Integral Equations on Time Scales

by Svetlin G. Georgiev

Integral Equations on Time Scales This book offers the reader an overview of recent developments of integral equations on time scales It also contains elegant analytical and numerical methods This book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses The students in mathematical and physical sciences will find many sections of direct relevance The book contains nine chapters and each chapter is pedagogically organized This book is specially designed f

Publisher : Atlantis Press

Author : Svetlin G. Georgiev

ISBN : 9789462392274

Year : 2016

Language: en

File Size : 2.52 MB

Category : Science Math

Atlantis Studies in Dynamical Systems
Series Editors: Henk Broer · Boris Hasselblatt

Svetlin G.Georgiev

Integral Equations
on Time Scales

Atlantis Studies in Dynamical Systems
Volume 5

Series editors
Henk Broer, Groningen, The Netherlands
Boris Hasselblatt, Medford, USA

The “Atlantis Studies in Dynamical Systems” publishes monographs in the area of
dynamical systems, written by leading experts in the field and useful for both
students and researchers. Books with a theoretical nature will be published
alongside books emphasizing applications.

More information about this series at http://www.atlantis-press.com

Svetlin G. Georgiev

Integral Equations on Time
Scales

Svetlin G. Georgiev
Department of Differential Equations
Sofia University
Sofia
Bulgaria

Atlantis Studies in Dynamical Systems
ISBN 978-94-6239-227-4
ISBN 978-94-6239-228-1
DOI 10.2991/978-94-6239-228-1

(eBook)

Library of Congress Control Number: 2016951703
© Atlantis Press and the author(s) 2016
This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any
means, electronic or mechanical, including photocopying, recording or any information storage and
retrieval system known or to be invented, without prior permission from the Publisher.
Printed on acid-free paper

Preface

Many problems arising in applied mathematics or mathematical physics, can be
formulated in two ways namely as differential equations and as integral equations.
In the differential equation approach, the boundary conditions have to be imposed
externally, whereas in the case of integral equations, the boundary conditions are
incorporated within the formulation, and this confers a valuable advantage to the
latter method. Moreover, the integral equation approach leads quite naturally to
the solution of the problem as an infinite series, known as the Neumann expansion,
the Adomian decomposition method, and the series solution method in which the
successive terms arise from the application of an iterative procedure. The proof
of the convergence of this series under appropriate conditions presents an interesting exercise in an elementary analysis.
This book encompasses recent developments of integral equations on time
scales. For many population models biological reasons suggest using their difference analogues. For instance, North American big game populations have discrete
birth pulses, not continuous births as is assumed by differential equations.
Mathematical reasons also suggest using difference equations—they are easier to
construct and solve in a computer spreadsheet. North American large mammal
populations do not have continuous population growth, but rather discrete birth
pulses, so the differential equation form of the logistic equation will not be convenient. Age-structured models add complexity to a population model, but make
the model more realistic, in that essential features of the population growth process
are captured by the model. They are used difference equations to define the population model because discrete age classes require difference equations for simple
solutions. The discrete models can be investigated using integral equations in the
case when the time scale is the set of the natural numbers. A powerful method
introduced by Poincaré for examining the motion of dynamical systems is that of a
Poincaré section. This method can be investigated using integral equations on the
set of the natural numbers. The total charge on the capacitor can be investigated
with an integral equation on the set of the harmonic numbers.
This book contains elegant analytical and numerical methods. This book is
intended for the use in the field of integral equations and dynamic calculus on time
v

vi

Preface

scales. It is also suitable for graduate courses in the above fields. This book contains
nine chapters. The chapters in this book are pedagogically organized. This book is
specially designed for those who wish to understand integral equations on time
scales without having extensive mathematical background.
The basic definitions of forward and backward jump operators are due to Hilger.
In Chap. 1 are given examples of jump operators on some time scales. The
graininess function, which is the distance from a point to the closed point on the
right, is introduced in this chapter. In this chapter, the definitions for delta derivative
and delta integral are given and some of their properties are deducted. The basic
results in this chapter can be found in [2]. Chapter 2 introduces the classification of
integral equations on time scales and necessary techniques to convert dynamic
equations to integral equations on time scales. Chapter 3 deals with the generalized
Volterra integral equations and the relevant solution techniques. Chapter 4 is
concerned with the generalized Volterra integro-differential equations and also
solution techniques. Generalized Fredholm integral equations are investigated in
Chap. 5. Chapter 6 is devoted on Hilbert–Schmidt theory of generalized integral
equations with symmetric kernels. The Laplace transform method is introduced in
Chap. 7. Chapter 8 deals with the series solution method. Nonlinear integral
equations on time scales are introduced in Chap. 9.
The aim of this book was to present a clear and well-organized treatment of the
concept behind the development of mathematics and solution techniques. The text
material of this book is presented in highly readable, mathematically solid format.
Many practical problems are illustrated displaying a wide variety of solution
techniques. Nonlinear integral equations on time scales and some of their applications in the theory of population models, biology, chemistry, and electrical
engineering will be discussed in a forthcoming book “Nonlinear Integral Equations
on Time Scales and Applications.”
The author welcomes any suggestions for the improvement of the text.
Paris, France
June 2016

Svetlin G. Georgiev

Contents

1 Elements of the Time Scale Calculus . . . . . . . . . . . . . . . . . . . . . .
1.1 Forward and Backward Jump Operators, Graininess Function
1.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Mean Value Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 The Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Hilger’s Complex Plane . . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Definition and Properties of the Exponential Function
1.5.3 Examples for Exponential Functions . . . . . . . . . . . . . .
1.6 Hyperbolic and Trigonometric Functions . . . . . . . . . . . . . . . .
1.7 Dynamic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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49
63
65
67
74

2 Introductory Concepts of Integral Equations on Time Scales . .
2.1 Reducing Double Integrals to Single Integrals . . . . . . . . . . . .
2.2 Converting IVP to Generalized Volterra Integral Equations . .
2.3 Converting Generalized Volterra Integral Equations to IVP . .
2.4 Converting BVP to Generalized Fredholm Integral Equation .
2.5 Converting Generalized Fredholm Integral Equation to BVP .
2.6 Solutions of Generalized Integral Equations and Generalized
Integro-Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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116
121

3 Generalized Volterra Integral Equations . . . . . . . . . . . . . . . . . . .
3.1 Generalized Volterra Integral Equations of the Second Kind .
3.1.1 The Adomian Decomposition Method . . . . . . . . . . . . .
3.1.2 The Modified Decomposition Method . . . . . . . . . . . . .
3.1.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . .
3.1.4 Differential Equations Method . . . . . . . . . . . . . . . . . . .
3.1.5 The Successive Approximations Method . . . . . . . . . . .

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vii

viii

Contents

3.2 Conversion of a Generalized Volterra Integral Equation
of the First Kind to a Generalized Volterra Integral
Equation of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . .
3.3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 Existence of Solutions of Generalized Volterra Integral
Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . .
3.3.3 Uniqueness of Solutions of Generalized Volterra Integral
Equations of Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.4 Existence and Uniqueness of Solutions of Generalized
Volterra Integral Equations of the First Kind . . . . . . . . . . .
3.4 Resolvent Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Application to Linear Dynamic Equations . . . . . . . . . . . . . . . . . . .
3.6 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Generalized Volterra Integro-Differential Equations . . . . . . . . .
4.1 Generalized Volterra Integro-Differential Equations
of the Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 The Adomian Decomposition Method . . . . . . . . . . . . .
4.1.2 Converting Generalized Volterra Integro-Differential
Equations of the Second Kind to Initial
Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Converting Generalized Volterra Integro-Differential
Equations of the Second Kind to Generalized
Volterra Integral Equations . . . . . . . . . . . . . . . . . . . . .
4.2 Generalized Volterra Integro-Differential Equations
of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
5 Generalized Fredholm Integral Equations . . . . . . . . . . . . . . . . . .
5.1 Generalized Fredholm Integral Equations of the Second Kind
5.1.1 The Adomian Decomposition Method . . . . . . . . . . . . .
5.1.2 The Modified Decomposition Method . . . . . . . . . . . . .
5.1.3 The Noise Terms Phenomenon . . . . . . . . . . . . . . . . . .
5.1.4 The Direct Computation Method . . . . . . . . . . . . . . . . .
5.1.5 The Successive Approximations Method . . . . . . . . . . .
5.2 Homogeneous Generalized Fredholm Integral Equations
of the Second Kind. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Fredholm Alternative Theorem . . . . . . . . . . . . . . . . . . . . . . . .
Z bZ b
jKðx; YÞj2 DXDY\1 . . . . . . .
5.3.1 The Case When

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5.3.2 The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.3.3 Fredholm’s Alternative Theorem . . . . . . . . . . . . . . . . . . . . . 274
5.4 The Schmidth Expansion Theorem and the Mercer
Expansion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
a

a

Contents

ix

5.4.1 Operator-Theoretical Notations . . . . . . . . . . . . . . . . . . . . . .
5.4.2 The Schmidt Expansion Theorem . . . . . . . . . . . . . . . . . . . .
5.4.3 Application to Generalized Fredholm Integral Equation
of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.4 Positive Definite Kernels. Mercer’s Expansion
Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Hilbert-Schmidt Theory of Generalized Integral Equations
with Symmetric Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Schmidt’s Orthogonalization Process . . . . . . . . . . . . . . . . . . .
6.2 Approximations of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . .
6.3 Inhomogeneous Generalized Integral Equations . . . . . . . . . . .
7 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . .
7.1.2 Properties of the Laplace Transform . . . . . . . . . . . . . .
7.1.3 Convolution and Shifting Properties
of Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Applications to Dynamic Equations . . . . . . . . . . . . . . . . . . . .
7.3 Generalized Volterra Integral Equations of the Second Kind .
7.4 Generalized Volterra Integral Equations of the First Kind . . .
7.5 Generalized Volterra Integro-Differential Equations of the
Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.6 Generalized Volterra Integro-Differential Equations
of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.7 Advanced Practical Exercises . . . . . . . . . . . . . . . . . . . . . . . . .
8 The
8.1
8.2
8.3

Series Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Generalized Volterra Integral Equations of the Second Kind .
Generalized Volterra Integral Equations of the First Kind . . .
Generalized Volterra Integro-Differential Equations of the
Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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298

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9 Non-linear Generalized Integral Equations . . . . . . . . . . . . . . . . . . . . . 395
9.1 Non-linear Generalized Volterra Integral Equations . . . . . . . . . . . . 395
9.2 Non-linear Generalized Fredholm Integral Equations . . . . . . . . . . . 396
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Chapter 1

Elements of the Time Scale Calculus

This chapter is devoted to a brief exposition of the time scale calculus that provide
the framework for the study of integral equations on time scales. Time scale calculus
is very interesting in itself, this challenging subject has been developing very rapidly
in the last decades. A detailed discussion of the time scale calculus is beyond the
scope of this book, for this reason the author confine to outlining a minimal set of
properties needed in the further proceeding. The presentation in this chapter follows
the book [2]. A deep and thorough insight into the time scale calculus, as well as the
discussion of the available bibliography on this issue, can be found in the book [2].

1.1 Forward and Backward Jump Operators, Graininess
Function
Definition 1 A time scale is an arbitrary nonempty closed subset of the real numbers.
We will denote a time scale by the symbol T .
We suppose that a time scale T has the topology that inherits from the real
numbers with the standard topology.
Example 1 [1, 2], R, N are time scales.
Example 2 [a, b), (a, b], (a, b) are not time scales.
Definition 2 For t ∈ T we define the forward jump operator σ : T −→ T as
follows
σ (t) = inf{s ∈ T : s > t}.
We note that σ (t) ≥ t for any t ∈ T .
© Atlantis Press and the author(s) 2016
S.G. Georgiev, Integral Equations on Time Scales, Atlantis Studies
in Dynamical Systems 5, DOI 10.2991/978-94-6239-228-1_1

1

2

1 Elements of the Time Scale Calculus

Definition 3 For t ∈ T we define the backward jump operator ρ : T −→ T by
ρ(t) = sup{s ∈ T : s < t}.
We note that ρ(t) ≤ t for any t ∈ T .
Definition 4 We set
inf Ø = sup T , sup Ø = inf T .
Definition 5 For t ∈ T we have the following cases.
If σ (t) > t, then we say that t is right-scattered.
If t < sup T and σ (t) = t, then we say that t is right-dense.
If ρ(t) < t, then we say that t is left-scattered.
If t > inf T and ρ(t) = t, then we say that t is left-dense.
If t is left-scattered and right-scattered at the same time, then we say that t is
isolated.
6. If t is left-dense and right-dense at the same time, then we say that t is dense.


Example 3 Let T = { 2n + 1 : n ∈ N }. If t = 2n + 1 for some n ∈ N , then
t2 − 1
n=
and
2
1.
2.
3.
4.
5.





σ (t) = inf{l ∈ N : 2l + 1 > 2n + 1} = 2n + 3 = t 2 + 2 f or n ∈ N ,



ρ(t) = sup{l ∈ N : 2l + 1 < 2n + 1} = 2n − 1 = t 2 − 2 f or n ∈ N , n ≥ 2.

For n = 1 we have


ρ( 3) = sup Ø = inf T = 3.
Since



t2 − 2 < t <



t2 + 2

for

n ≥ 2,


2n + 1, n ∈ N , n ≥ 2, is right-scattered and leftwe conclude that every point

scattered, i.e., every point 2n + 1, n ∈ N , n ≥ 2, is isolated.
Because




3 = ρ( 3) < σ ( 3) = 5,
we have that the point




Example 4 Let T =
1. t =

1
. Then
2

3 is right-scattered.

1
: n ∈ N ∪ {0} and t ∈ T be arbitrarily chosen.
2n

1.1 Forward and Backward Jump Operators, Graininess Function

3



 
1
1
1
1
1
, 0 : , 0 > , l ∈ N = inf Ø = sup T = ,
= inf
2
2l
2l
2
2


 
1
1
1
1
1
,0 : ,0 < ,l ∈ N = 0 < ,
= sup
ρ
2
2l
2l
2
2

σ

1
is left-scattered.
2
1
, n ∈ N , n ≥ 2. Then
2. t =
2n


 
1 1
1
1
1
1
:
>
,l ∈ N =
>
,
= inf
σ
2n
2l 2l
2n
2(n − 1)
2n


 
1
1
1
1
1
1
,0 : ,0 <
,l ∈ N =
<
.
= sup
ρ
2n
2l
2l
2n
2(n + 1)
2n
i.e.,

1
Therefore all points
, n ∈ N , n ≥ 2, are right-scattered and left-scattered,
2n
1
, n ∈ N , n ≥ 2, are isolated.
i.e., all points
2n
3. t = 0. Then
σ (0) = inf{s ∈ T : s > 0} = 0,
ρ(0) = sup{s ∈ T : s < 0} = sup Ø = inf T = 0.
Example 5 Let T =


n
: n ∈ N0 and t = , n ∈ N0 , be arbitrarily chosen.
3
3

n

1. n ∈ N . Then
σ
ρ

n

3
n

3




l
l
n
n
n+1
, 0 : , 0 > , l ∈ N0 =
> ,
3
3
3
3
3




n−1
l
l
n
n
, 0 : , 0 < , l ∈ N0 =
< .
3
3
3
3
3

= inf
= sup

n
Therefore all points t = , n ∈ N , are right-scattered and left-scattered, i.e., all
3
n
points t = , n ∈ N , are isolated.
3
2. n = 0. Then


l
1
l
σ (0) = inf
, 0 : , 0 > 0, l ∈ N0 = > 0,
3
3
3


l l
: , 0 < 0, l ∈ N0 = sup Ø = inf T = 0,
ρ(0) = sup
3 3

4

1 Elements of the Time Scale Calculus

i.e., t = 0 is right-scattered.


3
Exercise 1 Classify each point t ∈ T = { 2n − 1 : n ∈ N0 } as left-dense, leftscattered, right-dense, or right-scattered.

3
Answer. The points 2n − 1, n ∈ N , are isolated, the point −1 is right-scattered.
Definition 6 The numbers
H0 = 0,

Hn =

n

1
k=1

k

, n∈N,

will be called harmonic numbers.
Exercise 2 Let
H = {Hn : n ∈ N0 }.
Prove that H is a time scale. Find σ (t) and ρ(t).
Answer. σ (Hn ) = Hn+1 , n ∈ N0 , ρ(Hn ) = Hn−1 , n ∈ N , ρ(H0 ) = H0 .
Definition 7 The graininess function μ : T −→ [0, ∞) is defined by

μ(t) = σ (t) − t.

: n ∈ N . Let also, t = 2n+1 ∈ T for some n ∈ N .

Example 6 Let T = 2n+1
Then

σ (t) = inf 2l+1 : 2l+1 > 2n+1 , l ∈ N = 2n+2 = 2t.
Hence,



μ(t) = σ (t) − t = 2t − t = t or μ 2n+1 = 2n+1 , n ∈ N .
Example 7 Let T =

√



n + 1 : n ∈ N . Let also, t = n + 1 for some n ∈ N .

Then n = t 2 − 1 and
 √
√



l + 1 : l + 1 > n + 1, l ∈ N = n + 2 = t 2 + 1.
σ (t) =
Hence,
μ(t) = σ (t) − t =





t 2 + 1 − t or μ( n + 1) = n + 2 − n + 1, n ∈ N .


n
n
: n ∈ N0 . Let also, t = for some n ∈ N0 . Then n = 2t
Example 8 Let T =
2
2
and


n+1
l l
n
1
σ (t) = inf
: > , l ∈ N0 =
=t+ .
2 2
2
2
2

1.1 Forward and Backward Jump Operators, Graininess Function

Hence,
μ(t) = σ (t) − t = t +

5

n
1
1
1
= .
−t =
or μ
2
2
2
2

Example 9 Suppose that T consists of finitely many different points: t1 , t2 , . . . , tk .
Without loss of generality we can assume that
t1 < t 2 < . . . < t k .
For i = 1, 2, . . . , k − 1 we have
σ (ti ) = inf{tl ∈ T : tl > ti , l = 1, 2, . . . , k} = ti+1 .
Hence,
μ(ti ) = ti+1 − ti , i = 1, 2, . . . , k − 1.
Also,
σ (tk ) = inf{tl ∈ T : tl > tk , l = 1, 2, . . . , k} = inf Ø = sup T = tk .
Therefore
μ(tk ) = σ (tk ) − tk = tk − tk = 0.
From here,
k

μ(ti ) =

i=1

Exercise 3 Let T =
Answer. μ


3

k−1

μ(ti ) + μ(tk ) =

i=1

k−1

(ti+1 − ti ) = tk − t1 .

i=1


√
3
n + 2 : n ∈ N0 . Find μ(t), t ∈ T .



n + 2 = 3 n + 3 − 3 n + 2.

Definition 8 If f : T −→ R is a function, then we define the function
f σ : T −→ R by
f σ (t) = f (σ (t)) for any t ∈ T . i.e.,

f σ = f ◦ σ.

Below, for convenience, we will use the following notation σ k (t) = (σ (t))k ,
f (t) = ( f (t))k , k ∈ R.

Example 10 Let T = t = 2n+2 : n ∈ N , f (t) = t 2 + t − 1. Then
k

σ (t) = inf 2l+2 : 2l+2 > 2n+2 , l ∈ N = 2n+3 = 2t.

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