≤ 2 ε .

(5.5)

i=1

Moreover, u(T ) − u(a) − v(T ) + v(a) =

k ∑ (

) u(ti ) − u(ti−1 ) − v(ti ) + v(ti−1 ) ,

i=1

which together with (5.5) implies that ∥u(T ) − v(T ) − u(a) + v(a)∥ ≤ 2 ε and (ii) is true since ε > 0 is arbitrary.



5.6. Definition. Let U be SR-integrable and let u be its primitive on [a, b ]. For a ≤ S ≤ T ≤ b put ∫ T (SR) Dt U (τ, t) = u(T ) − u(S) . (5.6) S

The left-hand side of (5.6) is called the SR-integral of U over [S, T ].

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Furthermore, ∫ S ∫ (SR) Dt U (τ, t) = −(SR) T

T

Dt U (τ, t),

for S < T

(5.7)

S

and

∫

S

(SR)

Dt U (τ, t) = 0 .

(5.8)

S

Observe that the left-hand side of (5.6) is uniquely defined since the right hand side of (5.6) does not depend on the choice of the primitive by Lemma 5.5. 5.7. Remark. Let R, S, T ∈ [a, b ]. The preceding definition implies that ∫ T ∫ S ∫ T (SR) Dt U (τ, t) = (SR) Dt U (τ, t) + (SR) Dt U (τ, t) R

R

S

if U is SR-integrable on [a, b ] and u is its primitive. 5.8. Lemma. Let u : [a, b ] → X, a < c < b. Assume that U

is continuous at (c, c),

(5.9)

U

is SR −integrable on [a, c ] and u is its primitive,

(5.10)

U

is SR −integrable on [c, b ] and u is its primitive.

(5.11)

Then U is SR-integrable on [a, b ] and u is its primitive. Proof. Let ε > 0. (5.7)–(5.9) imply that there exists ξ ∈ R such that 0 < ξ ≤ 12 (b − a), } ∥U (τ, t) − U (c, c)∥ ≤ ε (5.12) if |τ − c | ≤ ξ, |t − c | ≤ ξ, (τ, t) ∈ Dom U, k ∑

∥u(ti ) − u(ti−1 ) − U (τi , ti ) + U (τi , ti−1 )∥ ≤ ε

(5.13)

i=1

for any sequence (t0 , τ1 , t1 , . . . , tk ) fulfilling

and

a = t0 ≤ τ1 ≤ t1 ≤ . . . ≤ τk ≤ tk = c, |ti − ti−1 | ≤ ξ

 

for i = 1, 2, . . . , k



(5.14)

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SR-integration of functions of a pair of coupled variables ℓ ∑

31

∥u(si ) − u(si−1 ) − U (σi , si ) + U (σi , si−1 )∥ ≤ ε

(5.15)

i=1

for any sequence (s0 , σ1 , s1 , . . . , σℓ , sℓ ) fulfilling c = s0 ≤ σ1 ≤ s1 ≤ . . . ≤ σℓ ≤ sℓ = b, |si − si−1 | ≤ ξ

 

for i = 1, 2, . . . , ℓ . 

(5.16)

Let the set {r0 , ρ1 , r1 , . . . , rm , ρm } fulfil a = r0 ≤ ρ1 ≤ r1 ≤ . . . ≤ ρm ≤ rm = b, |ri − ri−1 | ≤ ξ

 

for i = 1, 2, . . . , m . 

(5.17)

there is

p such that 2 ≤ p ≤ m−2 and rp = c, rp+1 = c

(5.18)

there is

p≥1

(5.19)

Then m > 2 and either

or such that rp−1 < c < rp .

If (5.18) holds then (cf. (5.11), (5.12), (5.14), (5.16)) m ∑

∥u(ri ) − u(ri−1 ) − U (ρi , ri ) + U (ρi , ri−1 )∥

i=1

=

p ∑

∥u(ri ) − u(ri−1 ) − U (ρi , ri ) + U (ρi , ri−1 )∥

              

      + ∥u(rj ) − u(rj−1 ) − U (ρj , rj ) + U (ρj , rj−1 )∥      j=p+1    ≤ 2ε. i=1

(5.20)

m ∑

Let (5.19) take place. Then m ∑ i=1

∥u(ri ) − u(ri−1 ) − U (ρi , ri ) + U (ρi , ri−1 )∥ = K1 + K2 + K3

(5.21)

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where

K1 =

p−1 ∑

∥u(ri ) − u(ri−1 ) − U (ρi , ri ) + U (ρi , ri−1 )∥

i=1

+ ∥u(c) − u(rp−1 ) − U (c, c) + u(c, rp−1 )∥ + ∥u(rp ) − u(c) − U (c, rp ) + U (c, c)∥ +

m ∑

∥u(ri ) − u(ri−1 ) − U (ρi , ri ) + U (ρi , ri−1 )∥,

i=p+1

K2 = ∥u(rp ) − u(rp−1 ) − U (ρp , rp ) + u(ρp , rp−1 )∥ − ∥u(rp ) − u(rp−1 ) − U (c, rp ) + U (c, rp−1 )∥ , K3 = ∥u(rp ) − u(rp−1 ) − U (c, rp ) + U (c, rp−1 )∥ − ∥u(c) − u(rp−1 ) − U (c, c) + U (c, rp−1 )∥ − ∥u(rp ) − u(c) − U (c, rp ) + U (c, c)∥ . By Definition 5.2 K1 ≤ ε

(5.22)

since the sequence {t0 , τ1 , t1 , . . . , τp−1 , tp−1 , c, c, c, tp , τp+1 , . . . , τm , tm } fulfils (5.15). Further, ∥u(rp ) − u(rp−1 ) − U (ρp , rp ) + U (ρp , rp−1 )∥ ≤ ∥u(rp ) − u(rp−1 ) − U (c, rp ) + U (c, rp−1 )∥ + ∥U (ρp , rp ) − U (ρp , cp )∥ + ∥U (ρp , rp−1 ) − U (c, rp−1 )∥ and by (5.7) K2 ≤ 2 ε .

(5.23)

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33

Finally, ∥u(rp ) − u(rp−1 ) − U (c, rp ) + U (c, rp−1 )∥ ≤ ∥u(rp ) − u(c) − U (c, rp ) + U (c, c)∥ + ∥u(c) − u(rp−1 ) − U (c, c) + U (c, rp−1 )∥ . Hence K3 ≤ 0 .

(5.24)

Putting (5.19)–(5.22) together we obtain that m ∑

∥u(ri ) − u(ri−1 ) − U (ρi , ri ) + U (ρi , ri−1 )∥ ≤ 3 ε .

(5.25)

i=1

(5.18) and (5.23) imply that U is SR-integrable on [a, b ] and u is its primitive.  5.9. Lemma. Let U be SR-integrable on [a, b ], u being its primitive. Let σ ∈ [a, b ]. Assume that U (σ, ·) : [σ − ξ0 , σ + ξ0 ] ∩ [a, b ] → X is continuous at σ. Then u is continuous at σ. Proof. Let ε > 0 and let ξ > 0 correspond to ε by Definition 5.2. Let a ≤ s¯ ≤ σ ≤ s ≤ c, s − s¯ ≤ ξ. There exists A = {t0 , τ1 , t1 , . . . , τk , tk } such that a = t0 ≤ τ1 ≤ t1 ≤ τ2 ≤ · · · ≤ τk ≤ tk = b, ti − ti−1 ≤ ξ

for i = 1, 2, . . . , k

is fulfilled and (¯ s, σ, s) = (ti−1 , τi , ti ) for some i = 1, 2, . . . , k. Then ∥u(s) − u(¯ s) − U (σ, s) + U (σ, s¯)∥ ≤ ε by (5.2).



5.10 . Remark. Let w : [a, b ] → X, U (τ, t) = w(τ ) t for τ, t ∈ [a, b ]. If U is ∫ b SR-integrable and if u is its primitive then (R) w(t) dt = u(b) − u(a), a ∫ b (R) w(t) dt denoting the Riemann integral of w over [a, b ]. This is a cona

sequence of Definition 5.2.

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Generalized Ordinary Differential Equations

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∫ b On the other hand, if (R) w(t) dt exists, y ∈ X and if a

∫

t

u(t) = (R)

w(s) ds + y

for t ∈ [a, b ]

a

then for every ε > 0 there exists ξ > 0 such that k

∑

(u(ti ) − u(ti−1 )) − U (τ, ti ) + U (τ, ti−1 ) ≤ ε

(

i=1

) if A = t0 , τ1 , t1 , τ2 , t2 , . . ., τk , tk is a set fulfilling a = t0 ≤ τ1 ≤ t1 ≤ τ2 ≤ · · · ≤ τk ≤ tk = b and ti − ti−1 ≤ ξ,

i = 1, 2, . . . , k ,

but U need not be SR-integrable. ∫ b If dim X < ∞ and if (R) w(t) dt exists then U is SR-integrable and a

u is its primitive. This is proved in Chapter 18 for the SKH-integration and it can be proved for the SR-integration in the same way (see also [Schwabik, Ye (2005)], Corollary 3.4.3 and Remark). 5.11 . Remark. If v : [a, b ] → X is continuous then the Riemann integral ∫ b (R) v(s) ds exists. Furthermore, observe that a

∫

(R)

t t¯

v(s) ds − v(τ ) (t − t¯) ≤ ε (t − t¯)

if a ≤ t¯ ≤ τ ≤ t ≤ b and if ∥v(s) − v(τ )∥ ≤ ε for s ∈ [t¯, t]. 5.12. Lemma. Let v : [a, b ]→X be continuous, U (τ, t) = v(τ ) t, ∫ t w(t) = (R) v(s) ds for t ∈ [a, b ] . a

Then U is SR -integrable, w is its primitive , dw (t) = v(t) for t ∈ [a, b ] . dt

(5.26) (5.27)

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Proof.

jk

35

For every ε > 0 there exists ξ > 0 such that ∥w(T ) − w(S) − v(τ ) (T − S)∥ ≤ ε (T − S)

(5.28)

if a ≤ S ≤ τ ≤ T ≤ b, T − S ≤ ξ since ∫

T

w(T ) − w(S) − v(τ ) (T − S) = (R)

[v(t) − v(τ )] dt . S

Both (5.26) and (5.27) are consequences of (5.28).



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Chapter 6

Generalized ordinary differential equations: Strong Riemann-solutions (concepts) 6.1. Notation. B(r) = {x ∈ X ; ∥x∥ ≤ r} for r ≥ 0,

Dom G ⊂ X × R2 ,

G : Dom G → X, [a, b ] ⊂ R, u : [a, b ] → X, ξ0 > 0, (u(τ ), τ, t) ∈ Dom G for τ, t ∈ [a, b ],

τ − ξ0 ≤ t ≤ τ + ξ0 .

6.2. Definition. u is an SR-solution of the generalized differential equation (GODE) d x = Dt G(x, τ, t) dt

(6.1)

on [a, b ] if ∫

T

u(T ) − u(S) = (SR)

Dt G(u(τ ), τ, t)

for [S, T ] ⊂ [a, b ] .

(6.2)

S

Briefly, u is an SR-solution of (6.1). 6.3. Lemma. Let u : [a, b ] → X, P ∈ [a, b ]. Then u

is an SR −solution of (6.1) on [a, b ]

(6.3)

if and only if ∫

T

u(T ) = u(P ) + (SR)

Dt G(u(τ ), τ, t) P

37

for T ∈ [a, b ] .

(6.4)

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Proof.

Let (6.3) hold. Then ∫

T

Dt G(u(τ ), τ, T )

u(T ) = u(a) + (SR) ∫

for T ∈ [a, b ] ,

a P

u(P ) = u(a) + (SR)

Dt G(u(τ ), τ, T ) . a

Hence (cf. Remark 5.7) ∫

T

u(T ) = u(P ) + (SR)

Dt G(u(τ ), τ, t)

if P ≤ T ≤ b ,

P ∫ P

u(T ) = u(P ) − (SR)

Dt G(u(τ ), τ, t) if a ≤ T ≤ P , T

and (6.4) holds by (5.7). Let (6.4) hold. Then ∫

S

u(S) = u(P ) + (SR)

Dt G(u(τ ), τ, t)

for S ∈ [a, b ] .

P

By (5.7) and Remark 5.7 ∫

T

u(T ) − u(S) = (SR)

∫ S Dt G(u(τ ), τ, t) − (SR) Dt G(u(τ ), τ, t)

P

∫

P

∫

T

= (SR)

P

Dt G(u(τ ), τ, t) + (SR) P

∫

Dt G(u(τ ), τ, t) S

T

= (SR)

Dt G(u(τ ), τ, t) S

and (6.3) is correct.



6.4. Lemma. The two assertions below are equivalent: u and

is an SR-solution of (6.1),

(6.5)

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Generalized ordinary differential equations: SR-solutions (concepts)

for every ε > 0 there exists ξ > 0 such that ∑

∥u(ti ) − u(ti−1 ) − G(u(τ ), τ, ti ) + G(u(τ ), τ, ti−1 )∥ ≤ ε

A

                            

(6.6)

Definition 5.2 implies that (6.2) and (6.6) are equivalent.



for every A = (t0 , τ1 , t1 , τ2 , t2 , . . ., τk , tk ) such that a = t0 ≤ τ1 ≤ t1 ≤ τ2 ≤ . . . ≤ τk ≤ tk = b , ti − ti−1 ≤ ξ Proof.

39

for i = 1, 2, . . . , k .

6.5 . Lemma. Let u be an SR-solution of (6.2) on [a, b ]. Assume that σ ∈ [a, b ] and that G(u(σ), σ, ·) is continuous at σ. Then u is continuous at σ. Proof.

This is a consequence of Definition 6.2 and Lemma 5.9.



6.6. Remark. Let u be an SR-solution of (6.1) on [a, b ]. By Definition 6.2 u is an SR-solution of (6.1) on any [c, d ] ⊂ [a, b ]. On the other hand, let a < c < b and let u be an SR-solution of (6.2) on [a, c] and on [c, b ]. Assume that G is continuous at (u(c), c, c). Then u is an SR-solution of (6.2) on [a, b ] by Lemma 5.8 since (i) u is continuous at c by Lemma 6.5 , (ii) if U (τ, t) = G(u(τ ), τ, t) then U is continuous at (c, c). 6.7 . Lemma. Let u be an SR-solution of (6.1) on [a, b ], [S, T ] ⊂ [a, b ]. Then u is an SR-solution of (6.1) on [S, T ]. Proof.

Lemma 6.7 is a consequence of Corollary 5.4.



6.8 . Definition. Let Dom g ⊂ B(r) × R2 , g : Dom g → X. A function u is a classical solution of x˙ = g(x, t, t)

(6.7)

on [a, b ] if (u(τ ), τ, τ ) ∈ Dom g

for τ ∈ [a, b ]

and if du (t) = g(u(t), t, t) for t ∈ [a, b ] . dt

(6.8)

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6.9. Lemma. Let [a, b ] ⊂ R, r> 0, g: B(r)× [a, b ]2 → X and u: [a, b ] → B(r). Let g and u be continuous. Put Dom G = B(r) × [a, b ] 2 , and

∫ G(x, τ, t) = (R)

t

g(x, τ, s) ds

f or (x, τ, t) ∈ Dom G .

(6.9)

a

Then

∫

∫

T

(SR)

T

Dt G(u(τ ), τ, t) = (R) S

g(u(s), s, s) ds .

(6.10)

S

Proof. Since g and u are continuous the integrals in (6.9), (6.10) exist. Let ε > 0. There exists ξ > 0 such that ∥g(u(τ ), τ, s) − g(u(τ ), τ, s¯)∥ ≤ ε,

(6.11)

∥g(u(s), s, s) − g(u(¯ s), s¯, s¯)∥ ≤ ε

(6.12)

for τ, s, s¯ ∈ [a, b ], |s − s¯| ≤ ξ. Moreover, (6.11) implies (cf. Remark 5.11) that ∫

(R)

t t¯

g(u(τ ), τ, s)ds − g(u(τ ), τ, τ )(t − t¯) ≤ ε (t − t¯) if a ≤ t¯≤ τ ≤ t ≤ b and t − t¯≤ ξ .

Hence

G(u(τ ), τ, t) − G(u(τ ), τ, t¯) − g(u(τ ), τ, τ ) (t − t¯) ∫ t



= (R) g(u(τ ), τ, s) ds − g(u(τ ), τ, τ ) (t − t¯) ≤ ε (t − t¯) t¯

if a ≤ t¯≤ τ ≤ t ≤ b and t − t¯≤ ξ .

          

(6.13)

Further, by (6.12) ∫

(R)

t¯

t

g(u(s), s, s)ds − g(u(τ ), τ, τ ) (t − t¯) ≤ ε (t − t¯) if a ≤ t¯≤ τ ≤ t ≤ b and t − t¯≤ ξ .

    

(6.14)

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41

Relations (6.13), (6.14) imply that ∫

G(u(τ ), τ, t) − G(u(τ ), τ, t¯) − (R)

t t¯

g(u(s), s, s) ds ≤ 2 ε (t − t¯) if t − t¯≤ ξ

and finally (cf. Lemma 6.4), G(u(τ ), τ, t) is SR-integrable on [a, b ], ∫ t (R) g(u(s), s, s) ds is its SR-primitive and (6.10) holds.  a

6.10. Theorem. Assume that [a, b ] ⊂ R, r > 0, g : B(r) × [a, b ] 2 → X and u : [a, b ] → B(r). Let g and u be continuous and let G be defined by (6.9). Then u

is a classical solution of (6.7)

(6.15)

if and only if u is an SR-solution of (6.1) . Proof.

If u is an SR-solution of (6.1) then ∫ T u(T ) − u(S) = (R) g(u(s), s, s) ds for [S, T ] ⊂ [a, b ]

(6.16)

S

by Lemma 6.9 and u is a classical solution of (6.7) since g and u are continuous (cf. Lemma 5.12). If u is a classical solution of (6.7) then (6.16) holds again (cf. Remark 5.11) and u is an SR-solution of (6.1) by Lemma 6.9. 

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Chapter 7

Functions ψ1, ψ2

+ 7.1. Notation. Let ψ1 , ψ2 : R+ 0 → R0 . Assume that

ψ1 , ψ 2

are nondecreasing, continuous,

ψ1 (σ) ≤ ψ2 (σ) for σ ∈ R+ 0 , ∞ ∑ 2 i ψ1 (2−i ) ψ2 (2−i ) < ∞ , i=1 ∞ ∑

(7.1) (7.2) (7.3)

ψ2 (2−i ) < ∞ .

(7.4)

i=1

Put Ψ(σ) =

∞ ∑

2 i ψ1 (σ2−i ) ψ2 (σ2−i ) for σ ∈ R+ 0 ,

(7.5)

i=1

A (σ) = Ψ(σ) exp

∞ ∑

ψ2 (σ2−j ) for σ ∈ R+ 0 ,

(7.6)

j=1

φ(σ) = 3 ψ1 (σ) ψ2 (σ) + A (σ) ψ1 (σ) for σ ∈ R+ 0 , ∞ ∑ Φ(σ) = 2i−1 φ(σ2−i ) for σ ∈ R+ 0 , i=1

B (σ) = Φ(σ) exp

∞ (∑

2 ψ2 (σ2−j ) + σ

)

for σ ∈ R+ 0 .

(7.7) (7.8) (7.9)

j=1

Observe that Ψ, A, φ are well defined (cf. (7.1)–(7.4)) but it remains to be proved that Φ, B are well defined as well.

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7.2. Remark. Let α > 12 , 0 < β ≤ α, α + β > 1, ψ1 (σ) = σ α , ψ2 (σ) = σ β for 0 ≤ σ ≤ 1, ψ1 (σ) = ψ2 (σ) = σ for σ ≥ 1. Then (7.1)–(7.4) hold and 2−α−β+1 , 1 − 2−α−β+1 ( 2−β ) A (σ) = Ψ(σ) exp σ β for 0 ≤ σ ≤ 1 . 1 − 2−β Ψ(σ) = σ α+β

7.3. Remark. (7.1)–(7.4) imply that Ψ, A, φ are continuous and nondecreasing. Moreover, ∞ ∑

2 i Ψ(σ 2−i ) ψ1 (σ2−i )

i=1

=

∞ ∑

2i

i=1

=

∞ ∑

2j ψ1 (σ2−i−j ) ψ2 (σ2−i−j ) ψ1 (σ2−i )

j=1

∞ ∑

2k ψ1 (σ2−k ) ψ2 (σ2−k )

∞ ∑

ψ1 (σ2−i ) < ∞

i=1

k=2

since

k−1 ∑

ψ1 (σ 2−i ) < ∞ by (7.2) and (7.4). Hence

i=1 ∞ ∑

−i

     

−i

2 A (σ2 ) ψ1 (σ2 )

i=1

≤

i

∞ ∑

i

−i

−i

2 Ψ(σ2 ) ψ1 (σ2 ) exp

∞ ∑

i=1

ψ1 (σ 2

−i−j

)<∞

j=1

    

(7.10)

and Φ and B are well defined, nondecreasing and continuous as well. + 7.4. Lemma. Let λ : R+ 0 → R0 be nondecreasing, continuous, ∞ ∑

2 i λ(2−i ) < ∞ ,

(7.11)

i=1

Λ(σ) =

∞ ∑ i=1

2 i λ(σ 2−i )

for σ ∈ R+ 0 .

(7.12)

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45

Then Λ

is nondecreasing, continuous, 1 lim Λ(σ) = 0 . σ → 0+ σ

(7.13) (7.14)

Proof. The series in (7.11) is uniformly convergent on [0, S] for 0 < S. Hence A is continuous and nondecreasing. Let k ∈ N. Then 2k Λ(2−k ) =

∞ ∑

2 i λ(2−i )

i=k+1

and

lim 2k Λ(2−k ) = 0 .

k→∞

Moreover, 1 Λ(σ) ≤ 2k+1 Λ(2−k ) σ and (7.14) is valid.

for 2−k−1 ≤ σ ≤ 2−k 

7.5. Remark. Lemma 7.4 applies if λ(σ) = ψ1 (σ) ψ2 (σ). Therefore 1 Ψ(σ) = 0 , σ 1 lim A (σ) = 0 . σ → 0+ σ lim

σ → 0+

(7.15) (7.16)

Moreover (cf. (7.10)), 1 Φ(σ) = 0 , σ → 0+ σ 1 lim B (σ) = 0 . σ → 0+ σ lim

(7.17) (7.18)

Observe that lim

σ → 0+

1 ψ1 (σ) ψ2 (σ) = 0 σ

(7.19)

by (7.15). 7.6. Lemma. Let σ ∈ R+ 0 . Then A (σ/2) 2 (1 + ψ2 (σ/2)) + 2 ψ1 (σ/2) ψ2 (σ/2) ≤ A (σ) . Proof.

We have 2 Ψ(σ/2) + 2 ψ1 (σ/2) ψ2 (σ/2) = Ψ(σ) for σ ∈ R+ 0 .

(7.20)

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Hence (

2 A (σ/2) exp −

∞ ∑

) ψ2 (σ2−j−1 ) + 2 ψ1 (σ/2) ψ2 (σ/2)

j=1

(

= A (σ) exp −

∞ ∑

) ψ2 (σ 2−j )

j=1

and 2 A (σ/2) exp ψ2 (σ/2) + 2 ψ1 (σ/2) ψ2 (σ/2) exp

∞ ∑

ψ2 (σ2−j ) = A (σ)

j=1

for σ ∈ R+ 0 . Therefore (7.20) is correct.



7.7. Lemma. There exists ξ1 > 0 such that B (σ) ≤ σ Proof.

for 0 ≤ σ ≤ ξ1 .

(7.21) 

(7.21) is a consequence of (7.18).

7.8. Lemma. Let 0 ≤ σ ≤ ξ1 . Then

}

B 2 (σ/2) + B (σ/2) 2 (1 + ψ2 (σ/2)) +A (σ/2) ψ1 (σ/2) + 3 ψ1 (σ/2) ψ2 (σ/2) ≤ B (σ) . Proof.

(7.22)

Φ fulfils 2 Φ(σ/2) + φ(σ/2) = Φ(σ)

for σ ∈ R+ 0.

Hence ∞ ( ∑ ) 2 B (σ/2) exp − ψ2 (σ2−j−1 ) − σ/2 + φ(σ/2) j=1 ∞ ( ∑ ) = B (σ) exp − ψ2 (σ2−j ) − σ , j=1

(

∞ ) (∑ ) 2 B (σ/2) exp ψ2 (σ/2) + σ/2 + φ(σ/2) exp ψ2 (σ2−j ) + σ j=1

= B (σ) , B (σ/2) 2 [1 + ψ2 (σ/2) + σ/2] + φ(σ/2) ≤ B (σ) for σ ∈ R+ 0 and (7.22) is valid by virtue of Lemma 7.7 and (7.8).



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Chapter 8

Strong Riemann solutions of generalized differential equations: a survey A theory of SR-solutions of generalized differential equations is developed in Chapters 9–13. Its brief survey is presented below. 8.1. Notation. Let X1 , X2 be vector spaces, Ω ⊂ X1 and let P be any set, f : Ω → X2 , F : Ω × P → X2 . Denote } ∆v f (x) = f (x + v) − f (x) , (8.1) ∆v F (x, p) = F (x + v, p) − F (x, p) for x, x + v ∈ Ω, p ∈ P. Let X be a Banach space with the norm ∥ · ∥, B(r) = {x ∈ X; ∥x∥ ≤ r} for 0 ≤ r < ∞. Let [a, b ] ⊂ R, R > 0, G : B(8R) × [a, b ] 2 → X. 8.2. Assumptions. Let G fulfil G

is continuous ,

(8.2)

∥G(x, τ, t) − G(x, τ, s)∥ ≤ ψ2 (t − s) ,

(8.3)

∥∆v (G(x, τ, t) − G(x, τ, s))∥ ≤ ∥v∥ψ1 (t − s) ,

(8.4)

∥∆w ∆v (G(x, τ, t) − G(x, τ, s))∥ ≤ ∥w∥ ∥v∥ ψ1 (t − s) , ∥G(x, τ, t) − G(x, τ, s) − G(x, σ, t) + G(x, σ, s)∥ ≤ ψ1 (t − s) ψ2 (τ − σ) , ∥∆v (G(x, τ, t) − G(x, τ, s) − G(x, σ, t) + G(x, σ, s))∥

(8.5) } (8.6) }

≤ ∥v∥ ψ1 (t − s) ψ2 (τ − σ) for x, x + v, x + w, x + v + w ∈ B(8R), t, s, τ, σ ∈ [a, b ], s ≤ t, σ ≤ τ. 47

(8.7)

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8.3. Proposition (Existence and uniqueness). There exists ξ5 > 0 having the following property: If b − a ≤ ξ5 , s ∈ [a, b ], y ∈ B(3R), then there exists an SR-solution u : [a, b ] → B(4R) of d x = Dt G(x, τ, t) (8.8) dt fulfilling u(s) = y. Moreover, u is unique. (See Theorems 12.10, 12.12.) 8.4. Proposition. Let b − a ≤ ξ5 and let u : [a, b ] → B(4R) be a solution of (8.8). Then lim (t − τ )−1 ∥u(t) − u(τ ) − G(u(τ ), τ, t) + G(u(τ ), τ, τ )∥ = 0 .

t→τ

(8.9)

(8.9) implies that ∂ (u(t) − G(u(τ ), τ, t))|t=τ = 0 ∂t

for τ ∈ [a, b ] .

(8.10)

E.g., if G(x, τ, ·) is nowhere differentiable then u is nowhere differentiable. (See Lemma 12.7 and Corollary 12.8.) 8.5. Proposition (Continuous dependence). b : R+ → R+ , Ω(σ) b There exists Ω → 0 for σ → 0 with the following property: Let G∗ : B(8R) × [a, b ] 2 → X. Assume that b − a ≤ ξ5 and that G∗ fulfils (8.2)– (8.7). Put } dist (G, G∗ ) (8.11) = sup {∥G(x, τ, t) − G∗ (x, τ, t)∥ ; x∈B(8R), τ, t ∈ [a, b ]} . Let y ∈ B(3R), s ∈ [a, b ]. Let v : [a, b ] → B(8R) be an SR-solution of (8.8), v(s) = y, and let v ∗ : [a, b ] → B(8R) be an SR-solution of d x = Dt G∗ (x, τ, t), dτ

v ∗ (s) = y .

(8.12)

for t ∈ [a, b ]

(8.13)

Then v(t), v ∗ (t) ∈ B(4R) and

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b ∥v(t) − v ∗ (t)∥ ≤ Ω(dist(G, G∗ ))

for t ∈ [a, b ] .

jk

49

(8.14)

Proof. (8.13) follows from Proposition 8.3 and (8.14) is a consequence b = 3 κ Ω. of Theorem 13.7 where Ω  2 8.6. Remark. Let e.g. [a, b ] ⊂ R,

1 2

b > 0, < α ≤ 1, 0 < β ≤ 1, α + β > 1, R

G(x, τ, t) = −h(x, τ + εα sin(τ /ε)) εα sin(t/ε) − λ p (τ + εα sin(τ /ε)) εβ sin (t/(λ ε)) , where 0 < ε ≤ 1, 0 ≤ λ ≤ 1, b × [a − 1, b + 1] → X, p : [a − 1, b + 1] → X , h : B(R) ∂ h(x, τ ) is Lipschitzian with respect to x, τ D1 h(x, τ ) = ∂x and D p (s) =

d p (s) is Lipschitzian . ds

Then any solution of the classical equation x˙ = h(x, t + εα sin(t/ε)) εα−1 cos(t/ε) + p(t + εα sin(t/ε)) εβ−1 cos(λ t/ε) is a solution of (8.8) and vice versa. Conditions (7.1)–(7.4) and (8.2)–(8.7) are fulfilled if ψ1 (σ) = κ σ α ,

ψ2 (σ) = κ σ β

for 0 ≤ σ ≤ 1

and ψ1 (σ) = ψ2 (σ) = κ σ

for 1 ≤ σ .

The existence and uniqueness of solutions of (8.8) are given by Proposition 8.4. Observe that β may be small if α is sufficiently close to 1.

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Chapter 9

Approximate solutions: boundedness

9.1 . Notation. Let [a, b ] ⊂ R, c = a + 2 (b − a), ξ2 ∈ R+ , b − a ≤ ξ2 , G : B(8R) × [a, b ] 2 → X. Assume that G fulfils (8.2)– (8.7). Further, assume that ψ2 (2 ξ2 ) + A(2 ξ2 ) < R,

b − a ≤ ξ2 ,

(9.1)

where the functions ψ1 and A were introduced in Notation 7.1. Extend G by putting e τ, t) = G(x, min{τ, b}, min{t, b}) for (τ, t) ∈ [a, c]2 , x ∈ B(8R) (9.2) G(x, and for S, t, T such that S, t, T ∈ [a, c] and S < T let G(S, t, T ) be a mapping from B(8R) to X, the value of G(S, t, T ) at x being denoted by x G(S, t, T ) and  if t ≤ S , 0   e S, t) − G(x, e S, S) if S ≤ t ≤ T , x G(S, t, T ) = G(x, (9.3)   e e S, S) if T ≤ t. G(x, S, T ) − G(x, 9.2 . Remark. Existence, uniqueness and continuous dependence of solutions of (8.8) will be proved for t ∈ [a, b ]. The proofs are based on the properties of the functions in the form } x (id + G(S, t, S + σ)) (id + G(S + σ, t, S + 2 σ)) (9.4) . . . (id + G(S + (k−1) σ, t, S + k σ)) for k ∈ N , where x (id + G(S, t, S + σ)) = x + x G (S, t, S + σ) . In Chapter 12 functions in the form (9.4) are needed such that S ∈ [a, b ], σ = (b − a) 2−j , k = 1, 2, . . . , 2j , j ∈ N, so that S + 2j σ = S + b − a and 51

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[S, S + b − a] ⊂ [a, c] (see (12.3)). This is the motivation for extending e G to G. e fulfils (8.2)–(8.7) on its domain 9.3. Lemma. G B(8R) × {(S, t, T ); [S, T ] ⊂ [a, c], t ∈ [a, c]} .

(9.5)

Moreover, G is continuous ,

(9.6)

∥x G(S, t, T )∥ ≤ ψ2 (t − S) ,

(9.7)

∥∆v (x G(S, t, T ))∥ ≤ ∥v∥ ψ1 (t − S) , ∥∆w ∆v (xG(S, t, T ))∥ ≤ ∥w∥ ∥v∥ ψ1 (t − S) ,



S+T ) + x G( , t, T ) − x G(S, t, T )

x G(S, t, S+T

2) ( 2 ) ( ≤ ψ1 t −2 S ψ2 t −2 S ,

(9.8) (9.9)   

( ) 

 S+T )+x G( , t, T ) − x G(S, t, T )

∆v x G(S, t, S+T

2 2  T −S T −S )ψ ( ) ≤ ∥v∥ ψ ( 1

2

2

(9.10)

(9.11)

2

for x, x + v, x + v + w, x + w ∈ B(8R), t ∈ [S, T ] ⊂ [a, c]. e fulfils (8.2)–(8.7) on its domain by (9.2) since ψ1 , ψ2 are nonProof. G decreasing. Relation (9.6) is a consequence of (8.2) and (9.7), (9.8), (9.9) follow directly from (8.3), (8.4), (8.5). If t ≤ S +2 T then the left hand side of (9.10) vanishes. Let

S+T 2

< t ≤ T. Then S+T x G(S, t, S+T 2 ) + x G( 2 , t, T ) − x G(S, t, T ) S+T e S, S+T ) − G(x, e S, S) + G(x, e = G(x, 2 2 , t) S+T S+T e e e − G(x, 2 , 2 ) − G(x, S, t) + G(x, S, S)

and (9.10) holds by (8.6). (9.11) is proved in an analogous way (cf. (8.7)). 

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9.4. Definition. Let x ∈ B(8R),

[S, T ] ⊂ [a, c],

t ∈ [a, c],

i ∈ N0 ,

j ∈ {0, 1, . . . , 2 i } .

Put x (id + G(S, t, T )) = x + x G(S, t, T ), Z(i, j) = S + j

T −S i

2

.

Assume that x (id + G(Z(i, 0), Z(i, 1), Z(i, 1))) (id + G(Z(i, 1), Z(i, 2), Z(i, 2))) . . .

      

 (id + G(Z(i, j − 2), Z(i, j − 1), Z(i, j − 1)))      (id + G(Z(i, j − 1), t, Z(i, j))) ∈ B(8R)

(9.12)

for j = 1, 2, . . . , 2 i and Z(i, j −1) < t ≤ Z(i, j). Then denote by x Vi (S, t, T ) the expression in (9.12), S < t ≤ T and put x Vi (S, t, T ) = x

for a ≤ t ≤ S ,

x Vi (S, t, T ) = x Vi (S, T, T ) for T < t ≤ c

(9.13) (9.14)

if x Vi (S, T, T ) exists. 9.5. Remark. Let ui (t) = x Vi (S, t, T ) for S ≤ t ≤ T. ui is called an approximate solution of (8.8), which is justified by (11.16), (12.3), Theorem 12.10. 9.6 . Lemma. Let t ∈ [a, c], i ∈ N0 , x ∈ B(8R) and let x Vi (S, T, T ) exist. Then  x Vi (S, t, T )  (9.15) = x (id + G(S, t, Z(i, 1))) (id + G(Z(i, 1), t, Z(i, 2)))  . . . (id + G(Z(i, 2 i −1), t, T )) for t ∈ [a, c] . Proof.

(9.15) is a consequence of (9.3) and Definition 9.4.



9.7 . Lemma. Let i ∈ N, t ∈ [S, T ] ⊂ [a, c], x ∈ B(8R) and let x Vi (S, T, T ) exist. Then S+T x Vi (S, t, T ) = x Vi−1 (S, t, S+T 2 ) Vi−1 ( 2 , t, T )

and

(9.16)

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x Vi−1 (S, t, S+T 2 ) = x Vi (S, t, T )

if t ≤ S+T 2

(9.17)

Proof. (9.17) is a consequence of Definition 9.4. If t ≤ S+T then (9.16) 2 is valid since S+T S+T x Vi−1 (S, t, S+T 2 ) Vi−1 ( 2 , t, T ) = z Vi−1 (S, t, 2 )



by (9.17). If t > S+T then (9.16) follows by Definition 9.4. 2 9.8. Definition. For [S, T ] ⊂ [a, c], t ∈ [a, c], x ∈ B(8R) and i ∈ N0 , put x Yi (S, t, T ) = x Vi (S, t, T ) − x − x G(S, t, T )

(9.18)

if x Vi (S, T, T ) exists. 9.9. Lemma. Let i ∈ N0 , t ∈ [S, T ] ⊂ [a, c], x ∈ B(8R) and let x Vi+1 (S, T, T ) exist. Then  S+T S+T x Yi+1 (S, t, T ) = x Vi (S, t, S+T 2 )Yi ( 2 , t, T )+x Yi (S, t, 2 )    ( ) S+T S+T S+T (9.19) + x Yi (S, t, 2 )+x+x G(S, t, 2 ) G( 2 , t, T )    + x G(S, t, S+T 2 ) − x G(S, t, T ) and



S+T ) Y ( , t, T ) ∥x Yi+1 (S, t, T )∥ ≤ x Vi (S, t, S+T

i 2 2

  

(9.20)

) ( ( )) ( 

T −S T −S T −S S+T  + x Yi S, t, 2 1 + ψ1 2 + 2 ψ1 ( 2 ) ψ2 ( 2 ) . Proof.

By (9.16) and by Definitions 9.4, 9.8, we have x Yi+1 (S, t, T )

[ ] S+T S+T = x Vi (S, t, S+T ) Y ( , t, T ) + id + G( , t, T ) i 2 2 2 − x − x G(S, t, T )

S+T = x Vi (S, t, S+T 2 ) Yi ( 2 , t, T ) S+T + x Yi (S, t, S+T 2 ) + x + x G(S, t, 2 ) ( S+T S+T + x Yi (S, t, S+T 2 ) + x + x G(S, t, 2 )) G( 2 , t, T )

− x − x G(S, t, T )

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S+T S+T since x Vi (S, t, S+T 2 ) ∈ B(8R) and x Vi (S, t, 2 ) G( 2 , t, T ) exists. Therefore   x Yi+1 (S, t, T )      S+T S+T   = x Vi (S, t, 2 ) Yi ( 2 , t, T )     S+T S+T + x Yi (S, t, 2 ) + x + x G(S, t, 2 ) (9.21)   [ ]    S+T S+T  + x Yi (S, t, S+T  2 ) + x + x G(S, t, 2 ) G( 2 , t, T )      − x − x G(S, t, T ) ,

Hence (9.19) holds. Moreover, (9.21) can be written in the form x Yi+1 (S, t, T ) S+T S+T = x Vi (S, t, S+T 2 ) Yi ( 2 , t, T ) + x Yi (S, t, 2 ) ) [( S+T S+T + x Yi (S, t, S+T 2 ) + x + x G(S, t, 2 ) G( 2 , t, T ) ( ) ] S+T − x + x G(S, t, S+T 2 ) G( 2 , t, T ) ) ] [( S+T S+T + x + x G(S, t, S+T 2 ) G( 2 , t, T ) − x G( 2 , t, T ) [ ] S+T + x G(S, t, S+T ) + x G( , t, T ) − x G(S, t, T ) . 2 2 T −S The first bracket does not exceed ∥x Yi (S, t, S+T 2 )∥ ψ1 ( 2 ) by (9.8), the T −S second one and the third one are estimated by ψ1 ( 2 ) ψ2 ( T −S 2 ). Therefore (9.20) is correct. 

9.10. Lemma. Let i ∈ N0 , t ∈ [S, T ] ⊂ [a, c], x ∈ B(8R) and let x Vi (S, T, T ) exist. Then ∥x Yi (S, t, T )∥ ≤ A(T − S) ∥x Vi (S, t, T ) − x∥ ≤ A(T − S) + ψ2 (T − S) Proof.

(9.22) (9.23)

Part 1. By Definitions 9.4, 9.8 x V0 (S, t, T ) = x G(S, t, T ),

and x Y0 (S, t, T ) = 0 .

(9.24)

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Lemma 9.10 is valid for i = 0 (cf. (9.7)). Part 2. Assume that there exists k ∈ N such that Lemma 9.10 is valid for i = 0, 1, . . . , k−1. Let t ∈ [S, T ] ⊂ [a, c], x ∈ B(8R) and let x Vk (S, T, T ) exist. By (9.20) } S+T ∥x Yk (S, t, T )∥ ≤ ∥x Vk−1 (S, t, S+T 2 ) Yk−1 ( 2 , t, T )∥ (9.25) ( ) T −S T −S T −S + ∥x Yk−1 (S, t, S+T )∥ 1 + ψ ( ) + 2 ψ ( ) ψ ( ) . 1 1 2 2 2 2 2 If t ≤ S+T then (cf. (9.18), (9.3)) 2 S+T S+T T −S x Vk−1 (S, t, S+T 2 ) Yk−1 ( 2 , t, T ) = 0, ∥x Yk−1 (S, t, 2 )∥ ≤ A( 2 )

and (9.22) holds by (7.20). If t > S+T then 2 S+T T −S ∥x Vk−1 (S, t, S+T 2 ) Yk−1 ( 2 , t, T )∥ ≤ A( 2 ), T −S ∥x Yk−1 (S, t, S+T 2 )∥ ≤ A( 2 )

and (9.22) holds by (7.20). Part 3. Parts 1 and 2 imply that (9.22) holds. Moreover, (9.23) follows by (9.22), (9.18), (9.8).  9.11. Theorem. Let i ∈ N0 , [S, T ] ⊂ [a, c], t ∈ [a, c] and x ∈ B(7R). Then ∥x Vi (S, t, T ) − x∥ < R . Proof.

(9.26)

Part 1. By Definition 9.4, (9.7), (9.1), x V0 (S, t, T ) = x + x G(S, t, T ) , ∥x V0 (S, t, T ) − x∥ ≤ ∥x G(S, t, T )∥ ≤ ψ2 (t−S) < R

if S ≤ t ≤ T. Moreover, x V0 (S, t, T ) = x

for a ≤ t < T,

x V0 (S, t, T ) = x V0 (S, T, T ) for T < t ≤ C . Hence Theorem 9.11 is valid if i = 0. Part 2. Assume that there exists k ∈ N such that (9.26) holds for i = 0, 1, . . . , k−1, S ≤ t ≤ T and that (9.26) does not hold for i = k and all

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x ∈ B(7R) and t ∈ [S, T ] ⊂ [a, c]. By (9.6) and Definition 9.4, G(S, ., T ) is b Tb] ⊂ [a, c] and x continuous. Hence there exist [S, b ∈ B(7R) such that b t, Tb) − x ∥b x Vk (S, b∥ < R

if Sb ≤ t < T

and b Tb, Tb) − x ∥b x Vk (S, b∥ = R

(9.27)

By Lemma 9.10 and (9.1) b Tb, Tb) − x b + ψ2 (Tb−S) b
for t ∈ [S, T ] ⊂ [a, c], x ∈ B(7R) .

By Definition 9.4 ∥x Vk (S, t, T ) − x∥ < R

for [S, T ] ⊂ [a, c], t ∈ [a, c], x ∈ B(7R) .

Part 3. Parts 1 and 2 imply that Theorem 9.11 is valid.



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Chapter 10

Approximate solutions: a Lipschitz condition

10.1. Notation. Observe that

∞ ∑

ψ1 (σ 2−j ) → 0 for σ → 0 (cf. (7.1), (7.2),

j=1

(7.4)). Let ξ3 ∈ R+ fulfil (cf. (7.21), (9.1), (7.1), (7.2), (7.4)) ξ3 ≤ min{ξ1 , ξ2 }

(10.1)

(ξ1 having been introduced by Lemma 7.7, ξ2 in (9.1)) , ∞ ∑ exp(2 ξ3 + ψ1 (2 ξ3 2−j )) ≤ 2 ,

(10.2)

j=1

A(2 ξ3 ) + ψ2 (2 ξ3 ) ≤ R .

(10.3)

Assume that G fulfils (8.2)–(8.7). 10.2. Theorem. Let [a, c] ⊂ R, c−a ≤ 2 ξ3 ,

(10.4)

[S, T ] ⊂ [a, c], i ∈ N0 , x, z ∈ B(6R), t ∈ [a, c] .

(10.5)

Then ∥x Vi (S, t, T ) − z Vi (S, t, T ) − x + z − x G(S, t, T ) + z G(S, t, T )∥ = ∥x Yi (S, t, T ) − z Yi (S, t, T )∥ ≤ ∥x − z∥ B(T − S) . The proof is postponed after Lemma 10.3. 59

            

(10.6)

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10.3. Lemma. Assume that there exists ℓ ∈ N0 such that ¯ t¯, T¯) − z¯ Yℓ (S, ¯ t¯, T¯)∥ ≤ ∥¯ ¯ ∥¯ x Yℓ (S, x − z¯∥ B(T¯ − S)

(10.7)

for ¯ T¯] ⊂ [a, c], [S,

x ¯, z¯ ∈ B(6R), t¯∈ [a, c] .

(10.8)

Then ∥x Yℓ+1 (S, t, T ) − z Yℓ+1 (S, t, T )∥ ≤ ∥x − z∥ B(T − S)

} (10.9)

for [S, T ] ⊂ [a, c], x, z ∈ B(6R), , t ∈ [a, c]. Proof.

Let [S, T ] ⊂ [a, c], x, z ∈ B(6R), , t ∈ [a, c].

Observe that (cf. (9.22), (9.6)) T −S ∥x Yℓ (S, t, S+T 2 )∥ ≤ A( 2 ), T −S ∥x G(S, t, S+T 2 )∥ ≤ ψ2 ( 2 )

  

(10.10)

and that (cf. (10.3)) S+T x Yℓ (S, t, S+T 2 ) + z + x G(S, t, 2 ) ∈ B(8R)

(10.11)

since ∥z∥ ≤ 6R. Write ℓ instead of i in (9.19), then write z instead of x and form the difference of these equations. In this way we obtain x Yℓ+1 (S, t, T ) − z Yℓ+1 (S, t, T ) = F1 + F2 + F3 + F4 + F5 , where S+T F1 = x Vℓ (S, t, S+T 2 ) Yℓ ( 2 , t, T ) S+T −z Vℓ (S, t, S+T 2 ) Yℓ ( 2 , t, T ),

  

S+T F2 = x Yℓ (S, t, S+T 2 ) − z Yℓ (S, t, 2 ), S+T S+T F3 = [x Yℓ (S, t, S+T 2 ) + x + x G(S, t, 2 )] G( 2 , t, T ) S+T − x G( S+T 2 , t, T ) + z G( 2 , t, T ) S+T S+T −[x Yℓ (S, t, S+T 2 ) + z + x G(S, t, 2 )] G( 2 , t, T ),

(10.12)

(10.13)

(10.14)         

(10.15)

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Approximate solutions: a Lipschitz condition S+T S+T F4 = [x Yℓ (S, t, S+T 2 ) + z+x G(S, t, 2 )] G( 2 , t, T )

61

} (10.16)

S+T S+T −[z Yℓ (S, t, S+T 2 ) + z + z G(S, t, 2 )] G( 2 , t, T ), S+T F5 = x G(S, t, S+T 2 ) + x G( 2 , t, T ) − x G(S, t, T )

} (10.17)

S+T −z G(S, t, S+T 2 ) − z G( 2 , t, T ) + z G(S, t, T ).

The following estimates are deduced from (10.13)–(10.17) by (10.7), (9.7)– (9.11), (9.23). In particular, by (10.7), (9.18) and (9.8), we have S+T T −S ∥F1 ∥ ≤ ∥x Vℓ (S, t, S+T 2 )−z Vℓ (S, t, 2 )∥ B( 2 ) ) T −S ( T −S ≤ ∥x − z∥ 1 + ψ2 ( T −S 2 ) + B( 2 ) B( 2 ) .

  

(10.18)

Further, by (10.7), we have ∥F2 ∥ ≤ ∥x − z∥ B( T −S 2 ).

(10.19)

By (9.5), (9.9), (9.21)

S+T ∥F3 ∥≤∥x−z∥ xYℓ (S, t, S+T )+xG(S, t, )

ψ1 ( T −S 2 2 2 ) ( T −S ) T −S ≤ ∥x−z∥ A( 2 ) + ψ2 ( T −S 2 ) ψ1 ( 2 )

  

(10.20)

since F3 = − ∆w ∆v (x G( S+T 2 , t, T ) where v = z − x,

S+T w = x Yℓ (S, t, S+T 2 ) + x G( 2 , t, T ) .

By (9.8), (10.6)

S+T ∥F4 ∥≤ x Yℓ (S, t, S+T 2 )−z Yℓ (S, t, 2 )

S+T + x G(S, t, S+T )−z G(S, t, )

ψ1 ( T −S 2 2 2 ) ( T −S ) T −S ≤ ∥x − z∥ B( 2 ) + ψ2 ( T −S 2 ) ψ1 ( 2 ) .

          

(10.21)

Finally, by (9.11) T −S ∥F5 ∥ ≤ ∥x − z∥ ψ1 ( T −S 2 ) ψ2 ( 2 ) .

(10.22)

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Now, (10.12) and (10.18)– (10.22) imply that  ∥x Yℓ+1 (S, t, T ) − z Yℓ+1 (S, t, T )∥     ( ) T −S T −S T −S 2 T −S ≤ ∥x − z∥ B( 2 ) 2 + ψ1 ( 2 ) + ψ2 ( 2 ) + B ( 2 )     T −S T −S T −S +3 ψ1 ( T −S ) ψ ( ) + A( ) ψ ( )) . 2 1 2 2 2 2 (10.9) holds by (10.23), (7.2) and (7.22).

(10.23)



Proof of Theorem 10.2 . (10.7) holds by (9.24) if ℓ = 0. Lemma 10.3 implies that (10.6) holds if ℓ ∈ N0 . 

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Chapter 11

Approximate solutions: convergence

11.1. Notation. Let 0 < ξ4 ≤ ξ3 , ψ2 (ξ4 ) + ψ1 ( ξ24 ) ψ2 ( ξ24 ) exp

∞ ( ∑ 1 2

        

) Cj (ξ4 ) < R

j=2

where Cj (z) = ψ2 (

z j

) + B(

z j)

(11.1)

       

for z > 0, j ∈ N

2 2 and B has been introduced in (7.9) , a < c ≤ a + ξ4 .

(11.2)

11.2. Lemma. Let [S, T ] ⊂ [a, c],

x ∈ B(7R) .

(11.3)

Then T −S ∥x V1 (S, t, T ) − x V0 (S, t, T )∥ ≤ 2 ψ1 ( T −S 2 ) ψ2 ( 2 ) .

Proof.

Let (11.3) be fulfilled. By Definition 9.4 for S ≤ t ≤ T ,

x V0 (S, t, T ) = x + x G(S, t, T ) x V1 (S, t, T ) = x + x G(S, t,

S+T 2

) for S ≤ t ≤

S+T 2

Hence Let

S+T 2

(11.4)

x V1 (S, t, T ) − x V0 (S, t, T ) = 0 < t. Then (cf. also (9.17), (9.16))

for t ≤

S+T 2

.

x V1 (S, t, T ) − x V0 (S, t, T ) S+T S+T = x V0 (S, S+T 2 , 2 ) V0 ( 2 , t, T ) − x V0 (S, t, T )

63

.

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( ) )( S+T = x + x G(S, S+T id + G( S+T 2 , 2 ) 2 , t, T ) − x − x G(S, t, T ) S+T = x + x G(S, S+T 2 , 2 ) ) ( S+T , ) G( S+T + x + x G(S, S+T 2 2 2 , t, T ) S+T − x G( S+T 2 , t, T ) + x G( 2 , t, T ) − x − x G(S, t, T )

and (11.4) holds since (cf. (9.7), (9.8), (9.10))

(

)

S+T S+T S+T , ) G( , t, T ) − x G( , t, T )

x + x G(S, S+T

2 2 2 2 S+T ≤ ψ1 ( S+T 2 ) ψ2 ( 2 ) ,



S+T S+T

x G(S, S+T 2 , 2 ) + x G( 2 , t, T ) − x G(S, t, T ) T −S ≤ ψ1 ( T −S 2 ) ψ2 ( 2 ) .



Lemma 11.2 is correct. 11.3. Lemma. Let i ∈ N. Assume that   

b b b b ∥b x Vi (S, t, Tb) − x b Vi−1 (S, t, Tb)∥ −1 ) ( i∑ b b b b b Cj (Tb − S) ≤ 2 i ψ1 ( T −iS ) ψ2 ( T −iS ) exp 12 2 2 j=1

 

(11.5)

for b Tb] ⊂ [a, c], x [S, b ∈ B(6R) . ◦ ( ∑ ) b = 1.) (Observe that for i = 1 we have exp 12 Cj (Tb − S)

(11.6)

j=1

Then ∥x Vi+1 (S, t, T ) − x Vi (S, t, T )∥ −S −S ≤ 2i+1 ψ1 ( T i+1 ) ψ2 ( T i+1 ) exp 2 2

i ( ∑ 1 2

j=1

Cj (T −S)

)

    

(11.7)

for x ∈ B(6R).

(11.8)

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Proof.

Let (11.8) be fulfilled. By (11.8) and (9.17) x Vi+1 (S, t, T ) − x Vi (S, t, T ) = 0

Let

65

S+T 2

for t ≤

S+T 2

.

(11.9)

< t. Then x Vi+1 (S, t, T ) − x Vi (S, t, T ) = U + V

(11.10)

where S+T S+T U = x Vi (S, S+T 2 , 2 ) Vi ( 2 , t, T ) S+T S+T − x Vi−1 (S, S+T 2 , 2 ) Vi ( 2 , t, T ), S+T S+T V = x Vi−1 (S, S+T 2 , 2 ) Vi ( 2 , t, T ) S+T S+T − x Vi−1 (S, S+T 2 , 2 ) Vi−1 ( 2 , t, T ) S+T S+T the term x Vi (S, S+T 2 , 2 ) Vi−1 ( 2 , t, T ) being well defined since S+T S+T x Vi (S, 2 , 2 ) ∈ B(7R). By (11.8), Lemma 9.7 and Theorem 9.11 S+T and x Vi−1 (S, S+T 2 , 2 ) ∈ B(7R).

x ∈ B(6R) Hence by (11.5)

−S −S ) ψ2 ( T i+1 ) exp ∥V∥ ≤ 2 i ψ1 ( T i+1 2 2

i ( ∑ 1 2

Cj (T −S)

(11.11)

j=2



S+T S+T S+T

x Vi (S, S+T 2 , 2 ) − x Vi−1 (S, 2 , 2 ) −S −S ≤ 2 i ψ1 ( T i+1 ) ψ2 ( T i+1 ) exp 2 2

)

i ( ∑ 1 2

j=2

    

)  Cj (T −S)   

(11.12)

and by (11.12), (10.6), (9.18) and (9.8) ∥U∥ ≤ 2

i

i ( ( ∑ )) ) 1 exp 2 Cj (T −S (1+ C1 (T −S) . (11.13)

−S −S ψ1 ( T i+1 )ψ2 ( T i+1 )

2

2

j=2

(11.10), (11.11) and (11.13) imply that ∥x Vi+1 (S, t, T ) − x Vi (S, t, T )∥ i ( ( ∑ )) ( ) −S −S ≤ 2 i ψ1 ( T i+1 )ψ2 ( T i+1 ) exp 12 Cj (T −S) 2 1+ 21 C1 ( T −S ) 2 2 2 j=2

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and (11.7) is correct.

11.4. Theorem. Let i ∈ N, t ∈ [S, T ] ⊂ [a, c] and let x ∈ B(6R). Then   ∥x Vi (S, t, T ) − x Vi−1 (S, t, T )∥   i−1 ( ∑ ) (11.14) T −S 1 ≤ 2 i ψ1 ( T −S Cj (T −S) .   i ) ψ2 ( i ) exp 2  2 2 j=1

Proof.



This is a consequence of Lemmas 11.2, 11.3.

11.5. Lemma. Let x ∈ B(6R), t ∈ [S, T ] ⊂ [a, c], k, ℓ ∈ N, k < ℓ. Then ∥x Vℓ (S, t, T ) − x Vk (S, t, T )∥ ℓ ∞ ( ∑ ) ∑ c −a ≤ 2 i ψ12 ( i ) exp 12 Cj (c −a) . 2 j=1 i=k+1 Proof.

    

(11.15)

The series ∞ ∑

Cj (c−a) =

j=1

∞ ( ∑ j=1

) c−a ψ1 ( c−a ) ) + B( 2j 2j

is convergent by (7.1), (7.2), (7.4) and (7.21). Lemma 11.5 follows from Theorem 11.4.  11.6. Definition. Put x V (S, t, T ) = lim x Vi (S, t, T ) i→∞

for x ∈ B(6R), t ∈ [S, T ] ⊂ [a, c]. The limit in (11.16) exists by (11.15) since the series ∞ ∑

2 i ψ12 ( c−a ) 2i i=1

is convergent by (7.1), (7.2), (7.3).

(11.16)

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11.7. Theorem. Let x, x ¯ ∈ B(6R), t ∈ [S, T ] ⊂ [a, c]. Then V

is continuous ,

(11.17)

x V (S, t, T ) ∈ B(7R) ,

(11.18)

x V (S, t, S+T 2 ) = x V (s, t, T )

for t ≤ S+T 2 ,

∥x V (S, t, T ) − x − x G(S, t, T )∥ ≤ A(2 (T − S)) , ∥x V (S, t, T ) − x ¯ V (S, t, T ) − x + x ¯ − x G(S, t, T ) + x ¯ G(S, t, T )∥ ≤ ∥x − x ¯∥ B(2(t−S)) .

(11.19) (11.20)     

(11.21)

Proof. V is the uniform limit of continuous Vi (cf. Lemma 11.5). Hence (11.17) is correct. (11.18) is a consequence of (9.23) and (9.1). (11.19) follows by (11.9). The limit procedure for i → ∞ in (9.22) (cf. also (9.18)) gives } ∥x V (S, t, T ) − x − x G(S, t, T )∥ ≤ A(T − S) (11.22) for x ∈ B(6R), t ∈ [S, T ] ⊂ [a, c] . By (9.3) x G(S, t, S+T 2 ) = x G(S, t, T ) for x ∈ B(8R), t ∈ [S, T ] ⊂ [a, c], t ≤ S+T 2 . Similarly ∥x V (S, t, T ) − x − x G(S, t, T )∥ ≤ A( T −j S ) 2 for x ∈ B(6R), t ≤ S + T −S , j ∈N. 2j

  

(11.23)

(11.20) follows by (11.23) since A is nondecreasing and 2 (t−S) ≥ T −S 2j

for

T −S

2j+1

≤ t−S ≤ T −S . 2j

The limit procedure for i → ∞ in (10.6) results in ∥x V (S, t, T ) − x ¯ V (S, t, T ) − x + x ¯ − x G(S, t, T ) + x ¯ G(S, t, T )∥ ≤ ∥x − x ¯∥ B(T − S) for x ∈ B(6R), t ∈ [S, T ] ⊂ [a, c] .

          

(11.24)

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(11.24) implies that ∥x V (S, t, T ) − x ¯ V (S, t, T ) − x + x ¯ − x G(S, t, T ) + x ¯ G(S, t, T )∥ ≤ ∥x − x ¯∥ B( T −j S ) 2 for x, x ¯ ∈ B(6R), t ≤ S + T −S , j ∈N, 2j

              

(11.25)

in a similar way as (11.22) implies that (11.23) holds. (11.21) follows by (11.25) since B is nondecreasing. The proof is complete.  11.8. Remark. Put u(t) = x V (S, t, T ), ui (t) = x Vi (S, t, T ) for t ∈ [S, T ], i ∈ N. (11.18) and (7.16) imply that u is an SR-solution of d x = G(x, τ, t) , dt

(11.26)

u(S) = x. Therefore ui may be called approximate solutions of (11.26). Let Dom g ⊂ X×R, G(x, τ, t) = g(x, τ ) t. Then ui are approximate solutions of the differential equation in the classical form x˙ = g(x, t) which are obtained by Euler’s method (going back to 1768).

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Chapter 12

Solutions

12.1 . Notation. By (7.1), (7.2), (7.4), (7.16), (7.18), (11.1) there exists ξ5 ∈ R+ such that } ξ5 ≤ ξ4 , ψ1 (ξ5 ) + B(2 ξ5 ) ≤ 31 , (12.1) ψ2 (ξ5 ) + A(2 ξ5 ) ≤ 23 R . Let [a, c] ⊂ R, c − a ≤ 2 ξ5 ,

b=

1 2

(a + c).

(12.2)

Let W be defined by x W (S, t) = x V (S, t, S + b − a)

(12.3)

for x ∈ B(6R), S ∈ [a, b ], t ∈ [S, S + b − a]. For σ ∈ [S, S+b−a] put ES,σ = {x W (S, σ) ; x ∈ B(5R)}.

(12.4)

12.2. Lemma. W is continuous. Proof. W is continuous since it is a restriction of a continuous V (cf. (11.17)).  12.3. Lemma. Let S, t ∈ [a, b ], S ≤ t, v, v¯ ∈ B(6R). Then ES,t ⊂ B(6R) ,

(12.5)

∥v W (S, t) − v − v G(S, t, S + b − a)∥ ≤ A(2 (t − S)) , ∥v W (S, t) − v¯ W (S, t) − v + v¯ − v G(S, t, S + b − a) +¯ v G(S, t, S + b − a)∥ ≤ ∥v − v¯∥ B(2(t − S)) , ∥v W (S, t) − v∥ ≤ ψ2 (t − S) + A(2(t − S)) , 69

(12.6) } (12.7) (12.8)

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}

∥v W (S, t) − v¯ W (S, t) − v + v¯ ∥

(12.9)

≤ ∥v − v¯∥ (ψ1 (t − S) + B(2 (t − S))) ,

}

∥v W (S, t) − v¯ W (S, t)∥

(12.10)

≥ ∥v − v¯∥ (1 − ψ1 (t − S) − B(2 (t−S))) .

Proof. (12.5) is a consequence of (12.4) and (11.18). Further, (12.6) and (12.7) follow by (12.3), (11.20), (11.21). Moreover, (12.8) holds by (12.6) since ∥v W (S, t) − v∥ ≤ ∥v G(S, t, S + b − a)∥ + A(2(T − S)) ≤ ψ2 (T − S) + A(2(T − S)) (cf. (9.7)). Similarly (12.9) holds by (12.7) and (9.8). 

(12.10) follows from (12.9). 12.4. Lemma. Let a ≤ τ ≤ t ≤ b.

(12.11)

Then x W (a, τ )W (τ, t) = x W (a, t), Proof.

x ∈ B(5R) .

(12.12)

x W (a, τ ) W (τ, t) is well defined since x W (a, τ ) = x V (a, τ, b) ∈ B(6R)

(cf. (12.3)). Let i, j, k, ℓ ∈ N0 ,

j ≤ i,

τ = a + k (b−a) 2−j ,

}

k ≤ ℓ ≤ 2j ,

(12.13)

t = a + ℓ (b−a) 2−j .

By Definition 9.4 x Vj (a, τ, b) Vj (τ, t, τ + b − a) = x Vj (a, t, b) and x Vi (a, τ, b) Vi (τ, t, τ + b − a) = x Vi (a, t, b) for i ∈ N, i > j .

} (12.14)

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The limit procedure for i → ∞ in (12.14) implies (cf. (12.3)) that (12.12) holds for the particular choice of τ, t in (12.13) since ∥v Vi (τ, t, τ + b − a) − v¯ Vi (τ, t, τ + b − a)∥ ≤ ∥v − v¯∥ (1 + ψ1 (T − S) + B(T − S)) for v, v¯ ∈ B(6R) (cf. (9.8), (10.6)). Hence (12.12) holds since W is continuous and for every i ∈ N, τ, t ∈ [a, b ], τ ≤ t there exist k, ℓ ∈ N0 , k ≤ ℓ such that a + (k−1)(b − a) 2−i ≤ τ ≤ a + k (b − a) 2−i and a + (ℓ−1) (b − a) 2−i ≤ t ≤ a + ℓ (b − a) 2−i .



12.5. Lemma. Let a ≤ S ≤ T ≤ b. Then W (S, T )

is a one-to-one map of B(5R) onto ES,T .

(12.15)

Moreover, if y ∈ B(5R) then there exists x ∈ B(6R) such that x W (a, S) = y .

(12.16)

Further B(6R) ⊂ ES,T .

(12.17)

Proof. W (S, T ) is a map of B(6R) onto ES,T by (12.4) and (12.8). W (S, T ) is a one-to-one map by (12.10), (12.1). Let Q : B(6R) → X be defined by x Q = x W (S, T ) − x . Lemma B.1 may be applied with R1 = 6R,

R2 = 5R,

ξ = min{ 13 , 32 R} .

Hence (cf. (B.3)) there exists x ∈ B(6R) such that x W (a, S) = x + x Q = y . (12.16) holds. Finally, (12.17) is a consequence of (12.8).



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12.6. Lemma. Let x ∈ B(6R), u(t) = x W (a, t) for t ∈ [a, b ]. Then ∥u(τ ) − u(t¯)∥ ≤ ψ2 (τ − t¯) + A(2 (τ − t¯)) and ∥u(t) − u(t¯) − G(u(τ ), τ, t) + G(u(τ ), τ, t¯)∥

(12.18) } (12.19)

≤ A(2(t − t¯)) (1 + ψ1 (t − t¯)) + 2ψ1 (t − t¯) ψ2 (t − t¯) if a ≤ t¯≤ τ ≤ t ≤ b . Proof.

(12.20)

Let (12.20) hold. Then (cf. (12.12)) u(t) = x W (a, t) = x W (a, t¯) W (t¯, t) = u(t¯) W (t¯, t) .

By (12.6), (9.7) ∥u(t) − u(t¯) − u(t¯) G(t¯, t, t¯+ b − a)∥ ≤ A(2 (t − t¯)),

 

∥u(t¯) G(t¯, t, t¯+ b − a)∥ ≤ ψ2 (t − t¯)



(12.21)

and (12.18) is correct. (9.3) and (9.2) imply that u(t¯) G(t¯, t, t¯ + b − a) = G(u(t¯), t¯, t) − G(u(t¯), t¯, t¯) since e τ, t) = G(x, τ, t) for τ, t ∈ [a, b ], x ∈ B(8R) . G(x, (12.20) and (12.21) imply that ∥u(t) − u(t¯) − G(u(t¯), t¯, t) + G(u(t¯), t¯, t¯)∥ ≤ A(2 (t−t¯)) .

(12.22)

Finally, u(t) − u(t¯) − G(u(τ ), τ, t) + G(u(τ ), τ, t¯) = A + B + C

(12.23)

where A = u(t) − u(t¯) − G(u(t¯), t¯, t) + G(u(t¯), t¯, t¯) , B = G(u(t¯), t¯, t) − G(u(t¯), t¯, t¯) − G(u(τ ), t¯, t) + G(u(τ ), t¯, t¯) , C = G(u(τ ), t¯, t) − G(u(τ ), t¯, t¯) − G(u(τ ), τ, t) + G(u(τ ), τ, t¯)

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and (cf. (12.21), (8.4), (12.18), (8.6)) ∥A∥ ≤ A(2(t − t¯)) ,

}

∥B∥ ≤ ∥u(τ ) − u(t¯)∥ ψ1 (τ − t¯) ≤ ψ1 (t − t¯) ψ2 (t − t¯) + A(2 (t − t¯)) ψ1 (t − t¯) , ∥C∥ ≤ ψ2 (t − t¯)ψ1 (t − t¯) .

(12.24) (12.25) (12.26) 

(12.19) holds by (12.1), (12.23)–(12.26).

12.7. Lemma. Let x ∈ B(6R), u(t) = xW (a, t) for t ∈ [a, b ], ε > 0. Then there exists ξ > 0 such that t − t¯ ∥u(t) − u(t¯) − G(u(τ ), τ, t) + G(u(τ ), τ, t¯)∥ ≤ ε b−a

(12.27)

if u(t) = x W (a, t)

for t ∈ [a, b ] ,

a ≤ t¯ ≤ τ ≤ t ≤ b, t − t¯ ≤ ξ . Proof.

(12.28) (12.29)

(7.1)–(7.4) and (7.16), (7.19) imply that

ψ1 (σ) → 0, σ −1 A(σ) → 0, σ −1 ψ1 (σ) ψ2 (σ) → 0 for σ → 0 + . Hence Lemma 12.7 is a consequence of Lemma 12.6.



12.8. Corollary. u is an SR-solution of (8.8). 12.9. Corollary. ∂ (u(t) − G(u(τ ), τ, t))|t=τ = 0 ∂t

for τ ∈ [a, b ] .

d u(t) | ∂ t=τ exists if and only if ∂t G(u(τ ), τ, t) |t=τ exists. dt 12.10. Theorem. Let y ∈ B(4R), s ∈ [a, b ]. Then

Hence

there exists x ∈ B(5R)

such that x W (a, s) = y .

(12.30)

Put u(t) = x W (a, t) for t ∈ [a, b ]. Then u

is an SR-solution of (8.8) on [a, b ] ,

∥u(t) − y∥ ≤ R Moreover, x is unique.

for t ∈ [a, b ] .

(12.31) (12.32)

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Proof. (12.30) holds by (12.16), (12.8). Moreover, (12.31) is true by Corollary 12.8. If t < s then y − u(t) = x W (a, t) W (t, s) − x W (a, t) and ∥u(t) − y∥ ≤ R by (12.8) and (12.1). If s < t then u(t) − y = x W (a, s) W (s, t) − x W (a, s) and again ∥u(t) − y∥ ≤ R by (12.8). Hence (12.32) holds. Let x ¯ ∈ B(5R) be such that x ¯ W (a, s) = y. Then by (12.9), (12.1) and (12.2) ∥x − x ¯∥ = ∥x W (a, s) − x ¯ W (a, s) − x + x ¯∥ 1 ≤ ∥x − x ¯∥ (ψ1 (ξ5 ) + B(ξ5 )) ≤ ∥x − x ¯∥ , 3 

i.e. x = x ¯.

12.11. Theorem. Let [S, T ] ⊂ [a, b ] and let w : [S, T ] → B(4R) be an SRsolution of (8.8). Then there exists a unique z ∈ B(5R) such that w(s) = z W (a, s)

for s ∈ [S, T ] .

(12.33)

Proof. Theorem 12.11 is a particular case of Theorem 16.1 since every SKH-solution of (8.8) is an SR-solution of (8.8).  12.12. Theorem. Let y ∈ B(3R), s ∈ [S, T ] ⊂ [a, b ]. Let v : [S, T ] → B(8R) be an SR-solution of (8.8), v(s) = y. Then ∥v(t)∥ ≤ ∥y∥ + 32 R

for t ∈ [S, T ]

and there exists a unique z ∈ B(∥y∥ + 23 R)

(12.34) }

such that v(t) = z W (a, t) for t ∈ [S, T ] . Proof.

Put Sb = inf{S1 ; S ≤ S1 ≤ s, ∥v(t)∥ ≤ 4R for S1 ≤ t ≤ s} , Tb = sup{T1 ; s ≤ T1 ≤ T, ∥v(t)∥ ≤ 4R for s ≤ t ≤ T1 } .

(12.35)

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Then Sb < s < Tb since v is continuous by Lemma 6.5. By Theorem 12.11 b Tb]. there exists z ∈ B(5R) such that v(t) = z W (a, t) for t ∈ [S, If Sb ≤ t ≤ s write v(t) − y = z W (a, t) − z W (a, t) W (t, s) and (cf. (12.8), (12.1)) ∥v(t) − y∥ ≤ ψ2 (s − t) + A(2s − 2t) ≤

2 R. 3

Hence ∥v(t)∥ ≤ ∥y∥ +

2 11 R≤ R, 3 3

which implies that Sb = S

and ∥v(t)∥ ≤ ∥y∥ +

2 R 3

for S ≤ t ≤ s .

(12.36)

If s ≤ t ≤ T write v(t) − y = z W (a, s) W (s, t) − z W (a, s) and by a similar argumentation Tb = T

and ∥v(t)∥ ≤ ∥y∥ +

2 R 3

for s ≤ t ≤ T .

(12.37)

Now, (12.34) and (12.35) hold by (12.36) and (12.37). The proof is complete. 

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Chapter 13

Continuous dependence

13.1 . Notation. Let (12.1), (12.2) hold and let η > 0. Assume that G : B(8R) × [a, b ] 2 → X and G∗ : B(8R) × [a, b ] 2 → X fulfil (8.2)–(8.7), ∥G∗ (x, τ, t) − G(x, τ, t)∥ ≤ η for x ∈ B(8R), τ, t ∈ [a, b ] .

(13.1)

e ∗ , G∗ , V ∗ , V ∗ , W ∗ , E ∗ in the same way as Starting from G∗ introduce G i S,T e G, Vi , V, W, Eσ were introduced starting from G. G, 13.2. Lemma. Let [S, S + τ ] ⊂ [a, b ], z ∈ B(6R). Then ∥z W (S, S + τ ) − z W ∗ (S, S + τ )∥ ≤ η + 2 A(2 τ ) .

(13.2)

Proof. (13.2) holds by (12.6), (9.3) and by an analogous inequality for W ∗ .  13.3. Lemma. Assume that ℓ ∈ N0 and that b Sb + 2 ℓ τ ) − z W ∗ (S, b Sb + 2 ℓ τ )∥ ∥z W (S, ≤ (η + 2 A(2τ ))2ℓ−1 exp

ℓ−1 (∑

) (ψ1 (2j τ ) + B (2j+1 τ ))

      

(13.3)

j=1

b Sb + 2 ℓ τ ] ⊂ [a, b ] and z ∈ B(5R), where for the case ℓ = 1 we take for [S, 0 ∑

(ψ1 (2j τ ) + B (2j+1 τ )) = 0 .

j=1

77

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Then ∥x W (S, S + 2ℓ+1 τ ) − x W ∗ (S, S + 2ℓ+1 τ )∥ ℓ (∑ ) ≤ (η + 2 A(2τ )) 2 ℓ exp (ψ1 (2j τ ) + B (2j+1 τ )) j=1

    (13.4)

  

for [S, S + 2ℓ+1 τ ] ⊂ [a, b ], x ∈ B(5R). Proof.

Let [S, S + 2ℓ+1 τ ] ⊂ [a, b ], x ∈ B(5R) .

Then (cf. (12.12)) x W (S, S + 2ℓ+1 τ ) − x W ∗ (S, S + 2ℓ+1 τ ) = J + K ,

(13.5)

where J = x W (S, S + 2 ℓ τ ) W (S + 2 ℓ τ, S + 2ℓ+1 τ ) − x W (S, S + 2 ℓ τ ) W ∗ (S + 2 ℓ , S + 2ℓ+1 τ ) , K = x W (S, S + 2 ℓ τ ) W ∗ (S + 2 ℓ τ, S + 2ℓ+1 τ ) − x W ∗ (S, S + 2 ℓ τ ) W ∗ (S + 2 ℓ τ, S + 2ℓ+1 τ ) . Observe that x W (S, S + 2 ℓ τ ) W ∗ (S + 2 ℓ τ, S + 2ℓ+1 τ ) is well defined since x W (S, S + 2 ℓ τ ) ⊂ B(6R). By (13.3) ∥J∥ ≤ (η + 2 A(2τ )) 2ℓ−1 exp

ℓ−1 (∑

) (ψ1 (2j τ ) + B (2j+1 τ ))

(13.6)

j=1

and by (13.3), (12.9) ∥K∥ ≤ (η + 2 A(2 τ )) 2

ℓ−1

[ exp

ℓ−1 (∑

(ψ1 (2 τ ) + B (2 j

j+1

 )]   τ )) × 

( ) × 1 + ψ1 (2 ℓ τ ) + B (2ℓ+1 τ ) . j=1

  



(13.4) is a consequence of (13.5)–(13.7). 13.4. Lemma. Let [S, T ] ⊂ [a, b ], x ∈ B(5R), ℓ ∈ N. Then ∥x W (S, T ) − x W ∗ (S, T )∥ ≤ (η + 2 A(2−ℓ+1 (T − S))) 2ℓ κ ,

(13.7)

  

(13.8)

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where κ = exp

∞ ∑

79

(ψ1 ((b − a) 2−j ) + B ((b − a) 2−j+1 )) .

(13.9)

j=1

Proof.

Lemmas 13.2 and 13.3 imply that    

∥x W (S, S + 2 ℓ τ ) − x W ∗ (S, S + 2 ℓ τ )∥ ≤ (η + 2 A(2 τ )) 2

ℓ−1

exp

ℓ−1 ∑ j=1

(ψ1 (2j τ ) + B (2j+1 τ ))   

(13.10)

for ℓ ∈ N, [S, S + 2 ℓ τ ] ⊂ [a, b ], x ∈ ES . (13.8) is a consequence of (13.10) since the series in (13.9) is convergent by (7.2), (7.4), (7.18) and τ ≤ (b−a) 2−ℓ .  13.5. Lemma. Define { } Ω(η) = min [η + 2 A(2 ( b − a)2−ℓ )] 2 ℓ ; ℓ ∈ N0 for η > 0 .

(13.11)

Then Ω(η) → 0 Proof.

for η → 0 + .

(13.12)

Let ε > 0. By (7.16) 2ℓ+1 A(2(b − a)2−ℓ ) → 0

for ℓ → ∞

and there exists m ∈ N such that 2m+1 A(2 (b − a) 2−m ) <

1 2

ε.

(13.13)

Let 0 < η ≤ 2−m−1 ε . Then [η + 2 A(2 (b − a) 2−m )] 2m < ε . 

Hence (13.12) is valid. 13.6. Lemma. Let S ∈ [a, b ], y ∈ B(5R). Then ∥y W (S, T ) − y W ∗ (S, T )∥ ≤ Ω(η) κ

for T ∈ [S, b ] .

(13.14)

Proof. (13.14) is a consequence of (13.8), (13.11), (13.12) since A is nondecreasing. 

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13.7 . Theorem. Let a ≤ S ≤ s ≤ T ≤ b, S < T, y ∈ B(3R). Assume that v : [S, T ] → B(8R) is an SR-solution of d x = Dt G(x, τ, t) , dt

(13.15)

v(s) = y and that v ∗ : [S, T ] → B(8R) is an SR-solution of d x = Dt G∗ (x, τ, t), dt

(13.16)

v ∗ (s) = y. Then ∥v(t) − v ∗ (t)∥ ≤ Proof.

3 2

Ω(η) κ

for t ∈ [S, T ].

(13.17)

By Theorem 12.12 there exist z, z ∗ ∈ B(∥y∥ + 32 R) such that v(t) = z W (a, t),

v ∗ (t) = z ∗ W ∗ (a, t)

for t ∈ [S, T ].

(13.18)

Lemma 13.6 implies that ∥q W (σ, τ ) − q W ∗ (σ, τ )∥ ≤ Ω(η)κ

} (13.19)

for q ∈ B(5R), a ≤ σ ≤ τ ≤ b. If s ≤ t ≤ T then (cf. Lemma 13.6) ∥v(t) − v ∗ (t)∥ = ∥y W (s, t) − y W ∗ (s, t)∥ ≤ Ω(η)κ.

(13.20)

Let S ≤ t < s. Then (cf. (13.18), (12.12)) y = z W (a, s) = z W (a, t) W (t, s) = v(t) W (t, s) and similarly y = v ∗ (t) W ∗ (t, s). Hence 0 = −v(t) W (t, s) + v ∗ (t)W (t, s) −v ∗ (t) W (t, s) + v ∗ (t) W ∗ (t, s), ∥v(t)−v ∗ (t)∥ ≤ ∥v(t)−v ∗ (t)−v(t)W (t, s)+v ∗ (t)W (t, s)∥ +∥v ∗ (t) W (t, s)−v ∗ (t) W ∗ (t, s)∥.

              

(13.21)

By (12.9) and (12.1) ∥v(t) − v ∗ (t) − v(t) W (t, s) + v ∗ (t) W (t, s)∥ ≤ ∥v(t) − v ∗ (t)∥ 31 ,

(13.22)

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by (13.14) ∥v ∗ (t) W (t, s) − v ∗ (t) W ∗ (t, s)∥ ≤ Ω(η) κ. (13.21)–(13.23) imply that ∥v(t) − v ∗ (t)∥ ≤

3 2

Ω(η) κ.

(13.23)

(13.24)

(13.17) holds by (13.20) and (13.24) and Theorem 13.7 is true.  13.8. Remark. The problem of the necessity of the condition ∞ ∑

2 i ψ1 (2−i ) ψ2 (2−i ) < ∞

(13.25)

i=1

in the above theory was studied by J. Jarn´ık (cf. [Jarn´ık (1961a)],[Jarn´ık (1961b)]). Assume that for ψ1 , ψ2 we have ψ1 = ψ2 ,

(13.26)

while ψ1 , ψ2 fulfil (7.1), (7.2), (7.4), ψ1 (σ1 + σ2 ) ≤ ψ1 (σ1 ) + ψ1 (σ2 ) and

∞ ∑

for σ1 , σ2 ≥ 0

2 i ψ1 (2−i ) ψ2 (2−i ) = ∞.

(13.27)

(13.28)

i=1

Then there exists a sequence of continuous functions ak : [0, 1] → R, bk : [0, 1] → R, k ∈ N, such that the functions ∫ t ∫ t Gk (x, t) = x ak (s) ds + bk (s) ds t ∈ [0, 1], |x| ≤ 1 0

0

fulfil |Gk (x, t2 ) − Gk (x, t1 )| ≤ ψ1 (|t2 − t1 |), |∆v (Gk (x, t2 ) − Gk (x, t1 ))| ≤ ∥v∥ ψ1 (|t2 − t1 |) for x, x + v ∈ [−1, 1], t1 , t2 ∈ [0, 1], Gk (x, t) → 0 Let uk be a solution of d x = Dt Gk (x, t), dt

uniformly for k → ∞.

uk (0) = 0.

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The sequence uk is divergent. If conditions (13.26), (13.27) are added to the theory which is presented in Chapters 8–13 then condition (13.25) becomes necessary. Observe that uk is also a solution of x˙ = x ak (t) + bk (t). For details see [Jarn´ık (1961a)]. On the other hand, there exist couples ψ1 , ψ2 such that (7.1), (7.2), (7.4), (13.28) are fulfilled and the continuous dependence holds. This occurs if ψ1 (σ) → 0 for σ → 0+ σ very slowly, e.g. if ψ1 (σ) → 1 for σ → 0 + . σ | ln σ| See [Jarn´ık (1961b)].

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Chapter 14

Strong Kurzweil Henstock-integration of functions of a pair of coupled variables 14.1 . Notation. Dom U ⊂ R2 , U : Dom U → X, [a, b ] ⊂ R, δ0 : [a, b ] → R+ , {(τ, t) ; τ − δ0 (τ ) ≤ t ≤ τ + δ0 (τ )} ⊂ Dom U, u : [a, b ] → X. If a ≤ σ < b and if u(σ−) =

lim

τ →σ,τ <σ

lim

τ →σ,τ >σ

u(τ ) exists, denote it by u(σ+). Similarly,

u(τ ) if a < σ ≤ b and if the limit exists.

14.2. Definition. A couple ([t¯, t], τ ) or ([¯ s, s], σ) is called a tagged interval, τ, σ being the tags. A finite set A = {([¯ si , si ], σi ) ; i = 1, 2, . . . , k}

(14.1)

is called a system in [a, b ] if σi ∈ [¯ si , si ] ⊂ [a, b ]

for i = 1, 2, . . . , k

and if the intervals [¯ si , si ], [¯ sj , sj ] are nonoverlapping for i ̸= j (i.e. their intersections [¯ si , si ] ∩ [¯ sj , sj ] contain at most one point). Let δ : [a, b ] → R+ . The system A is δ-fine if [t¯, t] ⊂ [τ − δ(τ ), τ + δ(τ )]

for ([t¯, t], τ ) ∈ A.

If ξ ∈ R+ and if t − t¯≤ ξ for ([t¯, t], τ ) ∈ A, then A is called ξ-fine. Let Q ⊂ [a, b ]. The system A is called Q-anchored if τ ∈ Q for ([t¯, t], τ ) ∈ A. The system A is called a partition of [a, b ] if k ∪

[¯ si , si ] = [a, b ].

i=1

83

(14.2)

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14.3. Remark. Any partition A of [a, b ] can be written in the form A = {([ti−1 , ti ], τi ) ; i = 1, 2, . . . , k}

(14.3)

where a = t0 ≤ τ1 ≤ t1 ≤ τ2 ≤ t2 ≤ · · · ≤ tk−1 ≤ τk ≤ tk = b.

(14.4)

The index i is frequently omitted so that the system A is written in the form A = {([t¯, t], τ )} (14.5) and e.g. (14.2) assumes the form ∪

[t¯, t] = [a, b ].

(14.6)

A

14.4. Lemma. (Cousin) Given δ : [a, b ] → R+ , there exists a δ-fine partition A = {([t¯, t], τ )} of [a, b ]. Lemma 14.4 can be found in almost all books on Kurzweil–Henstock integration. The proof can be based on the Heine–Borel covering theorem (e.g. [Kurzweil (1957)], Lemma 1.1.1; [Henstock (1988)], Theorem 3.1) or on the repeated bisection (e.g. [Henstock (1988)], Theorem 4.1; [Pfeffer (1993)], Proposition 1.2.4; [Kurzweil (2000)], Lemma 13.4) or on the concept of supremum (e.g. [Bartle, Sherbert (2000)], Theorem 5.5.5). The SKH-integral (Strong Kurzweil–Henstock) which is introduced in Definitions 14.5 and 14.10, and the SR-integral (Strong Riemann) are useful in the theory of GODEs. 14.5 . Definition. U is SKH-integrable (Strongly Kurzweil–Henstock integrable) on [S, T ] and u is an SKH-primitive of U on [S, T ] if   for every ε > 0 there exists δ : [S, T ] → R+ such that   k ∑ (14.7)  ∥u(ti ) − u(ti−1 ) − U (τi , ti ) + U (τi , ti−1 )∥ ≤ ε   i=1

for every δ-fine partition A = {([ti−1 , ti ], τi ); i = 1, 2, . . . , k} of [S, T ]. Briefly, U is called SKH-integrable and u is called a primitive of U. The couple (τ, t) is called a pair of coupled variables .

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14.6. Remark. If A is written in the form (14.5) then the inequality in (14.7) is replaced by ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε. A

14.7. Lemma. Assume that U is SKH-integrable on [a, b ], u is its primitive and ε > 0. Let δ : [a, b ] → R+ correspond to ε by Definition 14.5 and let [S, T ] ⊂ [a, b ]. Then ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε (14.8) A

for every δ-fine partition A = {([t¯, t], τ )} of [S, T ]. Proof.

The proof is analogous to the proof of Lemma 5.3.

Let A be a δ-fine partition of [S, T ]. By Lemma 14.4 there exist δ-fine partitions B = {([t¯, t], τ )} of [a, S] and C = {([t¯, t], τ )} of [T, b ]. Then ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ A

≤

∑

∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε

A∪B∪C

since A ∪ B ∪ C is a δ-fine partition of [a, b ]. (14.8) is correct.  14.8. Corollary. Let U be SKH-integrable on [a, b ], u being its primitive, [S, T ] ⊂ [a, b ]. Then U is SKH-integrable on [S, T ] and u is its primitive. 14.9 . Lemma. Let U be SKH-integrable on [a, b ], u being its primitive. Then (i) if y ∈ X, v(t) = u(t) + y for t ∈ [a, b ], then v is a primitive of U ; (ii) if v is a primitive of U, then v(t) − u(t) = v(a) − u(a) for t ∈ [a, b ]. The proof runs along the same lines as the proof of Lemma 5.5. 14.10 . Definition. Let U be SKH-integrable on [S, T ] and let u be its primitive. Put ∫ T (SKH) Dt U (τ, t) = u(T ) − u(S). (14.9) S

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∫

T

(SKH)

Dt U (τ, t) is called the strong Kurzweil–Henstock integral of U S

over [S, T ] or shortly, the SKH-integral of U over [S, T ]. ∫ T 14.11. Remark. By Lemma 14.9, (SKH) Dt U (τ, t) is independent of the S

choice of the primitive of U.

14.12. Remark. If ξ > 0 and δ : [S, T ] → R+ , δ(τ ) ≤ ξ/2 for τ ∈ [S, T ] then every δ-fine partition A = {([t¯, t], τ )} of [S, T ] is a ξ-fine partition of [S, T ]. Hence every SR-integrable U is SKH-integrable and ∫ T ∫ T (SKH) Dt U (τ, t) = (SR) Dt U (t, τ ) . S

S

14.13. Lemma. Let a < c < b and let the integrals ∫ c ∫ b (SKH) Dt U (τ, t), (SKH) Dt U (τ, t) a

exist.

(14.10)

c

Then U is SKH-integrable on [a, b ] and ∫ b ∫ c ∫ b (SKH) Dt U (τ, t) = (SKH) Dt U (τ, t) + (SKH) Dt U (τ, t). a

a

(14.11)

c

Proof. Let w1 : [a, c] → X be a primitive of U on [a, c], let w2 : [c, b ] → X be a primitive of U on [c, b ] and let ε > 0. By Lemma 14.4 and Definition 14.5 there exists δ : [a, b ] → R+ such that τ + δ(τ ) < c for τ < c, τ − δ(τ ) > c for τ > c , ∑ ∥w1 (t) − w1 (t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε

(14.12) (14.13)

A

for every δ-fine partition A = {([t¯, t], τ )} of [a, c] and ∑ ∥w2 (t) − w2 (t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε B

for every δ-fine partition B = {([t¯, t], τ )} of [c, b ]. Put  w1 (t) for a ≤ t ≤ c , u(t) = w2 (t) − w2 (c) + w1 (c) for c < t ≤ b .

(14.14)

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Let C = {([t¯, t], τ )} be a δ-fine partition of [a, b ]. (14.12) implies that either } there exist t¯< c and t > c such that (14.15) ([t¯, c ], c) ∈ C, ([c, t], c) ∈ C or there exist s¯ < c and s > c such that

} (14.16)

([¯ s, s], c) ∈ C. If (14.15) holds then ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ C

=

∑

∥w1 (t) − w1 (t¯) − U (τ, t) + U (τ, t¯)∥

C;[t¯,t] ⊂ [a,c]

           

∑

  + ∥w2 (t) − w2 (t¯) − U (τ, t) + U (τ, t¯)∥      C;[t¯,t] ⊂ [c,b ]     ≤ ε + ε.

(14.17)

If (14.16) takes place then u(s) − u(¯ s) − U (c, s) + U (c, s¯) = w2 (s) − w2 (c) + w1 (c) − w1 (¯ s) − U (c, s) − U (c, c) + U (c, c) + U (c, s¯), ∥u(s) − u(¯ s) − U (c, s) + U (c, s¯)∥ ≤ ∥w2 (s) − w2 (c) − U (c, s) + U (c, c)∥ + ∥w1 (c) − w1 (¯ s)∥ − U (c, c) + U (c, s¯)∥. Therefore ∑

∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ C [ ∑ ≤ ∥w1 (t) − w1 (t¯) − U (τ, t) + U (τ, t¯)∥ C;[t¯,t] ⊂ [a,c]

                 

] + ∥w1 (c) − w1 (¯ s) − U (c, c, + U (c, s¯)∥ [    + ∥w2 (s) − w2 (c) − U (c, s) + U (c, c)∥   ]  ∑   + ∥w2 (t) − w2 (t¯) − U (τ, t) + U (τ, t¯)∥      ¯ C;[t,t] ⊂ [c,b ]    ≤ 2 ε.

(14.18)

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By (14.17) and (14.18) ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ 2 ε. C



U is SKH-integrable on [a, b ] and u is its primitive. b : [−1, 1] → {0, 1} be defined by 14.14. Remark. Let H, H { { 0 for − 1 ≤ t ≤ 0, 0 for − 1 ≤ t < 0, b H(t) = H(t) = 1 for 0 < t ≤ 1, 1 for 0 ≤ t ≤ 1. Then

∫

0

(R) ∫

b dH = 0, H

−1 1

but (R)

∫

1

(R)

b dH = 1, H

0

b dH does not exist. On the other hand, H

−1

∫

1

(SKH) −1

b ) H(t)) = 1 Dt (H(τ

(cf. Lemma 14.13 and Remark 14.12). Of course, ∫ 0 ∫ 0 ∫ b ) H(t)) = (R) b dH(t) = (R) (SR) Dt (H(τ H(t) −1

−1

0

b dH H

etc.

−1

14.15. Theorem. The three conditions below are equivalent: ∫

T

(i) u(T ) − u(S) = (SKH)

Dt U (τ, t) for [S, T ] ⊂ [a, b ], S

(ii) U is SKH-integrable on [a, b ] and u is its primitive, (iii) for ε > 0 there exists δ : [a, b ] → R+ such that ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε A

if A = {([t¯, t], τ )} is a δ-fine partition of [a, b ]. ∫T Proof. Let (i) hold. Then (SKH) S Dt U (τ, t) exists for [S, T ] ⊂ [a, b ], i.e. U is SKH-integrable on [a, b ] and u : [a, b ] → X is an SKH-primitive of U. Hence (ii) is valid. Let (ii) hold. Then (iii) is valid by Definition 14.5. Let (iii) hold. Then (i) is true by Definitions 14.5 and 14.10.



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14.16. Remark. Let a ≤ S < T ≤ b and let U be SKH-integrable on [S, T ], u being its primitive. It is convenient to put  ∫ S ∫ T   (SKH) Dt U (τ, t) = −(SKH) Dt U (τ, t) = u(S)−u(T ),   T S (14.19) ∫ S    (SKH) Dt U (τ, t) = 0.  S

14.17. Lemma. Let U be SKH-integrable on [a, b ], u : [a, b ] → X being its primitive. Then u(T ) − u(σ) − U (σ, T ) + U (σ, σ) → 0

for T → σ.

(14.20)

Proof. Let ε > 0 and let δ : [a, b ] → R+ correspond to ε by Theorem 14.15 (iii). Let a ≤ σ < T ≤ σ + δ(σ) ≤ b. By Lemma 14.4 there exists a δ-fine partition T = {([t¯, t], τ )} of [a, b ] such that ([σ, t], σ) ∈ T . Then ∥u(T ) − u(σ) − U (σ, T ) + U (σ, σ)∥ ∑ ≤ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε . T

Similarly if a < σ − δ(σ) ≤ T < σ ≤ b then ∥u(T ) − u(σ) − U (σ, T ) + U (σ, σ)∥ ≤ ε . 

Hence (14.20) holds. The proof is complete. 14.18. Corollary. u(σ+) exists if and only if U (σ, σ+) exists and then u(σ+) − u(σ) = U (σ, σ+) − U (σ, σ) .

(14.21)

Let a ≤ S < T ≤ b and let u(S+) exist. The equation u(T ) − u(S) = u(T ) − u(S + ) + u(S + ) − u(S) can be written in the form ∫ T ∫ (SKH) Dt U (τ, t) = (SKH) S

T

∫ Dt U (τ, t)+(SKH)

S+

where

∫

S+

Dt U (τ, t) (14.22) S

S+

Dt U (τ, t) = U (S, S + ) − U (S, S).

(SKH) S

(14.23)

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Similarly u(σ−) exists if and only if U (σ, σ−) exists and u(σ) − u(σ−) = U (σ, σ) − U (σ, σ−) . Let u(T −) exist. Then ∫ T ∫ (SKH) Dt U (τ, t) = (SKH) S

∫ Dt U (τ, t)+(SKH)

T− S

where

∫

(14.24)

T

Dt U (τ, t) (14.25)

T−

T

(SKH) T−

Dt U (τ, t) = U (T, T ) − U (T, T −).

(14.26)

In particular, u is continuous at σ if and only if U (σ, .) is continuous at σ. 14.19 . Lemma. Let [a, b ] ⊂ R, let Φ : [a, b ] → R be nondecreasing and let U fulfil ∥U (τ, t) − U (τ, t¯)∥ ≤ Φ(t) − Φ(t¯)

for (τ, t), (τ, t¯) ∈ Dom U.

Assume that U is SKH-integrable and that u : [a, b ] → X is its primitive. Then ∥u(T ) − u(S)∥ ≤ Φ(T ) − Φ(s) Proof.

for T, S ∈ [a, b ].

(14.27)

Let a ≤ S < T ≤ b. For ε > 0 there exists δ : [a, b ] → R+ such that ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε, T

T being a δ-fine partition of [S, T ]. Hence ∑ ∥u(T ) − u(S)∥ ≤ ∥u(t) − u(t¯)∥ ≤

∑

T

∥U (τ, t) − U (τ, t¯)∥ + ε ≤ Φ(T ) − Φ(s) + ε

T

and (14.27) holds since ε is arbitrary positive. The proof is complete.



14.20. Theorem. Let U be SKH-integrable on [S, b ] for a < S < b, w ∈ X and ∫ b ( ) (14.28) lim −(SKH) Dt U (τ, t)+w−U (a, S)+U (a, a) = 0. S→a

S

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91

Then U is SKH-integrable on [a, b ] and ∫ (SKH)

b

Dt U (τ, t) = w. a

Proof.

Let  ∫ b   − (SKH) Dt U (τ, t) for a < S ≤ b, u(S) = S   − w for S = a.

(14.29)

Then ∫

b

u(b) = 0, u(b) − u(S) = (SKH)

Dt U (τ, t) for a < S ≤ B S

and u is an SKH-primitive of U on [S, b ] for a < S < b. Put ai = a + (b−a) 2−i for i ∈ N0 . There exist δi : [ai , b ] → R+ for i = 2, 3, 4, . . . such that ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε 2−i−2 (14.30) A

for a δi -fine system A in [ai+2 , b ] (cf. Lemma 14.4). By (14.28) there exists ξ, a < ξ < b, such that ∫ b

ε

for a < S ≤ ξ.

−U (a, S) + U (a, a) − (SKH) Dt U (τ, t) + w ≤ 2 S Hence ∥u(S) − u(a) − U (a, S) + U (a, a)∥ ≤

ε 2

(14.31)

since (cf. (14.29)) u(S) − u(a) − U (a, S) + U (a, a) ∫ b = −(SKH) Dt U (τ, t) + w − U (a, S) + U (a, a). S

There exists δ : [a, b ] → R fulfilling +

 ξ    δ(τ ) =

for τ = a,

min{(b − a) 2−i−2 , δi (τ )}    for ai+1 < τ ≤ ai , i ∈ N0 .

(14.32)

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If a < τ ≤ b then there exists a unique i ∈ N0 such that ai+1 < τ ≤ ai ,

ai+2 = ai+1 −

b−a < τ − δ(τ ), 2i+2

[τ − δ(τ ), b] ⊂ [ai+2 , b] .

  

(14.33)

Let A = {([t¯, t], τ )} be a δ-fine partition of [a, b ]. (14.32) implies that there exists ([a, S], a) ∈ A, a < S ≤ ξ. Let Ak = {([t¯, t], τ ) ∈ A ; ak+1 < τ ≤ ak } for k ∈ N0 . If ([t¯, t], τ ) ∈ Ak , then τ ∈ [ak+1 , ak ], [t¯, t] ⊂ [ak+2 , b ] and by (14.30) ∑ ∥u(t) − u(t¯) − U (τ, t) + u(τ, t¯)∥ ≤ ε 2−k−2 . (14.34) Ak

Moreover, A = {([a, S], a)} ∪

∞ ∪

Ak .

k=0

Hence (cf. (14.26), (14.24)) ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ A

= ∥u(S) − u(a) − U (S, a) + U (a, a)∥ ∞ ∑ ∑ + ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ k=0 Ak

which implies that U is SKH-integrable on [a, b ], that u is its primitive and ∫ b (SKH) Dt U (τ, t) = u(b) − u(a) = w . a  14.21 . Remark. Theorem 14.20 is an extension of Hake’s theorem (cf. [Schwabik, Ye (2005)], Theorem 3.4.5 and Remark) to the integration of functions of a pair of coupled variables. The next theorem is an analogue of Theorem 14.20 and can be proved in a similar way. 14.22. Theorem. Let U be SKH-integrable on [a, T ] for a < T < b, x ∈ X and let ∫ T ( ) lim − (SKH) Dt U (τ, t) + w + U (b, t) − U (b, b) = 0 . T →b

a

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Then U is SKH-integrable on [a, b ] and ∫ b (SKH) Dt U (τ, t) = w. a

14.23. Remark. Let f : [a, b ] → X and U (τ, t) = f (τ ) t

for (τ, t) ∈ [a, b ] 2 .

Then f is called SKH-integrable on [a, b ] if U is SKH-integrable on [a, b ] ∫ b ∫ b and (SKH) f (t) dt is written instead of (SKH) Dt U (τ, t). Furthermore, a

a

u is called an SKH-primitive of f if u is a primitive of U. Therefore f is SKH-integrable on [S, T ] and u is its primitive if and only if for every ε > 0 there exists δ : [a, b ] → R+ such that ∑ ∥f (τ )(t − t¯) − u(t) + u(t¯)∥ ≤ ε (14.35) A

for every δ-fine partition A = {([t¯, t], τ )} of [S, T ]. Moreover, ∫ b (SKH) f (t) dt = u(b) − u(a). a

14.24. Lemma. Let Q ⊂ [a, b ], |Q| = 0, h : Q → R+ 0 , ε > 0. Then there exists ξ : Q → R+ such that ∑ h(τ ) (t − t¯) ≤ ε A

if A = {([t¯, t], τ )} is a ξ-fine Q-anchored system in [a, b ]. Hint. Let Q0 = ∅, Qj = {t ∈ Q ; h(t) ≤ 2j }, j ∈ N. There exist open sets Gj ⊂ R, j ∈ N, such that Qj ⊂ Gj , |Gj | ≤ ε 2−2 j . Let (t − ξ(t), t + ξ(t)) ⊂ Gj for t ∈ Qj \ Qj−1 . Then ξ does the job. 14.25. Theorem. Let f : [a, b ] → X, u : [a, b ] → X. Then f is SKH -integrable and u is its primitive on [a, b ]

(14.36)

if and only if there exists Q ⊂ [a, b ] such that |Q| = 0, d u(t) = f (t) dt and

for t ∈ [a, b ] \ Q,

  

(14.37)

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for each ε > 0 there exists δ4 : [a, b ] → R+ such that ∑ ∥u(t) − u(t¯)∥ ≤ ε for each A

δ4 -fine Q-anchored system A = {([t¯, t], τ )} in [a, b ].

        

(14.38)

Proof. Let (14.37) and (14.38) hold. Then for every ε > 0 there exists δ1 : [a, b ] → R+ such that ∑

∥u(t) − u(t¯) − f (τ ) (t − t¯)∥+

([t¯,t],τ ) ∈ A, τ ∈ [a,b ]\Q

∑

∥u(t) − u(t¯)∥ ≤

([t¯,t],τ ) ∈ A, τ ∈Q

ε 2

for every δ1 -fine partition A = {([t¯, t], τ )} of [a, b ]. Moreover (cf. Lemma 14.24), there exists δ2 : [a, b ] → R+ such that ∑ C

∥f (τ )∥(t − t¯) ≤

ε 2

for every δ2 -fine Q-anchored system C = {([t¯, t], τ )} in [a, b ]. Hence ∑ ∥f (τ ) (t − t¯) − u(t) + u(t¯)∥ ≤ ε A

for a δ-fine partition A = {([t, t¯], τ )} of [a, b ] where δ(τ ) = δ1 (τ ) if τ ∈ [a, b ] \ Q and δ(τ ) = δ2 (τ ) if τ ∈ Q. On the other hand, let (14.36) hold. By Theorem 7.4.2 in [Schwabik, Ye (2005)] there exists Q such that (14.37) holds. Let ε > 0. By Lemma 14.24 there exists δ4 : [a, b ] → R+ which fulfils ∑ ∥f (τ )∥ (t − t¯) ≤ 21 ε (14.39) A

for every δ4 -fine Q-anchored system A = {([t¯, t], τ )} in [a, b ]. (14.38) is correct.  14.26 . Remark. The concept of a real-valued ACG∗ function was extended to Banach-valued functions in a natural way (see [Schwabik, Ye (2005)], Definition 7.1.5 (d)). By [Schwabik, Ye (2005)], Theorem 7.4.5, the function f : [a, b ] → X is SKH-integrable on [a, b ] and v : [a, b ] → X is its primitive if and only if v is continuous and ACG∗ on [a, b ] such that dv (t) = f (t) almost everywhere in [a, b ]. dt

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Observe that the set of primitives is the set of continuous ACG∗ functions. 14.27. Remark. If φ : [a, b ] → R is Lebesgue integrable, c ∈ [a, b ], ∫ t u(t) = (L) φ(s) ds (Lebesgue integral) for t ∈ [a, b ] c

then (14.37), (14.38) are fulfilled and, by Theorem 14.25, φ is SKH-integrable on [a, b ] and ∫ t ∫ t (SKH) φ(s) ds = (L) φ(s) ds, t ∈ [a, b ]. c

c

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Chapter 15

Generalized differential equations: Strong Kurzweil Henstock solutions

15.1. Notation. Let Dom G ⊂ X × R2 , δ0 : [a, b ] → R , +

G : Dom G → X,

[a, b ] ⊂ R,

u : [a, b ] → X,

(u(τ ), τ, t) ∈ Dom G for τ, t ∈ [a, b ], τ − δ0 (τ ) ≤ t ≤ τ + δ0 (τ ). 15.2. Definition. u is an SKH-solution of d x = Dt G(x, τ, t) dt

(15.1)

on [a, b ] if ∫ T u(T ) − u(S) = (SKH) Dt G(u(τ ), τ, t) for [S, T ] ⊂ [a, b ].

(15.2)

S

15.3. Theorem. The following three assertions are equivalent: (i)

u is an SKH-solution of (15.1),

(ii)

U is SKH-integrable on [a, b ] and u is its primitive, U (τ, t) = G(u(τ ), τ, t),

(iii)

for every ε > 0 there exists a δ : [a, b ] → R+ such that ∑ ∥u(t) − u(t¯) − G(u(τ ), τ, t) + G(u(τ ), τ, t¯)∥ ≤ ε, A

A = {([t¯, t], τ )} Proof.

being any δ-fine partition of [a, b ] .

The theorem is a consequence of Theorem 14.15. 97



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15.4. Theorem. If u is an SR-solution of (15.1) then it is an SKH-solution of (15.1). Proof. This is a consequence of Definitions 15.2 and 6.2 (cf. also Remark 14.12).  15.5. Lemma. Let u be an SKH-solution of (15.1) on [a, b ], [S, T ] ⊂ [a, b ]. Then u is an SKH-solution of (15.1) on [S, T ]. Proof.



This is a consequence of Corollary 14.8.

15.6. Lemma. Let a < c < b and let u be an SKH-solution of (15.1) on [a, c] and on [c, b ]. Then u is an SKH-solution of (15.1) on [a, b ]. Proof.



u is an SKH-solution of (15.1) on [a, b ] by Lemma 14.13.

15.7 . Lemma. Let u be an SKH-solution of (15.1)) on [a, b ], c ∈ [a, b ]. Assume that G(u(c), c, ·) is continuous at c. Then u is continuous at c. Proof.



u is continuous at c by Corollary 14.18.

15.8. Theorem. Let u : [a, b ] → X be an SKH-solution of (15.1) on [c, b ] for a < c < b. Assume that } u(S) − u(a) − G(u(a), a, S) + G(u(a), a, a) → 0 (15.3) for S → a, S > a . Then u Proof.

is an SKH-solution of (15.1) on [a, b ] .

(15.4)

Put U (τ, t) = G(u(τ ), τ, t) for τ, t ∈ [a, b ], |t − τ | ≤ δ0 (τ ) .

Then

∫

b

u(c) = u(b) −(SKH)

Dt U (τ, t) , c

since u is an SKH-solution of (15.1) on [c, b ]. Put w = u(b) − u(a). Then ∫ −(SKH)

b

U (τ, t) + w − U (a, c) + U (a, a) c

= u(c) − u(a) − G(u(a), a, c) + G(u(a), a, a) .

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Hence (14.28) is fulfilled by (15.3).

∫ b Theorem 14.20 implies that (SKH) Dt G(u(τ ), τ, t) exists and equals

u(b) − u(a). Theorem 15.8 is correct.

a



15.9. Remark. Theorem 15.8 is an extension of Hake’s theorem to SKHsolutions of (15.1), cf. Remark 14.21. b > 0. Assume that the functions 15.10. Lemma. Let R b × [−b, −a] 2 → X, G : B(R)

b × [a, b ] 2 → X , H : B(R)

b u : [−b, −a] → B(R),

b , w : [a, b ] → B(R)

δ : [−b, −a] → R ,

ρ : [a, b ] → R+

+

are coupled by the relations G(x, τ, t) = H(x, σ, s), τ = −σ,

t = −s

u(τ ) = w(σ),

δ(τ ) = ρ(σ) ,

b τ, t ∈ [a, b ] . for x ∈ B(R),

Then u is an SKH-solution of (15.1) on [−b, −a] if and only if w is an SKH-solution of d x = Ds H(x, σ, s) (15.5) ds on [a, b ]. Proof.

Let the sets

{t0 , τ1 , t1 , τ2 . . . , τk , tk } ⊂ [−b, −a],

{s0 , σ1 , s1 , σ2 , . . . , σk , sk } ⊂ [a, b ]

be coupled by si = −ti for i = 0, 1, . . . , k,

σi = −τi for i = 1, 2, . . . , k .

Then {([ti−1 , ti ], τi ) ; i = 1, 2, . . . , k} is a δ-fine partition of [−b, −a] if and only if {([si , si−1 ], σi ) ; i = 1, 2, . . . , k} is a ρ-fine partition of [a, b ]. Moreover, ∥u(ti ) − u(ti−1 ) − G(u(τi ), τi , ti ) + G(u(τi ), τi , ti−1 )∥ = ∥w(si−1 ) − w(si ) − H(w(σi ), σi , si−1 ) + H(w(σi ), σi , si )∥ and the lemma is correct.



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15.11. Remark. Let Dom g ⊂ X× R, u : [a, b ] → X and let (u(t), t) ∈ Dom g for t ∈ [a, b ]. Then u is called an SKH-solution of d x = g(x, t) dt

(15.6)

on [a, b ] if (cf. Remark 14.23) ∫ T u(T ) − u(S) = (SKH) g(u(t), t) dt for [S, T ] ⊂ [a, b ] .

(15.7)

S

15.12. Theorem. u is an (SKH)-solution of (15.6) on [a, b ] if and only if  there exists Q ⊂ [a, b ] such that |Q| = 0,  (15.8) d  u(t) = g(u(t), t) for t ∈ [a, b ] \ Q dt and for each ε > 0 there exists δ4 : [a, b ] → R+ such that ∑ ∥u(t) − u(t¯)∥ ≤ ε for each δ4 −fine A

Q−anchored system A = {([t¯, t], τ )} in [a, b ] . This is a consequence of Theorem 14.25.

        

(15.9)

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Chapter 16

Uniqueness

16.1. Notation. Let a, b, c fulfil (12.2). Assume that G : B(8R) × [a, c] 2 → X

fulfils (8.2)–(8.7) .

(16.1)

16.2. Theorem. Let [S, T ] ⊂ [a, b ] and let w : [S, T ] → B(4R) be an SKHsolution of (8.8). Then there exists a unique z ∗ ∈ B(5R) such that w(s) = z ∗ W (a, s) Proof.

for s ∈ [S, T ] .

(16.2)

By Theorem 12.10 there exists a z ∗ ∈ B(5R) such that w(S) = z ∗ W (a, S) .

(16.3)

z ∗ is unique by (12.8) and (12.1). Let 0 < ε < 31 , S < s ≤ T. By Theorem 15.3 there exists δ1 : [S, T ] → R+ such that ∑ ∥w(t) − w(t¯) − G(w(τ ), τ, t) + G(w(τ ), τ, t¯)∥ ≤ ε (16.4) A

for every δ1 -fine partition A = {([t¯, t], τ )} of [S, s]. Moreover, by Lemma 12.7 there exists η, 0 < η < s − S such that   ∥x W (a, t) − x W (a, t¯)   −G(x W (a, τ ), τ, t) + G(x W (a, τ ), τ, t¯)∥ (16.5)    −1 ≤ ε (t − t¯) (b − a) if a ≤ t¯≤ τ ≤ t ≤ s, [t¯, t] ⊂ [τ − η2 , τ + η2 ] . 101

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Put

{ } δ2 (τ ) = min δ1 (τ ), η2 , 12 (τ − S), 12 (s − τ ) {

for S < τ < s ,

}

        (16.6)

      

δ2 (S) = min δ1 (S), η2 , { } δ2 (s) = min δ1 (s), η2 .

By Lemma 14.4 there exists a δ2 -fine partition C = {([ti−1 , ti ], τi ) ; i = 1, 2, . . . , k} of [S, s], i.e. S = t0 ≤ τ1 ≤ t1 ≤ τ2 ≤ . . . ≤ tk−1 ≤ τk ≤ tk = s, t0 < t1 < · · · < tk−1 < tk . (16.6) implies that S < τ − δ2 (τ ) < τ + δ2 (τ ) < s if S < τ < T so that S = t0 = τ1 < t1 , tk−1 < τk = tk = s .

(16.7)

By Theorem 12.10 there exist zi ∈ B(5R) such that zi W (a, τi ) = w(τi ),

i = 1, 2, . . . , k .

Observe that z ∗ = z1 .

(16.8)

ui (t) = zi W (a, t) for t ∈ [a, b ], i = 1, 2, . . . , k .

(16.9)

Put By Theorem 12.10, ui are SR-solutions of (8.8) fulfilling ui (τi ) = w(τi ),

i = 1, 2, . . . , k .

(16.10)

Partition C is modified to D = {([ti−1 , τi ], τi ), ([τi , ti ], τi ),

i = 1, 2, . . . , k}

which again is a δ2 -fine partition of [S, s]. Put Λi = ∥w(ti ) − w(τi ) − G(w(τi ), τi , ti ) + G(w(τi ), τi , τi )∥,

} (16.11)

i = 1, 2, . . . , k − 1 ,

λi = ∥w(ti ) − w(τi+1 ) − G(w(τi+1 ), τi+1 , ti ) + G(w(τi+1 ), τi+1 , τi+1 )∥, i = 1, 2, . . . , k − 1 .

} (16.12)

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Then k−1 ∑

(Λi + λi ) ≤ ε

(16.13)

i=1

since D is δ2 -fine. Moreover, (cf. (16.5)) ∥ui (ti ) − ui (τi ) − G(ui (τi ), τi , ti ) + G(ui (τi ), τi )∥ ≤ε

ti − τ i , b−a



i = 1, 2, . . . , k − 1 ,

∥ui+1 (ti ) − ui+1 (τi+1 ) −G(ui+1 (τi+1 ), τi+1 , ti ) + G(ui+1 (τi+1 ), τi+1 , τi+1 )∥ τi+1 − ti ≤ε , b−a

 

i = 1, 2, . . . , k − 1 .

(16.14)

          

(16.15)

(16.11) and (16.14) imply that ∥w(ti ) − ui (ti )∥ ≤ Λi + ε

ti − τ i , b−a

i = 1, 2, . . . , k − 1

(16.16)

since ui (τi ) = w(τi ). Similarly (16.12) and (16.15) imply that ∥w(ti ) − ui+1 (ti )∥ ≤ λi + ε

τi+1 − ti , b−a

i = 1, 2, . . . , k − 1

(16.17)

since ui+1 (τi+1 ) = w(τi+1 ). By (16.13), (16.16), (16.17) ∥ui (ti ) − ui+1 (ti )∥ ≤ ε + ε

τi+1 − τi 2 ≤ , b−a 3

i = 1, 2, . . . , k − 1

(16.18)

since ε < 13 . By (12.10), where S = a, t = ti , v = ui (a) = zi , v¯ = ui+1 (a) = zi+1 , v W (a, ti ) = ui (ti ),

v¯ W (a, ti ) = ui+1 (ti ) ,

we have ∥u( ti ) − ui+1 (ti )∥ ∥ui (a) − ui+1 (a)∥ ≤ 1 − ψ2 (t − S) − B (2 (t − S)) ( 3 τi+1 − τi ) ≤ Λ i + λi + ε , i = 1, 2, . . . , k − 1 , 2 b−a

        

(16.19)

and ∥u1 (a) − uk (a)∥ ≤

3 2ε = 3ε 2

(16.20)

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(cf. (16.13), (12.1)), (12.2)). Finally, u1 (a) = z1 = z ∗ (cf. (16.8)) , uk (a) = zk , ∥zk − z ∗ ∥ ≤ 3 ε , zk W (a, s) = uk (s) = w(s) and (cf. (12.8)) ∥z ∗ W (a, s) − zk W (a, s)∥ ≤

4 ∗ ∥z − zk ∥ ≤ 4 ε , 3

i.e., ∥z ∗ W (a, s) − w(s)∥ ≤ 4 ε , which proves (16.2).



16.3. Remark. Every (local) SKH-solution u of (8.8) is by Theorem 16.2 an SR-solution of (8.8) and is uniquely determined by the Cauchy condition u(s) = y.

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Chapter 17

Differential equations in classical form

17.1. Notation. Dom g ⊂ X × R, g : Dom g → X, [a, b ] ⊂ R, u : [a, b ] → X, (u(τ ), τ ) ∈ Dom g for τ ∈ [a, b ]. 17.2. Definition. u is an SKH-solution of x˙ = g(x, t) if

(17.1)

∫ T u(T ) − u(S) = (SKH) g(u(t), t) dt for [S, T ] ⊂ [a, b ] .

(17.2)

S

17.3. Theorem. The seven conditions below are equivalent. (i) u is an SKH-solution of (17.1), ∫ T (ii) u(T ) − u(S) = (SKH) Dt U (τ, t), where S

U (τ, t) = g(u(τ ), τ ) t for τ, t ∈ [a, b ], a ≤ S < T ≤ b, (iii) U is SKH-integrable on [a, b ], u being its primitive, (iv) for every ε > 0 there exists a δ : [a, b ] → R+ such that ∑ ∥u(t) − u(t¯) − g(u(τ ), τ ) (t − t¯)∥ ≤ ε, A

A being any δ-fine partition of [a, b ], (v) there exists Q ⊂ [a, b ], |Q| = 0 such that du (t) = g(u(t), t) dt 105

for t ∈ [a, b ] \ Q

(17.3)

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and for every ε > 0

there exists a δ1 : [a, b ] → R+

such that ∑ ¯ A ∥u(t) − u(t )∥ ≤ ε if A = {([t¯, t], τ ]} is a δ1 − fine Q − anchored system in [a, b ] ,

                

(17.4)

(vi) g(u(t), t) is SKH-integrable on [a, b ] and u is its primitive, (vii) u is continuous and ACG∗ on [a, b ] and du (t) = g(u(t), t) dt almost everywhere. Proof. (i) and (ii) are equivalent since U (τ, t) = g(τ ) t. (ii)–(iv) are equivalent by Theorem 15.3. (iii) and (v) are equivalent by Theorem 14.25. (ii) and (vi) are equivalent since ∫ (SKH)

T

S

∫ g(u(s), s) ds = (SKH)

T

Dt U (τ, t)

for [S, T ] ⊂ [a, b ]

S



by Remark 14.23. (vi) and (vii) are equivalent by Remark 14.26.

17.4 . Theorem. Assume that a < T ≤ b, that u : [a, b ] → X is an SKHsolution of (17.1) on [S, T ] for a < S < T and that u(a) = lim u(S) .

(17.5)

S→a

Then u is an SKH-solution of (17.1) on [a, T ]. Proof.

Put Dom G = (Dom g) × R, G(x, τ, t) = g(x, τ ) t.

By Theorem 17.3 (i) and (ii), u is an SKH-solution of the GODE d x = Dt G(x, τ, t) dt

(17.6)

on [S, T ] for a < S < T. Then condition (15.3) is fulfilled since G(u(a), a, c) → G(u(a), a, a)

for c → a

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and (17.5) holds. By Theorem 15.8 u is an SKH-solution of (17.6) on [a, T ]. By Theorem 17.3 (i) and (ii), u is an SKH-solution of (17.1) on [a, T ].  17.5 . Lemma. Let u : [a, b ] → X, a < c < b and let the restrictions u|[a,c] , u|[c,b ] be SKH-solutions of (17.1) on [a, c] and on [c, b ] respectively. Then u is an SKH-solution of (17.1) on [a, b ]. Proof. u is an SKH-solution of (17.1) by Lemma 15.6 and Theorem 17.3 (i), (ii).  17.6. Theorem. Let Dom G ⊂ X × R2 , G : Dom G → X, δ0 : [a, b ] → R+ , u : [a, b ] → X, Dom g ⊂ X × R, g : Dom g → X, (u(τ ), τ, t) ∈ Dom G (u(τ ), τ ) ∈ Dom g

for τ, t ∈ [a, b ], |t − τ | ≤ δ0 (τ ) ,

for τ ∈ [a, b ] ,

(17.7) (17.8)

(u(τ ), τ ) ∈ Dom g for τ ∈ [a, b ], Q ⊂ [a, b ], |Q| = 0. Assume that ∂ G(u(τ ), τ, t)|t=τ = g(u(τ ), τ ) ∂t

for τ ∈ [a, b ] \ Q,

(17.9)

and for every ε > 0 there exists δ : [a, b ] → R+ such that ∑

∥G(u(τ ), τ, t) − G(u(τ ), τ, t¯)∥ ≤ ε ,

(17.10)

A

A = {([t¯, t], τ )} being a δ-fine Q-anchored system in [a, b ]. Then u is an SKH-solution of (17.1) if and only if u is an SKH-solution of the GODE d (17.11) x = Dt G(x, τ, t) . dt Proof.

Put W (τ, t) = g(u(τ ), τ ) t − G(u(τ ), τ, t)

(17.12)

for τ, t ∈ [a, b ], |t − τ | ≤ δ0 (τ ). Let A = {([t¯, t], τ )} be a δ0 -fine partition of [a, b ]. Then ∑ ∥W (τ, t) − W (τ, t¯)∥ ≤ Θ1 + Θ2 + Θ3 (17.13) A

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where Θ1 =

∑

∥g(u(τ ), τ )(t − t¯) − G(u(τ ), τ, t) + G(u(τ ), τ, t¯)∥ ,

A;τ ∈Q /

Θ2 =

∑

∥g(u(τ ), τ ) (t − t¯)∥ ,

τ ∈Q

Θ3 =

∑

∥G(u(τ ), τ, t) − G(u(τ ), τ, t¯)∥ .

τ ∈Q

Let ε > 0. By (17.9) there exists δ1 : [a, b ] → R+ such that ∥G(u(τ ), τ, t) − G(u(τ ), τ, t¯) − g(u(τ ), τ ) (t − t¯)∥ ≤ ε

t − t¯ b−a

if τ ∈ [a, b ] \ Q, t, t¯∈ [a, b ], τ − δ1 (τ ) ≤ t¯ ≤ τ ≤ t ≤ τ + δ1 (τ ). Hence ∥Θ1 ∥ ≤ ε if A is δ1 − fine .

(17.14)

By Lemma 14.24 there exists δ2 : [a, b ] → R+ such that ∥Θ2 ∥ ≤ ε if A is δ2 − fine .

(17.15)

By (17.10) there exists δ3 : [a, b ] → R+ such that ∥Θ3 ∥ ≤ ε if A is δ3 − fine .

(17.16)

Let δ4 (τ ) = min{δ1 (τ ), δ2 (τ ), δ3 (τ )} for τ ∈ [a, b ]. By (17.13)–(17.16) ∑ ∥W (τ, t) − W (τ, t¯)∥ ≤ 3 ε if A is δ4 − fine . A

Hence

∫ (SKH)

b

Dt W (τ, t) = 0 a



and Theorem 17.6 is correct. The next lemma is a consequence of Lemma 15.10. b > 0. Assume that functions 17.7. Lemma. Let R b × [−b, −a] → X, g : B(R)

b × [a, b ] → X , f : B(R)

b u : [−b, −a] → B(R),

b , w : [a, b ] → B(R)

δ : [−b, −a] → R+ ,

δb : [a, b ] → R+

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are coupled by the relations g(x, τ ) = −f (x, σ), b δ(τ ) = δ(σ)

u(τ ) = w(σ),

b σ = −τ, τ ∈ [a, b ] . for x ∈ B(R),

Then u is an SKH-solution of (17.1) on [−b, −a] if and only if w is an SKH-solution of x˙ = f (x, t) on [a, b ].

(17.17)

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Chapter 18

On a class of differential equations in classical form b ∈ R+ , 18.1. Notation. Let α, β, η, ν, R b × (R \ {0}) → X, h : B(R)

b × (R \ {0}) → X . H : B(R)

Assume that 1 ≤ α < 1 + 21 β,

η = min{1, 2 + β − 2 α} ,

(18.1)

h, D1 H, D2 H are continuous (cf. Notation 3.1) , D2 H(x, t) =

(18.2)

∂ b t ∈ R \ {0} , H(x, t) = h(x, t) for x ∈ B(R), ∂t

∥h(x, t)∥ ≤ ν, ∥∆v h(x, t)∥ ≤ ∥v∥ ν

(18.3) } (18.4)

b t ∈ R \ {0} , for x, x + v ∈ B(R), ∥H(x, t)∥ ≤ ν, ∥∆v H(x, t)∥ ≤ ∥v∥ ν, ∥∆v D1 H(x, t)∥ ≤ ∥v∥ ν b t ∈ R \ {0} , for x, x + v ∈ B(R),  b 0
} (18.5)

(18.6)

18.2. Remark. In this chapter couples (β, α) ∈ (R+ )2 are found such that SKH-solutions of x˙ = g(x, t)

(18.7)

exist on intervals [−µ, µ], µ > 0. We may expect that α may be large if β 111

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is large since x˙ = |t|−α sin |t|−β is a particular case of (18.7) and ∫

1

τ −α sin τ −β dτ

lim

t → 0+

t

exists if α < 1 + β. 18.3. Lemma. Let ξ ∈ R, ξ > − 1, 0 ≤ S < T ≤ 1. Then ∫

T

σξ d σ ≤ S

2+ξ (T − S)min{1,1+ξ} . 1+ξ

(18.8)

2+β −α (T − S) η , 1+β −α

(18.9)

3 + β − 2α (T − S) η . 2 + β − 2α

(18.10)

In particular, ∫

T

σ β−α d σ ≤ S

∫

T

σ 1+β−2α d σ ≤ S

Proof.

If ξ ≥ 0 then ∫

T

σ ξ dσ ≤ (T − S) . S

If −1 < ξ < 0 then ∫

∫

T

T −S

σ ξ dσ ≤ S

σ ξ dσ ≤

0

1 (T − S)1+ξ 1+ξ

and (18.8) holds. (18.9) and (18.10) follow from (18.8).



b be a (classical) 18.4 . Lemma. Let 0 < a < b ≤ 1 and let u : [a, b ] → B(R) solution of x˙ = t−α h(x, t−β ) . Then

∫

T

u(T ) − u(S) = S

σ −α h(u(σ), σ −β ) dσ

(18.11)

(18.12)

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for a ≤ S < T ≤ b and u(T ) − u(S) =

1 1+β−α S H(u(S), S −β ) β 1 − T 1 + β − α H(u(T ), T −β ) β ∫ 1 + β − α T β−α + σ H(u(σ), σ −β ) dσ β S ∫ 1 T 1+β−2α + σ D1 H(u(σ), σ −β ) h(u(σ), σ −β ) dσ β S

                                    

(18.13)

for a ≤ S < T ≤ b, D1 H being the differential of H(x, t) with respect to x. Proof. (18.12) is an immediate consequence of (18.11). Further, (18.13) holds since d 1+β−α (t H(u(t), t−β )) = (1 + β − α) tβ−α H(u(t), t−β ) dt + t1+β−α D1 H(u(t), t−β ) u(t) ˙ − β t−α h(u(t), t−β ) and u(t) ˙ = t−α h(u(t), t−β ) .



b be a solution of 18.5. Lemma. Let 0 < a < b ≤ 1 and let u : [a, b ] → B(R) (18.11). Then ∥u(T ) − u(S)∥ ≤

2ν η T + κ1 (T − S) η β

for a ≤ S < T ≤ b

(18.14)

where κ1 =

ν ν2 3 + β − 2 α . (2 + β − α) + 2 β β 2+β −2α

(18.15)

Moreover, ∥u(S)∥ ≤ ∥u(T )∥ +

2ν η T + κ1 T η . β

(18.16)

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Proof.

Let a ≤ S < T ≤ b. By (18.13), (18.9), (18.10) ∫ T 2ν 1+β−α ν ∥u(T ) − u(S)∥ ≤ T + (1 + β − α) σ β−α dσ β β S ∫ ν 2 T 1+β−2 α + σ dσ β S 2 ν 1+β−α ≤ T + κ1 (T − S) η β

and (18.14) holds since 0 < η ≤ 2 + β − 2 α < 1 + β − α. (18.16) is a consequence of (18.14).  18.6. Lemma. Let 0 < c ≤ d ≤ 1, y ∈ X, ∥y∥ +

2ν η b. d + κ1 d η ≤ R β

(18.17)

Then there exists an SKH-solution u of (18.7) on [0, d ] such that u(c) = y. Moreover, lim u(t) exists. t → 0+

Proof. By classical results on ordinary differential equations there exist b such that u is a classical solution P, Q, 0 < P < c < Q ≤ d, u : [P, Q] → B(R) of (18.7), u(c) = y. By Lemma 18.5 ( ) ( ) 2ν 2ν η ∥u(t) − y∥ ≤ + κ1 t ≤ + κ1 d η for c ≤ t ≤ Q β β and

( ∥u(t) − y∥ ≤

) 2ν + κ1 c η β

for P ≤ t ≤ c .

By standard arguments and by (18.17) and Lemma 18.5 there exists a classical solution u of (18.7) such that u(t) is defined for 0 < t ≤ d, u(c) = y and ( )  2ν   ∥u(t)∥ ≤ ∥y∥ + + κ1 d η for 0 < t ≤ d ,  β ( ) (18.18)  2ν η   ∥u(t) − u(S)∥ ≤ + κ1 S for 0 < t ≤ S ≤ d . β u is an SKH-solution of (18.7) on [a, d ] for 0 < a < d. (18.18) implies that lim u(t) exists. Define u(0) = lim u(t). Then u is an SKH-solution

t→0+

t→0+

of (18.7) on [0, d ] by Theorem 15.8.



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18.7. Lemma. Let ν 1+β−α d ≤ β

0 < d ≤ 21 ,

1 2

(18.19)

b u : [0, d ] → B(R) b be SKH-solutions of (18.7). and let u : [0, d ] → B(R), Then } ∥u(T ) − u(T )∥ ≤ 3 ∥u(S) − u(S) ∥ exp(2 κ1 (T − S) η ) (18.20) for 0 ≤ S < T ≤ d . Proof.

u fulfils (cf. (18.13)) 1 1+β−α T H(u(T ), T −β ) β 1 = u(S) + S 1+β−α H(u(S), S −β ) β ∫ 1 + β − α T β−α + σ H(u(σ), σ −β ) dσ β S ∫ 1 T 1+β−2 α + σ D1 H(u(σ), σ −β ) h(u(σ), σ −β ) dσ . β S

u(T ) +

Subtracting the same equation with u replaced by u we obtain 1 1+β−α T [H(u(T ), T −β ) − H(u(T ), T −β )] β 1 = u(S) − u(S) + S 1+β−α [H(u(S), S −β ) − H(u(S), S −β )] β ∫ 1 + β − α T β−α + σ [H(u(σ), σ −β ) − H(u(σ), σ −β )] dσ β S ∫ 1 T 1+β−2 α + σ [D1 H(u(σ), σ −β ) h(u(σ), σ −β ) β S

u(T ) − u(T ) +

− D1 H(u(σ), σ −β )h(u(σ, σ −β )] dσ . Hence (cf. (18.5), (18.19)) 1 2

∥u(T ) − u(T )∥ ≤

3 2

∫

∥u(S) − u(S)∥ T

ν σ β−α ∥u(σ) − u(σ)∥ dσ (1 + β − α) β S ∫ 2 ν 2 T 1+β−2 α + σ ∥u(σ) − u(σ)∥ dσ . β S +

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Lemma A.1 implies (cf. also (18.9), (18.10), (18.15)) that (18.20) holds.  b and let (18.16) hold. Then there exists 18.8. Lemma. Let y ∈ X, ∥y∥ < R b an SKH-solution u1 : [0, d ] → B(R) of (18.7) on [0, d ] fulfilling u1 (0) = y. Proof. By Lemma 18.6 there exists an SKH-solution vi of (18.7) on [0, d ], vi (d 2−i ) = y for i ∈ N. By Lemma 18.5 ∥vi+1 (d 2−i ) − vi+1 (d 2−i−i )∥ ≤ ( Hence ∥vi+1 (d 2−i ) − vi (d 2−i )∥ ≤

(

2ν + κ1 ) (d 2−i ) η . β

) 2ν + κ1 (d 2−i ) η β

since vi+1 (d 2−i−1 ) = y = vi (d 2−i ). By Lemma 18.7 ( ) ( ) 2ν ∥vi+1 (t) − vi (t)∥ ≤ 3 + κ1 d 2−i ) η exp(2 κ1 d η β for d 2−i ≤ t ≤ d, i ∈ N .

  

(18.21)

By Lemma 18.5 (

) 2ν ∥vi+1 (t) − vi+1 (d 2 )∥ ≤ + κ1 (d 2−i ) η , β ( ) 2ν −i ∥vi (t) − vi (d 2 )∥ ≤ + κ1 (d 2−i ) η , β −i

∥vi+1 (t) − vi (t)∥ ( ) 2ν ≤2 + κ1 (d 2−i ) η + ∥vi+1 (d 2−i ) − vi (d 2−i )∥ β ( ) 2ν ≤3 + κ1 (d 2−i ) η β for 0 < t ≤ d 2−i . The above inequalities hold for t = 0 as well since lim vi (t) exists for t→0+

i ∈ N. Hence

∥vi+1 (t) − vi (t)∥ ≤ 3 κ1 (d 2−i ) η for 0 ≤ t ≤ d 2−i , i ∈ N .

(18.22)

(18.21) together with (18.22) implies that the sequence vi , i ∈ N, is convergent uniformly on [0, d ]. Put u1 (t) = lim vi (t) for t ∈ [0, d ]. Obviously, u1 i→∞

is continuous and u1 (0) = y.

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It remains to prove that u1 is an SKH-solution of (18.7). It is sufficient to prove that ∫ T u1 (T ) − u1 (S) = (SKH) g(u1 (t), t) dt (18.23) S

for 0 ≤ S ≤ T ≤ d (cf. (18.6) and Theorem 17.3 (i), (ii)). Equality (18.23) holds for 0 < S ≤ T ≤ d since g(u1 (t), t) is continuous for ∫ d 0 < t ≤ d. The integral (SKH) g(u1 (t), t) dt exists by Theorem 14.20 0

where U (τ, t) = g(u1 (τ ), τ ) t for τ ∈ [0, d ], t ∈ R, and w = u1 (d) − u1 (0) since U (u1 (0), t) = 0 and ∫ d ( ) lim − (SKH) g(u1 (t), t) dt + u1 (d) − u1 (0) S → 0+ S ( ) = lim −u1 (d)+u1 (S)+u1 (d)−u1 (0) = lim (u1 (S)−u1 (0)) = 0. S→0+

S→0+

(18.23) holds for 0 ≤ S ≤ T ≤ d and Lemma 18.8 is correct.



b and let (18.19) hold. Then there exists 18.9. Theorem. Let y ∈ X, ∥y∥ < R b an SKH-solution u : [−d, d ] → B(R) of (18.7) on [−d, d ] fulfilling u(0) = y. Moreover, u is unique. b Proof. By Lemma 18.8 there exists an SKH-solution u1 : [0, d ] → B(R) of (18.7) such that u1 (0) = y. Let b τ ∈ [0, 1] . f (x, τ ) = −g(x, −τ ) for x ∈ B(R), Lemma 18.8 may be applied to the equation x˙ = f (x, t)

(18.24)

since (cf. (18.6)) f (x, t) = −(−t)−α h(x, (−t)−β ) and h fulfils (18.2)–(18.5). Hence there exists an SKH-solution w of (18.23) on [0, d ] such that w(0) = y. Put u2 (τ ) = w(−τ ) for τ ∈ [−d, 0]. By Lemma 17.7 (where a = 0, b = d) u2 is an SKH-solution of (18.7) on [−d, 0], u2 (0) = y. Let  u1 (t) if t ∈ [0, d ] , u(t) = u2 (t) if t ∈ [−d, 0] . u is an SKH solution of (18.7) on [−d, d ] by Lemma 15.6. The uniqueness of u1 is a consequence of Lemma 18.7. Similarly, also u2 is unique. Hence u is unique. 

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Chapter 19

Integration and Strong Integration

19.1. Notation. Let Dom U ⊂ R2 , U : Dom U → X. 19.2. Definition. U is KH-integrable on [a, b ] if there exist δ0 : [a, b ] → R+ and Γ(a, b) ∈ X such that {(τ, t) ∈ [a, b ] 2 ; τ − δ0 (τ ) ≤ t ≤ τ + δ0 (τ )} ⊂ Dom U and for every ε > 0 there exists δ : [a, b ] → R+ such that ∑ ∥Γ(a, b) − (U (τ, t) − U (τ, t¯))∥ ≤ ε

(19.1)

(19.2)

A

for every δ-fine partition A = {([t¯, t], τ )} of [a, b ]. b b) ̸= Γ(a, b) 19.3 . Remark. Γ(a, b) is unique. Indeed, if there exists Γ(a, + b b ) ≤ δ(τ ) for and for every ε > 0 there exists δ : [a, b ] → R such that δ(τ τ ∈ [a, b ] and ∑ b b) − ∥Γ(a, (U (τ, t) − U (τ, t¯))∥ ≤ ε A

b for every δ-fine partition A of [a, b ], then a contradiction is obtained e.g. 1 b b)∥). Γ(a, b) is denoted by for ε = 3 ∥Γ(a, b) − Γ(a, ∫ b (KH) Dt U (τ, t) a

and called the Kurzweil–Henstock integral (KH-integral) of U over [a, b ]. 19.4. Theorem. Let U be SKH-integrable on [a, b ], u : [a, b ] → X being its primitive. Then U is KH-integrable on [a, b ] and ∫ T ∫ T (KH) Dt U (τ, t) = (SKH) Dt U (τ, t) . S

S

119

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Proof. Let U be SKH-integrable on [a, b ] and let u be its primitive. Then for ε > 0 there exists δ : [a, b ] → R+ such that ∑ ∥u(t) − u(t¯) − U (τ, t) + U (τ, t¯)∥ ≤ ε A

for any δ-fine partition A = {([t¯, t], τ )} of [a, b ]. Hence ∑ ∥u(b) − u(a) − (U (τ, t) − U (τ, t¯)∥ ≤ ε A



and the theorem is valid.

19.5. Remark. Theorem 19.4 can be converted if dim X < ∞ (cf. Corollary 19.14). 19.6. Lemma. Let U be KH-integrable on [a, b ]. Then U

is KH-integrable on [S, T ]

for [S, T ] ⊂ [a, b ] .

(19.3)

Moreover, ∫

∫

T

(KH) S

∫

Tb

Dt U (τ, t) = (KH)

Dt U (τ, t) + (KH) S

T

Tb

Dt U (τ, t)

(19.4)

for a ≤ S ≤ Tb ≤ T ≤ b. Proof. Let ε > 0 and let δ : [a, b ] → R+ correspond to ε by Definition 19.2, [S, T ] ⊂ [a, b ]. There exist a δ-fine partition A = {([t¯, t], τ )} of [a, S] and a δ-fine partition C = {([t¯, t], τ )} of [T, b ]. If B1 = {([t¯, t], τ )} and B2 = {([t¯, t], τ )] are δ-fine partitions of [S, T ], then A ∪ B1 ∪ C and A ∪ B2 ∪ C are δ-fine partitions of [a, b ]. Hence

∑

(U (τ, t) − U (τ, t¯)) ≤ ε ,

Γ(a, b) −

Γ(a, b) −

A∪B1 ∪C

∑

A∪B2 ∪C

(U (τ, t) − U (τ, t¯)) ≤ ε ,

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and

121

∑

∑

(U (τ, t) − U (τ, t¯)) − (U (τ, t) − U (τ, t¯)) ≤ 2 ε .

B1

(19.5)

B2

(19.5) implies that there exists Γ(S, T ) such that

∑

(U (τ, t) − U (τ, t¯)) ≤ 2 ε

Γ(S, T ) − B

for every δ-fine partition B = {([t¯, t], τ )} of [S, T ]. (19.3) is correct. Let a ≤ S ≤ Tb ≤ T ≤ b. By (19.3) U is KH-integrable on [S, Tb], on [Tb, T ] and on [S, T ]. Therefore there exists δ : [S, T ] → R+ such that

 ∑

 (U (τ, t) − U (τ, t¯)) ≤ ε

Γ(S, Tb) − (19.6) B1  for every δ−fine partition B1 = {([b t, t], τ )} of [S, Tb],

 ∑

b

 (U (τ, t) − U (τ, t¯)) ≤ ε

Γ(T , T ) − (19.7) B2  for every δ−fine partition B2 = {([b t, t], τ )} of [Tb, T ], and

∑

(U (τ, t) − U (τ, t¯)) ≤ ε

Γ(S, T ) − B

for every δ-fine partition B = {([b t, t], τ )} of [S, T ].

  

(19.8)

Let B1 be a δ-fine partition of [S, Tb] and let B2 be a δ-fine partition of b [T , T ]. Then B = B1 ∪ B2 is a δ-fine partition of [S, T ] and (19.6)–(19.8) imply that ∥Γ(S, Tb) + Γ(Tb, T ) − Γ(S, T )∥ ≤ 3 ε . (19.9) (19.4) holds by (19.9) since ε > 0 is arbitrary.  19.7 . Remark. Let U be KH-integrable on [a, b ]. Then u : [a, b ] → X is called a KH-primitive of U on [a, b ] or simply a primitive of U if ∫ T u(T ) − u(S) = (KH) Dt U (τ, t) for [S, T ] ⊂ [a, b ]. S

∫T

Dt U (τ, t) exists for T ∈ [a, b ]. Put ∫ T v(T ) = (KH) Dt U (τ, t) for T ∈ [a, b ].

By Lemma 19.6 (KH)

a

a

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Then v is a primitive of U. On the other hand, if u is a primitive of U then u(T ) − u(a) = v(T ) for T ∈ [a, b ]. 19.8. Remark. Similarly to Remark 14.16 we put ∫ S ∫ T (KH) Dt U (τ, t) = −(KH) Dt U (τ, t) for a ≤ S ≤ T ≤ b. T

S

19.9. Lemma. Let U be KH-integrable on [a, b ], ε > 0 and let δ : [a, b ] → R+ correspond to ε by Definition 19.2. Then

∑

(Γ(¯ s, s) − U (σ, s) + U (σ, s¯)) ≤ 2 ε (19.10)

D

for every δ-fine system D = {([¯ s, s], σ)} in [a, b ]. Proof. Let D = {([¯ si , si ], σi ); i = 1, 2, . . . , k} be a δ-fine system in [a, b ]. For simplicity assume that a = s¯1 < s1 < s¯2 < s2 < · · · < s¯k < sk = b. By Lemma 19.6, U is KH-integrable on [si−1 , s¯i ] for i = 2, 3, . . . , k. Hence there exist δi : [si−1 , s¯i ] → R+ such that δi (τ ) ≤ δ(τ ) for τ ∈ [si−1 , s¯i ] and ∑ ε ∥Γ(si−1 , s¯i ) − (U (ρ, r) − U (ρ, r¯))∥ ≤ k Ei

for every δi -fine partition Ei = {([¯ r, r], ρ)} of [si−1 , s¯i ]. By Lemma 14.4 there k ∪ exists a δi -fine partition Ebi = {([¯ r, r], ρ)} of [si−1 , s¯i ]. Then D ∪ ( Ebi ) is a i=2

δ-fine partition of [a, b ], k

∑

(Γ(¯ si , si ) − U (σi , s¯i ) + U (σi , si ))

i=1

+

k ∑ (

Γ(si−1 , s¯i ) −

∑

)

(U (ρ, r) − U (ρ, r¯)) ≤ ε ,

Ebi

i=2

ε ∑

(U (ρ, r) − U (ρ, r¯)) ≤

Γi (si−1 , s¯i ) − k Ebi

and (19.10) holds.



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19.10. Lemma. Let φ: [a, b ] → R be nondecreasing, let δ0 : [a, b ] → R+ and let U be KH-integrable on [a, b ]. Assume that ∥U (τ, t) − U (τ, t¯)∥ ≤ φ(t) − φ(t¯)

for τ, t, t¯∈ [a, b ], t¯≤ τ ≤ t ,

and τ − δ0 (τ ) ≤ t¯ ≤ τ ≤ t ≤ τ + δ0 (τ ) . Then ∫

(KH)

T

Dt U (τ, t) ≤ φ(T ) − φ(S)

for a ≤ S < T ≤ b.

(19.11)

S

Proof. Let ε > 0, [S, T ] ⊂ [a, b ]. By Definition 19.2 and Lemmas 19.6, 14.4 there exists a δ0 -fine partition T = {([t¯, t], τ )} of [S, T ] such that ∫ T

∑

Dt U (τ, t) − (U (τ, t) − U (τ, t¯)) ≤ ε .

(KH) S

Hence

T

∫

(KH)

T

Dt U (τ, t) ≤ φ(T ) − φ(S) + ε

S

and (19.11) holds, since ε is arbitrary positive.



19.11. Remark. Let U : [a, b ] 2 → X. Put −U ∗ (−τ, −t) = U (τ, t)

for (τ, t) ∈ [a, b ] 2 .

Then U ∗ : [−b, −a] 2 → X. Let T = {([ti−1 , ti ], τi ), i = 1, 2, . . . , k} be a partition of [a, b ]. Then  k ∑    (U (τi , ti ) − U (τi , ti−1 ))   i=1 (19.12) k ∑   ∗ ∗  =− (U (−τi , −ti ) − U (−τi , −ti−1 )) .   i=1

Put T ∗ = {([−ti , −ti−1 ], τi ); i = 1, 2, . . . , k}. Then T ∗ is a partition of [−b, −a]. By (19.11) k ∑ i=1

(U (τi , ti ) − U (τi , ti−1 )) =

k ∑ i=1

(U ∗ (−τi , −ti−1 ) − U ∗ (−τi , −ti )) .

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Let δ : [a, b ] → R+ , δ ∗ : [−b, −a] → R+ , δ ∗ (−τ ) = δ(τ ) for τ ∈ [a, b ]. If T is δ-fine, then T ∗ is δ ∗ -fine and vice versa. We may conclude that ∫ (KH)

∫ Dt U ∗ (τ, t) = (KH)

−a

−b

b

Dt U (τ, t) for a < b

(19.13)

a

if one of the integrals exists. If U (τ, t) = ξ(t) Φ(t), put ξ ∗ (−τ ) = ξ(τ ) and Φ∗ (−t) = − Φ(t). Then −U ∗ (−τ, −t) = U (τ, t) and ∫

−a

(KH) −b

Dt ξ ∗ (τ ) Φ∗ (t) = (KH)

∫

b

Dt ξ(τ ) Φ(t)

(19.14)

a

if one of the integrals exists, and ∫ (SKH)

−a

−b

∗

∫

∗

b

Dt ξ (τ ) Φ (t) = (SKH)

Dt ξ(t) Φ(t)

(19.15)

a

if one of the integrals exists. 19.12. Lemma. Let n ∈ N, X = Rn , ∥x∥ =

n ∑

|xi |

for x = (x1 , x2 , . . . , xn ) ∈ Rn

(19.16)

i=1

and L = {−1, 1}n .

(19.17)

For L ∈ L (i.e. L = (L1 , L2 , . . . , Ln ), |Li | = 1) put QL = {x ∈ Rn ; Li xi ≥ 0

for i = 1, 2, . . . , n} .

(19.18)

for z = (z1 , z2 , . . . , zn ) ∈ QL ,

(19.19)

if Z ⊂ QL is finite ,

(19.20)

if Z ⊂ QL is finite .

(19.21)

Then ∥z∥ =

n ∑

Li zi

i=1

∑ ∑

z = ∥z∥

z∈Z

∑

z∈Z

z∈Z

∥z∥ ≤ 2n max L∈L

∑ z∈Z∩QL

z

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Proof. (19.19) follows by (19.16)–(19.18). Further, (19.20) is a consequence of (19.19). Finally, let Z ⊂ Rn be finite. Then

∑ ∑ ∑ ∑

∑

∥z∥ ≤ ∥z∥ ≤ z

z∈Z

L∈L z∈Z∩QL

≤ 2 max n

L∈L

∑

L∈L

z .

z∈Z∩QL

z∈Z∩QL



Hence (19.21) holds.

19.13. Lemma. Assume that n ∈ N and that U : Dom U → Rn is KH-integrable on [a, b ]. Given ε > 0, let δ : [a, b ] → R+ be such that ∑ ∥Γ(a, b) − (U (τ, t) − U (τ, t¯)∥ ≤ ε A

for every δ-fine partition A = {([t¯, t], τ )} of [a, b ]. Then

∑

∥Γ(t¯, t) − (U (τ, t) − U (τ, t¯))∥ ≤ 2n ε

(19.22)

A

for every δ-fine partition A = {([t¯, t], τ )} of [a, b ]. Proof. Let ε > 0, let δ correspond to ε by Definition 19.2, let the partition A = {([¯ s, s], σ)} of [a, b ] be δ-fine and let Z = {z = Γ(¯ s, s) − U (σ, s) + U (σ, s¯) ; ([¯ s, s], σ) ∈ A} . Then



∑

z ≤ ε

z∈Z∩QL



and (19.22) is true by (19.21).

19.14 . Corollary. Let U : Dom U → Rn be KH-integrable on [a, b ]. Then U is SKH-integrable on [a, b ] and ∫ b ∫ b (SKH) Dt U (τ, t) = (KH) Dt U (τ, t) . a

a

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Chapter 20

A class of Strong Kurzweil Henstock-integrable functions

The main result of Chapter 20 implies that the product ψ(τ ) φ(t) is SKHintegrable if ψ is a regulated function and φ is a function of bounded variation. 20.1. Notation. Let [c, d ] ⊂ [a, b ] ⊂ R, U : [a, b ] 2 → X, Ψ : [a, b ] → R, Φ : [a, b ] → R, ε >0. Assume that Ψ

is a regulated function ,

(20.1)

Φ

is nondecreasing, Φ(b) − Φ(a) > 0 ,

(20.2)

∥U (τ, t) − U (τ, s)∥ ≤ |Φ(t) − Φ(s)|

for t, s, τ ∈ [a, b ] , }

∥U (τ, t) − U (τ, s) − U (σ, t) + U (σ, s)∥ ≤ (Ψ(τ ) − Ψ(σ)) (Φ(t) − Φ(s))

(20.3)

(20.4)

for t, s, τ, σ ∈ [a, b ] .

Define sets A(ε) = A(ε)(ε, c, d), B(ε) = B(ε)(ε, c, d) by { } A(ε) = τ ; c ≤ τ < d, |Ψ(τ +) − Ψ(τ )| ≥ ε { } ∪ τ ; c < τ ≤ d, |Ψ(τ ) − Ψ(τ −)| ≥ ε , B(ε) = {τ ; c ≤ τ < d, |Ψ(τ +) − Ψ(τ )| < ε} ∩ {τ ; c < τ ≤ d, |Ψ(τ ) − Ψ(τ −)| < ε} .

} (20.5) } (20.6)

Obviously, B(ε) = [c, d ] \ A(ε) .

(20.7)

and (20.1) implies that A(ε) is a finite set. Denote by m(ε) the number of elements of A(ε). 127

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20.2. Definition. A partition T = {([t¯, t], τ )} of [c, d ] is called simple if τ = t¯ or τ = t for ([t¯, t], τ ) ∈ T and if c < t¯< t < d whenever c < τ < d. 20.3 . Remark. Let δ : [c, d ] → R+ . Let T = {([t¯, t], τ )} be a partition of [c, d ] such that c < t¯< t < d whenever c < τ < d. Modify T in the following way: keep elements ([t¯, t], τ ) such that τ = t¯ or τ = t and replace every element ([t¯, t], τ ) such that t¯< τ < t by two elements ([t¯, τ ], τ ), ([τ, t], τ ). Denote the modified set by Ω(T ). Then Ω(T ) is a simple partition of [c, d ] which is δ-fine if T is δ-fine. The operation Ω may be used in order to prove that

(i) there exists a δ-fine simple partition of [c, d ] (cf. Lemma 14.4), (ii) U is KH-integrable if and only if there exists Γ(a, b) and if for ε > 0 there exists δ : [a, b ] → R+ such that

∑( )

U (τ, t) − U (τ, t¯) ≤ ε

Γ(a, b) − T

for every δ-fine simple partition T of [a, b ]. 20.4 . Definition. Let T = {([t¯, t], τ )}, R = {([¯ r, r], ρ)} be partitions of [c, d ]. The partition R is called a refinement of T if for every ([¯ r, r], ρ) ∈ R there exists a ([t¯, t], τ ) ∈ T such that [¯ r, r] ⊂ [t¯, t]. 20.5. Remark. Let T = {([t¯, t], τ )}, S = {([¯ s, s], σ)} be partitions of [a, b ], + δ : [a, b ] → R . Then there exists a δ-fine simple partition of [a, b ] which is a refinement of both T and S (by Remark 20.3 (i) there exists a δ-fine simple partition of every nondegenerate interval [t¯, t] ∩ [¯ s, s]). 20.6 . Remark. Let T = {([t¯, t], τ )}, R = {([¯ r, r], ρ)} be simple partitions of [a, b ], R being a refinement of T . For [t¯, t], τ ) ∈ T , denote by R(t¯, t) the set of [¯ r, r], ρ) ∈ R such that [¯ r, r] ⊂ [t¯, t]. Then R(t¯, t) is a simple partition of [t¯, t].

The next lemma plays a crucial role in the proof that U is integrable.

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129

20.7. Lemma. Let κ > 0 and δ : [c, d ] → R+ fulfil |Ψ(τ )| ≤ κ

for τ ∈ [a, b ],

(20.8)

[τ − δ(τ ), τ + δ(τ )] ∩ [c, d ] ⊂ B(ε)

if τ ∈ B(ε) ,

(20.9) }

|Ψ(t) − Ψ(τ )| < ε

(20.10)

if τ ∈ B(ε) and t ∈ [τ − δ(τ ), τ + δ(τ )] ∩ [c, d ] , [τ − δ(τ ), τ + δ(τ )] ∩ [c, d ] ∩ A(ε) = {τ }

if τ ∈ A(ε),

(20.11) }

|Ψ(t) − Ψ(τ +)| ≤ ε

(20.12)

if τ ∈ A(ε) and τ < t ≤ min{τ + δ(τ ), b} , }

|Ψ(τ −) − Ψ(t)| ≤ ε

(20.13)

if τ ∈ A(ε) and max{τ − δ(τ ), a} ≤ t < τ , ε m(ε) ε Φ(τ −) − Φ(τ − δ(τ )) ≤ m(ε) Φ(τ + δ(τ )) − Φ(τ +) ≤

if τ ∈ A(ε) and τ < b ,

   

if τ ∈ A(ε) and τ > a .

  

(20.14)

Let T = {([t¯, t], τ )}, R = {([¯ r, r], ρ)} be δ-fine simple partitions of [c, d ], R being a refinement of T . Then ∑ ) ∑( )

U (ρ, r) − U (ρ, r¯)

U (τ, t) − U (τ, t¯) − T

R

( ) ≤ ε Φ(b) − Φ(a) (2 + 4 κ) .

  

(20.15)

Proof. A constant κ and a function δ fulfilling (20.8)–(20.14) exist since Ψ fulfils (20.1) and Φ fulfils (20.2). The proof consists of five parts. Part 1. Let ([τ, t], τ ) ∈ T and τ ∈ A(ε). If τ < σ ≤ t then σ ∈ B(ε)

and

[σ − δ(σ), σ + δ(σ)] ⊂ B(ε)

by (20.11). Hence there exists ([τ, r1 ], τ ) ∈ R(τ, t) and U (τ, t) − U (τ, τ ) = U (τ, r1 ) − U (τ, τ ) + U (τ, t) − U (τ, r1 ).

(20.16)

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130

On the other hand, ∑ (U (ρ, r) − U (ρ, r¯)) R(τ,t)

= U (τ, r1 ) − U (τ, τ ) +

∑ (U (ρ, r) − U (ρ, r¯)) , R∗

      

(20.17)

where R∗ = R(τ, t) \ {([τ, r1 ], τ )}. By (20.16), (20.17), (20.4), (20.14)

 ∑

 (U (ρ, r) − U (ρ, r¯))

U (τ, t) − U (τ, τ ) −     R(τ,t)   ∑



U (τ, r) − U (τ, r¯) − U (ρ, r) + U (ρ, r¯) ≤ (20.18)  R∗     ( )  ε   ≤ 2 κ Φ(b) − Φ(a) m(ε) since Ψ(ρ) − Ψ(τ ) ≤ 2 κ, ∑(

) r) ≤ Φ(τ + δ(τ )) − Φ(τ +) ≤ Φ(r) − Φ(¯

R

Part 2. Similarly, if τ ∈ A(ε), ([t¯, τ ], τ ) ∈ T then

∑

(U (ρ, ρ)−U (ρ, r¯))

U (τ, τ )−U (τ, t¯)− R(t¯,τ )

≤

( ) ε 2 κ Φ(b)−Φ(a) . m(ε)

ε . m(ε)

      

Part 3. Let ([τ, t], τ ) ∈ T , τ ∈ B(ε). By (20.9), (20.10), (20.4)

 ∑ ( )

 U (ρ, r) − U (ρ, r¯)

U (τ, t) − U (τ, τ ) −     R(τ,t)   

∑ (

 ) 

≤ U (τ, r) − U (τ, r¯) − U (ρ, r) + U (ρ, r¯)  R(τ,t)    ∑    ≤ |Ψ(ρ) − Ψ(τ )| (Φ(r) − Φ(¯ r)) ≤ ε (Φ(t) − Φ(τ )) .   

(20.19)

(20.20)

R(τ,t)

Part 4. Let ([t¯, τ ], τ ) ∈ T , τ ∈ B(ε). Then similarly to Part 3

∑ ( )

U (ρ, r) − U (ρ, r¯) ≤ ε (Φ(τ ) − Φ(t¯)) . (20.21)

U (τ, τ ) − U (τ, t¯) − R(t¯,τ )

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Part 5. ∑[ ] ∑[ ] U (τ, t)−U (τ, τ )+U (τ, τ )−U (τ, t¯) − U (ρ, r)−U (ρ, r¯)      T R    ∑ [ ( )]  ∑   = 1 U (τ, t)−U (τ, τ ) − U (ρ, r) − U (ρ, r¯)     R(τ,t)  ∑  ( )] ∑ [ ¯ + U (τ, τ )−U (τ, t ) − U (ρ, r) − U (ρ, r ¯ ) 2   R(t¯,τ )  ∑ (  )] ∑ [   + U (τ, t)−U (τ, τ ) − U (ρ, r) − U (ρ, r ¯ )  3    R(τ,t)   ∑ ( )] ∑ [  ¯  + U (ρ, r) − U (ρ, r¯) ,   4 U (τ, τ )−U (τ, t )− 

jk

131

(20.22)

R(t¯,τ )

∑ ∑ where 2 runs over 1 runs over ([τ, t], τ ) ∈ T∑ such that τ ∈ A(ε), runs over ([τ, t], τ ) ∈ T such that ([t¯, τ ], τ ) ∈ T such that τ ∈ A(ε), 3 ∑ τ ∈ B(ε), 4 runs over ([t¯, τ ], τ ) ∈ T such that τ ∈ B(ε). (20.22) holds since A(ε) ∪ B(ε) = [a, b ], A(ε) ∩ B(ε) = ∅. The number of elements of A(ε) does not exceed m(ε). (20.18)–(20.22) imply that (20.15) holds. The proof is complete.  20.8. Theorem. U is KH-integrable, Proof. Let [c, d ] = [a, b ]. By Lemma 20.7, for ε > 0 there is δ : [a, b ] → R+ such that (20.15) holds. Let T = {([t¯, t], τ )}, S = {([¯ s, s], σ)} be δ-fine simple partitions of [a, b ]. By Remark 20.5 then there exists a δ-fine simple partition R = {([¯ r, r], ρ)} of [a, b ] which is a refinement of both T and S. (20.15) implies that

∑( ∑ ( ) ∑ )

U (τ, t) − U (τ, t¯) − U (ρ, r) − U (ρ, r¯)

T

T

∑( ) ∑

U (σ, s) − U (σ, s¯) −

S

R(t¯,τ )∪R(τ,t)

( ) ≤ ε (2 + 4 κ) Φ(b) − Φ(a) , ∑ ( )

U (σ, r) − U (σ, r¯)

S R(¯ s,σ)∪R(σ,s)

( ) ≤ ε (2 + 4 κ) Φ(b) − Φ(a) ,

and

∑(  ) ∑( )

 U (τ, t) − U (τ, t¯) − U (σ, s) − U (σ, s¯)

T S ( )  (20.23) ≤ ε (4 + 8 κ) Φ(b) − Φ(a)

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132

since ∑ T

∑

(

) ∑ U (ρ, r)−U (ρ, r¯) =

∑

(

) U (σ, r)−U (σ, r¯) .

S R(¯ s,σ)∪R(σ,s)

R(t¯,τ )∪R(τ,t)

U is KH-integrable by (20.23). The proof is complete.

∫

 T

20.9. Theorem. Define u : [a, b ] → X by u(T ) = (KH) U is SKH-integrable and u is its primitive.

Dt U (τ, t). Then a

Moreover, ∥u(d) − u(c)∥ ≤ Φ(d) − Φ(c)

for a ≤ c < d ≤ b .

(20.24)

Proof. Let [c, d ] = [a, b ] and let δ : [a, b ] → R+ fulfil (20.9)–(20.14). Let T = {([t¯, t], τ )} be a δ-fine partition of [a, b ]. Let ([t¯, t], τ ) ∈ T be given and let R = {([¯ r, r], ρ) be a δ-fine simple partition of [τ, t]. If τ ∈ A(ε) then by (20.18)  ∑(

)

U (τ, t)−U (τ, τ )−  U (ρ, r)−U (ρ, r¯)  R ( )  ε  2κ Φ(b)−Φ(a) ≤ m(ε) and



U (τ, t) − U (τ, τ ) − u(t) + u(τ ) ≤

( ) ε 2 κ Φ(b) − Φ(a) . m(ε)

(20.25)

) ∑ ( since ¯) can be arbitrarily close to u(t) − u(τ ). If R U (ρ, r) − U (ρ, r τ ∈ B(ε) then by (20.20)  ∑(

)

U (τ, t)−U (τ, τ )−  U (ρ, r)−U (ρ, r¯) R ( )  ≤ ε Φ(t)−Φ(τ ) and

( )

U (τ, t) − U (τ, τ ) − u(t) + u(τ ) ≤ ε Φ(t) − Φ(τ ) .

Let ([t¯, τ ], τ ) ∈ T . Then similarly

U (τ, t¯)−U (τ, τ )−u(t¯) + u(τ ) ≤

( ) ε 2 κ Φ(b)−Φ(a) m(ε) for τ ∈ A(ε)

and

(20.26)   

(20.27)

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( )

U (τ, t¯)−U (τ, τ )−u(t¯) + u(τ ) ≤ ε Φ(τ )−Φ(t¯)

133

}

for τ ∈ B(ε) .

(20.28)

(20.25)–(20.28) implies that ∑ ( ) ∥u(t) − u(t¯) − U (τ, t) + U (τ, bart)∥ ≤ ε (4 κ + 1) Φ(b) − Φ(a) . sopT

Hence U is SKH-integrable and u is its primitive. (20.24) is a consequence of Lemma 14.19 and (20.3). 20.10 . Remark. The limit U (τ, τ +) =

lim

t→τ,t>τ

since U (τ, .) is a function of bounded variation.



U (τ, t), a ≤ τ < b, exists

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Chapter 21

Integration by parts

21.1. Notation. Let [a, b ] ⊂ R, U : [a, b ] 2 →X. Put V (τ, t) = U (t, t) + U (τ, τ ) − U (τ, t) − U (t, τ ),

(21.1)

τ, t ∈ [a, b ] .

W (τ, t) = U (t, τ ), Define

τ, t ∈ [a, b ] ,

∫

∫ b Dτ U (τ, t) = (KH) Dt W (τ, t) (21.2) a a ∫ b if the right-hand side exists (i.e. (KH) Dτ U (τ, t) exists if for ε > 0 there b

(KH)

a

exists δ : [a, b ] → R+ such that ∫ b

∑

(U (τ, t) − U (¯ τ , t)) ≤ ε Dτ U (τ, t) −

(KH) a

T

for every δ-fine partition T

∗

∗

= {([¯ τ , τ ], t)} of [a, b ]).

Similarly, define ∫ (SKH)

∫

b

Dτ U (τ, t) = (SKH) a

b

Dt W (τ, t)

(21.3)

a

if the right-hand side exists. For [c, d ], [e, f ] ⊂ [a, b ] put b ([c, d] × [e, f ]) = U (d, f ) − U (d, e) − U (c, f ) + U (c, e) . U 21.2. Lemma. Let t¯, τ, t ∈ [a, b ]. Then V (τ, t) − V (τ, t¯) = −U (τ, t) + U (τ, t¯) − U (t, τ ) + U (t¯, τ ) + U (t, t) − U (t¯, t¯) . 135

(21.4) } (21.5)

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136

Let T = {([t¯, t], τ )} be a partition of [a, b ]. Then  ∑( ) ∑( ) ∑ U (τ, t)−U (τ, t¯) + U (t, τ )−U (t¯, τ ) − (U (t, t)−U (t¯, t¯))   T T T ∑ (21.6)  =− (V (τ, t) − V (τ, t¯)).   T

Proof. (21.5).

(21.5) is a consequence of (21.1).

Further, (21.6) follows by 

21.3. Theorem. If two of the integrals ∫ b ∫ b (SKH) Dt U (τ, t), (SKH) Dτ U (t, τ ), a

∫

b

(SKH)

a

Dτ V (τ, t) a

exist, then the third one exists as well and  ∫ b ∫ b  (SKH) Dt U (τ, t) + (SKH) Dτ U (τ, t) − U (b, b) + U (a, a)    a

a

∫

   

b

= −(SKH)

Dτ V (τ, t) .

(21.7)

a

Proof. Let Ui : [a, b ] 2 → X be SKH-integrable, i = 1, 2. Definition 19.2 implies that U1 + U2 is SKH-integrable and that ∫ b ∫ b ∫ b ( ) (SKH) Dt U1 (τ, t)+(SKH) Dt U2 (t, τ ) = (SKH) Dt U1 (t, t)+U2 (τ, t) . a

a

a

∑( ) Also, U (t, t)−T (t¯, t¯) = U (b, b) − U (a, a). Hence (21.7) can be deduced T

by (21.6). The proof is complete.



b is additive in the following sense: 21.4. Remark. U if [c, d ], [d, e], [f, g] ⊂ [a, b ] then b ([c, e] × [f, g]) = U b ([c, d ] × [f, g]) + U b ([d, e] × [f, g]) , U

(21.8)

b ([f, g] × [c, e]) = U b ([f, g] × [c., d ]) + U b ([f, g] × [d, e]) . U

(21.9)

21.5 . Lemma. Let a ≤ σ < r < s ≤ b, a ≤ s¯ < r¯ < σ ≤ b. Then b ([r, s] × [σ, r]) + U b ([σ, s] × [r, s]) , V (σ, s) = V (σ, r) + U

(21.10)

b ([¯ b ([¯ V (¯ s, σ) = V (¯ r, σ) + U s, σ] × [¯ s, r¯]) + U s, r¯] × [¯ r, σ]) .

(21.11)

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Proof.

137

(21.10) holds (cf. Remark 21.4) since ( ) ( ) [σ, s] 2 = [σ, r] 2 ∪ [r, s] × [σ, r] ∪ [σ, s] × [r, s] .

Similarly, (21.11) follows from ( ) ( ) [¯ s, σ] 2 = [¯ r, σ] 2 ∪ [¯ s, σ] × [¯ s, r¯] ∪ [¯ s, r¯] × [¯ r, σ] .



21.6 . Lemma. Assume that Ψ : [a, b ] → R

is a regulated function ,

(21.12)

Φ : [a, b ] → R

is nondecreasing ,

(21.13)

∥U (τ, t) − U (τ, t¯)∥ ≤ |Φ(t) − Φ(t¯)| ,

τ, t¯, t ∈ [a, b ] ,

(21.14) }

∥U (τ, t) − U (τ, t¯) − U (σ, t) + U (σ, t¯)∥ ≤ |Ψ(τ ) − Ψ(σ)| |Φ(t) − Φ(t¯)| ,

(21.15)

τ, σ, t¯, t ∈ [a, b ] .

Then there exists κ > 0 such that |Ψ(τ )| ≤ κ,

τ ∈ [a, b ] .

(21.16)

Further, b ([c, d ] × [e, f ])∥≤ |Ψ(d) − Ψ(c)| |Φ(f ) − Φ(e)| ∥U

} (21.17)

for [c, d ], [e, f ] ⊂ [a, b ], b ([τ, t] 2 ) V (τ, t) = U

for a ≤ τ < t ≤ b ,

∥V (σ, s) − V (σ, r)∥ ≤ |Ψ(s) − Ψ(r)| (Φ(r) − Φ(σ)) + |Ψ(s) − Ψ(σ)| (Φ(s) − Φ(r)) for a ≤ σ ≤ r ≤ s ≤ b , ∥V (¯ s, σ) − V (¯ r, σ)∥ ≤ |Ψ(σ) − Ψ(s)| (Φ(¯ r) − Φ(s)) + |Ψ(¯ r) − Ψ(¯ s)| (Φ(σ) − Φ(¯ r)) for a ≤ s¯ ≤ r¯ ≤ σ ≤ b . Moreover, the limits

(21.18)       

(21.19)

      

(21.20)

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lim V (e, t) = Θright (e),

t→e,t>e

lim

t→g,t
V (g, t) = Θleft (g),

a≤e
(21.21)

a
(21.22)

exist and ∥Θright (e)∥ ≤ |Ψ(e+) − Ψ(e)| (Φ(e+) − Φ(e)) ,

(21.23)

∥Θleft (g)∥ ≤ |Ψ(g) − Ψ(g−)| (Φ(g) − Φ(g−)) .

(21.24)

Observe that then Ψ(e+) − Ψ(e) > 0 and Φ(e+) − Φ(e) > 0 , then Ψ(e) − Ψ(e−) > 0 and Φ(e) − Φ(e−) > 0 .

if Θright (e) ̸= 0 if Θℓ (e) ̸= 0

Proof. (21.16) holds by (21.12). Further, (21.17) holds by (21.15), (21.4), (21.1) whereas (21.18) is valid by (21.4), (21.1). And (21.19) holds by (21.10), (21.17). Similarly, (21.20) follows from (21.11). It can be deduced by (21.19), (21.12), (21.13) that the limit in (21.21) exists. (21.23) holds b ([σ, t] 2 ) for σ < t ≤ b. (21.22) and (21.24) are by (21.17) since V (σ, t) = U proved by similar arguments. The proof is complete.  21.7. Theorem. Let (21.12)–(21.15) hold. Define v : [a, b ] → X by v(a) = 0, v(t) = Θright (a) +

∑( ) Θright (τ )+Θleft (τ ) + Θleft (t),

a<τ
for a < t ≤ b .

        

(21.25)

Then V is SKH-integrable and v is its primitive. Proof. If Φ(b) − Φ(a) = 0 then v(t) = 0 for a ≤ t ≤ b and V (τ, t) = 0 for τ, t ∈ [a, b ] (cf. (21.12), (21.13), (21.17), (21.18)) and the theorem holds. Assume that Φ(b) − Φ(a) > 0 . Observe that the sum in (21.25) is well defined since ( ) ∥Θright (τ )∥ + ∥Θleft (τ )∥ ≤ 2 κ Φ(τ +) − Φ(τ −)

(21.26)

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139

by (21.23), (21.24). Let ε > 0. Put Λ = {λ; a ≤ λ < b, |Ψ(λ+) − Ψ(λ)| ≥ ε} ∪ {λ; a < λ ≤ b, |Ψ(λ) − Ψ(λ−)| ≥ ε} .

  

(21.27)

Then Λ has a finite number m of elements .

(21.28)

There exists κ ∈ R+ such that |Ψ(λ)| ≤ κ

for λ ∈ [a, b ].

(21.29)

(21.28) and (21.29) are consequences of the basic properties of regulated functions. Let δ: [a, b ] → R+ fulfil a+δ(a) < b, b−δ(b) > a, a < t−δ(t) < t+δ(t) < b for a < t < b ,

(21.30)

[λ−δ(λ), λ+δ(λ)] ∩ [a, b ] ∩ Λ = {λ} for λ ∈ Λ , Ψ(t) − Ψ(λ+) ≤ ε for λ < t ≤ λ + δ(λ), λ, t ∈ [a, b ] , Ψ(λ−) − Ψ(t) ≤ ε for λ − δ(λ) ≤ t < λ, λ, t ∈ [a, b ] ,

(21.31)

ε for λ ∈ Λ, a < λ , κm ε Φ(λ + δ(λ)) − Φ(λ+) ≤ for λ ∈ Λ, λ < b , κm

Φ(λ−) − Φ(λ−δ(λ)) ≤

(21.32) (21.33) (21.34) (21.35)

[a, b ] ∩ [λ − δ(λ), λ + δ(λ)] ⊂ [a, b ] \ Λ for λ ∈ [a, b ] \ Λ , Ψ(t) − Ψ(λ−) ≤ ε if λ−δ(λ) ≤ t < λ, λ, t ∈ [a, b ] , Ψ(t) − Ψ(λ+) ≤ ε if λ < t ≤ λ+δ(λ), λ, t ∈ [a, b ] ,

(21.36)

Φ(λ−) − Φ(t) ≤ ε

for λ − δ(λ) ≤ t < λ, a < λ ≤ b ,

(21.39)

Φ(t) − Φ(λ+) ≤ ε

for λ < t < λ + δ(λ), a ≤ λ < b .

(21.40)

(21.37) (21.38)

There exists a δ which fulfils (21.30)–(21.40) since Λ is finite, Ψ is a regulated function and Φ is nondecreasing.

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Let S = {([¯ s, s], σ)} be a δ-fine simple partition of [a, b ]. Put S1 = {([σ, s], σ); |Ψ(σ+)−Ψ(σ)| ≥ ε} , (21.41) ( ) ( ) S2 = {([s, σ], σ); σ ∈ [a, b ] \ Λ or σ ∈ Λ, |Ψ(σ+)−Ψ(σ)| < ε } , (21.42) S3 = {([σ, s], σ); |Ψ(σ)−Ψ(σ−)| ≥ ε} , (21.43) ( ) ( ) S4 = {([s, σ], σ); σ ∈ [a, b ] \ Λ or σ ∈ Λ, |Ψ(σ)−Ψ(σ−)| < ε } . (21.44) Obviously, S=

4 ∪

Sj ,

Si ∩ Sj = ∅ for i ̸= j .

j=1

Let ([σ, s], s) ∈ S1 , σ < r ≤ s. Then (cf. (21.19)) ( ( ∥V (σ, s)−V (σ, r)∥ ≤ Ψ(s)−Ψ(r) Φ(r)−Φ(σ)) + Ψ(s)−Ψ(σ) Φ(s)−Φ(r)) . The limiting process for r → σ gives (cf. (21.21), (21.38), (21.35))  ∥V (σ, s)−Θright (σ)∥     ( (  ≤ Ψ(s)−Ψ(σ+) Φ(σ+)−Φ(σ))+ Ψ(s)−Ψ(σ) Φ(s)−Φ(σ+)) (21.45)    ( ) ( 2ε ε  ≤ 2 ε Φ(σ+) − Φ(σ) + .  ≤ 2 ε Φ(σ+) − Φ(σ) + 2 κ κm m Moreover (cf. (21.25), (21.24), (21.23)) ∥v(s) − v(σ) − Θright (σ)∥ ∑ [ ] ∥Θleft (λ)∥+∥Θright (λ)∥ + ∥Θleft (s)∥ ≤ σ<λ<s ∑ [ ( ( ) )] Ψ(λ)−Ψ(λ−) Φ(λ)−Φ(λ−) + Ψ(λ+)−Ψ(λ) Φ(λ+)−Φ(λ) ≤ σ<λ<s

Hence

( ) + Ψ(s) − Ψ(s−) Φ(s)−Φ(s−) . ( ) ∥v(s) − v(σ) − Θright (σ)∥ ≤ ε Φ(s) − Φ(σ)

(21.46)

since σ ∈ Λ and if σ < λ ≤ σ + δ(σ) then λ ∈ [a, b ] \ Λ (cf. (21.31)) and Ψ(λ) − Ψ(λ−) < ε,

Ψ(λ+) − Ψ(λ) < ε .

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By (21.45), (21.47) ( ) 2ε ∥v(s) − v(σ) − V (σ, s)∥ ≤ 3 ε Φ(b) − Φ(a) + m and

∑

( ) ∥v(s) − v(σ) − V (σ, s)∥ ≤ 3 ε Φ(b) − Φ(a) + 2 ε

(21.47)

S1

(cf. (21.28)). ( ) Let [¯ s, σ], σ ∈ S3 , s¯ ≤ r¯ < σ. Then (cf. (21.1), (21.20)), ∥V (σ, s¯) − V (σ, r¯)∥ = ∥V (¯ s, σ) − V (¯ r, σ)∥ ( ( ) ) ≤ Ψ(σ) − Ψ(s) Φ(¯ r) − Φ(s) + Ψ(¯ r) − Ψ(¯ s) Φ(σ) − Φ(¯ r) . The limiting process for r¯ → σ gives (cf. (21.22), (21.34), (21.37))

( )

V (σ, s¯) − Θ (σ) ≤ 2 κ ε + 2 ε Φ(σ) − Φ(σ−) . (21.48) left κm Moreover (cf. (21.25), (21.24), (21.23)) ∥v(σ)−v(¯ s)−Θleft (σ)∥ ≤ ∥Θright (¯ s)∥ +

∑ (



)

Θ (λ) + Θ

(λ) left right

s¯<λ<σ

( ) ≤ Ψ(¯ s+) − Ψ(¯ s) Φ(¯ s+) − Φ(¯ s) +

∑ ( ( ( ) )) Ψ(λ)−Ψ(λ−) Φ(λ)−Φ(λ−) + Ψ(λ+)−Ψ(λ) Φ(λ+)−Φ(λ) .

s¯<λ<σ

Hence

( ) ∥v(σ) − v(¯ s) − Θleft (σ)∥ ≤ ε Φ(σ) − Φ(¯ s) .

(21.49)

By (21.47), (21.48) ( ) 2ε ∥v(σ) − v(¯ s) − V (σ, s¯)∥ ≤ 3 ε Φ(σ) − Φ(¯ s) + m and

∑

( ) ∥v(σ) − v(¯ s) − V (σ, s¯)∥ ≤ 3 ε Φ(b) − Φ(a) + 2 ε .

(21.50)

S3

The relations (21.47) and (21.50) give ∑ ( ) ∥v(s) − v(¯ s) − V (σ, s) + V (σ, s¯)∥ ≤ 6 ε Φ(b) − Φ(a) + 4 ε . (21.51) S1 ∪S3

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Let ([σ, s], σ) ∈ S2 . Then σ < s ≤ σ + δ(σ) and either σ ∈ [a, b ] \ Λ or σ ∈ Λ, Ψ(λ+) − Ψ(λ) < ε .

(21.52)

In both cases (cf. (21.36), (21.36)) λ ∈ [a, b ] \ Λ for σ < λ <s .

(21.53)

Hence (cf. (21.23)–(21.25))     

∥v(s) − v(σ)∥

∑( ) (21.54) ≤ ∥Θright (σ)∥ + ∥Θright (λ)∥ + ∥Θleft (λ)∥ + ∥Θleft (s)∥   σ<λ<s  ( )  ≤ ε Φ(s) − Φ(σ) since Ψ(σ+) − Ψ(σ) < ε, Ψ(λ+) − Ψ(λ) < ε,

Ψ(λ) − Ψ(λ−) < ε , Ψ(s) − Ψ(s−) < ε .

Moreover (cf. (21.4), (21.15), (21.52), (21.38)) b ([σ, s]×[σ, s]) ∥V (σ, s)∥ ≤ ∥U [ ]( ) ≤ Ψ(s) − Ψ(σ+) + Ψ(σ+) − Ψ(σ) Φ(s) − Φ(σ) ( ) ≤ 2 ε Φ(s) − Φ(σ) since Ψ(s) − Ψ(σ+) ≤ ε by (21.38) and Ψ(σ+) − Ψ(σ) < ε

      

(21.55)

by (21.52). The relations (21.54) and (21.55) imply that ( ) ∥v(s) − v(σ) − V (σ, s)∥ ≤ 3 ε Φ(s) − Φ(σ) for ([σ, s], σ) ∈ S2 . (21.56)

Similarly ( ) ∥v(¯ s) − v(σ) − V (σ, s¯)∥ ≤ 3 ε Φ(σ) − Φ(¯ s)

for ([¯ s, σ], σ) ∈ S4 . (21.57)

By (21.56) and (21.57) ( ) ∥v(s) − v(¯ s) − V (σ, s) + V (σ, s¯)∥ ≤ 3 ε Φ(s) − Φ(¯ s) for ([¯ s, s], σ) ∈ S2 ∪ S4

} (21.58)

and ∑ S2 ∪S4

( ) ∥v(s) − v(¯ s) − V (σ, s) + V (σ, s¯)∥ ≤ 3 ε Φ(b) − Φ(a) .

(21.59)

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Finally, (21.51) and (21.59) give ∑ ( ( ) ) ∥v(s) − v(¯ s) − V (σ, s) + V (σ, s¯)∥ ≤ ε 9 Φ(b)−Φ(a) + 4 . S

V is SKH-integrable and v is its primitive. The proof is complete.  ∫ b 21.8. Remark. Let (21.12)–(21.15) hold. Then (SKH) Dτ U (τ, t) exists a

and ∫ (SKH)

b

Dt U (τ, t) a

∫ = U (b, b) − U (a, a) −(SKH)

∫

b

b

Dτ U (τ, t) +(SKH)

a

Dt V (τ, t) . a

This is a consequence of Theorems 21.3 and 21.7. 21.9 . Remark. A similar topic was treated in [Schwabik (2001)] and [Monteiro, Tvrd´ y (2011)] by different methods.

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Chapter 22

A variant of Gronwall inequality

22.1. Notation. Let [a, b ] ⊂ R and Φ : [a, b ] → R. Assume that Φ is nondecreasing,

} (22.1)

Φ(τ )−Φ(τ −) < 1 for a<τ ≤b, Φ(τ +)−Φ(τ ) < 1 for a≤τ 0}. countable and may be written as Q(S, T ) = {σj }j∈N . The sums ∑ ∑ (Φ(σj )−Φ(σj −)) , (Φ(σj +)−Φ(σj )), ∑

Q(S,T )

Q(S,T )

Q(S, T ) is       

(Φ(σj +)−Φ(σj −))

Q(S,T )

(22.2)

are convergent and independent of the choice of the sequence {σj }j∈N . Therefore they are denoted by ∑ ( ) ∑ ( ) ∑ ( ) Φ(σ+)−Φ(σ) , Φ(σ)−Φ(σ−) , Φ(σ+)−Φ(σ−) . S<σ
S<σ
S<σ
Similarly we interpret some analogous cases and the products ∏ ( ∏ ( ) ) 1 + Φ(σ+) − Φ(σ) , 1 + Φ(σ−) − Φ(σ) , S<σ
etc .

S<σ
Define ΦJ (a) = 0, ΦJ (T ) = Φ(a+)−Φ(a)+

∑

(Φ(τ +)−Φ(τ −)) + Φ(T )−Φ(T −)

a<τ
ΦC (T ) = Φ(T ) − ΦJ (T )

for a < T ≤ b ,

for T ∈ [a, b ] . 145

        

(22.3)

(22.4)

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ΦJ is called the jump function, ΦC is continuous, ΦC (a) = Φ(a), and both ΦJ and ΦC are nondecreasing. (22.3) and (22.4) imply that Φ(T )−Φ(S) = ΦC (T ) − ΦC (S) + Φ(S + ) − Φ(S)+

∑

(Φ(σ+)−Φ(σ−))

a<σ
+Φ(T ) − Φ(T −),

a≤S ≤T ≤b.

          

(22.5)

Define         

Z(S, S) = 1,

Z(S, T ) = exp(ΦC (T )−ΦC (S)) (1 + Φ(S + ) − Φ(S)) ∏ [ ] (1+Φ(σ−)−Φ(σ))−1 (1+Φ(σ+)−Φ(σ))  ×     S<σ
(22.6)

The possibly infinite product in (22.6) is convergent as we will show in Remark 22.8. Hence (cf. also (22.1)) Z(S, T ) is well defined. 22.2 . Remark. Let [S, T ] ⊂ [a, b ], ξ : [S, T ] → R. As in Remark 14.23, ∫ T ∫ T ∫ T ξ dΦ or ξ(τ ) dΦ(τ ) are abbreviations for (KH) Dt ξ(τ ) Φ(t), i.e. S

S

∫

S

∫

T

∫

T

ξ dΦ =

T

ξ(τ ) dΦ(τ ) = (KH)

S

Dt ξ(τ ) Φ(t) .

S

S

Moreover, ∫

∫

T

(SKH)

T

Dt ξ(τ ) Φ(t) = (KH) S

Dt ξ(τ ) Φ(t) S

by Corollary 19.14. The aim of this chapter is to prove Theorems 22.3 and 22.5. 22.3. Theorem. Let ξ : [a, b ] → R+ , ∫ ξ(T ) = 1 +

T

ξ dΦ a

for T ∈ [a, b ] ,

(22.7)

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147

and let ρ : [a, b ] → R+ 0 , η > 0, and ∫

T

ρ(T ) ≤ η +

ρ dΦ

for T ∈ [a, b ] .

(22.8)

a

Then ρ(T ) ≤ η ξ(T )

for T ∈ [a, b ] .

22.4. Proposition. The solution ξ of (22.7) is unique. Proof. Assume that Theorem 22.3 is valid and let (22.7) and (22.8) hold. Then by Theorem 22.3, ζ(T ) ≤ ξ(T ) and ξ(T ) ≤ ζ(T ) for T ∈ [a, b ].  22.5. Theorem. Put ξ(T ) = Z(a, T )

for T ∈ [a, b ] .

(22.9)

Then ξ is a solution of (22.7). 22.6. Remark. Let Φ(τ ) = Φ(τ −) for a < τ ≤ b. Then ( ) ξ(T ) ≤ exp Φ(T ) − Φ(a) for T ∈ [a, b ] . This is a consequence of (22.6) since ( ) ∏ ( ) Z(S, T ) = exp ΦC (T ) − ΦC (S) 1 + Φ(σ+) − Φ(σ) (

) ≤ exp Φ(T ) − Φ(a) .

S≤σ
Proof of Theorem 22.3 By (22.7) and (22.8) ∫

T

(ρ − η ξ) dΦ,

ρ(T ) − η ξ(T ) ≤

T ∈ [a, b ] .

(22.10)

a

∫T Put M = {µ ≥ a ; a (ρ − η ξ) dΦ ≤ 0 for T ∈ [a, µ]} and m = sup M. Then a ∈ M and there exists ν ∈ [a, b ] such that either a<ν ≤b

and M = {t ; a ≤ t < ν} ,

(22.11)

or M = [a, ν] .

(22.12)

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Let (22.11) take place. Then (cf. (14.24), (14.25), (22.10)) ∫

∫

ν

∫

ν−

(ρ − η ξ) dΦ =

ν

(ρ − η ξ) dΦ+

a

a

(ρ − η ξ) dΦ ν−

∫

ν

≤ 0 + (ρ(ν) − η ξ(ν)) (Φ(ν) − Φ(ν−)) ≤

(ρ − η ξ) dΦ (Φ(ν) − Φ(ν−)) , a

which implies that

∫

ν

(ρ − η ξ) dΦ ≤ 0 a

since 0 = Φ(ν) − Φ(ν−) < 1 by (22.1). Therefore (22.12) holds. Let (22.12) take place. Assume that ν < b. Then ∫

ν

ρ(ν) − η ξ(ν) ≤

(ρ − η ν) dΦ ≤ 0 a

and there exists λ, ν < λ ≤ b, such that Φ(λ) − Φ(ν+) ≤

1 2

.

Put Θ = sup{ρ(τ ) − η ξ(τ ) ; ν ≤ τ ≤ λ}. Then ∫ Θ ≤ sup

ν≤τ ≤λ

∫

∫

τ

(ρ − η ξ) dΦ ≤ sup

ν≤τ ≤λ

a

∫

ν+

≤

ν≤τ ≤λ

(ρ − η ξ) dΦ ν

τ

(ρ − η ξ) dΦ + sup ν

τ

(ρ − η ξ) dΦ ν+

≤ 0 + Θ (Φ(λ) − Φ(ν+)) ≤ Θ 12 . Hence Θ ≤ 0, ∫

λ

(ρ−η ξ) dΦ = a

∫

ν

(ρ−η ξ) dΦ + a

∫

λ

(ρ−η ξ) dΦ ≤ Θ (Φ(λ)−Φ(ν+)) ≤ 0 , ν

which contradicts the assumption ν < b. Therefore M = [a, b ] and (22.9) holds. The proof is complete.  The proof of Theorem 22.5 is preceded by several lemmas. Some elementary inequalities are summarized in the following lemma:

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22.7. Lemma.

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149

Let λ ∈ R, |λ| ≤ 12 . Then | exp (λ) − 1 − λ| ≤ λ2 ,

(22.13)

|(1 + λ)−1 −1 + λ| ≤ 2 λ2 ,

(22.14)

| ln(1+λ)− λ| ≤ λ2 .

(22.15)

Proof. (22.13)–(22.15) can be deduced from the well-known expansions of functions exp (λ), (1 + λ)−1 , ln (1 + λ), where ln(1 + λ) is the natural logarithm of (1 + λ).  22.8. Remark. The product in (22.6) is convergent since its convergence is equivalent with the convergence of the series ∑ [

] ln(1 + Φ(σ+) − Φ(σ)) − ln(1 + Φ(σ) − Φ(σ−)) .

S<σ
(22.15) implies that the series is convergent since the number of σ such that Φ(σ+) − Φ(σ) > 21 or Φ(σ) − Φ(σ−) > 21 is finite (see also (22.1)). Hence there exists κ > 1 such that 1 ≤ Z(S, T ) ≤ κ

for a ≤ S < T ≤ b .

(22.16)

Moreover, by (22.6) Z(R, T ) = Z(R, S) Z(S, T )

for a ≤ R ≤ S < T ≤ b .

(Z(S, T ))−1 exists by (22.16). The domain of Z may be extended to [a, b ] 2 by putting Z(S, T¯) = (Z(T¯, S))−1

for T¯ < S .

(22.17)

Then Z(R, T ) = Z(R, S) Z(S, T ) for R, S, T ∈ [a, b ] .

(22.18)

22.9. Lemma. Let a ≤ R < S < T ≤ b,

Φ(T ) − Φ(S + ) ≤ 31 ,

Φ(S−) − Φ(R) ≤ 31 .

(22.19)

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Then  Z(S, T )−Z(S, S+)   ( )( ) ( ) = 1+Φ(S+)−Φ(S) Φ(T )−Φ(S+) + 1+Φ(S+)−Φ(S) R1 , (22.20)  ( )2  where |R1 | ≤ 5 Φ(T ) − Φ(S+) , Z(S, R)−Z(S, S−) ( )( ) ( ) = 1+Φ(S−)−Φ(S) Φ(R)−Φ(S−) + 1+Φ(S−)−Φ(S) R2 , ( )2 where |R2 | ≤ 5 Φ(S−) − Φ(R) .

Proof.

    

(22.21)

By (22.6)

Z(S, S+) = 1 + Φ(S+) − Φ(S) , ( )( ) Z(S, T ) − Z(S, S+) = 1 + Φ(S+) − Φ(S) A − 1 ,

(22.22)

where

( ) A = exp ΦC (T ) − ΦC (S) ∏ [( )−1 ( )] × 1+Φ(σ−)−Φ(σ) 1+Φ(σ+)−Φ(σ) S<σ
( )−1 × 1+Φ(T −)−Φ(T ) , ∑ [ ln A = ΦC (T )−ΦC (S) + ln(1+Φ(σ+)−Φ(σ)) +

∑ [

S<σ
] ln(1+Φ(σ+)−Φ(σ)− ln(1+Φ(σ−)−Φ(σ))

S<σ
] − ln(1 + Φ(T −) − Φ(T )) ∑ [ ] = ΦC (T )−ΦC (S)+ Φ(σ+)−Φ(σ−) + Φ(T )−Φ(T −) + R3 S<σ
= Φ(T )−Φ(S+) + R3 and |R3 | ≤ 2

( ∑ [( S<σ
)2 ) )2 ] ( )2 ( + Φ(T )−Φ(T −) Φ(σ) − Φ(σ−) + Φ(σ+) −Φ(σ)

( )2 ≤ 2 Φ(T ) − Φ(S+) .

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151

Hence ( )[ ( ) ] Z(S, T ) − Z(S, S+) = 1 + Φ(S+) − Φ(S) exp Φ(T )−Φ(S+)+R3 − 1 ( )[ ] = 1 + Φ(S+) − Φ(S) Φ(T )−Φ(S+)+R1 ( )( ) ( ) = 1 + Φ(S+) − Φ(S) Φ(T )−Φ(S+) + 1 + Φ(S+) − Φ(S) R1 , where ( )2 |R1 | ≤ |R3 | + Φ(T ) − Φ(S+) + R3 ( )2 ( )2 [ ( )]2 ≤ 2 Φ(T ) − Φ(S+) + Φ(T ) − Φ(S+) 1+4 Φ(T )−Φ(S+) ( )2 [ ( )2 4 ] ≤ Φ(T ) − Φ(S+) 2 + (1+ )2 ≤ 5 Φ(T ) − Φ(S+) . 9 Hence (22.20) holds. (22.21) can be proved in a similar way starting from the equations ( )−1 Z(S−, S) = 1+Φ(S−)−Φ(S) , Z(S, S−) = 1+Φ(S−)−Φ(S) , (22.23) ( )−1 Z(S, R) = Z(R, S) ( ( )( ) = exp ΦC (S)−ΦC (R) 1+Φ(R+)−Φ(R) )−1 ∏[ ] (1+Φ(σ−)−Φ(σ))−1(1+Φ(σ+)−Φ(σ)) (1+Φ(S−)−Φ(S))−1 . × R<σ<S

The proof is complete.



22.10. Lemma. Let (22.19) hold. Then  |Z(S, T ) − 1 − Φ(T ) + Φ(S)|    ( )( ) ( ) ≤ Φ(S+)−Φ(S) Φ(T ) − Φ(S+) + 1+Φ(S+)−Φ(S) R1 , (22.24)   ( )2  |R1 | ≤ 5 Φ(T ) − Φ(S+) for a ≤ S < T ≤ b,  |Z(S, R) − 1 − Φ(R) + Φ(S)|    ( )( ) ( ) ≤ Φ(S)−Φ(S−) Φ(S−) − Φ(R) + 1+Φ(S−)−Φ(S) R2 , (22.25)   ( )2  |R2 | ≤ 5 Φ(S−) − Φ(R) for a ≤ R < S ≤ b . Proof.

(22.24) follows by (22.22), (22.20) since

Z(S, T ) − 1 − Φ(T ) + Φ(S) = Z(S, S+)−1−Φ(S+)+Φ(S)+Z(S, T )−Z(S, S+)−Φ(T )+Φ(S+) . Similarly, (22.25) follows by (22.23) and (22.21).



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152

Proof of Theorem 22.5. By (22.9), (22.11), (22.18) }

ξ(T ) − ξ(S) − ξ(S) (Φ(T ) − Φ(S)) ( ) = Z(a, S) Z(S, T ) − 1 − Φ(T ) + Φ(S)

for a ≤ S < T ≤ b ,

ξ(R) − ξ(S) − ξ(S) (Φ(R) − Φ(S)) ( ) = Z(a, S) Z(S, R) − 1 − Φ(R) + Φ(S)

for a ≤ R < S ≤ b .

(22.26) } (22.27)

Put { 1} M (1) = τ ∈ [a, b ]; Φ(τ +)−Φ(τ ) > , 2 { 1} N (1) = τ ∈ [a, b ]; Φ(τ )−Φ(τ −) > , 2 { 1 1} for k ∈ N , M (k+1) = τ ∈ [a, b ]; k+1 < Φ(τ +)−Φ(τ ) ≤ k 2 2 { 1 1} N (k+1) = τ ∈ [a, b ]; k+1 < Φ(τ )−Φ(τ −) ≤ k for k ∈ N . 2 2 The sets M (k), N (k) are finite. Furthermore, Φ(τ +) = Φ(τ ) for τ ∈ [a, b ]\

∪

M (k) ,

(22.28)

N (k) .

(22.29)

k∈N

Φ(τ −) = Φ(τ ) for τ ∈ [a, b ]\

∪

k∈N

Let ε > 0 and let δ: [a, b ] → R+ fulfil the following conditions ∑( )[ ( )2 ] 1+Φ(τ +)−Φ(τ ) Φ(τ +δ(τ ))−Φ(τ +)+5 Φ(τ +δ(τ ))−Φ(τ +)   τ ∈M (k) (22.30)   k−2 ≤ ε2 for k ∈ N , ∑(

)[ ( )2 ] 1+Φ(τ −)−Φ(τ ) Φ(τ −)−Φ(τ −δ(τ ))+5 Φ(τ −)−Φ(τ −δ(τ ))  

τ ∈N (k)

≤ ε2

k−2

for k ∈ N .

 

(22.31)

(Conditions (22.30), (22.31) can be fulfilled since the sets M (k), N (k) are

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153

finite.) Furthermore, let ( ) [ ] 1+Φ(τ +)−Φ(τ ) 5 Φ(τ +δ(τ ))−Φ(τ ) ≤

ε Φ(b)−Φ(a) ∪( ) for τ ∈ [a, b ] \ M (k) ∪ N (k) .

      

(22.32)

k∈N

Let T = {([t¯, t], τ )} be a simple δ-fine partition of [a, b ]. Then (22.16), (22.24), (22.26) and (22.30) imply that ∑  |ξ(t) − ξ(τ ) − ξ(τ ) (Φ(t) − Φ(τ ))|     ([t¯,t],τ )∈T   τ ∈ M (k) (22.33) ∑  ≤κ |Z(t, τ ) − 1 − (Φ(t) + Φ(τ ))| ≤ κ ε 2−k−2 for k ∈ N.      ([τ,t],τ )∈T τ ∈M (k)

Similarly ∑ |ξ(t¯) − ξ(τ ) − ξ(τ ) (Φ(t¯) − Φ(τ ))| ≤ κ ε 2−k−2

 for k ∈ N.  

([τ,t],τ )∈T τ ∈N (k)

Let T1 be the set of ([τ, t], τ ) ∈ T such that τ ∈ [a, b ]\ be the set of ([t¯, τ ], τ ) ∈ T such that τ ∈ [a, b ]\

∪

∪

(22.34)

M (k) and let T2

k∈N

N (k). Then by (22.16)

k∈N

and (22.24) ∑ ∑ |ξ(t) − ξ(τ ) − ξ(τ ) (Φ(t) − Φ(τ ))| ≤ κ R1 T1

≤κ

T1 ∑ ( )( ) 5 Φ(τ +δ(τ ))−Φ(τ ) Φ(t)−Φ(τ +) T1

≤ κε

∑ Φ(t)−Φ(τ +) T1

Similarly ∑

Φ(b)−Φ(a)

≤ κε.

|ξ(t¯) − ξ(τ ) − ξ(τ ) (Φ(t¯) − Φ(τ ))| ≤ κ ε .

T2

(22.33)–(22.36) imply that ∑ |ξ(t) − ξ(t¯) − ξ(τ ) (Φ(t) − Φ(t¯))| ≤ 3 κ ε . T

                

(22.35)

(22.36)

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Obviously ξ(a) = 1. Therefore ξ fulfils (22.9) The proof is complete.



22.11. Remark. If Φ(τ −) = Φ(τ ) for a < τ ≤ b then ( ) ξ(T ) ≤ exp Φ(T ) − Φ(a) for T ∈ [a, b ] . This follows from (22.5) and (22.4) since 1 + α ≤ exp (α) for α ∈ R+ . Hence the estimate ( ) ρ(T ) ≤ η exp Φ(T ) − Φ(a) is valid for S from Theorem 22.3.

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Chapter 23

Existence of solutions of a class of generalized ordinary differential equations The aim of this chapter is to prove existence, uniqueness and continuous dependence theorems for the GODE d x = Dt F (x, τ, t) . dt In particular, the theorems are valid in the case of the equation k ∑ d x = Dt fi (x) ψi (τ ) ψi+k (t) , dt i=1

(23.1)

where fi : X → X and D fi fulfil a Lipschitz condition and ψi : [a, b ] → R, |ψi (τ ) − ψi (σ)| ≤ |Φ(τ ) − Φ(σ)|,

i = 1, 2, . . . , 2k,

τ, σ ∈ [a, b ] .

23.1. Notation. Let [a, b ] ⊂ R, Φ : [a, b ] → R, F : X × [a, b ] 2 → X. Let Φ be nondecreasing and continuous from the left. Assume that } ∥F (x, τ, t) − F (x, τ, s)∥ ≤ (1 + ∥x∥) |Φ(t) − Φ(s)| (23.2) for x ∈ X, τ, s, t ∈ [a, b ] , } ∥∆v (F (x, τ, t) − F (x, τ, s))∥ ≤ ∥v∥ |Φ(t) − Φ(s)| (23.3) for x, v ∈ X, τ, s, t ∈ [a, b ] ,  ∥F (x, τ, t) − F (x, τ, s) − F (x, σ, t) + F (x, σ, s)∥   (23.4) ≤ (1 + ∥x∥) |Φ(τ ) − Φ(σ)| |Φ(t) − Φ(s)|   for x ∈ X, τ, σ, s, t ∈ [a, b ] , ( )  ∥∆v F (x, τ, t) − F (x, τ, s) − F (x, σ, t) + F (x, σ, s) ∥   (23.5) ≤ ∥v∥ |Φ(τ ) − Φ(σ)| |Φ(t) − Φ(s)|   for x, v ∈ X, τ, σ, s, t ∈ [a, b ] . 155

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156

Observe that the one-sided limits Φ(s+) = lim Φ(t), a ≤ s < b,

Φ(s−) = lim Φ(t) = Φ(s), a < s ≤ b ,

t→s,t>s

F (x, σ, s+)= lim F (x, σ, t), t→s,t>s

t→s,t<s

x ∈ X, σ ∈ [a, b ], a ≤ s < b ,

F (x, σ, s−)= lim F (x, σ, t) = F (x, σ, s) t→s,t<s

x ∈ X, σ ∈ [a, b ], a < s ≤ b , etc.

exist. Moreover, ( )( ) ∥F (x, σ, t) − F (x, σ, s+)∥ ≤ 1 + ∥x∥ Φ(t) − Φ(s+)

}

for x ∈ X, σ ∈ [a, b ], a ≤ s < b

(23.6)

and several similar inequalities are valid. 23.2. Lemma. Let [c, d ] ⊂ [a, b ] and let w : [c, d ] → X be a function of bounded variation. Then ∫ d the integral (SKH) Dt F (w(τ ), τ, t) exists . (23.7) c

Define z : [c, d ] → X by ∫ z(T ) = (SKH)

T

Dt F (w(τ ), τ, t) . c

Then z has a bounded variation and is continuous from the left .

(23.8)

Proof. Put U (τ, t) = F (w(τ ), τ, t) for τ, t ∈ [c, d ]. It can be deduced by (23.2)–(23.5) that U fulfils (20.3), (20.4) where Ψ, Φ are suitable nondecreasing functions (since w is bounded). (23.7) and (23.8) are true by Theorem 20.9.  23.3. Lemma. Let [c, d ] ⊂ [a, b ], yb ∈ X. Assume that Φ(d) − Φ(c+) ≤

1 2

.

Then there exists a function vc,d : [c, d ] → X such that ∫ T vc,d (T ) = yb + (SKH) Dt F (vc,d (τ ), τ, t) , for T ∈ [c, d ] ,

(23.9)

(23.10)

c

vc,d has a bounded variation and is continuous from the left .

(23.11)

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Proof.

157

Define

 v0 (T ) = yb for T ∈ [c, d ] ,   ∫ T (23.12)  vi+1 (T ) = yb +(SKH) Dt F (vi (τ ), τ, t) for T ∈ [c, d ], i ∈ N0 .  c

By Lemma 23.2 the integrals in (23.12) exist and vi , i ∈ N0 , are functions of bounded variation continuous from the left. By (23.12), Remark 20.10 and Corollary 14.18 vi (c) = yb,

vi (c+) = yb+

for i ∈ N ,

(23.13)

where yb+ = yb + F (b y , c, c+) − F (b y , c, c) .

(23.14)

By (23.12) and (23.2) ∫

T

vi+1 (T ) = yb+ + (SKH)

Dt F (vi (τ ), τ, t) for T ∈ [c, d ], i ∈ N ,

(23.15)

c+

∫

T

v1 (T ) − v0 (T ) = (SKH)

Dt F (b y , τ, t) for T ∈ [c, d ] ( ) ∥v1 (T ) − v0 (T )∥ ≤ (1 + ∥b y ∥) Φ(T ) − Φ(c) for T ∈ [c, d ] . c

(23.16)

By (23.15), (23.3) (cf. Theorem 20.9) ∫ T

[ ]

∥vi+2 (T ) − vi+1 (T )∥ ≤ (SKH) Dt F (vi+1 (τ ), τ, t)−F (vi (τ ), τ, t) c+ { }( ) ≤ max ∥vi+1 (τ )−vi (τ )∥; τ ∈ [c, T ] Φ(T )−Φ(c+) for c < T ≤ d . Furthermore,

( ) ∥v1 (t) − v0 (t)∥ ≤ (1 + ∥b y ∥) Φ(T ) − Φ(c)

for c ≤ t ≤ T ≤ d

and (by induction)

( )( )i  ∥vi+1 (T ) − vi (T )∥ ≤ (1 + ∥b y ∥) Φ(d) − Φ(c) Φ(T ) − Φ(c+)  for c < T ≤ d, i ∈ N0 .



(23.17)

Hence (cf. (22.1)) {vi (T )} is uniformly convergent for c ≤ T ≤ d when i → ∞ .

(23.18)

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Put vc,d (T ) = lim vi (T ), i→∞

T ∈ [c, d ] .

The limit procedure for i → ∞ in (23.12) gives ∫ T vc,d (T ) = yb + (SKH) Dt F (vc,d (τ ), τ, t) , for c ≤ T ≤ d,

(23.19)

c

(23.10) holds by (23.19). (23.11) holds by Lemma 23.2. The proof is complete.  23.4. Theorem. Let y ∈ X. Then there exists u : [a, b ] → X such that ∫ T u(T ) = y + (SKH) Dt F (u(τ ), τ, t) for T ∈ [a, b ] . (23.20) a

Equivalently, u is a solution of the GODE d x = Dt F (x, τ, t) dt

(23.21)

on [a, b ] and u(a) = y. Moreover, u has a bounded variation and is continuous from the left . Proof.

(23.22)

Let δ : [a, b ] → R+ be such that Φ(τ + δ(τ )) − Φ(τ +) ≤ 21 for a ≤ τ < b

and Φ(τ ) − Φ(τ − δ(τ )) ≤ 21 for a < τ ≤ b . Let R = {([rj−1 , rj ], ρj )}, a = r0 < r1 < . . . < rℓ = b, be a δ-fine partition of [a, b ]. Let {si ; i = 0, 1, . . . , k} = {rj ; j = 0, 1, . . . , ℓ} ∪ {ρj ; j = 0, 1, . . . , ℓ} and s0 < s1 < . . . < sk . Then Φ(si ) − Φ(si−1 +) ≤ 21

for i = 1, 2, . . . , k−1 .

Put y0 = y. By Lemma 23.3 there exists u1 : [s0 , s1 ] → X of bounded variation, continuous from the left and such that ∫ T Dt F (u1 (τ ), τ, t) for s0 ≤ T ≤ s1 . u1 (T ) = y0 + (SKH) s0

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159

Put y1 = u1 (s1 ) By Lemma 23.3 there exists u2 : [s1 , s2 ] → X of bounded variation, continuous from the left and such that ∫ T u2 (T ) = y1 + (SKH) Dt F (u2 (τ ), τ, t) for s1 ≤ T ≤ s2 . s1

By induction there exist functions ui : [si−1 , si ] → X, i = 3, 4, . . . , k, of bounded variation, continuous from the left and yi ∈ X, i = 2, 3, . . . , k−1, such that yi−1 = ui−1 (si−1 ) and ∫ T ui (T ) = yi−1 + (SKH) Dt F (ui (τ ), τ, t) for si−1 ≤ T ≤ si , i = 3, 4, . . . , k . si−1

Define u : [a, b ] → X by u(a) = y,

u(T ) = ui (T ) for si−1 < T ≤ si ,

i = 1, 2, . . . , k .

Then u(s1 ) = u1 (s1 ) = y1 ,

∫

T

u(T ) = u(s1 ) + (SKH)

Dt F (u(τ ), τ, t)

for s1 ≤ T ≤ s2 ,

s1

∫

T

u(T ) = y + (SKH)

Dt F (u(τ ), τ, t)

for a ≤ T ≤ s2 ,

a

u(s2 ) = u2 (s2 ) = y2 ,

. . . etc.

Then u fulfils (23.20). By Lemma 23.2, the functions ui , i = 1, 2, . . . , k, have bounded variation and are continuous from the left. Therefore u fulfils (23.22). Moreover, by Definition 15.2, u is a solution of (23.21) if and only if it fulfils (23.20) and u(a) = y. The proof is complete.  23.5. Lemma. Let u fulfil (23.20). Then u has a bounded variation and is continuous from the left , (23.23) ( ) ( ) ∥u(T )∥ ≤ ∥y∥ + Φ(T ) − Φ(a) exp Φ(T ) − Φ(a) for T ∈ [a, b ] . (23.24) Proof. There exists κ > 0 such that ∥u(τ )∥ ≤ κ for τ ∈ [a, b ] since u is locally bounded by the definition of SKH-integral. Hence ∫ T ( ) ∥u(T ) − u(S)∥ ≤ (SKH) Dt F (u(τ ), τ, t)∥ ≤ (1 + κ) Φ(T ) − Φ(S) S

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160

for a ≤ S ≤ T ≤ b and (23.23) holds. Let ε > 0 and a < T ≤ b. Then there exists δ: [a, T ] → R+ such that ∫ T

∑[ ]

Dt F (u(τ ), τ, t) − F (u(σ), σ, s) − F (u(σ), σ, s¯) < ε

(SKH) a

S

for all δ-fine partitions S = {([¯ s, s], σ)} of [a, T ]. Hence ∑( )( ) ∥u(T ) − u(a)∥ ≤ 1 + ∥u(σ)∥ Φ(s) − Φ(¯ s) + ε S

≤ Φ(T ) − Φ(a) +

∑

( ) ∥u(σ∥ Φ(s) − Φ(¯ s) + ε ,

S

and ∥u(T )∥ ≤ ∥u(a)∥ +

∑(

)( ) 1 + ∥u(σ)∥ Φ(s) − Φ(¯ s) + ε .

(23.25)

S

The function ∥u∥ has a bounded variation since u is a function of bounded ∫ T variation. Hence the integral ∥u(τ )∥ dΦ(τ ) exists and taking S in a

(23.25) sufficiently fine, we get ∫ T ∥u(T )∥ ≤ ∥u(a)∥ + Φ(T )−Φ(a)+ ∥u(τ )∥dΦ(τ )+2 ε a

for T ∈ [a, b ] .

The relations (23.20) and (23.25) imply that ∫ T ∥u(T )∥ ≤ ∥y∥ + Φ(T ) − Φ(a) + ∥u(τ )∥ dΦ(τ )

  

(23.26)

for T ∈ [a, b ]

a

since ε > 0 can be arbitrary. Thus, by Theorem 22.3, ( ) ∥u(T )∥ ≤ ∥y∥ + Φ(T ) − Φ(a) ξ(T ) , where ξ is defined in (22.7) and, by Remark 22.6, fulfils ( ) ∥ξ(T )∥ ≤ exp Φ(T ) − Φ(a) for T ∈ [a, b ] , since Φ is continuous from the left. This implies that (23.24) is true. The proof is complete.  23.6. Lemma. Let y, y ∈ X, let u fulfil (23.9) and let v fulfil ∫ T v(T ) = y + Dt F (v(τ ), τ, t) for T ∈ [a, b ] . a

(23.27)

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161

Then ( ) ∥u(T ) − v(T )∥ ≤ ∥y − y∥ exp Φ(T ) − Φ(a) Proof.

for T ∈ [a, b ] .

(23.28)

By (23.9) and (23.27),

∫ T [ ] u(T )−v(T ) = y−y+(SKH) Dt F (u(τ ), τ, t) − F (v(τ ), τ, t) f or T ∈ [a, b ] . a

Let T ∈ [a, b ]. There exists δ: [a, b ] → R+ such that ∫

(SKH)

[ ] Dt F (u(τ ), τ, t) − F (v(τ ), τ, t)

T

a

∑[ ]

− F (u(σ), σ, s) − F (u(σ), σ, s¯) − F (v(σ), σ, s) + F (v(σ), σ, s¯) ≤ ε S

whenever S = {([¯ s, s], σ)} is a δ-fine partition of [a, T ]. Hence ∥u(T ) − v(T )∥ ∑

≤ ∥y−y∥+ F (u(σ), σ, s)−F (u(σ), σ, s¯)−F (v(σ), σ, s)+F (v(σ), σ, s¯) +ε S

≤ ∥y−y∥ +

∑

( ) ∥u(σ − v(σ)∥ Φ(s)−Φ(¯ s) + ε .

S

Consequently, ∫

T

∥u(T ) − v(T )∥ ≤

∥u(σ) − v(σ)∥ d Φ(s) + ε

for T ∈ [a, b ] .

a

Now, Theorem 22.3 and Remark 22.6 yield ( ) ∥u(T ) − v(T )∥ ≤ (ε + ∥y − y∥) exp Φ(T ) − Φ(a)

for T ∈ [a, b ] .

(23.27) holds as ε > 0 is arbitrary. The proof is complete.



23.7. Remark. Let u fulfil (23.9). Then u is unique. This is a consequence of Lemma 23.6. 23.8. Lemma. Let y ∈ X and let u fulfil (23.20). Let F : X × [a, b ]2 → X fulfil (23.2)–(23.5) with F replaced by F . Assume that u: [a, b ] → X, ∫

T

u(T ) = y + (SKH)

Dt F (u(τ ), τ, t) a

for T ∈ [a, b ] ,

(23.29)

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that Φ∗ : [a, b ] → R is nondecreasing and continuous from the left and that ∥F (x, τ, t) − F (x, τ, s) − F (x, τ, t) + F (x, τ, s )∥ ( ) ≤ (1 + ∥x∥) Φ∗ (t) − Φ∗ (s) for x ∈ X, s, t ∈ [a, b ], s
      

(23.30)

Then ( ) ∥u(T ) − u(T )∥ ≤ C Φ∗ (b) − Φ∗ (a) for T ∈ [a, b ] , where

(23.31)

( ) ( ) C = ∥y∥ + Φ(b) − Φ(a) exp 2 (Φ(b) − Φ(a)) .

Proof.

By Lemma 23.5, u and u are functions of bounded variation and ∫ b continuous from the left. The integral (SKH) Dt F (u(τ ), τ, t) exists by a

Lemma 23.2. Therefore ∫ u(T ) − u(T ) = (SKH)

b

[ ] Dt F (u(τ ), τ, t) −F (u(τ ), τ, t)

a

∫

b

+ (SKH)

     

  ]  Dt F (u(τ ), τ, t) −F (u(τ ), τ, t) for T ∈ [a, b ] .   [

(23.32)

a

Let ε > 0 and T ∈[a, b ]. There exists δ: [a, b ]→ R+ such that

(SKH) ≤

∫

∑

T

] [ Dt F (u(τ ), τ, t) −F (u(τ ), τ, t)

a

∥F (u(σ), σ, s)−F (u(σ), σ, s¯)−F (u(σ), σ, s)+F (u(σ), σ, s¯)∥ + ε

S

≤

∑( )( ) 1 + ∥u(σ)∥ Φ∗ (s) − Φ∗ (¯ s) + ε S

if S = {([¯ s, s], σ)} is a δ-fine partition of [a, T ]. Hence ∫ T ∫ T

] [ ( )

(SKH) Dt F (u(τ ), τ, t)−F (u(τ ), τ, t) ≤ 1+∥u(τ )∥ dΦ∗ (τ ) (23.33) a

a

since the integral on the right hand side exists and ε > 0 is arbitrary. By

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(23.33) and (23.24),

(SKH)

∫

T

   

[ ] Dt F (u(τ ), τ, t) −F (u(τ ), τ, t)

a

) ( )( )   ≤ ∥y∥ + Φ(b) − Φ(a) exp Φ(b) − Φ(a) Φ∗ (b) − Φ∗ (a) .  (

(23.34)

F fulfils (23.3) (with F replaced by F ). Therefore

(SKH)

∫

T

[

] Dt F (u(τ ), τ, t)−F (u(τ ), τ, t) ≤

a

∫

T

∥u(τ )−u(τ )∥ dΦ(τ ) a

which together with (23.32)–(23.34) implies that ∥u(T ) − u(T )∥ ( ) ( )( ) ≤ ∥y∥+Φ(b)−Φ(a) exp Φ(b) − Φ(a) Φ∗ (b) − Φ∗ (a) ∫

T

∥u(τ )−u(τ )∥ dΦ(τ ) for T ∈ [a, b ] .

+ a

(23.31) holds by Theorems 22.3, 22.5 and Remark 22.6. The proof is complete.  23.9 . Remark. Theorem 23.4 and Lemmas 23.6, 23.8 describe the existence, uniqueness and continuous dependence (on y and F ) of solutions to the GODE (23.9). In particular, they can be applied in the case of the equation (23.1). 23.10. Lemma. Define F ◦ : X × [a, b ] → X by ∫ t ◦ F (x, t) = (SKH) Ds F (x, σ, s) .

(23.35)

a

Let v: [a, b ] → X be a function of bounded variation and continuous from the left. Then ∫ T ∫ T ◦ (SKH) Dt F (v(τ ), t) = (SKH) Dt F (v(τ ), τ, t) for T ∈ [a, b ] . (23.36) a

a

Proof. F ◦ is well defined by (23.35) due to Theorem 20.9 and assumptions (23.2) and (23.4). Furthermore, by (23.2) and (23.4), we have ( ( ) ∥F ◦ (x, t)−F ◦ (x, s)∥ ≤ 1+∥x∥) Φ(t)−Φ(s) for x ∈ X, t, s ∈ [a, b ] . (23.37)

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Similarly, ( ) ( ) ∥∆v F ◦ (x, t) − F ◦ (x, s) ∥ ≤ ∥v∥ Φ(t) − Φ(s) for x, v ∈ X, t, s ∈ [a, b ] . Hence F ◦ fulfils (23.2),(23.4) (with F replaced by F ◦ ). Analogically, F ◦ fulfils also (23.3), (23.5) since it depends only on x, t. The integrals in (23.36) exist by Lemma 23.2. Let ε > 0, a < T ≤ b. There exists δ1 : [a, b ] →R+ such that ∫ s

∑

(SKH) Dt F (v(τ ), τ, t)−F (v(σ), σ, s)+F (v(σ), σ, s¯) ≤ ε, (23.38) s¯

S

∫ s

∑

(SKH) Dt F ◦ (v(τ ), t)−F ◦ (v(σ), s)+F ◦ (v(σ), s¯) ≤ ε

(23.39)

s¯

S

whenever S = {([¯ s, s], σ)} is δ1 -fine partition of [a, T ]. By (23.38), (23.39), (23.35) ∫ s ∫ s

∑

Dt F (v(τ ), τ, t)−(SKH) Dt F ◦ (v(τ ), t)

(SKH) s¯

S

≤

s¯

∑

F (v(σ), σ, s)−F (v(σ), σ, s¯)−F ◦ (v(σ), s)+F ◦ (v(σ), s¯)

+2 ε S

∫ ∑

≤ F (v(σ), σ, s)−F (v(σ), σ, s¯)−(SKH)

s

Dt F (v(τ ), τ, t) +2 ε .

s¯

S

By (23.38) ∫ T ∫ ∥(SKH) Dt F (v(τ ), τ, t) − (SKH) a

T

Dt F ◦ (v(τ ), t)∥ ≤ 3 ε

if a < T ≤ b.

a

(23.22) holds since ε > 0 may be arbitrary. The proof is complete.



23.11. Theorem. Let y ∈ X, u: [a, b ] → X, u(a) = y. Then u is a solution of the GODE d x = Dt F (x, τ, t) dt if and only if it is a solution of the GODE d x = Dt F ◦ (x, t) . dt Proof. Theorem 23.11 is a consequence of Lemma 23.10 (cf. Theorem 23.4). 

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Chapter 24

A convergence process as a source of discontinuities in the theory of differential equations 24.1. Motivation. Let [a, b ] ⊂ R, q : X × [a, b ] → X, κ b ∈ R+ , p : X → X. Assume that q

is continuous,

∥q(0, t)∥ ≤ κ b,

∥q(x, t) − q(y, t)∥ ≤ κ b ∥x − y∥ ,

∥p(x) − p(y)∥ ≤ κ b ∥x − y∥ . Then for t, s ∈ [a, b ], s < t, there exists a map Q(s, t): X → X such that y Q(s, t) is its value at y and that the function t 7→ y Q(s, t) is a solution of the classical equation x˙ = q(x, t) such that y Q(s, s) = y. The function y Q(s, .) is uniquely defined, Q(s, t) is an automorphism of X and y Q(s, r) Q(r, t) = y Q(s, t)

for s, r, t ∈ [a, b ] .

Define H : R → {0, 1} by H(t) =

 0

for t ≤ 0 ,

1

for t > 0 .

(24.1)

Let αj , β, γj ∈ [a, b ], αj ≤ β ≤ γj , αj < γj for j ∈ N. Assume that αj → β, γj → β for j → ∞. Let uj : [a, b ] → X be the solution of the classical equation x˙ = q(x, t) + p(x)

χj (t) , βj − αj 165

uj (a) = y .

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χj being the characteristic function of [αj , γj ]. Let v P (s, .) be the unique solution of x˙ = p(x) such that for s ∈ R, v ∈ X .

v P (s, s) = v

Define z : [a, b ] → X by  y Q(a, t) for a ≤ t ≤ β, z(t) = y Q(a, β)(id + P (0, 1)) Q(β, t) for β < t ≤ b . Then uj (t) → z(t) Put

for j → ∞, t ∈ [a, b ], t ̸= β .

( ) F (x, t) = q(x, t) + x P (0, 1) H(t − β),

x ∈ X, t ∈ [a, b ] .

Then z is a solution of the GODE d x = Dt F (t, x) . dt Observe that F (x, .) is continuous from the left. The aim of this chapter is to extend the above result to a situation such that the set of discontinuities of the limit solution z may be countable. 24.2 . Notation. Let [a, b ]⊂R, Ψ: [a, b ]→R, Q: X×[a, b ]→X, p i : X → X, αi , βi , γi ∈ [a, b ], κi ∈ R+ for i ∈ N. Assume that Ψ

is nondecreasing and continuous,

(24.2)

a < βi < b, βi ̸= βj for i ̸= j, i, j ∈ N ,



Q(x, t) − Q(x, s) ≤ (1 + ∥x∥) Ψ(t) − Ψ(s) ,

}

for x ∈ X, t, s ∈ [a, b ] , ( ) ∆v Q(x, t) − Q(x, s) ≤ ∥v∥ Ψ(t) − Ψ(s) ,

}

for x, v ∈ X, t, s ∈ [a, b ] , ∞ ∑ i=1

κi < ∞ ,

(24.3) (24.4)

(24.5)

(24.6)

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∥p i (x)∥ ≤ κi (1 + ∥x∥),

167

}

∥p i (x) − p i (y)∥ ≤ κi (∥x − y∥) ,

(24.7)

for x, y ∈ X, i ∈ N .

24.3. Lemma. Let w ∈ X and i ∈ N. Then there exists a unique solution Pi (w, .): R+ 0 → X of the classical differential equation x˙ = p i (x)

(24.8)

( ) ∥Pi (w, t)∥ ≤ ∥w∥ + κi t exp(κi t) for t ∈ R+ 0 , ( )[ ] ∥Pi (w, 1) − w∥ ≤ κi ∥w∥ + 1 exp(κi ) − 1 .

(24.9)

such that Pi (w, 0) = w. Moreover,

(24.10)

Let w ∈ X. Then ∥Pi (w, t) − Pi (w, t)∥ ≤ ∥w − w∥ exp(κi t) for t ∈ R+ 0 , [ ] ∥Pi (w, 1) − w − Pi (w, 1) + w∥ ≤ ∥w −w∥ exp(κi ) − 1 .

(24.11) (24.12)

Sketch of the proof. By standard methods the solution Pi (w, .) exists for 0 ≤ t < α, where α > 0, and fulfils ∫ t Pi (w, t) = w + p i (Pi (w, τ )) dτ for 0 ≤ t < α . 0

By (24.7), ∫

t

∥Pi (w, t)∥ ≤ ∥w∥ +

( ) κi 1 + ∥Pi (w, τ )∥ dτ

for 0 ≤ t < α .

0

By Gronwall inequality (Lemma A.1) ∥Pi (w, t)∥ ≤ (∥w∥ + κi t) exp(κi t)

for 0 ≤ t < α

which implies that Pi (w, t) is defined on R+ 0 and (24.9) holds. Further, ∫ t ∫ t [ ] Pi (w, t)−w = p i (Pi (w, τ )−w) dτ + p i (Pi (w, τ ))−p i (Pi (w, τ )−w) dτ 0

0

for 0 ≤ t < ∞ . By (24.7), ∫ t ∫ t ( ) ∥Pi (w, t)−w∥ ≤ κi 1+∥Pi (w, τ )−w∥ dτ + κi ∥w∥dτ 0

0

for 0 ≤ t < α .

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(24.10) holds by Gronwall inequality. Similarly,

∫ t [ ]

∥Pi (w, t)−Pi (w, t)∥ ≤ ∥w − w∥ + p i (Pi (w, τ )) − p i (Pi (w, τ ) dτ 0 ∫ t ≤ ∥w − w∥ + κi ∥Pi (w, τ )) − Pi (w, τ )∥ dτ for 0 ≤ t < α . 0

and (24.11) is correct by Gronwall inequality. Finally,

∫ t [ ]

∥Pi (w, t) − w − Pi (w, t) + w∥ ≤ p i (Pi (w, τ )) − p i (Pi (w, τ ) dτ 0

∫ ≤ κi

t

∥Pi (w, τ )−Pi (w, τ )∥ dτ

for t ∈ R+

0



and (24.12) holds by (24.11). 24.4. Theorem. Let y ∈ X. Put (cf. (24.1)) ) ∑∞ ( F1 (x, t) = Q(x, t) + i=1 Pi (x, 1) − x H(t−βi )

Φ1 (t) = Ψ(t) +

∑∞ ( i=1

for x ∈ X, t ∈ [a, b ] ,

) exp(κi ) − 1 H(t−βi )

} (24.13) } (24.14)

for t ∈ [a, b ] . Then ∥F1 (x, t) − F1 (x, s)∥ ≤ (1 + ∥x∥) |Φ1 (t) − Φ1 (s)| for x ∈ X, t, s ∈ [a, b] ,

( ) ∥∆v F1 (x, t) − F1 (x, s) ∥ ≤ ∥v∥ |Φ1 (t) − Φ1 (s)|

for x, v ∈ X, t, s ∈ [a, b] and there exists u1 : [a, b ] → X fulfilling ∫ T u1 (T ) = y + (SKH) Dt F1 (u1 (τ ), t)

for T ∈ [a, b ].

} (24.15) } (24.16)

(24.17)

a

Moreover, u1 u1

is a function of bounded variation continuous from the left , (24.18)

is unique , (24.19) ( ) ( ) ∥u1 (T )∥ ≤ ∥y∥+Φ1 (T )−Φ1 (a) exp Φ1 (T )−Φ1 (a) for T ∈[a, b ] . (24.20)

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Proof.

∞ ∑ [

169

] exp(κi ) − 1 < ∞ by (24.6). Hence F1 , Φ1 are well defined

i=1

and fulfil (23.2), (23.3). Conditions (23.4), (23.5) are irrelevant since F1 does not depend on τ. By Theorem 23.4 there exists a solution u1 : [a, b ] → X of (24.17), u1 (a) = y. It fulfils (24.18) and is unique by Lemma 23.6. Moreover, (24.20) holds by Lemma 23.5.  24.5. Notation. Let ζ > 0, a < αi < βi < γi < b for i ∈ N. There exists k ∈ N such that ∞ ∑ (exp(κi ) − 1) ≤ ζ (24.21) i=k+1

and a permutation m of {1, 2, . . . , k} such that βm(1) < βm(2) < . . . < βm(k) ,

(24.22)

a < αm(1) < βm(1) < γm(1) < . . . < αm(k) < βm(k) < γm(k) < b .

(24.23)

Assume that k ∑

(γi − αi ) ≤ ζ ,

(24.24)

i=1 k ∑ (

) Ψ(γi ) − Ψ(αi ) ≤ ζ,

i=1

∞ ∑ ( ) Ψ(γi ) − Ψ(αi ) ≤ ζ .

(24.25)

i=k+1

Define QM : X × [a, b ] → X and Ψ∗ : [a, b ] → R as follows: QM (x, t) = Q(x, t),

Ψ∗ (t) = 0

QM (x, t) = QM (x, αm(1) ),

Ψ∗ (t) = Ψ(t) − Ψ(αm(1) )

if a ≤ t ≤ αm(1) ,

QM (x, t) = QM (x, γm(1) ) + Q(x, t) − Q(x, γm(1) ),

if αm(1) ≤ t ≤ γm(1) ,

Ψ∗ (t) = Ψ∗ (γm(1) ) if γm(1) ≤ t ≤ αm(2) ,

QM (x, t) = QM (x, αm(2) ),

Ψ∗ (t) = Ψ∗ (αm(2) ) + Ψ(t) − Ψ(αm(2) ) if αm(2) ≤ t ≤ γm(2) ,

QM (x, t) = QM (x, γm(2) ) + Q(x, t) − Q(x, γm(2) ),

Ψ∗ (t) = Ψ∗ (γm(2) ) if γm(2) ≤ t ≤ αm(3) ,

...

...

...

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Ψ∗ (t) = Ψ∗ (γm(k) )

QM (x, t) = Q(x, γm(k) ) + Q(x, t) − Q(x, γm(k) ),

if γm(k) ≤ t ≤ b . Put ∞ ∑ (

F2 (x, t) = QM (x, t) +

  

) Pi (x, 1) − x H(t − βi )

i=k+1

for x ∈ X, t ∈ [a, b ] , Φ∗1 (t)

∗

= Ψ (t)+

∞ ∑

(

) exp(κi ) − 1 H(t − βi )

i=k+1

for x ∈ X, t ∈ [a, b ] .

 

(24.26)

    

(24.27)

24.6. Remark. The number k and the permutation m having the properties mentioned in Notation 24.5 exist by (24.6) and (24.3), respectively. Moreover (cf. (24.25)), Ψ∗

is nondecreasing and continuous,,

Ψ∗ (a) = 0,

Φ∗1

is nondecreasing and continuous from the left,

Ψ∗ (b) < ζ, (24.28) } (24.29)

Φ∗1 (b) − Φ∗1 (a) < 2 ζ . By (24.10) and (24.12), ( )( ) ∥Pi (x, 1) −x∥ ≤ 1 + ∥x∥ exp(κi ) − 1 , ( ) ( ) ∥∆v P1 (x, 1) − x ∥ ≤ ∥v∥ exp(κi ) − 1

(24.30) (24.31)

for x ∈ X. Hence ( ) ∥F2 (x, t) −F2 (x, s)∥ ≤ 1 + ∥x∥ |Φ1 (t) − Φ1 (s)| for x ∈ X, t, s ∈ [a, b ] ,

( ) ∥∆v F2 (x, t) −F2 (x, s) ∥ ≤ ∥v∥ |Φ1 (t) − Φ1 (s)|

for x, v ∈ X, t, s ∈ [a, b ] .

} (24.32) } (24.33)

24.7. Theorem. Let y ∈ X. Then there exists u2 : [a, b ] → X fulfilling ∫ u2 (T ) = y +(SKH)

T

Dt F2 (u2 (τ ), t) a

for T ∈ [a, b ] .

(24.34)

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Moreover, u2

has a bounded variation and is continuous from the left ,

u2

is unique , ( ) ( ) ∥u2 (T )∥ ≤ ∥y∥ + Φ1 (T ) − Φ1 (a) exp Φ1 (T )−Φ1 (a) for T ∈ [a, b ] .

(24.35) (24.36) } (24.37)

The proof is omitted since it follows the same arguments as the proof of Theorem 24.4 (cf. (24.32), (24.33)).  ∥u1 (T ) − u2 (T )∥ ≤ 2 C1 ζ, where ( ) ( ) C1 = ∥y∥ + Φ1 (b) − Φ1 (a) exp 2 (Φ1 (b) − Φ1 (a) .

24.8. Theorem.

Hint : Theorem 24.8 is a consequence of Lemma 23.8, where F = F1 , F = F2 , Φ∗ = Φ∗1 and C = C1 (cf. (24.29)) since ( )( ) ∥F1 (x, t) − F1 (x, s) − F2 (x, t) + F2 (x, t)∥ ≤ 1 + ∥x∥ Φ∗1 (t) − Φ∗1 (s) for x ∈ X, s, t ∈ [a, b ], s < t , Φ∗1 (b) − Φ∗1 (a) ≤ 2 ζ

.

24.9. Lemma. For i ∈ N put   0  ωi (t) = (γi − αi )−1   0 ∫ t Ωi (t) = ωi (τ ) dτ for

if a ≤ t ≤ αi , (24.38)

if αi < t < γi , if γi ≤ t ≤ b , t ∈ [a, b ] ,

(24.39)

a

Φ3 (t) = Ψ(t)+

∞ ∑ (

) exp(κi ) − 1 Ωi (t)

for t ∈ [a, b ] ,

(24.40)

for x ∈ X, t ∈ [a, b ] ,

(24.41)

for x ∈ X, t ∈ [a, b ] ,

(24.42)

i=1

F3 (x, t) = Q(x, t) +

∞ ∑

pi (x) Ωi (t)

i=1

F4 (x, t) = QM (x, t) +

k ∑

pi (x) Ωi (t)

i=1

Φ∗3 (t) = Ψ∗ (t)+

∞ ∑ ( i=k+1

) exp(κi )−1

∫

t

ωi (s) ds a

for t ∈ [a, b ] . (24.43)

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(For QM , Ψ∗ (t) see Notation 24.5, for k see (24.21).) Then ∥F3 (x, t) − F3 (x, s)∥ ≤ (1 + ∥x∥) |Φ3 (t) − Φ3 (s)| for x ∈ X, t, s ∈ [a, b] ,

( ) ∥∆v F3 (x, t) − F3 (x, s) ∥ ≤ ∥v∥ |Φ3 (t) − Φ3 (s)|

for x, v ∈ X, t, s ∈ [a, b] , ∥F4 (x, t) − F4 (x, s)∥ ≤ (1 + ∥x∥) |Φ3 (t) − Φ3 (s)| for x ∈ X, t, s ∈ [a, b] ,

( ) ∥∆v F4 (x, t) − F4 (x, s) ∥ ≤ ∥v∥ |Φ3 (t) − Φ3 (s)|

} (24.44) } (24.45) } (24.46) } (24.47)

for x, v ∈ X, t, s ∈ [a, b] , Φ3 (b) − Φ3 (a) = Φ1 (b) − Φ1 (a) .

(24.48)

Hint : The series in (24.40), (24.41) are convergent by (24.7) since 0 ≤ Ωi (t) ≤ 1. (24.44)–(24.47) are valid by (24.4), (24.5) and (24.37). Finally, (24.48) holds since Ωi (b) = 1 for i ∈ N.  24.10. Theorem. Let y ∈ X. Then there exists u3 : [a, b ] → X fulfilling ∫ u3 (T ) = y +(SKH)

T

Dt F3 (u3 (τ ), t)

for T ∈ [a, b ] .

(24.49)

a

Moreover, u3

has a bounded variation and is continuous from the left ,

u3

is unique , ( ) ( ) ∥u3 (T )∥ ≤ ∥y∥ + Φ3 (T ) − Φ3 (a) exp Φ3 (T )−Φ3 (a) for T ∈ [a, b ] .

(24.50) (24.51) } (24.52)

The proof is omitted since it follows the same arguments as the proof of Theorem 24.4.  Similarly, the proofs of Theorems 24.11, 24.12 are omitted (cf. the proofs of Theorems 24.4 and 24.8 together with (24.48) and observe that Φ∗3 (b)−Φ∗3 (a) ≤ 2 ζ and Φ3 (b)−Φ3 (a) = Φ1 (b)−Φ1 (a)).

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24.11. Theorem. Let y ∈ X. Then there exists u4 : [a, b ] → X fulfilling ∫ T u4 (T ) = y +(SKH) Dt F4 (u4 (τ ), t) for T ∈ [a, b ] . (24.53) a

Moreover, u4

has a bounded variation and is continuous from the left ,

u4

is unique , ( ) ( ) ∥u4 (T )∥ ≤ ∥y∥ + Φ1 (T ) − Φ1 (a) exp Φ1 (T )−Φ1 (a)

(24.54) (24.55) } (24.56)

for T ∈ [a, b ] . 24.12. Theorem.

∥u3 (T ) − u4 (T )∥ ≤ 2 C1 ζ.

24.13. Lemma.

}

for t ∈ A, where

u4 (t) = u3 (t)

(24.57)

A=[a, αm(1) ]∪[γm(1) , αm(2) ]∪. . .∪[γm(k−1) , αm(k) ]∪[γm(k) , b] , |[a, b ] \ A| < ζ. Proof.

(24.58)

Observe that (cf. (24.26), (24.42))

F4 (x, t) − F4 (x, s) = F3 (x, t) − F3 (x, s) for t, s ∈ [a, αm(1) ] or t, s ∈ [γm(j) , αm(j+1) ], j =1, 2, . . . , k−1, or t, s ∈ [γm(k) , b] .

      

(24.59)

By (24.26), u3 (γm(j) ) = Pj (αm(j) , 1) for j = 1, 2, . . . , k

(24.60)

and, by (24.42), ∫

t

u4 (t) = u4 (αm(j) ) + (SKH)

Ds F4 (u4 (σ), s) αm(j)

∫

t

= u4 (αm(j) ) + (SKH)

[ ]−1 dσ, pj (u4 (σ)) γm(j) − αm(j)

αm(j)

for j = 1, 2, . . . , k. Lemma 24.3 gives u4 (t) = Pj (u4 (αj ), t) for αm(j) ≤ t ≤ γm(j) , j = 1, 2, . . . , k .

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In particular, u4 (γm(j) ) = Pj (u4 (αj ), γm(j) )

for j = 1, 2, . . . , k .

(24.61)

(24.59)–(24.61) imply successively that u4 (t) = u3 (t) for t ∈ [a, αm(1) ],

u4 (γm(1) ) = u2 (γm(1) ),

u4 (t) = u3 (t) for t ∈ [γm(1) , αm(2) ],

u4 (γm(2) ) = u2 (γm(2) ),

...

...

...

u4 (t) = u3 (t) for t ∈ [γm(k−1) , αm(k) ],

u4 (γm(k) ) = u2 (γm(k) ),

u4 (t) = u3 (t) for t ∈ [γm(k) , b] . Hence (24.57) is true. (24.58) is a consequence of (24.24). The proof is complete.  24.14. Theorem. There is A ⊂ [a, b ] such that ∥u1 (T ) − u3 (T )∥ ≤ 2 C1 ζ

for T ∈ A

and

|[a, b ] \ A| < ζ .

Proof follows from Theorems 24.8, 24.12 and Lemma 24.13.

(24.62) 

24.15 . Remark. Let there exist sequences (αi,j , j ∈ N), (γi,j , j ∈ N) for i ∈ N such that a < αi,j ≤ βi ≤ γi,j < b, αi,j < γi,j , and αi,j → βi , γi,j → βi

for j → ∞ and i ∈ N .

For ℓ ∈ N put ζℓ = 2−ℓ . There exist k = k(ℓ) such that (24.21) holds with ζ = ζℓ , the permutation m = m(ℓ) such that (24.22) holds and αm(1) , γm(1) , . . . , αm(k) , γm(k) such that (24.23)–(24.25) hold. Define ∗ ∗ QM = QM,ℓ , Ψ = Ψℓ in the same way as in Notation 24.5, F2 = F2,ℓ by (24.26), Φ∗1 by (24.27), ωi = ωi,ℓ , Ωi = Ωi,ℓ , Φ3 = Φ3,ℓ by (24.38)–(24.40), F3 , F4 by (24.41), (24.42), Φ∗3 = Φ∗3,ℓ by (24.43). Then all the objects introduced from Notation 24.5 till Lemma 24.13 — in particular, αi = αi,ℓ , γi = γi,ℓ for i ∈ N, k = k(ℓ), the permutation m = mℓ , QM = QM,ℓ , Ψ∗ = Ψ∗ℓ , ωi = ωi,ℓ , Ωi = Ωi,ℓ , Φ3 = Φ3,ℓ , u3 = u3,ℓ , A = Aℓ - are dependent on ℓ. But the difference Φ3,ℓ (b) − Φ3,ℓ (a) is independent of ℓ since Φ3,ℓ (b) − Φ3,ℓ (a) = Φ(b) − Φ(a)

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(cf. (24.48)) and C1 is also independent of ℓ (cf. Theorem 24.8). By Theorem 24.10, ( ) ( ) ∥u3,ℓ (T )∥ ≤ ∥y∥+Φ1 (b)−Φ1 (a) exp 2(Φ1 (b)−Φ1 (a)) for T ∈ [a, b ], ℓ ∈ N , i.e. functions u3,ℓ are bounded uniformly. By Theorem 24.14, ∥u3,ℓ (T ) − u1 (T )∥ ≤ 4 C1 2−ℓ and

[a, b ] \ Aℓ ≤ 2−ℓ

for T ∈ Aℓ , ℓ ∈ N

for ℓ ∈ N .

Let n ∈ N. Then −ℓ

∥u3,ℓ − u1 (T )∥ ≤ 4 C1 2

for T ∈

∞ ∩

Aℓ , ℓ = n, n+1, n+2, . . .

j=n

and

∞ ∞ ∩ ∑ [a, b ] \ [a, b ] \ Aj ≤ 2−n+1 . Aj ≤ j=n

j=n

Hence u3,ℓ → u1

almost everywhere.

The above results can be summarized: Let Q(., a) be continuous. Then the functions Q, F3,ℓ are continuous while F1 The function u3,ℓ is a solution to d x = Dt F3,ℓ (x, t), dt

u3,ℓ (a) = y ,

on [a, b ] and u1 is a solution to d x = Dt F1 (x, t), dt

u1 (a) = y .

on [a, b ]. The functions u3,ℓ are uniformly bounded and u3,ℓ → u1 almost everywhere as ℓ → ∞.

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Chapter 25

A class of Strong Riemann-integrable functions

Let [a, b ] ⊂ R, Q: [a, b ]2 → X. Assume that Q is continuous,

(25.1)

∥Q(τ, t) − Q(τ, s)∥ ≤ ψ2 (|t − s|)

for t, s, τ ∈ [a, b ],

∥Q(τ, t) − Q(τ, s) − Q(σ, t) + Q(σ, s)∥ ≤ ψ1 (|τ − σ) ψ2 (|t − s|) for t, s, τ, σ ∈ [a, b ],

(25.2) } (25.3)

ψ1 , ψ2 being introduced in Chapter 7. The purpose of this chapter is to prove that Q is SR-integrable on [a, b ]. The chapters 25 and 26 are preparatory for for the which is formulated in Theorem 27.5. 25.1. Notation. Put c = 2 b − a and extend Q to [a, b ] × [a, c] in such a way that Q(τ, t) = Q(τ, b) if τ ∈ [a, b ], t ∈ [b, c], i.e. Q(τ, t) = Q(τ, min{t, b})

for τ ∈ [a, b ], t ∈ [a, c].

(25.4)

Then the inequalities in (25.2), (25.3) are valid if τ, σ ∈ [a, b ], t, s ∈ [a, c] and Q is continuous on its new domain. Put   if a ≤ t ≤ S,  0 Q(S, t, T ) = Q(S, t) − Q(S, S) if S < t ≤ T,   Q(S, T ) − Q(S, S) if T < t ≤ b , 177

(25.5)

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µi = (b − a) 2−i , νi = (T − S) 2−i , 2 ∑

(25.6)    

i

Ri (S, t, T ) =

Q(S + (j−1) νi , t, S + j νi )

(25.7)

  for i ∈ N0 , S ∈ [a, b ], t ∈ [a, c], T ∈ [S, c],  j=1

2 ∑ i

Pi (t) =

Q(a + (j−1) µi , t, a + j µi )

for i ∈ N0 , t ∈ [a, c].

(25.8)

j=1

Observe that for t ∈ [b, c], i ∈ N0 ,

(25.9)

Pi (t) = Ri (a, t, b) for t ∈ [a, c], i ∈ N0 .

(25.10)

Pi (t) = Pi (b)

25.2. Theorem. The limit on the right-hand side of P (t) = lim Pi (t)

(25.11)

i→∞

exists for each t ∈ [a, b ]. Furthermore, for ε > 0 there exists ξ > 0 such that   ∥P (t) − P (s) − Q(s, t) + Q(s, s)∥ ≤ ε |t − s| (25.12) for t, s ∈ [a, b ], |t − s| ≤ ξ .  Moreover, Q is SR-integrable on [a, b ] and P is its primitive. Proof.

(25.13)

After some manipulation (25.7) gives 2 ∑ i

Ri+1 (S, t, T ) − Ri (S, t, T ) =

Li,j (t)

j=1

for S ∈ [a, b ], t ∈ [a, c], i ∈ N0 , where Li,j (t) = Q(S + (2j−2) νi+1 , t, S + (2j−1) νi+1 ) + Q(S + (2j−1) νi+1 , t, S + 2j νi+1 ) − Q(S + (j−1) νi , t, S + j νi ).

    (25.14)

         

(25.15)

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If t ≤ S + (2j−1) νi+1 then (cf. (25.5)) Q(S + (2j−1) νi+1 , t, S + 2j νi+1 ) = 0 and Q(S+(2j−2) νi+1 , t, S+(2j−1) νi+1 ) = Q(S+(j−1) νi , t, S+j νi ). Therefore Li,j (t) = 0

for a ≤ t ≤ S + (2j−1) νi+1 , i ∈ N, j = 1, 2, . . . , 2i .

(25.16)

If S + (2j−1) νi+1 ≤ t < S + 2j νi+1 then Li,j (t) = Q(S + (2j−2) νi+1 , S + (2j−1) νi+1 ) − Q(S + (2j−2) νi+1 , S + (2j−2) νi+1 ) + Q(S + (2j−1) νi+1 , t) − Q(S + (2j−1) νi+1 , S + (2j−1) νi+1 ) − Q(S + (j−1) νi , t) + Q(S + (j−1) νi , S + (j−1) νi )

                        

and − Q(S + (j−1) νi , t) + Q(S + (j−1) νi , S + (j−1) νi ) = − Q(S+(2j−2) νi+1 , t) + Q(S+(2j−2) νi+1 , S+(2j−2)νi+1 )

(25.17)

} (25.18)

since (j−1) νi = (2j−2) νi+1 . (25.16), (25.17), (25.3) imply that       = ∥Q(S+(2j−2)νi+1 , S+(2j−1)νi+1 ) − Q(S+(2j−2)νi+1 , t)     (25.19) − Q(S+(2j−1)νi+1 , S+(2j−1)νi+1 ) + Q(S+(2j−1)νi+1 , t)∥      ≤ ψ1 (νi+1 ) ψ2 (νi+1 )     for S+(2j−1)νi+1 ≤ t ≤ S+2jνi+1 , i ∈ N0 , j = 1, 2, . . . , 2i .

∥Li,j (t)∥

Further (cf. (25.5)), ∥Li,j (t)∥ = ∥Li,j (S + 2j νi+1 )∥ for S + 2j νi+1 ≤ t ≤ c .

(25.20)

(25.17)–(25.20) imply that ∥Li,j (t)∥ ≤ ψ1 (νi+1 ) ψ2 (νi+1 ) for t ∈ [a, c], i ∈ N0 , j = 1, 2, . . . , 2i . (25.21)

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By (25.10), (25.14), (25.19)  

∥Ri+1 (S, t, T ) − Ri (S, t, T )∥ ≤ 2i ψ1 (νi+1 ) ψ2 (νi+1 )



for i ∈ N0 , t ∈ [a, c], ∥Ri+1 (S, t, T ) − R0 (S, t, T )∥ ≤

i ∑

j

2 ψ1 (νj+1 ) ψ2 (νj+1 )

j=0

≤

1 2

Ψ(T − S)

for i ∈ N0 , t ∈ [a, c] .

(25.22)

    

(25.23)

If S ≤ t ≤ T then (cf. (25.5)–(25.7), (25.23))

R0 (S, t, T ) = Q(S, t) − Q(S, S), ∥Ri+1 (S, t, T ) − Q(S, t) + Q(S, S)∥ ≤

(25.24) 1 Ψ(T − S). 2

(25.25)

By (25.8), (25.5) Pi (a) = 0 for i ∈ N0 . By (25.10), (25.22)

∥Pi+1 (t) − Pi (t)∥ ≤ 2i ψ1 (µi+1 ) ψ2 (µi+1 ) for i ∈ N0 , t ∈ [a, c] ,

(25.26)

which implies that

the limit in (25.11) exists

since the series m ≤ 2i , i ≥ ℓ,

∞ ∑

2i ψ1 (µi+1 ) ψ2 (µi+1 ) is convergent.

(25.27)

Let i, ℓ, m ∈ N0 ,

i=0

S = a + m µi ,

T = S + (b − a) 2−ℓ ,

S ≤ t ≤ T.

(25.28)

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181

Observe that νi−ℓ = µi . Then (cf. (25.7)) Ri−ℓ (S, t, T ) =

i−ℓ 2∑

Q(S + (j−1) νi−ℓ , t, S + j νi−ℓ )

j=1

=

i−ℓ 2∑

Q(a + (m+j−1) µi , t, a + (m+j) µi )

j=1

=

i−ℓ m+2 ∑

Q(a +(j−1) µi , t, a + j µi )

                      

j=m+1

=

i−ℓ m+2 ∑

−

j=1 m ∑

                       

Q(a + (j−1) µi , t, a + j µi )

Q(a + (j−1) µi , t, a + j µi ) .

(25.29)

j=1

By (25.8) 2 ∑

       

i

Pi (t) =

Q(a + (j−1) µi , t, a + j µi )

j=1

=

i−ℓ m+2 ∑

      

Q(a + (j−1) µi , t, a + j µi )

j=1

(25.30)

since t ≤ T = a + (m + 2i−ℓ ) µi which implies that Q(a + (j−1) µi , t, a + j µi ) = 0

if a + (j−1) µi ≥ T,

i.e. if j ≥ m + 1 + 2i−ℓ .

Moreover, m ∑

Q(a + (j−1) µi , t, a + j µi ) =

j=1

m ∑

Q(a + (j−1) µi , S, a + j µi )

j=1

since t ≥ S and Q(a + (j−1) µi , t, a + j µi ) = Q(a + (j−1) µi , a + j µi , a + j µi ) = Q(a + (j−1) µi , S, a + j µi ) for a + j µi ≤ S, Therefore

m ∑ j=1

i.e. for j ≤ m .

Q(a + (j−1) µi , t, a + j µi ) = Pi (S) .

(25.31)

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(25.29)–(25.31) imply that Ri−ℓ (S, t, T ) = Pi (t) − Pi (S) .

(25.32)

since Q(a + (j−1) µi , t, a+j µi ) = 0 if a + j µi > T, i.e. j > m + 2i−ℓ . Now, (25.32) and (25.23) give } ∥Pi (t) − Pi (S) − Q(S, t) + Q(S, S)∥ ≤ 12 Ψ(T − S) (25.33) for S ≤ t ≤ T, S, T fulfilling (25.28). If k ∈ N0 , S = a + m µi then S = a + 2k m µi+k . Hence ∥Pi+k (t) − Pi+k (S) − Q(S, t) + Q(S, S)∥ ≤

1 Ψ(T − S) . 2

The limit procedure for k → ∞ (cf. also (25.27)) implies that ∥P (t) − P (S) − Q(S, t) + Q(S, S)∥ ≤

1 Ψ(T − S) 2

(25.34)

if S, t, T fulfil (25.28) for suitable i, ℓ, m. But P and Q are continuous and (25.34) is valid if S, T ∈ [a, b ], S ≤ T , t = T. Hence } ∥P (t) − P (S) − Q(S, t) + Q(S, S)∥ ≤ 21 Ψ(t − S) (25.35) for S, t ∈ [a, b ], S ≤ t. If a ≤ t¯≤ S ≤ b then by (25.34) ∥P (S) − P (t¯) − Q(t¯, S) + Q(t¯, t¯)∥ ≤

1 Ψ(S − t¯). 2

(25.36)

By (25.3) ∥ − Q(t¯, S) + Q(t¯, t¯) + Q(S, S) − Q(S, t¯)∥ ≤ ψ1 (S − t¯) ψ2 (S − t¯) , which together with (25.36) implies that ∥P (t¯) − P (S) − Q(S, t¯) + Q(S, S)∥ ≤

1 2

Ψ(S − t¯) + ψ1 (S − t¯) ψ2 (S − t¯) .

Moreover (cf. (7.15), (7.19)), 1 Ψ(σ) → 0, σ

1 ψ1 (σ) ψ2 (σ) → 0 for σ → 0 + . σ

  

(25.37)

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Hence (cf. (25.35), (25.37)) for ε > 0 there exists ξ > 0 such that ∥P (t) − P (S) − Q(t, S) + Q(S, S)∥ ≤ ε (t − S) for a ≤ S < t ≤ min{S + ξ, b} and ∥P (t¯) − P (S) − Q(t¯, S) + Q(S, S)∥ ≤ ε (S − t¯) for a ≤ t¯< S ≤ min{t¯+ ξ, b}. Therefore (25.12) is correct. (25.13) is a consequence of (25.12). The proof is complete. 

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Chapter 26

On equality of two integrals

26.1. Notation. Let b ¯ : [a, b ]2 → X, Ψ: b R+ → R+ . [a, b ] ⊂ R, R>0, U : [a, b ]2 → X, U 0 0 26.2. Lemma. Assume that U is SR-integrable on [a, b ] ,

(26.1)

b Ψ(σ) →0 σ

(26.2)

for σ → 0+ ,

b ∥U (τ, t)−U (τ, t¯)−U (τ, t)+U (τ, t¯)∥ ≤ Ψ(t− t¯) for a ≤ t¯≤ τ ≤ t ≤ b. (26.3) Then U is SR-integrable on [a, b ] , ∫ T ∫ T (SR) Dt U (τ, t) = (SR) Dt U (τ, t) a

(26.4) for T ∈ [a, b ] .

(26.5)

a

Proof. Let u : [a, b ] → X be an SR-primitive of U. Let a < T ≤ b, ε > 0. There exists ξ1 > 0 such that b Ψ(σ) ≤ εσ

for 0 < σ ≤ ξ1 .

Moreover, there exists ξ2 , 0 < ξ2 ≤ ξ1 , such that ∑ ∥u(s) − u(¯ s) − U (σ, s) + U (σ, s¯)∥ ≤ ε S

for an arbitrary ξ2 -fine partition S = {([¯ s, s], σ)} of [a, b ]. Let R= {([¯ r, r], ρ)} 185

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be a ξ2 -fine partition of [a, b ]. Then ∑ ∥u(r) − u(¯ r) − U (ρ, r) + U (ρ, r¯)∥ R

≤

∑ R

+

∥u(r) − u(¯ r) − U (ρ, r) + U (ρ, r¯)∥

∑

∥U (ρ, r) − U (ρ, r¯) − U (ρ, r) + U (ρ, r¯)∥

R

≤ ε+

∑

b − r¯) ≤ ε (1 + b − a) Ψ(r

R

i.e. U is SR-integrable and (26.5) holds.



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Chapter 27

A class of generalized ordinary differential equations with a restricted right hand side 27.1. Definition. The right-hand side of the GODE d = Dt F (x, τ, t) dt is called restricted if F is independent of τ. 27.2. Remark. GODEs of this type were met in Theorem 23.11. In this chapter an analogous result will be proved for the class of GODEs from Chapters 8–13. b = 8R. Let ψ1 , ψ2 , Ψ be functions 27.3. Notation. Let [a, b ] ⊂ R, R > 0, R which were introduced in Chapter 7 and let G : B(8R) × [a, b ]2 → X fulfil (8.2)–(8.7). Put 1 b Ψ(σ) = Ψ(σ) + 2 ψ1 (σ) ψ2 (σ) for σ ∈ R+ 0 , 2 ∫ T ◦ G (x, T ) = (SR) Dt G(x, τ, t) for x ∈ B(8R), T ∈ [a, b ] .

(27.1) (27.2)

a

27.4. Lemma. G(x, ., .) is SR-integrable (x being kept fixed). Moreover,  ∥G◦ (x, T ) − G◦ (x, S) − G(x, S, T ) + G(x, S, S)∥    b (27.3) ≤ Ψ(|T −S|) + ψ1 (|T −S|) ψ2 (|T −S|)    for x ∈ B(8R), S, T ∈ [a, b ]. Proof. The SR-integrability of G(x, ., .) and (27.3) are consequences of Theorem 25.2 where Q(S, T ) = G(x, S, T ), P (T ) = G◦ (x, T ) for S, T ∈ [a, b ].  187

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27.5. Theorem. b is an SR-solution of the GODE (i) If u: [a, b ] → B(R) d x = Dt G(x, τ, t) dt

(27.4)

then it is an SR-solution of d x = Dt G◦ (x, t) . dt

(27.5)

b is an SR-solution of the GODE (27.5) then it (ii) If v: [a, b ] → B(R) is an SR-solution of (27.4) as well. Proof. (i) is a consequence of the definition of an SR-solution of a GODE and of Lemmas 26.2, 27.4 (where U (τ, t) = G(u(τ ), τ, t), U = G◦ (u(τ ), t)) since σ1 Ψ(σ) → 0, σ1 ψ1 (σ) ψ2 (σ)→ 0 for σ→0+ (cf. (7.17), (7.19)). The proof of (ii) is analogous. 

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Appendix A

Some elementary results

+ A.1. Lemma (Gronwall). Let S > 0, η > 0, ω : [0, S] → R+ 0 , ξ : [0, S] → R0 , ω, ξ being continuous. Assume that ∫ t ξ(t) ≤ η + ω(s) ξ(s) ds if 0 ≤ t ≤ S. (A.1) 0

Then ξ(t) ≤ η exp

(∫

t

) ω(s) ds

if 0 ≤ t ≤ S.

(A.2)

0

Comment. (A.2) is proved by integration of

η+

∫

ω(t) ξ(t)

t 0

ω(s) ξ(s) ds

≤ ω(t)

if 0 ≤ t ≤ S.

Lemma A.1 is valid if ω is Lebesgue integrable and if ξ is bounded, measurable. A.2. Lemma. Let λ : R → X. Assume that

  

λ is continuous, ∫ ∥λ(t)∥ ≤ 1,

 

(A.3)

Λ(τ ) dτ = 0 for t ∈ R

(A.4)

1

λ(t + 1) = λ(t) for t ∈ R,

λ(τ ) dτ = 0. 0

Then there exists Λ : R → X such that dΛ (t) = λ(t), dt

∫

t+1

Λ(t + 1) = Λ(t), t

and ∥Λ(t)∥ ≤ 41 , t ∈ R. 189

(A.5)

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∫ Proof.

∫

t+1

1

λ(τ ) dτ = 0 for t ∈ R by (A.3). Put

λ(τ ) dτ = t

0

∫

∫

t

¯ Λ(t) =

1

¯ − Λ(t) = Λ(t)

λ(τ ) dτ, 0

¯ dt . Λ(t) 0

Then (A.4) is fulfilled. Moreover, ∫ t ¯ ¯ Λ(t) − Λ(s) = λ(σ) dσ,

¯ ¯ Λ(t+1) = Λ(t) for t ∈ R ,

s

¯ ¯ ∥Λ(t)) − Λ(s)∥ ≤ |t − s| for s, t ∈ R , ∫ 1 ¯ ¯ − Λ(σ) dσ , Λ(t) = Λ(t) 0

∫ ¯ = Λ(t)) − Hence

1 t− 2

∫ ∥Λ(t)∥ ≤

and (A.5) holds.

1 t+ 2

∫ ¯ dσ = Λ(σ)

1 t+ 2

1 t− 2

1 t+ 2 1 t− 2

|t − σ| dσ =

1 4

¯ ¯ dσ . [Λ(t)) − Λ(σ)]

for t ∈ R 

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Appendix B

Trifles from functional analysis

B.1. Lemma. Let Q : B(R2 ) → X,

0 < R 1 < R2 , and assume that ∥x Q∥ ≤

2 3

(R2 − R1 )

for x ∈ B(R1 ) ,

∥x Q − x b Q∥ ≤ 31 ∥x − x b∥

(B.1)  

for x, x b ∈ B(R2 ) , 

(B.2)

x Q being the value of Q at x. Then for every z ∈ B(R1 ) there exists x ∈ B(R2 ) such that

Proof.

x + x Q = z.

(B.3)

x0 = z

(B.4)

xℓ+1 = z − xℓ Q.

(B.5)

Let z ∈ B(R1 ). Put

and if ℓ ∈ N0 , xℓ ∈ B(R2 ), put

Then (cf. (B.2)) x1 − x0 = −z Q , ∥x1 − x0 ∥ ≤

2 (R2 − R1 ), 3

(B.6) ∥x ℓ+1 − x ℓ ∥ ≤

2 1 (R2 − R1 ) ( )ℓ , 3 3

2 1 1 1 (R2 − R1 ) ( )ℓ (1 + + ( )2 + · · · ) 3 3 3 3 1 ℓ ≤ ( ) (R2 − R1 ) for m > ℓ , 3

∥xm − xℓ ∥ ≤

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Generalized Ordinary Differential Equations

192

∥xm − x0 ∥ ≤ R2 − R1 , xm ∈ B(R2 ) for m ∈ N0 and (xm , m = 0, 1, 2, . . . ) is a Cauchy sequence. Define x = lim xm . m→∞



Then x fulfils (B.3) by the limit procedure in (B.5).

B.2. Remark. Let [a, b ] ⊂ R, u : [a, b ] → X. u is called a function of bounded variation on [a, b ] if there exists κ ∈ R+ such that k ∑

∥u(ti+1 ) − u(ti )∥ ≤ κ

i=1

for any k ∈ N and any sequence a = t1 < t1 < . . . < tk ≤ b. The infimum of such κ is called the variation of u on [a, b ] and denoted by var (u, [a, b ]) or var u. If u is a function of bounded variation on [a, b ] then u(τ +) = lim u(t) exists for a ≤ τ < b and u(τ −) = lim u(t) exists t→τ,t>τ

t→τ,t<τ

for a < τ ≤ b. Let ε > 0. The set of τ such that either ∥u(τ +) − u(τ )∥ ≥ ε or ∥u(τ ) − u(τ −)∥ ≥ ε is finite. B.3. Remark. Let [a, b ] ⊂ R, v : [a, b ] → X. v is called a regulated function on [a, b ] if v(τ +) =

lim

v(t)

exists for a ≤ τ < b

lim

v(t)

exists for a < τ ≤ b .

t→τ,t>τ

and v(τ −) =

t→τ,t<τ

Let v be a regulated function. Then v is bounded and for every ε > 0 the number of τ such that either ∥v(τ +) − v(τ )∥ ≥ ε or ∥v(τ ) − v(τ −)∥ ≥ ε is finite.

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Bibliography

[Bartle, Sherbert (2000)] Bartle, R. G. and Sherbert, D. R., Introduction to Real Analysis. Wiley, New York, 2000. [Gordon (1994)] Gordon, R. A., The integrals of Lebesgue, Denjoy, Perron, and Henstock. American Mathematical Society, 1994. [Henstock (1988)] Henstock, R., Lectures on the Theory of Integration. World Scientific, Singapore, 1988. [Henstock (1991)] Henstock, R., The General Theory of Integration. Clarendon Press, Oxford, 1991. [Jarn´ık (1961a)] Jarn´ık, J., On some assumptions of the theorem on the continuous depenˇ dence on a parameter. Cas. Pˇest. Mat. 86 (1961), 404–414. [Jarn´ık (1961b)] Jarn´ık, J., On a certain modification of the theorem on the continuous ˇ dependence on a parameter. Cas. Pˇest. Mat. 86 (1961), 415–424. [Jarn´ık (1965)] Jarn´ık, J., Dependence of solutions of a class of differential equations of the second order on a parameter. Czechoslovak Math. J., 15 (90), (1965), 124–160. [Kapitza (1951a)] Kapitza, P. L., Pendulum with vibrating suspension (in Russian). Uspechi fiz. nauk, XLIV, 1 (1951), 7–20. [Kapitza (1951b)] Kapitza, P. L., Dynamic stability of a pendulum with an oscilatting point of suspension (in Russian). Zhurnal eksperimental’noi i teoreticheskoi fiziki, 21, 5 (1951), 588–597. Translation in : Collected Papers by P. L. Kapitza, vol. 2, 714–726, Pergamon Press, London, 1965. 193

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Generalized Ordinary Differential Equations

[Kurzweil (1957)] Kurzweil, J., Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J., 7 (82),(1957), 418–449. [Kurzweil (1958)] Kurzweil, J., On integration by parts. Czechoslovak Math. J.,8 (82),(1958), 356–359. [Kurzweil (1980)] Kurzweil, J., Nichtabsolut konvergente Integrale. Teubner-Verlag, Leipzig, 1980. [Kurzweil (2000)] Kurzweil, J., Henstock-Kurzweil Integration : its relation to topological vector spaces. World Scientific Publishing Co. Inc., River Edge, NJ, 2000. [Lee P.Y. (1989)] Lee, P. Y., Lanzhou Lectures on Henstock Integration. World Scientific, Singapore, 1989. [Lee P.Y., V´ yborn´ y (2000)] Lee, P. Y. and V´ yborn´ y, R., The Integral : An Easy Approach after Kurzweil and Henstock. Cambridge Univ. Press, Cambridge, 2000. [Lojasiewicz (1955)] Lojasiewicz, S., Sur un effet asymptotique dans les ´equations diff´erentielles dont les seconds membres contiennent des termes p´eriodiques de pulsation et d’amplitude tendant a l’infini. Ann. Pol. Math. 1 (1955), 388–413. [Mitropolskij (1971)] Mitropolskij, Yu. A, Averaging in Nonlinear Mechanics (in Russian). Naukova Dumka, Kiev 1971. [Monteiro, Tvrd´ y (2011)] Monteiro, G. and Tvrd´ y, M. On Kurzweil-Stieltjes integral in Banach space. Math. Bohem., to appear. [Pfeffer (1993)] Pfeffer, W. F., The Riemann Approach to Integration : Local geometric theory. Cambridge University Press, 1993. [Schwabik (1992)] ˇ Generalized Ordinary Differential Equations. World ScienSchwabik, S., tific, Singapore, 1992. [Schwabik (2001)] ˇ A note on integration by parts for abstract Perron-Stieltjes Schwabik, S., integrals. Math. Bohem. 126 2001, 613–629. [Schwabik, Ye (2005)] ˇ and Ye, G., Topics in Banach space Integration. World SciSchwabik, S. entific, Singapore, 2005.

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Symbols

D1 h(x, t), 12 D2 h(x, t), 12 D2 h(x), 12 D h(x), 12

N, 12 N0 , 12

[a, b ], 12 A (σ), 43

[p, q](x), 16 Ψ(σ), 43 ψ1 , ψ2 , 43

Ω, 48, 79

B(r), 12, 37 B (σ), 43

R, 12 R+ , 12 R+ 0 , 12

∆v F (x, p), 47 ∆v f (x), 47 dist (G, G∗ ), 48 D p ◦ q(x), 16

U (σ, ·), 27 b , 133 U

ES,σ , 69

W , 69

Φ(σ), 43 φ(σ), 43

∥x∥, 27 X, 27 x Vi (S, t, T ), 53 xYi (S, t, T ), 54

G, 47, 51 G, 51 e 51 G,

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Subject index

ACG∗ function, 94 additive function, 136 approximate solution, 53

Q-anchored, 83 refinement of a partition, 128 regulated function, 192

classical solution, 39 simple partition, 127 SKH-integrable, 84, 93 SKH-integral, 86 SKH-primitive, 84, 93 SKH-solution, 97, 105 SR-solution, 37 SR-integrable, 27 SR-primitive, 27 system in, 83

δ-fine, 83 function of bounded variation, 192 GODE, 37 Hake’s theorem, 92 jump function, 146

tag, 83 tagged interval, 83

KH-integrable, 119 KH-integral, 119

variation, 192 pair of coupled variables, 27, 84 partition, 83 primitive, 84

ξ- fine, 83

197

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