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July 1, 2011 9:49 book˙Pachpatte

ATLANTIS STUDIES IN MATHEMATICS FOR ENGINEERING AND SCIENCE

VOLUME 9

SERIES EDITOR: C.K. CHUI


Atlantis Studies in Mathematics for Engineeringand Science

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Multidimensional IntegralEquations and Inequalities

B.G. Pachpatte

57, Shri Niketen Coloney,Aurangabad, India

AMSTERDAM – PARIS


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Atlantis Studies in Mathematics for Engineering and Science

Volume 1: Continued Fractions: Volume 1: Convergence Theory - L. Lorentzen, H. WaadelandVolume 2: Mean Field Theories and Dual Variation - T. SuzukiVolume 3: The Hybrid Grand Unified Theory - V. Lakshmikantham, E. Escultura, S. LeelaVolume 4: The Wavelet Transform - R.S. PathakVolume 5: Theory of Causal Differential Equations - V. Lakshmikantham, S. Leela, Z. DriciVolume 6: The Omega Problem of all members of the United Nations - E.N. ChukwuVolume 7: Boundary Element Methods with Applications to Nonlinear Problems - G. Chen, J. ZhouVolume 8: Nonlinear Hybrid Continuous/Discrete-Time Models - M. Akhmet

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Dedicated to the memory of my parents.


Preface

Integral equations that involve functions of two or more independent variables occur fre-

quently in the study of many problems in partial differential equations which arise from

various dynamic models. The study of various multidimensional integral equations and

inequalities attracted the attention of many researchers and there exists a very vast litera-

ture. The aim of this monograph is to provide the readers, representative overview of the

important recent developments, focusing on some selected multidimensional integral equa-

tions and inequalities. It is assessable to any one having a reasonable background in real

analysis, partial differential equations and acquaintance with their related areas.

The material included in the monograph is recent and hard to find in other books. It is

self-contained and all results are presented in an easy-to-read, informal style and it could

also serve as a textbook for an advanced graduate course. It will be an invaluable reading

for pure and applied mathematicians, physists, engineers, computer scientists and will also

be most valuable as a source of reference in the field.

It is impossible to thank all the individuals who have influenced me directly or indirectly

during the writing of this book, without their constant encouragement it would still have

been remained no more than an idea. In particular, I wish to express my deep and sincere

gratitude to Professor Charles Chui, Editor AMES who offered invaluable suggestions for

the improvement of the presentation. Also, I am grateful to Professor Jan van Mill and

Arjen Sevenster for their support and interest in the present work. It is a pleasure to ac-

knowledge the fine collaboration and assistance provided by the editorial and production

staff of Atlantis Press. Last, but not least, I would like to thank to my family members for

their understanding, patience and long-lasting inspiration.

B.G. Pachpatte

vii


Contents

Preface vii

Introduction 1

1. Integral equations in two variables 9

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Basic integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Volterra-type integral equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Volterra-Fredholm-type integral equation . . . . . . . . . . . . . . . . . . . . . . 211.5 Integrodifferential equations of hyperbolic-type . . . . . . . . . . . . . . . . . . . 271.6 Fredholm-type integrodifferential equation . . . . . . . . . . . . . . . . . . . . . . 371.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2. Integral inequalities and equations in two and three variables 59

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.2 Integral inequalities in two variables . . . . . . . . . . . . . . . . . . . . . . . . . 592.3 Integral inequalities in three variables . . . . . . . . . . . . . . . . . . . . . . . . 652.4 Integral equation in two variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.5 Integral equation in three variables . . . . . . . . . . . . . . . . . . . . . . . . . . 782.6 Hyperbolic-type Fredholm integrodifferential equation . . . . . . . . . . . . . . . 832.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 892.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3. Mixed integral equations and inequalities 97

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.2 Volterra-Fredholm-type integral inequalities I . . . . . . . . . . . . . . . . . . . . 973.3 Volterra-Fredholm-type integral inequalities II . . . . . . . . . . . . . . . . . . . . 1083.4 Integral equation of Volterra-Fredholm-type . . . . . . . . . . . . . . . . . . . . . 1163.5 Volterra-Fredholm-type integral equations . . . . . . . . . . . . . . . . . . . . . . 1223.6 General Volterra-Fredholm-type integral equations . . . . . . . . . . . . . . . . . 1283.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4. Parabolic-type integrodifferential equations 143

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.2 Basic integral inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434.3 Integrodifferential equation of Barbashin-type . . . . . . . . . . . . . . . . . . . . 1474.4 General integral equation of Barbashin-type . . . . . . . . . . . . . . . . . . . . . 155

ix


x Multidimensional Integral Equations and Inequalities

4.5 Integrodifferential equation of the type arising in reactor dynamics . . . . . . . . . 1614.6 Initial-boundary value problem for integrodifferential equations . . . . . . . . . . 1694.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5. Multivariable sum-difference inequalities and equations 191

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.2 Sum-difference inequalities in two variables . . . . . . . . . . . . . . . . . . . . . 1915.3 Sum-difference inequalities in three variables . . . . . . . . . . . . . . . . . . . . 1985.4 Multivariable sum-difference inequalities . . . . . . . . . . . . . . . . . . . . . . 2045.5 Sum-difference equations in two variables . . . . . . . . . . . . . . . . . . . . . . 2115.6 Volterra-Fredholm-type sum-difference equations . . . . . . . . . . . . . . . . . . 2225.7 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2285.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Bibliography 237

Subject Index 243


Introduction

In studying mathematical models of various dynamic equations, it is often desirable not

only to prove the existence of a solution satisfying the given initial or boundary conditions

but also to ensure that the solution in question possesses certain qualitative properties. It is

well known that the beginning of the qualitative theory of differential equations is directly

connected with the classical works of H. Poincare, A.M. Lyapunov and G.D. Birkhoff

on problems of ordinary and classical mechanics. The theory of partial differential equa-

tions and many physical, chemical and biological phenomena give rise to multidimensional

integral and integrodifferential equations and their study provide results of enormous im-

portance, revealing deep and fundamental connections. The classical book: Partial integral

operators and integro-differential equations by J.M. Appell, A.S. Kalitvin and P.P. Zabrejko

[5] contains an overview of many contributions to such equations, including comprehensive

list of references.

First, it will be helpful to summarize briefly certain important multivariable integral and

integrodifferential equations arising while studying some specific problems, which greatly

stimulated the present work. However, the sample results we are going to discuss are

certainly far from being exhaustive.

In [28, p. 20] C. Corduneanu pointed out that, by means of the substitution u =

vexp(−∫ x

0 b0(y, t)dy), the following hyperbolic equation

uxt +a0(x, t)ux +b0(x,t)ut = c0(x, t,u), (1)

considered on the semi-strip 0 � x � �, 0 � t < ∞, with the given characteristic data

u(x,0) = u1(x),u(0, t) = u0(t), (2)

takes the form

vxt +a(x, t)vx = c(x, t,v), (3)

1

2 Multidimensional Integral Equations and Inequalities

where a(x, t) and c(x,t,v) are like a0(x,t) and c0(x,t,u) and the data on the characteristics

preserve their form

v(x,0) = u1(x)exp(∫ x

0b0(y,0)dy

)

= v1(x), v(0,t) = u0(t). (4)

Furthermore, by taking z(x,t) = vxt(x, t) it is easy to observe that the equation (3) with

characteristic data (4) takes the form

z(x,t)+a(x,t)(

∂∂ x

v1(x)+∫ t

0z(x,τ)dτ

)

= c(

x,t,u0(t)+ v1(x)−u1(0)+∫ x

0

∫ t

0z(y,τ)dτ dy

)

. (5)

In [22], J.R. Cannon and Y. Lin described and analyzed a typical boundary value problem

with pseudo-parabolic equation:

uxxt = Au+F in QT , (6)

u(x,0) = φ(x), 0 � x � 1, (7)

u(0,t) = f (t), 0 � t � T, (8)

ux(0,t) = g(t), 0 � t � T, (9)

where

Au = a1ut +a2uxt +a3uxx +a4ux +a5u, (10)

QT = (0,1)× (0,T ], T > 0 and f , g, φ , F and ai (i = 1, . . . ,5) are given functions. In [22],

the above problem is studied by reducing it to the following equivalent integrodifferential

equation

u(x,t) = G(x,t)+∫ x

0

∫ η

0

∫ t

0[Au(ξ ,τ)+F(ξ ,τ)]dτ dξ dη , (11)

in QT , where

G(x,t) = φ(x)−φ(0)−φ ′(0)x +g(t)x+ f (t). (12)

For more details, see [22] and the references therein.

In [4], G. Andrews studied the partial differential equation of the form

utt(x,t)−auxxt(x,t) = F(x, t,u(x,t)), (13)

for x ∈ [0,L], t ∈ [0,T ]; L > 0, T > 0, with the initial conditions

u(x,0) = φ(x), ut(x,0) = ψ(x), x ∈ [0,L], (14)

Introduction 3

and boundary conditions

u(0,t) = u(L,T ) = 0, t ∈ [0,T ]. (15)

Under appropriate conditions, in [4] it is shown that the problem (13)–(15) can be reduced

to an integral equation of the form

u(x,t) = f (x, t)+∫ t

0

∫ s

0

∫ L

0G(x,y,s− τ)F(y,τ,u(y,τ))dydτ ds, (16)

where G(x,y,t) is the Green’s function for the heat equation wt(x, t) = auxx(x,t) with zero

Dirichlet boundary data, a is a positive constant and L > 0,T > 0 are finite but can be

arbitrarily large constants. For more details, see [4].

D.L. Lovelady [64] studied the hyperbolic-type Fredholm integrodifferential equation

∂ 2

∂ s∂ tu(s,t,z) = f (s,t,z)+H

(

z,u(s,t,z),∂∂ s

u(s,t,z),∂∂ t

u(s,t,z))

+∫ 1

0K

(

z,r,u(s,t,r),∂∂ s

u(s,t,r),∂∂ t

u(s,t,r))

dr, (17)

with the given data

u(s,0,z) = σ(s,z), u(0,t,z) = τ(t,z), (18)

for s, t ∈ [0,∞), z ∈ [0,1]. It is easy to observe that the problem (17)–(18) contains as a

special case the integral equation of the form

u(s, t,z) = h(s,t,z)+∫ s

0

∫ t

0

∫ 1

0L(z,v,w,r,u(v,w,r))dr dwdv. (19)

In [31], O. Diekmann analyzed a model of spatio-temporal development of an epidemic.

The model considered leads to the following nonlinear integral equation of the form

u(t,x) = f (t,x)+∫ t

0

∫

ΩS0(ξ )A(τ,x,ξ )g(u(t − τ))dξ dτ, (20)

for (t,x) ∈ [0,∞)×Ω, where Ω is a bounded domain in Rn. Detailed descriptions and

analysis of the above model and of related ones may be found in Diekmann [31,32] and

Thieme [128] which contain additional relevant references. The integral equation (20)

appears to be Volterra-type in t, and of Fredholm-type with respect to x and hence it can be

viewed as a mixed Volterra-Fredholm-type integral equation.

In [8], E.A. Barbashin first initiated the study of the integrodifferential equations of the

form

∂∂ t

u(t,x) = c(t,x)u(t,x)+∫ b

ak(t,x,y)u(t,y)dy+ f (t,x), (21)


which arise in mathematical modeling of many applied problems (see [5]). The equation

(21) has been studied by many authors and is now known in the literature as integrodiffer-

ential equation of Barbashin-type or simply Barbashin equation (see [5, p. 1]). For detailed

account on the study of such equations, see [5] and the references cited therein.

Another significant source of parabolic-type integrodifferential equations is provided by

the study of C.V. Pao [121,122] related to an integrodifferential system arising in reactor

dynamics of the form

p′(t) = p(t)∫

Ωβ (x)u(t,x)dx (t > 0, x ∈ Ω), (22)

∂ u∂ t

−Lu = f (t,x, p(t)− p∗), (23)

p∗ � 0 is a constant, with the given boundary and initial conditions

α1(x)∂u∂τ

+α2(x)u = 0 (t > 0, x ∈ ∂Ω), (24)

u(0,x) = u0(x) (x ∈ Ω), (25)

p(0) = p0, (26)

where

Lu =n

∑i, j=1

ai j(x)uxix j +n

∑i=1

ai(x)uxi , (27)

on the bounded domain Ω in Rn. For detailed account on the study of such equations, see

[37,39,57,69,125,134]. Solving the equation (22) by using (26) and then substituting it into

(23), we get

∂ u∂ t

−Lu = f(

t,x, p0 exp(∫ t

0

∫

Ωβ (x)u(s,x)dxds

)

− p∗)

, (28)

for t > 0, x ∈ Ω. We note that the study of equations like (28) with (24), (25) is interesting

in itself.

Integral and integrodifferential equations of the type (5), (11), (16), (19), (20), (21), (28)

are remarkable in terms of simplicity, the large number of results to which they lead, and

the variety of applications which can be related to them. When dealing with the above

noted equations, the basic considerations give rise to the questions to be answered are:

(i) under what conditions the equations under considerations have solutions?

(ii) how can we find the solutions or closely approximate them?

(iii) what are their nature?

Introduction 5

The study of such questions related to the above noted equations is a challenging task and

requires special attention for handling such problems. Although, there is an enormous liter-

ature on all these equations, some of them are still in an elementary stage of development.

In practice, it is often difficult to obtain explicitly the solutions to nonlinear equations and

thus need a new insight to handle the qualitative properties of their solutions. In general,

existence theorems for equations of the above noted forms are proved by the use of one

of the three fundamental methods: the method of successive approximations, the method

based on the theory of nonexpansive and monotone mappings and the theory exploiting

the compactness of the operator often by the use of the well known fixed point theorems.

The method of inequalities which provides explicit estimates on unknown functions has

been a significant source in the study of many qualitative properties of solutions of various

differential, integral and finite difference equations. It enable us to obtain valuable infor-

mation about solutions without the need to know in advance the solutions explicitly. In

many cases while studying the behavior of solutions, the method which works very effec-

tively to establish existence does not yield other properties of the solutions in ready fashion

and one often needs some new ideas and methods in the analysis. It is easy to observe that

the explicit estimates available on various inequalities in [82,85,87,134] and the references

cited therein are not directly applicable to study the qualitative behavior of solutions of the

equations of the above noted forms and their discrete versions. Moreover, in [22, p. 378]

J.R. Cannon and Y. Lin pointed out that in the study of certain basic qualitative properties

of solutions, the equation (11) can be dealt with in a more satisfactory manner than dealing

directly with the equations (6)–(9).

During the last decade or so the above noted facts inspired the author a new line of thought,

which resulted in a series of recent papers [88–118] dealing with the qualitative theory

related to the equations of the above noted types. The literature related to the above types

of cited integral equations and inequalities is now very extensive, it is scattered in various

journals encompassing different subject areas. There is thus an urgent need of a book that

brings readers to the forefront of current research in this prosperous field.

This monograph is an attempt to organize in a systematic way the recent progress related

to the equations and inequalities of the above noted types, in the hope that it will further

broaden developments and the scope of applications. The field is vast and has not stabilized

as yet so that it is extremely difficult to produce a work that traces all the relevant contri-

butions. We mostly focus on certain recent advances not covered in earlier monographs,

which reflect our taste and as well as those we consider potentially applicable in a wide


range of problems.

The exposition consists of five chapters and references. The first chapter presents a large

number of basic results related to certain integral and integrodifferential equations in two

variables. The tools employed in the analysis are based on the applications of Banach fixed

point theorem; with Bielecki-type norm and integral inequalities with explicit estimates.

In the second chapter, we consider some integral inequalities with explicit estimates in-

volving functions of two and three independent variables. In this chapter, the reader will

also find the study of some important qualitative properties of solutions of certain inte-

gral and integrodifferential equations in two and three independent variables. Chapter 3 is

devoted to present some fundamental mixed Volterra-Fredholm-type integral inequalities

which can be used as tools in certain applications. It also contains the results on existence,

uniqueness and other properties related to certain mixed Volterra-Fredholm-type integral

equations. Chapter 4 is concerned with certain parabolic type integrodifferential equations

which arise in mathematical modeling of many applied problems. The basic problems of

existence, uniqueness and other qualitative properties of solutions are dealt with by using

different techniques. Chapter 5 is dedicated to the theory of multivariable sum-difference

inequalities and equations in the hope that it will provide a clue to effective methods for

dealing with the discrete dynamics theory for its treatment. Each chapter contains a sec-

tion on miscellanea, indicating adequate sources, intended to stimulate the reader’s interest.

Throughout, we let R and N denote the set of real and natural numbers respectively and

Ia = [0,a] (a > 0), R+ = [0,∞), R1 = [1,∞), N0 = {0, 1, 2, . . .}, Nα,β = {α,α +1, . . . ,α +

n = β} (α ∈N0, n∈N) are the given subsets of R, and Rn the real n-dimensional Euclidean

space with appropriate norm denoted by | · |. The derivatives of a function u(t), t ∈ R are

denoted by u(i)(t) for i = 1, . . . ,n. The partial derivatives of a function z(x,y) for x, y ∈ R

with respect to x, y and xy are denoted by

D1z(x,y) or∂∂x

z(x,y) (or zx(x,y)), D2z(x,y) or∂∂ y

z(x,y) (or zy(x,y)),

and

D1D2z(x,y) = D2D1z(x,y) or∂ 2

∂x∂yz(x,y) (or (or zxy(x,y))

respectively. For any function w(n), n ∈ N0, we define the operator Δ by

Δw(n) = w(n+1)−w(n) and Δiw(n) = Δ(Δi−1w(n)) for i � 2.

For any function z(m,n), m, n ∈ N0, we define the operators

Δ1z(m,n) = z(m+1,n)− z(m,n), Δ2z(m,n) = z(m,n+1)− z(m,n)

Introduction 7

and

Δ2Δ1z(m,n) = Δ2(Δ1z(m,n)).

The class of continuous functions and the class of discrete functions from the set S1 to the

set S2 are denoted by C(S1,S2) and D(S1,S2) respectively. We use the usual conventions that

the empty sums and products are taken to be 0 and 1 respectively. Furthermore, we shall

assume that all the integrals, sums and products involved exist on the respective domains

of their definitions and are finite, and hence converge. We note that the results we establish

for scalar equations can be extended without any difficulty to the case of vector valued

functions. The notation, definitions and symbols used in the text are standard or otherwise

explained.


Chapter 1

Integral equations in two variables

1.1 Introduction

Integral equations in two and more variables have been treated by many investigators and

various methods have been proposed for the study of different aspects of their solutions.

This intensively investigated area is in a process of continuous development, reflected in the

great number of books and papers dedicated to it, see [5–10,13–18,20,23–27,33,34,36,38–

45,49,50,52–55,59–66,68,74–140] and the references cited therein. Nevertheless, there

are still many aspects related to certain such equations, which we believe need to be en-

lightened. In response to the growing use of such equations in many applications, in this

chapter we study some important qualitative properties of solutions of certain integral and

integrodifferential equations in two variables. The fundamental tools employed in the anal-

ysis are based on applications of the Banach fixed point theorem and certain recent integral

inequalities with explicit estimates on the unknown functions.

1.2 Basic integral inequalities

In this section we present some basic integral inequalities with explicit estimates needed in

the sequel. In our considerations here and subsequent chapters we shall use the notation

E = R+×R+, E0 = Ia× Ib, E1 ={(x,y,s) : 0 � s � x < ∞, y ∈ R+

}and E2 =

{(x,y,s, t) ∈

E2 : 0 � s � x < ∞, 0 � t � y < ∞}

.

We start with the following inequality established in [111].

Theorem 1.2.1. Let u ∈C(E,R+); q, D1q ∈C(E1,R+); r, D1r, D2r, D2D1r ∈C(E2,R+)

and c � 0 is a constant. If

u(x,y) � c+∫ x

0q(x,y,ξ )u(ξ ,y)dξ +

∫ x

0

∫ y

0r(x,y,σ ,τ)u(σ ,τ)dτ dσ , (1.2.1)

9


for x, y ∈ R+, then

u(x,y) � cP(x,y)exp(∫ x

0

∫ y

0R(s,t)dt ds

)

, (1.2.2)

for x, y ∈ R+, where

P(x,y) = exp(Q(x,y)), (1.2.3)

in which

Q(x,y) =∫ x

0

[

q(η,y,η)+∫ η

0D1q(η,y,ξ )dξ

]

dη , (1.2.4)

and

R(x,y) = r(x,y,x,y)P(x,y)+∫ x

0D1r(x,y,σ ,y)P(σ ,y)dσ +

∫ y

0D2r(x,y,x,τ)P(x,τ)dτ

+∫ x

0

∫ y

0D2D1r(x,y,σ ,τ)P(σ ,τ)dτ dσ . (1.2.5)

Proof. Define a function z(x,y) by

z(x,y) = c+∫ x

0

∫ y

0r(x,y,σ ,τ)u(σ ,τ)dτ dσ , (1.2.6)

then (1.2.1) can be restated as

u(x,y) � z(x,y)+∫ x

0q(x,y,ξ )u(ξ ,y)dξ . (1.2.7)

From the hypotheses, it is easy to observe that z(x,y) is nonnegative and nondecreasing for

x, y ∈ R+. Treating (1.2.7) as a one-dimensional integral inequality for any fixed y ∈ R+

and a suitable application of the inequality given in [87, Theorem 1.2.1, Remark 1.2.1,

p. 11] yields

u(x,y) � P(x,y)z(x,y). (1.2.8)

From (1.2.6) and (1.2.8), we have

z(x,y) � c+∫ x

0

∫ y

0r(x,y,σ ,τ)P(σ ,τ)z(σ ,τ)dτ dσ . (1.2.9)

Now a suitable application of the inequality given in [87, Theorem 2.2.1, Remark 2.2.1,

p. 66] to (1.2.9) yields

z(x,y) � cexp(∫ x

0

∫ y

0R(s,t)dt ds

)

. (1.2.10)

Using (1.2.10) in (1.2.8), we get the required inequality in (1.2.2).

Next, we shall state the following versions of the inequalities in Theorems 2.5.7 and 2.5.1

given in [87] for completeness.

Integral equations in two variables 11

Theorem 1.2.2. Let u, a, b, c, f , g ∈C(E,R+) and

u(x,y) � a(x,y)+b(x,y)∫ x

0

∫ y

0f (s, t)u(s,t)dt ds

+ c(x,y)∫ ∞

0

∫ ∞

0g(s,t)u(s, t)dt ds, (1.2.11)

for x, y ∈ R+. If

p =∫ ∞

0

∫ ∞

0g(s, t)d(s,t)dt ds < 1, (1.2.12)

then

u(x,y) � B(x,y)+M D(x,y), (1.2.13)


B(x,y) = a(x,y)+b(x,y)A(x,y)∫ x

0

∫ y

0f (s, t)a(s,t)dt ds, (1.2.14)

D(x,y) = c(x,y)+b(x,y)A(x,y)∫ x

0

∫ y

0f (s,t)c(s,t)dt ds, (1.2.15)

A(x,y) = exp(∫ x

0

∫ y

0f (s,t)b(s,t)dt ds

)

, (1.2.16)

and

M =1

1− p

∫ ∞

0

∫ ∞

0g(s,t)B(s, t)dt ds. (1.2.17)

As a consequence of Theorem 1.2.2, we have the following inequality.

Theorem 1.2.3. Let u, p, q, r ∈C(E0,R+). Suppose that

u(x,y) � p(x,y)+q(x,y)∫ a

0

∫ b

0r(s,t)u(s,t)dt ds, (1.2.18)

for (x,y) ∈ E0. If

d =∫ a

0

∫ b

0r(s,t)q(s,t)dt ds < 1, (1.2.19)

then

u(x,y) � p(x,y)+q(x,y){

11−d

∫ a

0

∫ b

0r(s,t)p(s,t)dt ds

}

, (1.2.20)

for (x,y) ∈ E0.


Theorem 1.2.4. Let u, p, q, r ∈C(E0,R+) and

u(x,y) � c+∫ x

0

∫ y

0p(s,t)

[

u(s,t)+∫ s

0

∫ t

0q(σ ,τ)u(σ ,τ)dτ dσ

+∫ a

0

∫ b

0r(σ ,τ)u(σ ,τ)dτ dσ

]

dt ds, (1.2.21)

for (x,y) ∈ E0, where c � 0 is a constant. If

d =∫ a

0

∫ b

0r(σ ,τ)exp

(∫ σ

0

∫ τ

0[p(s,t)+q(s,t)]dt ds

)

dτ dσ < 1, (1.2.22)

then

u(x,y) � c1−d

exp(∫ x

0

∫ y


)

, (1.2.23)

for (x,y) ∈ E0.

Another useful inequality proved in [92] is embodied in the following theorem.

Theorem 1.2.5. Let u, a ∈C(E,R+), b, D1b, D2b, D2D1b, e ∈C(E2,R+) and c � 0 is a

constant. If

u(x,y) � c+∫ x

0

∫ y

0

{

a(s,t)u(s, t)+b(x,y,s,t)u(s, t)

+∫ s

0

∫ t

0e(s,t,m,n)u(m,n)dndm

}

dt ds, (1.2.24)

for x, y ∈ R+, then

u(x,y) � cexp(∫ x

0

∫ y

0[a(s, t)+K(s, t)]dt ds

)

, (1.2.25)


K(x,y) = b(x,y,x,y)+∫ x

0D1b(x,y,m,y)dm+

∫ y

0D2b(x,y,x,n)dn

+∫ x

0

∫ y

0D2D1b(x,y,m,n)dndm+

∫ x

0

∫ y

0e(x,y,m,n)dndm. (1.2.26)

Proof. Define a function z(x,y) by the right hand side of (1.2.21). Then z(x,0) = z(0,y) =

c, u(x,y) � z(x,y), z(x,y) is nondecreasing in x and y and (see [87, p. 65])

D2D1z(x,y) = a(x,y)u(x,y)+b(x,y,x,y)u(x,y)

+∫ x

0D1b(x,y,m,y)u(m,y)dm+

∫ y

0D2b(x,y,x,n)u(x,n)dn

+∫ x

0

∫ y

0D2D1b(x,y,m,n)u(m,n)dndm+

∫ x

0

∫ y

0e(x,y,m,n)u(m,n)dndm


� a(x,y)z(x,y)+b(x,y,x,y)z(x,y)

+∫ x

0D1b(x,y,m,y)z(m,y)dm+

∫ y

0D2b(x,y,x,n)z(x,n)dn

+∫ x

0

∫ y

0D2D1b(x,y,m,n)z(m,n)dndm+

∫ x

0

∫ y

0e(x,y,m,n)z(m,n)dndm

�[

a(x,y)+b(x,y,x,y)+∫ x

0D1b(x,y,m,y)dm+

∫ y

0D2b(x,y,x,n)dn

+∫ x

0

∫ y

0D2D1b(x,y,m,n)dndm+

∫ x

0

∫ y

0e(x,y,m,n)dndm

]

z(x,y)

= [a(x,y)+K(x,y)]z(x,y), (1.2.27)

where K(x,y) is given by (1.2.26). Now by following the proof of Theorem 4.2.1 given in

[82], from (1.2.27), we get

z(x,y) � cexp(∫ x

0

∫ y

0[a(s,t)+K(s,t)]dt ds

)

. (1.2.28)

Using (1.2.28) in u(x,y) � z(x,y), we get the required inequality in (1.2.25).

1.3 Volterra-type integral equation

Consider the integral equation of the form

u(x,y) = f (x,y)+∫ x

0g(x,y,ξ ,u(ξ ,y))dξ +

∫ x

0

∫ y

0h(x,y,σ ,τ,u(σ ,τ))dτ dσ , (1.3.1)

for x, y ∈ R+, where f ∈C(E,Rn), g ∈C(E1 ×Rn,Rn), h ∈C(E2 ×R

n,Rn) are the given

functions and u is the unknown function to be fond. The origin of equation (1.3.1) can be

traced back to the important observation in [28, p. 20] and the driven equation in (5). In

this section, we present some basic qualitative properties of solutions of equation (1.3.1)

under some suitable conditions on the functions f , g, h (see [115]). Let S be the space of

functions z ∈C(E,Rn) which fulfil the condition

|z(x,y)| = O(exp(λ (x + y))), (1.3.2)

where λ > 0 is a constant. In the space S we define the norm

|z|S = sup(x,y)∈E

[|z(x,y)|exp(−λ (x+ y))] . (1.3.3)


It is easy to see that S with norm defined in (1.3.3) is a Banach space. We note that the

condition (1.3.2) implies that there exists a constant N � 0 such that

|z(x,y)| � N exp(λ (x+ y)).

Using this fact in (1.3.3), we observe that

|z|S � N. (1.3.4)

To carry out the proof of the existence and uniqueness of solutions of equation (1.3.1) and

some other equations considered in the subsequent chapters, we shall make use of the above

class S of functions without further mention.

We start with the following theorem which ensures the existence of a unique solution to

equation (1.3.1).

Theorem 1.3.1. Suppose that

(i) the functions g, h in equation (1.3.1) satisfy the conditions

|g(x,y,ξ ,u)−g(x,y,ξ ,u)| � a(x,y,ξ )|u−u|, (1.3.5)

|h(x,y,σ ,τ,u)−h(x,y,σ ,τ,u)| � b(x,y,σ ,τ)|u−u|, (1.3.6)

where a ∈C(E1,R+), b ∈C(E2,R+),

(ii) for λ as in (1.3.2),

(a1) there exists a nonnegative constant α such that α < 1 and∫ x

0a(x,y,ξ )exp(λ (ξ + y))dξ +

∫ x

0

∫ y

0b(x,y,σ ,τ)exp(λ (σ + τ))dτ dσ

� α exp(λ (x + y)), (1.3.7)

(a2) there exists a nonnegative constant β such that∣∣∣∣ f (x,y)+

∫ x

0g(x,y,ξ ,0)dξ +

∫ x

0

∫ y

0h(x,y,σ ,τ,0)dτ dσ

∣∣∣∣ � β exp(λ (x + y)), (1.3.8)

where f , g, h are the functions in equation (1.3.1).

Under the assumptions (i) and (ii) the equation (1.3.1) has a unique solution u(x,y) on E

in S.

Proof. Let u ∈ S and define the operator T by

(Tu)(x,y) = f (x,y)+∫ x


∫ x

0

∫ y

0h(x,y,σ ,τ,u(σ ,τ))dτ dσ . (1.3.9)

Now we shall show that T maps S into itself. Evidently, Tu is continuous on E and Tu∈Rn.

We verify that (1.3.2) is fulfilled. From (1.3.9) and using the hypotheses and (1.3.4), we

have

|(Tu)(x,y)| �∣∣∣∣ f (x,y)+

∫ x

0g(x,y,ξ ,0)dξ +

∫ x

0

∫ y


∣∣∣∣


+x∫

0

|g(x,y,ξ ,u(ξ ,y))−g(x,y,ξ ,0)|dξ +x∫

0

y∫

0

|h(x,y,σ ,τ,u(σ ,τ))−h(x,y,σ ,τ,0)|dτ dσ

� β exp(λ (x+ y))+∫ x

0a(x,y,ξ )|u(ξ ,y)|dξ +

∫ x

0

∫ y

0b(x,y,σ ,τ)|u(σ ,τ)|dτ dσ

� β exp(λ (x+ y))+ |u|S[∫ x

0a(x,y,ξ ) exp(λ (ξ + y))dξ

+∫ x

0

∫ y

0b(x,y,σ ,τ) exp(λ (σ + τ))dτ dσ

]

� [β +Nα ]exp(λ (x+ y)). (1.3.10)

From (1.3.10) it follows that Tu ∈ S. This proves that T maps S into itself.

Now, we verify that the operator T is a contraction map. Let u, v ∈ S. From (1.3.9) and

using the hypotheses, we have

|(Tu)(x,y)− (Tv)(x,y)| �∫ x

0|g(x,y,ξ ,u(ξ ,y))−g(x,y,ξ ,v(ξ ,y))|dξ

+∫ x

0

∫ y

0|h(x,y,σ ,τ,u(σ ,τ))−h(x,y,σ ,τ,v(σ ,τ))|dτ dσ

�∫ x

0a(x,y,ξ )|u(ξ ,y)− v(ξ ,y)|dξ +

∫ x

0

∫ y

0b(x,y,σ ,τ)|u(σ ,τ)− v(σ ,τ)|dτ dσ

� |u− v|S[∫ x

0a(x,y,ξ )exp(λ (ξ + y))dξ +

∫ x

0

∫ y

0b(x,y,σ ,τ)exp(λ (σ + τ))dτ dσ

]

� α|u− v|S exp(λ (x+ y)). (1.3.11)

From (1.3.11), we obtain

|Tu−T v|S � α|u− v|S.

Since α < 1, it follows from Banach fixed point theorem (see [28, p. 37] and [151, p. 372])

that T has a unique fixed point in S. The fixed point of T is however a solution of equation

(1.3.1). The proof is complete.

Remark 1.3.1. We note that, Theorem 1.3.1 given above provides a simple way to estab-

lish the existence and uniqueness for solutions of equation (1.3.1) in the space of continuous

functions. The norm defined by (1.3.3) is a variant of Bielecki’s norm [12,29], first used

for the study of solutions of ordinary differential equations and has the role of improving

the length of the interval on which the existence is assured. For a number results on the

existence and uniqueness of solutions of special and general versions of equation (1.3.1)

by using various techniques, see [5,134] and the references therein.

A slight variant of Theorem 1.3.1 is given in the following theorem.


Theorem 1.3.2. Suppose that the functions g,h in equation (1.3.1) satisfy the conditions

(1.3.5), (1.3.6) and assume that

supx,y∈R+

[∫ x

0a(x,y,ξ )dξ +

∫ x

0

∫ y

0b(x,y,σ ,τ)dτ dσ

]

� α < 1. (1.3.12)

Then the equation (1.3.1) has a unique solution u ∈C(E,Rn).

Proof. Consider the space C(E,Rn) endowed with a norm ‖ · ‖ defined by

‖u‖ = supx,y∈R+

|u(x,y)|, (1.3.13)

for u ∈C(E,Rn). It is well known that C(E,Rn) is a Banach space with norm (1.3.13). For

any u, v ∈C(E,Rn), one can easily verify from the hypotheses that, the operator T defined

by (1.3.9) for any u ∈C(E,Rn) satisfies

‖Tu−T v‖ � α‖u− v‖. (1.3.14)

This shows that T is a contraction. Therefore, (1.3.1) has a unique solution u ∈C(E,Rn).

The following theorem deals with the estimate on the solution of equation (1.3.1).

Theorem 1.3.3. Suppose that the functions f , g, h in equation (1.3.1) satisfy the condi-

tions

|g(x,y,ξ ,u)−g(x,y,ξ ,u)| � q(x,y,ξ )|u−u|, (1.3.15)

|h(x,y,σ ,τ,u)−g(x,y,σ ,τ,u)| � r(x,y,σ ,τ)|u−u|, (1.3.16)

where q, D1q ∈C(E1,R+) and r, D1r, D2r, D2D1r ∈C(E2,R+) and

c = supx,y∈R+

∣∣∣∣ f (x,y)+

∫ x

0g(x,y,ξ ,0)dξ +

∫ x

0

∫ y


∣∣∣∣ < ∞. (1.3.17)

If u(x,y) is any solution of equation (1.3.1) on E, then

|u(x,y)| � cP(x,y)exp(∫ x

0

∫ y

0R(s,t)dt ds

)

, (1.3.18)

for (x,y) ∈ E, where P(x,y) and R(x,y) are given by (1.2.3) and (1.2.5).

Proof. Using the fact that u(x,y) is a solution of equation (1.3.1) and hypotheses, we have

|u(x,y)| �∣∣∣∣ f (x,y)+

∫ x

0g(x,y,ξ ,0)dξ +

∫ x

0

∫ y


∣∣∣∣

+x∫

0

|g(x,y,ξ ,u(ξ ,y))−g(x,y,ξ ,0)|dξ +x∫

0

y∫

0

|h(x,y,σ ,τ,u(σ ,τ))−h(x,y,σ ,τ,0)|dτ dσ

� c+∫ x

0q(x,y,ξ )|u(ξ ,y)|dξ +

∫ x

0

∫ y

0r(x,y,σ ,τ)|u(σ ,τ)|dτ dσ . (1.3.19)

Now, an application of Theorem 1.2.1 to (1.3.19) yields (1.3.18).


Remark 1.3.2. In Theorem 1.3.3, if we assume that

(i) the function Q(x,y) given by (1.2.4) is such that Q(x,y) < ∞ and

(ii)∫ ∞

0

∫ ∞

0R(s,t)dt ds < ∞,

then the solution u(x,y) of equation (1.3.1) is bounded on E.

A variant of Theorem 1.3.3 is embodied in the following theorem.

Theorem 1.3.4. Suppose that the functions g, h in equation (1.3.1) satisfy the conditions

(1.3.15), (1.3.16) and assume that∫ x

0|g(x,y,ξ , f (ξ ,y))|dξ +

∫ x

0

∫ y

0|h(x,y,σ ,τ, f (σ ,τ))|dτ dσ � d, (1.3.20)

for x, y ∈ R+, where f is the function involved in (1.3.1) and d � 0 is a constant. If u(x,y)

is any solution of equation (1.3.1) on E, then

|u(x,y)− f (x,y)| � d P(x,y)exp(∫ x

0

∫ y

0R(s,t)dt ds

)

, (1.3.21)


Proof. Let w(x,y) = |u(x,y)− f (x,y)| for (x,y) ∈ E. Using the fact that u(x,y) is a solu-

tion of equation (1.3.1) and the hypotheses, we have

w(x,y) �∫ x

0|g(x,y,ξ , f (ξ ,y))|dξ +

∫ x

0

∫ y

0|h(x,y,σ ,τ, f (σ ,τ))|dτ dσ

+∫ x

0|g(x,y,ξ ,u(ξ ,y))−g(x,y,ξ , f (ξ ,y))|dξ

+∫ x

0

∫ y

0|h(x,y,σ ,τ,u(σ ,τ))−h(x,y,σ ,τ, f (σ ,τ))|dτ dσ

� d +∫ x

0q(x,y,ξ )w(ξ ,y)dξ +

∫ x

0

∫ y

0r(x,y,σ ,τ)w(σ ,τ)dτ dσ . (1.3.22)


We call the function u ∈ C(E,Rn) an ε-approximate solution to equation (1.3.1), if there

exists a constant ε � 0 such that∣∣∣∣u(x,y)−

{

f (x,y)+∫ x


∫ x

0

∫ y

0h(x,y,σ ,τ,u(σ ,τ))dτ dσ

}∣∣∣∣ � ε,

for x, y ∈ R+.

The next theorem deals with the estimate on the difference between two approximate solu-

tions of equation (1.3.1).


Theorem 1.3.5. Let u1(x,y) and u2(x,y) be respectively, ε1- and ε2-approximate solutions

of equation (1.3.1) on E. Suppose that the functions g, h in equation (1.3.1) satisfy the

conditions (1.3.15), (1.3.16). Then

|u1(x,y)−u2(x,y)| � (ε1 + ε2)P(x,y)exp(∫ x

0

∫ y

0R(s, t)dt ds

)

, (1.3.23)


Proof. Since u1(x,y) and u2(x,y) are respectively, ε1- and ε2-approximate solutions to

equation (1.3.1), we have∣∣∣∣ui(x,y)−

{

f (x,y) +∫ x

0g(x,y,ξ ,ui(ξ ,y))dξ

+∫ x

0

∫ y

0h(x,y,σ ,τ,ui(σ ,τ))dτ dσ

}∣∣∣∣ � εi, (1.3.24)

for i = 1, 2. From (1.3.24) and using the elementary inequalities

|v− z| � |v|+ |z|, |v|− |z| � |v− z|, (1.3.25)

we observe that

ε1 + ε2 �∣∣∣∣∣u1(x,y)−

{

f (x,y)+∫ x

0g(x,y,ξ ,u1(ξ ,y))dξ +

∫ x

0

∫ y

0h(x,y,σ ,τ,u1(σ ,τ))dτ dσ

}∣∣∣∣∣

+∣∣∣∣u2(x,y)−

{

f (x,y)+∫ x

0g(x,y,ξ ,u2(ξ ,y))dξ +

∫ x

0

∫ y


}∣∣∣∣

�∣∣∣∣∣[u1(x,y)−u2(x,y)]

−[{

f (x,y)+∫ x

0g(x,y,ξ ,u1(ξ ,y))dξ +

∫ x

0

∫ y


}

−{

f (x,y)+∫ x

0g(x,y,ξ ,u2(ξ ,y))dξ +

∫ x

0

∫ y


}]∣∣∣∣∣

� |u1(x,y)−u2(x,y)|−∣∣∣∣

∫ x

0{g(x,y,ξ ,u1(ξ ,y))−g(x,y,ξ ,u2(ξ ,y))}dξ

+∫ x

0

∫ y

0{h(x,y,σ ,τ,u1(σ ,τ))−h(x,y,σ ,τ,u2(σ ,τ))}dτ dσ

∣∣∣∣ . (1.3.26)

Let w(x,y) = |u1(x,y)− u2(x,y)|, (x,y) ∈ E. From (1.3.26) and using conditions (1.3.15),

(1.3.16), we have

w(x,y) � (ε1 + ε2)+∫ x

0|g(x,y,ξ ,u1(ξ ,y))−g(x,y,ξ ,u2(ξ ,y))|dξ

+∫ x

0

∫ y

0|h(x,y,σ ,τ,u1(σ ,τ))−h(x,y,σ ,τ,u2(σ ,τ))|dτ dσ

� (ε1 + ε2)+∫ x


∫ x

0

∫ y

0r(x,y,σ ,τ)w(σ ,τ)dτ dσ . (1.3.27)

Now an application of Theorem 1.2.1 to (1.3.27) yields (1.3.23).


Remark 1.3.3. In case, if u1(x,y) is a solution of equation (1.3.1), then we have ε1 = 0

and from (1.3.23) we see that u1(x,y) → u2(x,y) as ε2 → 0. Moreover, from (1.3.23), the

uniqueness of solutions of equation (1.3.1) follows if εi = 0 (i = 1, 2).

We next consider the following variants of equation (1.3.1)

u(x,y) = f (x,y)+∫ x

0g(x,y,ξ ,u(ξ ,y),μ)dξ

+∫ x

0

∫ y

0h(x,y,σ ,τ,u(σ ,τ),μ)dτ dσ , (1.3.28)

and

u(x,y) = f (x,y)+∫ x

0g(x,y,ξ ,u(ξ ,y),μ0)dξ

+∫ x

0

∫ y

0h(x,y,σ ,τ,u(σ ,τ),μ0)dτ dσ , (1.3.29)

for (x,y) ∈ E, where f ∈C(E,Rn), g ∈C(E1 ×Rn ×R,Rn), h ∈C(E2 ×R

n ×R,Rn), and

μ, μ0 are parameters.

The following theorem shows the dependency of solutions of equations (1.3.28) and

(1.3.29) on parameters.

Theorem 1.3.6. Suppose that the functions g, h in equations (1.3.28), (1.3.29) satisfy the

conditions

|g(x,y,ξ ,u,μ)−g(x,y,ξ ,u,μ)| � q(x,y,ξ )|u−u|, (1.3.30)

|g(x,y,ξ ,u,μ)−g(x,y,ξ ,u,μ0)| � p1(x,y,ξ )|μ −μ0|, (1.3.31)

|h(x,y,σ ,τ,u,μ)−h(x,y,σ ,τ,u,μ)| � r(x,y,σ ,τ)|u−u|, (1.3.32)

|h(x,y,σ ,τ,u,μ)−h(x,y,σ ,τ,u,μ0)| � p2(x,y,σ ,τ)|μ −μ0|, (1.3.33)

where p1, q, D1q ∈C(E1,R+), p2, r, D1r, D2r, D2D1r ∈C(E2,R+) and∫ x

0p1(x,y,ξ )dξ � M1, (1.3.34)

∫ x

0

∫ y

0p2(x,y,σ ,τ)dτ dσ � M2, (1.3.35)

in which M1, M2 are nonnegative constants. Let u1(x,y) and u2(x,y) be the solutions of

equations (1.3.28) and (1.3.29) respectively. Then

|u1(x,y)−u2(x,y)| � (M1 +M2)|μ −μ0|P(x,y)exp(∫ x

0

∫ y

0R(s,t)dt ds

)

, (1.3.36)



Proof. Let w(x,y) = |u1(x,y)− u2(x,y)|,(x,y) ∈ E. Using the facts that u1(x,y) and

u2(x,y) are the solutions of equations (1.3.28) and (1.3.29) and hypotheses, we have

w(x,y) �∫ x

0|g(x,y,ξ ,u1(ξ ,y),μ)−g(x,y,ξ ,u2(ξ ,y),μ)|dξ

+∫ x

0|g(x,y,ξ ,u2(ξ ,y),μ)−g(x,y,ξ ,u2(ξ ,y),μ0)|dξ

+∫ x

0

∫ y

0|h(x,y,σ ,τ,u1(σ ,τ),μ)−h(x,y,σ ,τ,u2(σ ,τ),μ)|dτ dσ

+∫ x

0

∫ y

0|h(x,y,σ ,τ,u2(σ ,τ),μ)−h(x,y,σ ,τ,u2(σ ,τ),μ0)|dτ dσ

�∫ x


∫ x

0p1(x,y,ξ )|μ −μ0|dξ

+∫ x

0

∫ y

0r(x,y,σ ,τ)w(σ ,τ)dτ dσ +

∫ x

0

∫ y

0p2(x,y,σ ,τ)|μ −μ0|dτ dσ

�(M1 +M2)|μ −μ0|

+∫ x


∫ x

0

∫ y

0r(x,y,σ ,τ)w(σ ,τ)dτ dσ . (1.3.37)

Now an application of Theorem 1.2.1 to (1.3.37) yields (1.3.36), which shows the depen-

dency of solutions of equations (1.3.28) and (1.3.29) on parameters.

Remark 1.3.4 We note that the method employed in this section can be extended to study

the integrodifferential equation of the form

D2D1u(x,y) = f (x,y,u(x,y),Gu(x,y),Hu(x,y)), (1.3.38)

with the given data

u(x,0) = σ(x),u(0,y) = τ(y), (1.3.39)


Gu(x,y) =∫ x

0g(x,y,ξ ,u(ξ ,y))dξ , (1.3.40)

Hu(x,y) =∫ x

0

∫ y

0h(x,y,m,n,u(m,n))dndm, (1.3.41)

under some suitable conditions on the functions f , g,h, σ , τ and by making use of a suitable

variant of the inequality given in [87, Theorem 2.5.1, p. 96].


1.4 Volterra-Fredholm-type integral equation

We shall deal in this section with the Volterra-Fredholm-type integral equation in the form

u(x,y) = h(x,y)+∫ x

0

∫ y

0F(x,y,s,t,u(s, t))dt ds+

∫ ∞

0

∫ ∞

0G(x,y,s,t,u(s,t))dt ds, (1.4.1)

for x, y∈R+, where h∈C(E,Rn), F ∈C(E2×Rn,Rn), G∈C(E2×R

n,Rn). The equations

of the form (1.4.1) are of particular interest, since the special versions of the same arise in

a variety of applications, see [5,134]. It is our aim here to give some important qualita-

tive properties of solutions of equation (1.4.1) under various assumptions on the functions

involved therein (see [89]).

Our first result concerning the existence and uniqueness of solutions of equation (1.4.1) is

given in the following theorem.

Theorem 1.4.1. Assume that

(i) the functions F, G in equation (1.4.1) satisfy the conditions

|F(x,y,s,t,u)−F(x,y,s,t,u)| � k(x,y,s, t)|u−u|, (1.4.2)

|G(x,y,s,t,u)−G(x,y,s,t,u)| � r(x,y,s,t)|u−u|, (1.4.3)

where k ∈C(E2,R+), r ∈C(E2,R+),


(b1) there exist nonnegative constants α1, α2 such that α1 +α2 < 1 and∫ x

0

∫ y

0k(x,y,s,t)exp(λ (s+ t))dt ds � α1 exp(λ (x+ y)), (1.4.4)

∫ ∞

0

∫ ∞

0r(x,y,s,t)exp(λ (s+ t))dt ds � α2 exp(λ (x+ y)), (1.4.5)

(b2) there exists a nonnegative constant β such that

|h(x,y)|+∫ x

0

∫ y

0|F(x,y,s,t,0)|dt ds+

∫ ∞

0

∫ ∞

0|G(x,y,s,t,0)|dt ds

� β exp(λ (x+ y)), (1.4.6)

where h, F, G are the functions in equation (1.4.1).

Then the equation (1.4.1) has a unique solution u(x,y) on E in S.

Proof. Let u ∈ S and define the operator T by

(Tu)(x,y) = h(x,y)+∫ x

0

∫ y

0F(x,y,s, t,u(s,t))dt ds

+∫ ∞

0

∫ ∞

0G(x,y,s,t,u(s,t))dt ds. (1.4.7)

for (x,y)∈E. The proof that T maps S into itself and is a contraction map can be completed

by closely looking at the proof of Theorem 1.3.1 with suitable modifications. Here, we omit

the details.


Remark 1.4.1. We note that Theorem 1.4.1 yields existence and uniqueness of solutions

of equation (1.4.1) in S. One can also formulate existence and uniqueness result similar to

that of Theorem 1.3.2 for the solution u ∈C(E,Rn) of equation (1.4.1).

The following theorem which asserts only the uniqueness of solution of equation (1.4.1) on

E in Rn can be obtained by using the inequality in Theorem 1.2.2.

Theorem 1.4.2. Suppose that the functions F, G in equation (1.4.1) satisfy the conditions

|F(x,y,s,t,u)−F(x,y,s,t,u)| � b(x,y) f (s,t)|u−u|, (1.4.8)

|G(x,y,s,t,u)−G(x,y,s,t,u)| � c(x,y)g(s,t)|u−u|, (1.4.9)

where b, f , c, g ∈ C(E,R+). Let p, D(x,y) be as in (1.2.12), (1.2.15) respectively. Then

the equation (1.4.1) has at most one solution on E in Rn.

Proof. Let u1(x,y) and u2(x,y) be two solutions of equation (1.4.1). Then, we have

u1(x,y)−u2(x,y) =∫ x

0

∫ y

0{F(x,y,s, t,u1(s,t))−F(x,y,s,t,u2(s,t))}dt ds

+∫ ∞

0

∫ ∞

0{G(x,y,s,t,u1(s, t))−G(x,y,s,t,u2(s,t))}dt ds. (1.4.10)

From (1.4.10) and using hypotheses, we have

|u1(x,y)−u2(x,y)| � b(x,y)∫ x

0

∫ y

0f (s,t)|u1(s,t)−u2(s,t)|dt ds

+ c(x,y)∫ ∞

0

∫ ∞

0g(s, t)|u1(s, t)−u2(s,t)|dt ds. (1.4.11)

Here, it is easy to observe that B(x,y) and M defined in (1.2.14) and (1.2.17) reduces to

B(x,y) = 0 and M = 0. Now an application of Theorem 1.2.2 (with a(x,y) = 0) to (1.4.11)

yields |u1(x,y)− u2(x,y)| � 0, and hence u1(x,y) = u2(x,y). Thus there is at most one

solution to equation (1.4.1) on E in Rn.

The following theorem concerning the estimate on the solution of equation (1.4.1) holds.

Theorem 1.4.3. Assume that the functions F, G in equation (1.4.1) satisfy the conditions

|F(x,y,s,t,u)| � b(x,y) f (s,t)|u|, (1.4.12)

|G(x,y,s,t,u)| � c(x,y)g(s,t)|u|, (1.4.13)

where b, f , c, g ∈C(E,R+). Let p, D(x,y) be as in (1.2.12), (1.2.15) respectively and

M1 =1

1− p

∫ ∞

0

∫ ∞

0g(s,t)B1(s, t)dt ds, (1.4.14)

where B1(x,y) is defined by the right hand side of (1.2.14) by replacing a(x,y) by |h(x,y)|.If u(x,y) is any solution of equation (1.4.1), then

|u(x,y)| � B1(x,y)+M1D(x,y), (1.4.15)

for (x,y) ∈ E.


Proof. Using the fact that u(x,y) is a solution of equation (1.4.1) and the hypotheses, we

have

|u(x,y)| � |h(x,y)|+∫ x

0

∫ y

0|F(x,y,s,t,u(s, t))|dt ds +

∫ ∞

0

∫ ∞

0|G(x,y,s,t,u(s,t))|dt ds

� |h(x,y)|+b(x,y)∫ x

0

∫ y

0f (s,t)|u(s, t)|dt ds + c(x,y)

∫ ∞

0

∫ ∞

0g(s, t)|u(s,t)|dt ds.

(1.4.16)


In the next theorem we give an estimate on the solution of equation (1.4.1) assuming that

the functions F, G satisfy the Lipschitz type conditions (1.4.8), (1.4.9).

Theorem 1.4.4. Suppose that the functions F,G in equation (1.4.1) satisfy the conditions

(1.4.8), (1.4.9) respectively. Let p, D(x,y) be as in (1.2.12), (1.2.15) respectively and

h0(x,y) =∫ x

0

∫ y

0|F(x,y,s,t,h(s,t))|dt ds+

∫ ∞

0

∫ ∞

0|G(x,y,s,t,h(s,t))|dt ds, (1.4.17)

M2 =1

1− p

∫ ∞

0

∫ ∞

0g(s,t)B2(s, t)dt ds, (1.4.18)

where B2(x,y) is defined by the right hand side of (1.2.14) by replacing a(x,y) by h0(x,y).

If u(x,y) is any solution of equation (1.4.1), then

|u(x,y)−h(x,y)| � B2(x,y)+M2D(x,y), (1.4.19)

for (x,y) ∈ E.

The proof can be completed by following the proof of Theorem 1.3.4. Here, we omit the

details.

The following theorem deals with the estimate on the difference between the solutions of

equation (1.4.1) and the system of Volterra integral equations

v(x,y) = h(x,y)+∫ x

0

∫ y

0F(x,y,s,t,v(s,t))dt ds, (1.4.20)

for (x,y) ∈ E, where the functions h, F are as given in equation (1.4.1).

Theorem 1.4.5. Suppose that the functions F,G in equation (1.4.1) satisfy the conditions

(1.4.8), (1.4.9) respectively and G(x,y,s,t,0) = 0. Let v(x,y) be a solution of equation

(1.4.20) such that |v(x,y)| � Q, where Q � 0 is a constant. Let p,D(x,y) be as in (1.2.12),

(1.2.15) respectively and

a(x,y) = Qc(x,y)∫ ∞

0

∫ ∞

0g(s,t)dt ds, (1.4.21)


M3 =1

1− p

∫ ∞

0

∫ ∞

0g(s,t)B3(s, t)dt ds, (1.4.22)

where B3(x,y) is defined by the right hand side of (1.2.14) by replacing a(x,y) by a(x,y).

If u(x,y) is a solution of equation (1.4.1), then

|u(x,y)− v(x,y)| � B3(x,y)+M3D(x,y), (1.4.23)

for (x,y) ∈ E.

Proof. Using the facts that u(x,y) and v(x,y) are the solutions of equations (1.4.1) and

(1.4.20) on E, we observe that

u(x,y)− v(x,y) =∫ x

0

∫ y

0{F(x,y,s, t,u(s,t))−F(x,y,s, t,v(s, t))}dt ds

+∫ ∞

0

∫ ∞

0

{G(x,y,s, t,u(s,t))−G(x,y,s,t,v(s,t))

+G(x,y,s, t,v(s, t))−G(x,y,s,t,0)}

dt ds. (1.4.24)

From (1.4.24) and using the hypotheses, we have

|u(x,y)− v(x,y)| � a(x,y)+b(x,y)∫ x

0

∫ y

0f (s,t)|u(s,t)− v(s,t)|dt ds

+ c(x,y)∫ ∞

0

∫ ∞

0g(s,t)|u(s,t)− v(s,t)|dt ds. (1.4.25)


The following theorems provide conditions for continuous dependence of solutions of

equation (1.4.1) on functions involved on the right hand side and continuous dependence

of solutions of equations of the form (1.4.1) on parameters.

Consider the equation (1.4.1) and the system of Volterra-Fredholm-type integral equations

w(x,y) = h(x,y)+∫ x

0

∫ y

0F(x,y,s,t,w(s,t))dt ds

+∫ ∞

0

∫ ∞

0G(x,y,s,t,w(s,t))dt ds, (1.4.26)

for (x,y) ∈ E, where h ∈C(E,Rn), F ∈C(E2 ×Rn,Rn), G ∈C(E2 ×R

n,Rn).

Theorem 1.4.6. Suppose that the functions F, G in equation (1.4.1) satisfy the conditions

(1.4.8), (1.4.9) respectively. Let w(x,y) be a given solution of equation (1.4.26) on E and

assume that

|h(x,y)−h(x,y)|+∫ x

0

∫ y

0|F(x,y,s,t,w(s,t))−F(x,y,s,t,w(s, t))|dt ds

+∫ ∞

0

∫ ∞

0|G(x,y,s,t,w(s, t))−G(x,y,s,t,w(s, t))|dt ds � ε , (1.4.27)


where h, F, G and h, F, G are functions in equations (1.4.1) and (1.4.26) and ε > 0 is an

arbitrary small constant. Let p, D(x,y) be as in (1.2.12), (1.2.15) respectively and

M4 =1

1− p

∫ ∞

0

∫ ∞

0g(s,t)B4(s, t)dt ds, (1.4.28)

where B4(x,y) is defined by the right hand side of (1.2.14) by replacing a(x,y) by ε . Then

the solution u(x,y) of equation (1.4.1) on E depends continuously on the functions involved

on the right hand side of equation (1.4.1).

Proof. Let u(x,y) be a solution of (1.4.1) on E and w(x,y) be a given solution of equation

(1.4.26). Then

u(x,y)−w(x,y) = h(x,y)−h(x,y)

+∫ x

0

∫ y

0

{F(x,y,s,t,u(s,t))−F(x,y,s, t,w(s,t))

+F(x,y,s, t,w(s,t))−F(x,y,s,t,w(s,t))}

dt ds

+∫ ∞

0

∫ ∞

0{G(x,y,s,t,u(s,t))−G(x,y,s, t,w(s,t))

+G(x,y,s,t,w(s,t))−G(x,y,s,t,w(s,t))}

dt ds. (1.4.29)

From (1.4.29) and using the hypotheses, we have

|u(x,y)−w(x,y)| � |h(x,y)−h(x,y)|

+∫ x

0

∫ y

0|F(x,y,s,t,u(s,t))−F(x,y,s,t,w(s,t))|dt ds

+∫ x

0

∫ y

0|F(x,y,s,t,w(s,t))−F(x,y,s,t,w(s, t))|dt ds

+∫ ∞

0

∫ ∞

0|G(x,y,s,t,u(s, t))−G(x,y,s,t,w(s,t))|dt ds

+∫ ∞

0

∫ ∞

0|G(x,y,s, t,w(s,t))−G(x,y,s,t,w(s,t))|dt ds

� ε +b(x,y)∫ x

0

∫ y

0f (s,t)|u(s,t)−w(s, t)|dt ds

+c(x,y)∫ ∞

0

∫ ∞

0g(s, t)|u(s,t)−w(s,t)|dt ds. (1.4.30)

Now, an application of Theorem 1.2.2 to (1.4.30) yields

|u(x,y)−w(x,y)| � B4(x,y)+M4D(x,y), (1.4.31)

for (x,y) ∈ E. From (1.4.31), it follows that the solutions of equation (1.4.1) depends

continuously on the functions involved on the right hand side of equation (1.4.1).


Remark 1.4.2. From (1.4.31), it is easy to observe that if B4(x,y) and D(x,y) are bounded

for (x,y) ∈ E and ε → 0, then |u(x,y)−w(x,y)| → 0 on E.

We next consider the following systems of Volterra-Fredholm-type integral equations

z(x,y) = h(x,y)+∫ x

0

∫ y

0F(x,y,s, t,z(s,t),μ)dt ds

+∫ ∞

0

∫ ∞

0G(x,y,s, t,z(s,t),μ)dt ds, (1.4.32)

and

z(x,y) = h(x,y)+∫ x

0

∫ y

0F(x,y,s, t,z(s,t),μ0)dt ds

+∫ ∞

0

∫ ∞

0G(x,y,s, t,z(s,t),μ0)dt ds, (1.4.33)

for (x,y) ∈ E, where h ∈C(E,Rn), F ∈C(E2 ×Rn ×R,Rn), G ∈C(E2 ×R

n ×R,Rn) and

μ, μ0 are parameters.

Theorem 1.4.7. Suppose that the functions F, G in equations (1.4.32), (1.4.33) satisfy

the conditions

|F(x,y,s, t,z,μ)−F(x,y,s,t,z,μ)| � b(x,y) f (s, t)|z− z|, (1.4.34)

|F(x,y,s,t,z,μ)−F(x,y,s, t,z,μ0)| � r1(x,y,s,t)|μ −μ0|, (1.4.35)

|G(x,y,s,t,z,μ)−G(x,y,s,t,z,μ)| � c(x,y)g(s, t)|z− z|, (1.4.36)

|G(x,y,s,t,z,μ)−G(x,y,s,t,z,μ0)| � r2(x,y,s,t)|μ −μ0|, (1.4.37)

where b, f , c, g ∈ C(E,R+), and r1, r2 ∈ C(E2,R+). Let p, D(x,y) be as in (1.2.12),

(1.2.15) respectively and

a0(x,y) = |μ −μ0|[∫ x

0

∫ y

0r1(x,y,s, t)dt ds+

∫ ∞

0

∫ ∞

0r2(x,y,s, t)dt ds

]

, (1.4.38)

M5 =1

1− p

∫ ∞

0

∫ ∞

0g(s,t)B5(s, t)dt ds, (1.4.39)

where B5(x,y) is defined by the right hand side of (1.2.14) by replacing a(x,y) by a0(x,y).

Let z1(x,y) and z2(x,y) be the solutions of equations (1.4.32) and (1.4.33) respectively.

Then

|z1(x,y)− z2(x,y)| � B5(x,y)+M5D(x,y), (1.4.40)

for (x,y) ∈ E, which shows the dependency of solutions of equations (1.4.32) and (1.4.33)

on parameters.

The proof can be completed by following the proof of Theorem 1.3.6 and using Theo-

rem 1.2.2. Here, we leave the details to the reader.


Remark 1.4.3. We note that, one can use our approach here, to study the equations of the

form (1.4.1) involving more than two independent variables. The details of the formulation

of such results are very close to those of given above with suitable modifications. We omit

the details.

1.5 Integrodifferential equations of hyperbolic-type

In this section, first we shall deal with the following Volterra-Fredholm-type integrodiffer-

ential equation

D2D1u(x,y) = f (x,y,u(x,y),Gu(x,y),Hu(x,y)), (1.5.1)

with the given data

u(x,0) = α(x), u(0,y) = β (y), u(0,0) = 0, (1.5.2)

for x ∈ Ia, y ∈ Ib, where

Gu(x,y) =∫ x

0

∫ y

0g(x,y,s,t,u(s,t))dt ds, (1.5.3)

Hu(x,y) =∫ a

0

∫ b

0h(x,y,s,t,u(s,t))dt ds, (1.5.4)

f ∈C(E0 ×R3,R), g, h ∈C(E2

0 ×R,R), α ∈C(Ia,R), β ∈C(Ib,R). The problem of exis-

tence and uniqueness of solutions to (1.5.1)–(1.5.2) can be dealt with the method employed

in earlier sections. Here, we shall discuss the error evaluation of two approximate solutions

of (1.5.1)–(1.5.2) and convergence properties of solutions of approximate problems (see

[117]).

Let u ∈C(E0,R); D2D1u(x,y) exists and satisfies the inequality

|D2D1u(x,y)− f (x,y,u(x,y),Gu(x,y),Hu(x,y))| � ε,

for a given constant ε � 0, where it is supposed that (1.5.2) holds. Then we call u(x,y) an

ε-approximate solution of equation (1.5.1) with the given data (1.5.2).

The following theorem deals with the estimate on the difference between the two approxi-

mate solutions of equation (1.5.1).


tions

| f (x,y,u,v,w)− f (x,y,u,v,w)| � k(x,y) [|u−u|+ |v− v|+ |w−w|] , (1.5.5)


|g(x,y,σ ,τ,u)−g(x,y,σ ,τ,u)| � e(x,y)q(σ ,τ)|u−u|, (1.5.6)

|h(x,y,σ ,τ,u)−h(x,y,σ ,τ,u)| � m(x,y)r(σ ,τ)|u−u|, (1.5.7)

where k, e, q, m, r ∈ C(E0,R+). Let p(x,y) = max{k(x,y),e(x,y),m(x,y)} for (x,y) ∈ E0

and d be as in (1.2.22). Let ui(x,y) (i = 1, 2) be respectively εi-approximate solutions of

equation (1.5.1) on E0 with

ui(x,0) = αi(x), ui(0,y) = βi(y), ui(0,0) = 0, (1.5.8)

where αi ∈C(Ia,R), βi ∈C(Ib,R) such that

|α1(x)−α2(x)+β1(y)−β2(y)| � δ , (1.5.9)

where δ � 0 is a constant. Then

|u1(x,y)−u2(x,y)| �(εab+δ )

1−dexp

(∫ x

0

∫ y

0[p(s, t)+q(s,t)]dt ds

)

, (1.5.10)

for (x,y) ∈ E0, where ε = ε1 + ε2.

Proof. Since ui(x,y) (i = 1, 2) for (x,y) ∈ E0 are respectively, εi-approximate solutions

of equation (1.5.1) with (1.5.8), we have

|D2D1ui(x,y)− f (x,y,ui(x,y),Gui(x,y),Hui(x,y))| � εi. (1.5.11)

Integrating both sides of (1.5.11) on E0 and using (1.5.8), we have

εixy �∫ x

0

∫ y

0|D2D1ui(s,t)− f (s, t,ui(s,t),Gui(s,t),Hui(s,t))|dt ds

�∣∣∣∣

∫ x

0

∫ y

0{D2D1ui(s,t)− f (s,t,ui(s, t),Gui(s,t),Hui(s,t))}dt ds

∣∣∣∣

=∣∣∣∣ui(x,y)−αi(x)−βi(y)−

∫ x

0

∫ y

0f (s,t,ui(s,t),Gui(s,t),Hui(s, t))

∣∣∣∣ . (1.5.12)

From (1.5.12) and using the elementary inequalities in (1.3.25), we observe that

(ε1 + ε2)xy �∣∣∣∣u1(x,y)− [α1(x)+β1(y)]−

∫ x

0

∫ y

0f (s,t,u1(s,t),Gu1(s,t),Hu1(s,t))dt ds

∣∣∣∣

+∣∣∣∣u2(x,y)− [α2(x)+β2(y)]−

∫ x

0

∫ y


∣∣∣∣

�∣∣∣∣

{

u1(x,y)− [α1(x)+β1(y)]−∫ x

0

∫ y

0f (s,t,u1(s, t),Gu1(s, t),Hu1(s,t))dt ds

}

−{

u2(x,y)− [α2(x)+β2(y)]−∫ x

0

∫ y

0f (s,t,u2(s,t),Gu2(s,t),Hu2(s, t))dt ds

}∣∣∣∣


� |u1(x,y)−u2(x,y)|− |[α1(x)+β1(y)]− [α2(x)+β2(y)]|

−∣∣∣∣

∫ x

0

∫ y


−∫ x

0

∫ y

0f (s,t,u2(s,t),Gu2(s,t),Hu2(s, t))dt ds

∣∣∣∣ . (1.5.13)

Let z(x,y) = |u1(x,y)− u2(x,y)| for (x,y) ∈ E0. From (1.5.13) and using the hypotheses,

we observe that

z(x,y) � εxy+ |α1(x)−α2(x)+β1(y)−β2(y)|

+∫ x

0

∫ y

0| f (s, t,u1(s,t),Gu1(s,t),Hu1(s,t))

− f (s,t,u2(s, t),Gu2(s,t),Hu2(s,t))|dt ds

� εxy+δ +∫ x

0

∫ y

0k(s,t)

[

z(s, t)+ e(s,t)∫ s

0

∫ t

0q(σ ,τ)z(σ ,τ)dτ dσ

+m(s,t)∫ a

0

∫ b

0r(σ ,τ)z(σ ,τ)dτ dσ

]

dt ds

� [εab+δ ]+∫ x

0

∫ y

0p(s,t)

[

z(s,t)+∫ s

0

∫ t

0q(σ ,τ)z(σ ,τ)dτ dσ

+∫ a

0

∫ b

0r(σ ,τ)z(σ ,τ)dτ dσ

]

dt ds. (1.5.14)


Remark 1.5.1. In case if u1(x,y) is a solution of problem (1.5.1)–(1.5.2), then we have

ε1 = 0 and from (1.5.10), we see that u2(x,y) → u1(x,y) as ε2 → 0 and δ → 0. Furthermore,

if we put ε1 = ε2 = 0, α1(x) = α2(x), β1(y) = β2(y) in (1.5.10), then the uniqueness of

solutions of problem (1.5.1)–(1.5.2) is established.

Now we consider the problem (1.5.1)–(1.5.2) together with the following problem

D2D1v(x,y) = f (x,y,v(x,y),Gv(x,y),Hv(x,y)), (1.5.15)

v(x,0) = α(x), v(0,y) = β (y), v(0,0) = 0, (1.5.16)

where G, H are as defined in (1.5.3), (1.5.4) and f ∈ C(E0 ×R3,R), α ∈ C(Ia,R), β ∈

C(Ib,R).

In the next theorem we provide conditions concerning the closeness of solutions of prob-

lems (1.5.1)–(1.5.2) and (1.5.15)–(1.5.16).



tions (1.5.5)–(1.5.7) and there exist constants ε � 0, δ � 0 such that

| f (x,y,u,v,w)− f (x,y,u,v,w)| � ε, (1.5.17)

|α(x)−α(x)+β (y)−β(y)| � δ , (1.5.18)

where f , α, β and f , α , β are as in problems (1.5.1)–(1.5.2) and (1.5.15)–(1.5.16). Let

p(x,y) and d be as in Theorem 1.5.1. Let u(x,y) and v(x,y) be respectively the solutions of

problems (1.5.1)–(1.5.2) and (1.5.15)–(1.5.16) on E0. Then

|u(x,y)− v(x,y)| � (εab+δ )1−d

exp(∫ x

0

∫ y


)

, (1.5.19)

for (x,y) ∈ E0.

Proof. Let w(x,y) = |u(x,y)− v(x,y)| for (x,y) ∈ E0. Using the facts that u(x,y) and

v(x,y) are the solutions of problems (1.5.1)–(1.5.2) and (1.5.15)–(1.5.16) and hypotheses,

we observe that

w(x,y) � |α(x)−α(x)+β (y)−β(y)|

+∫ x

0

∫ y

0| f (x,y,u(s,t),Gu(s,t),Hu(s,t))− f (x,y,v(s,t),Gv(s,t),Hv(s,t))|dt ds

+∫ x

0

∫ y

0| f (x,y,v(s, t),Gv(s,t),Hv(s,t))− f (x,y,v(s,t),Gv(s,t),Hv(s,t))|dt ds

� δ +∫ x

0

∫ y

0k(s,t)

[

w(s, t)+ e(s,t)∫ s

0

∫ t

0q(σ ,τ)w(σ ,τ)dτ dσ

+m(s,t)∫ a

0

∫ b

0r(σ ,τ)w(σ ,τ)dτ dσ

]

+∫ x

0

∫ y

0εdt ds

� εab+δ +∫ x

0

∫ y

0p(s,t)

[

w(s, t)+∫ s

0

∫ t

0q(σ ,τ)w(σ ,τ)dτ dσ

+∫ a

0

∫ b

0r(σ ,τ)w(σ ,τ)dτ dσ

]

dt ds. (1.5.20)


Remark 1.5.2. The result given in Theorem 1.5.2 relates the solutions of problems

(1.5.1)–(1.5.2) and (1.5.15)–(1.5.16) in the sense that if f is close to f , α is close to α ,

β is close to β , then the solutions to problems (1.5.1)–(1.5.2) and (1.5.15)–(1.5.16) are

also close together.

Next we consider the problem (1.5.1)–(1.5.2) together with the sequence of problems

D2D1u(x,y) = fk(x,y,u(x,y),Gu(x,y),Hu(x,y)), (1.5.21)


u(x,0) = αk(x),u(0,y) = βk(y),u(0,0) = 0, (1.5.22)

where G, H are as defined in (1.5.3), (1.5.4) and fk ∈ C(E0 ×R3,R), αk ∈ C(Ia,R), βk ∈

C(Ib,R) for k = 1, 2, . . ..

The following result is an immediate consequence of Theorem 1.5.2.

Theorem 1.5.3. Suppose that the functions f , g,h in equation (1.5.1) satisfy the condi-

tions (1.5.5)–(1.5.7) and there exist constants εk � 0,δk � 0 (k = 1, 2, . . .) such that

| f (x,y,u,v,w)− fk(x,y,u,v,w)| � εk, (1.5.23)

|α(x)−αk(x)+β (y)−βk(y)| � δk, (1.5.24)

with εk → 0 and δk → 0 as k → ∞, where f , α, β and fk, αk, βk are as in problems

(1.5.1)–(1.5.2) and (1.5.21)–(1.5.22). Let p(x,y) and d be as in Theorem 1.5.1. If uk(x,y)

(k = 1, 2, . . .) and u(x,y) are respectively the solutions of problems (1.5.21)–(1.5.22) and

(1.5.1)–(1.5.2) on E0, then uk(x,y) → u(x,y) as k → ∞.

Proof. For k = 1, 2, . . ., the conditions of Theorem 1.5.2 hold. As an application of The-

orem 1.5.2 yields

|uk(x,y)−u(x,y)| � (εkab+δk)1−d

exp(∫ x

0

∫ y


)

, (1.5.25)

for (x,y) ∈ E0. The required result follows from (1.5.25).

Remark 1.5.3. We note that the result obtained in Theorem 1.5.3 provide sufficient con-

ditions that ensures solutions to problems (1.5.21)–(1.5.22) will converge to solutions to

problem (1.5.1)–(1.5.2).

Next, we shall study some fundamental properties of solutions related to the following

neutral type hyperbolic integrodifferential equation (see [27])

D2D1u(x,y) = f (x,y,u(x,y),D2D1u(x,y),Mu(x,y)), (1.5.26)

with the given data

u(x,0) = σ(x), u(0,y) = τ(y), u(0,0) = 0, (1.5.27)


Mu(x,y) =∫ x

0

∫ y

0g(x,y,m,n,u(m,n),D2D1u(m,n))dndm, (1.5.28)

f ∈ C(E ×Rn ×R

n ×Rn,Rn), g ∈ C(E2 ×R

n ×Rn,Rn), σ , τ ∈ C(R+,Rn). Obviously,

M0 =∫ x

0∫ y

0 g(x,y,m,n,0,0)dndm. By a solution of equation (1.5.26) with the given data


(1.5.27) (IBVP (1.5.26)–(1.5.27) for short), we mean a function u ∈C(E,Rn) which satisfy

the equations (1.5.26) and (1.5.27). For z, D2D1z ∈ C(E,Rn), we denote by |z(x,y)|0 =

|z(x,y)|+ |D2D1z(x,y)|. Let S0 be the space of functions z, D2D1z ∈C(E,Rn) which fulfil

the condition

|z(x,y)|0 = O(exp(λ (x + y))), (1.5.29)

for (x,y) ∈ E, where λ > 0 is a constant. As in Section 1.3, in the space S0 we define the

norm

|z|S0 = sup(x,y)∈E

[|z(x,y)|0 exp(−λ (x + y))] . (1.5.30)

It is easy to see that S0 is a Banach space and

|z|S0 � N, (1.5.31)

where N � 0 is a constant.

Now we formulate the following theorem concerning the existence of a unique solution to

IBVP (1.5.26)–(1.5.27).


(i) the functions f , g in equation (1.5.26) satisfy the conditions

| f (x,y,u,v,w)− f (x,y,u,v,w)| � k(x,y) [|u−u|+ |v− v|]+ |w−w|, (1.5.32)

|g(x,y,m,n,u,v)−g(x,y,m,n,u,v)| � h(x,y,m,n) [|u−u|+ |v− v|] , (1.5.33)

where k ∈C(E,R+), h ∈C(E2,R+),

(ii) for λ as in (1.5.29)

(c1) there exists a nonnegative constant α such that α < 1 and

L(x,y)+∫ x

0

∫ y

0L(s,t)dt ds � α exp(λ (x+ y)), (1.5.34)

for (x,y) ∈ E, where

L(x,y) = k(x,y)exp(λ (x + y))+∫ x

0

∫ y

0h(x,y,m,n)exp(λ (m+n))dndm, (1.5.35)

(c2) there exists a nonnegative constant β such that

|σ(x)|+ |τ(y)|+ | f (x,y,0,0,M0)|+∫ x

0

∫ y

0| f (s, t,0,0,M0)|dt ds � β exp(λ (x+ y)).

(1.5.36)

Then the IBVP (1.5.26)–(1.5.27) has a unique solution u(x,y) on E in S0.


Proof. Let u(x,y) ∈ S0 and define the operator T by

(Tu)(x,y) = σ(x)+ τ(y)+∫ x

0

∫ y

0f (s,t,u(s,t),D2D1u(s,t),Mu(s,t))dt ds. (1.5.37)

From (1.5.37), we observe that

D2D1(Tu)(x,y) = f (x,y,u(x,y),D2D1u(x,y),Mu(x,y)). (1.5.38)

Evidently Tu is continuous on E and Tu ∈ Rn. From (1.5.37), (1.5.38), using the hypothe-

ses and (1.5.31), we have

|(Tu)(x,y)|0 � |σ(x)|+ |τ(y)|

+∫ x

0

∫ y

0| f (s,t,u(s,t),D2D1u(s,t),Mu(s, t))− f (s, t,0,0,M0)|dt ds

+∫ x

0

∫ y

0| f (s, t,0,0,M0)|dt ds

+| f (x,y,u(x,y),D2D1u(x,y),Mu(x,y))− f (x,y,0,0,M0)|+ | f (x,y,0,0,M0)|

� β exp(λ (x+ y))+∫ x

0

∫ y

0

{

k(s,t)|u(s,t)|0 +∫ s

0

∫ t

0h(s,t,m,n)|u(m,n)|0dndm

}

dt ds

+k(x,y)|u(x,y)|0 +∫ x

0

∫ y

0h(x,y,m,n)|u(m,n)|0dndm

� β exp(λ (x + y))+ |u|S0

{

k(x,y)exp(λ (x+ y))+∫ x

0

∫ y

0h(x,y,m,n)exp(λ (m+n))dndm

+∫ x

0

∫ y

0

{

k(s,t)exp(λ (s+ t))+∫ s

0

∫ t

0h(s,t,m,n)exp(λ (m+n))dndm

}

dt ds

}

� β exp(λ (x+ y))+N{

L(x,y)+∫ x

0

∫ y

0L(s,t)dt ds

}

� [β +Nα ]exp(λ (x+ y)). (1.5.39)

From (1.5.39), it follows that Tu ∈ S0. This proves that the operator T maps S0 into itself.

Let u(x,y),v(x,y) ∈ S0. From (1.5.37), (1.5.38) and using the hypotheses, we have

|(Tu)(x,y)− (Tv)(x,y)|0

�∫ x

0

∫ y

0| f (s,t,u(s,t),D2D1u(s,t),Mu(s, t))− f (s, t,v(s,t),D2D1v(s,t),Mv(s,t))|dt ds

+| f (x,y,u(x,y),D2D1u(x,y),Mu(x,y))− f (x,y,v(x,y),D2D1v(x,y),Mv(x,y))|


�∫ x

0

∫ y

0

{

k(s,t)|u(s,t)− v(s,t)|0 +∫ s

0

∫ t

0h(s,t,m,n)|u(m,n)− v(m,n)|0dndm

}

dt ds

+k(x,y)|u(x,y)− v(x,y)|0 +∫ x

0

∫ y

0h(x,y,m,n)|u(m,n)− v(m,n)|0dndm

� |u− v|S0

{

k(x,y)exp(λ (x+ y))+∫ x

0

∫ y

0h(x,y,m,n)exp(λ (m+n))dndm

+∫ x

0

∫ y

0

{

k(s,t)exp(λ (s+ t))+∫ s

0

∫ t

0h(s,t,m,n)exp(λ (m+n))dndm

}

dt ds

}

= |u− v|S0

{

L(x,y)+∫ x

0

∫ y

0L(s,t)dt ds

}

� α|u− v|S0 exp(λ (x + y)). (1.5.40)

From (1.5.40), we have

|Tu−T v|S0 � α|u− v|S0 .

Since α < 1, it follows from Banach fixed point theorem (see [9, p. 37], [16, p. 372]) that

T has a fixed point in S0. The fixed point of T is however a solution of IBVP (1.5.26)–

(1.5.27). The proof is complete.

The following theorem contains the estimate on the solution of IBVP (1.5.26)–(1.5.27).


| f (x,y,u,v,w)| � γ [|u|+ |v|]+ |w|, (1.5.41)

|g(x,y,m,n,u,v)| � q(x,y,m,n) [|u|+ |v|] , (1.5.42)

|σ(x)|+ |τ(y)| � δ , (1.5.43)

where γ, δ are nonnegative constants such that γ < 1 and q, D1q, D2q, D2D1q∈C(E2,R+).

If u(x,y) is any solution of IBVP (1.5.26)–(1.5.27) on E, then

|u(x,y)|0 � δ1− γ

exp(∫ x

0

∫ y

0

[γ

1− γ+K(s, t)

]

dt ds)

, (1.5.44)

for (x,y) ∈ E, where K(x,y) is defined by the right hand side of (1.2.26), replacing b and e

by 11−γ q.


Proof. Using the fact that u(x,y) is a solution of IBVP (1.5.26)–(1.5.27) and the hypothe-

ses, we have

|u(x,y)|0 � |σ(x)|+ |τ(y)|

+∫ x

0

∫ y

0| f (s,t,u(s, t),D2D1u(s,t),Mu(s,t))|dt ds+ | f (x,y,u(x,y),D2D1u(x,y),Mu(x,y))|

� δ +∫ x

0

∫ y

0

{

γ|u(s,t)|0 +∫ s

0

∫ t

0q(s,t,m,n)|u(m,n)|0dndm

}

dt ds

+γ|u(x,y)|0 +∫ x

0

∫ y

0q(x,y,m,n)|u(m,n)|0dndm. (1.5.45)


|u(x,y)|0 � δ1− γ

+1

1− γ

∫ x

0

∫ y

0

{

γ‖u(s,t)|0 +q(x,y,s,t)|u(s,t)|0

+∫ s

0

∫ t

0q(s,t,m,n)|u(m,n)|0dndm

}

dsdt. (1.5.46)

Now, a suitable application of Theorem 1.2.5 to (1.5.46) yields (1.5.44).

Remark 1.5.4. We note that, if the estimate obtained in (1.5.44) is bounded, then the

solution u(x,y) of IBVP (1.5.26)–(1.5.27) is bounded on E.

The next result deals with the continuous dependence of solutions of equation (1.5.26) on

given initial boundary values.

Theorem 1.5.6. Assume that the functions f , g in equation (1.5.26) satisfy the conditions

| f (x,y,u,v,w)− f (x,y,u,v,w)| � d [|u−u|+ |v− v|]+ |w−w|, (1.5.47)

|g(x,y,m,n,u,v)−g(x,y,m,n,u,v)| � p(x,y,m,n) [|u−u|+ |v− v|] , (1.5.48)

where d is a nonnegative constant such that d < 1 and p, D1 p, D2 p, D2D1 p ∈C(E2,R+).

Let u1(x,y) and u2(x,y) be the solutions of equation (1.5.26) with the given initial boundary

conditions

u1(x,0) = σ1(x), u1(0,y) = τ1(y), u1(0,0) = 0, (1.5.49)

u2(x,0) = σ2(x), u2(0,y) = τ2(y), u2(0,0) = 0, (1.5.50)

respectively, where σ1, σ2,τ1,τ2 ∈C(R+,Rn) and

|σ1(x)+ τ1(y)−σ2(x)− τ2(y)| � μ, (1.5.51)

where μ � 0 is a constant. Then

|u1(x,y)−u2(x,y)|0 � μ1−d

exp(∫ x

0

∫ y

0

[d

1−d+K0(s,t)

]

dt ds)

, (1.5.52)

for (x,y) ∈ E, where K0(x,y) is defined by the right hand side of (1.2.26), replacing b and

e by 11−d p.


Proof. From the hypotheses, we have

|u1(x,y)−u2(x,y)|0 � |σ1(x)+ τ1(y)−σ2(x)− τ2(y)|

+∫ x

0

∫ y

0| f (s,t,u1(s,t),D2D1u1(s, t),Mu1(s, t))

− f (s,t,u2(s,t),D2D1u2(s,t),Mu2(s,t))|dt ds

+| f (x,y,u1(x,y),D2D1u1(x,y),Mu1(x,y))

− f (x,y,u2(x,y),D2D1u2(x,y),Mu2(x,y))|

� μ +∫ x

0

∫ y

0

{

d|u1(s,t)−u2(s,t)|0

+∫ s

0

∫ t

0p(s, t,m,n)|u1(m,n)−u2(m,n)|0dndm

}

dsdt

+d|u1(x,y)−u2(x,y)|0

+∫ x

0

∫ y

0p(x,y,m,n)|u1(m,n)−u2(m,n)|0dndm. (1.5.53)


|u1(x,y)−u2(x,y)|0

� μ1−d

+1

1−d

∫ x

0

∫ y

0

{

d|u1(s,t)−u2(s,t)|0 + p(x,y,s, t)|u1(s, t)−u2(s,t)|0

+∫ s

0

∫ t

0p(x,y,m,n)|u1(m,n)−u2(m,n)|0dndm

}

dt ds. (1.5.54)

Now, a suitable application of Theorem 1.2.5 to (1.5.54) yields the bound (1.5.52), which

shows the dependency of solutions of equation (1.5.26) on given initial boundary condi-

tions.

Remark 1.5.5. We note that the inequalities in Theorems 1.2.4 and 1.2.5 can be used

to study some other properties related to the solutions of equations (1.5.1)–(1.5.2) and

(1.5.26)–(1.5.27), similar to those obtained in earlier sections. Here, we do not discuss the

details.


1.6 Fredholm-type integrodifferential equation

The main objective of the present section is to give some fundamental qualitative properties

of solutions of the Fredholm-type integral equation

u(x,y) = f (x,y)+∫ a

0

∫ b

0g(x,y,s, t,u(s,t),D1u(s,t),D2u(s,t))dt ds, (1.6.1)

where f , g are given functions and u is the unknown function to be fond (see [96]).

Throughout, we assume that f ∈ C(E0,R), g ∈ C(E20 × R

3,R), and Di f ∈ C(E0,R),

Dig ∈ C(E20 ×R

3,R) for i = 1, 2. By a solution of equation (1.6.1) we mean a function

u ∈ C(E0,R) which is continuously differentiable with respect to x and y for (x,y) ∈ E0

and satisfies the equation (1.6.1). For z, D1z, D2z ∈ C(E0,R) we denote by |z(x,y)|1 =

|z(x,y)|+ |D1z(x,y)|+ |D2z(x,y)|. Let S1 be the space of functions z, D1z, D2z ∈C(E0,R)

which fulfil the condition

|z(x,y)|1 = O(exp(λ (x + y))), (1.6.2)

for (x,y) ∈ E0, where λ is a positive constant. In the space S1 we define the norm

|z|S1 = max(x,y)∈E0

[|z(x,y)|1 exp(−λ (x+ y))]. (1.6.3)

It is easy to see that S1 is a Banach space and

|z|S1 � N, (1.6.4)


In the following theorem we give conditions under which a solution of equation (1.6.1)

exists on E0 in S1.


(i) the function g in equation (1.6.1) and its derivatives Dig for i = 1, 2 satisfy the conditions

|g(x,y,s,t,u,v,w)−g(x,y,s,t,u,v,w)|

� r(x,y,s,t)[|u−u|+ |v− v|+ |w−w|], (1.6.5)

and

|Dig(x,y,s, t,u,v,w)−Dig(x,y,s,t,u,v,w)|

� ri(x,y,s,t)[|u−u|+ |v− v|+ |w−w|], (1.6.6)



(e1) there exists a nonnegative constant α such that α < 1 and∫ a

0

∫ b

0[r(x,y,s, t)+ r1(x,y,s, t)+ r2(x,y,s,t)] exp(λ(s+ t))dt ds

� α exp(λ (s+ t)), (1.6.7)

for (x,y) ∈ E0,

(e2) there exists a nonnegative constant β such that

| f (x,y)|1 +∫ a

0

∫ b

0|g(x,y,s,t,0,0,0)|1dt ds � β exp(λ(x+ y)), (1.6.8)

for (x,y) ∈ E0.

Then the equation (1.6.1) has a unique solution u(x,y) on E0 in S1.

Proof. Let u(x,y) ∈ S1 and define the operator T by

(Tu)(x,y) = f (x,y)+∫ a

0

∫ b

0g(x,y,s,t,u(s, t),D1u(s, t),D2u(s,t))dt ds. (1.6.9)

Differentiating both sides of (1.6.9) partially with respect to x and y, we have

Di(Tu)(x,y) = Di f (x,y)+∫ a

0

∫ b

0Dig(x,y,s,t,u(s,t),D1u(s,t),D2u(s, t))dt ds, (1.6.10)

for i = 1, 2. Evidently Tu,Di(Tu) for i = 1, 2 are continuous on E0 and Tu, Di(Tu) ∈ R.

From (1.6.9), (1.6.10), (1.6.4) and using the hypotheses, we have

|(Tu)(x,y)|1 � | f (x,y)|1

+∫ a

0

∫ b

0|g(x,y,s,t,u(s, t),D1u(s,t),D2u(s,t))−g(x,y,s, t,0,0,0)|dt ds

+∫ a

0

∫ b

0|g(x,y,s, t,0,0,0)|dt ds

+∫ a

0

∫ b

0|D1g(x,y,s,t,u(s, t),D1u(s,t),D2u(s,t))−D1g(x,y,s,t,0,0,0)|dt ds

+∫ a

0

∫ b

0|D1g(x,y,s,t,0,0,0)|dt ds

+∫ a

0

∫ b

0|D2g(x,y,s,t,u(s, t),D1u(s,t),D2u(s,t))−D2g(x,y,s,t,0,0,0)|dt ds

+∫ a

0

∫ b

0|D2g(x,y,s,t,0,0,0)|dt ds

� | f (x,y)|1 +∫ a

0

∫ b

0|g(x,y,s, t,0,0,0)|1dt ds+

∫ a

0

∫ b

0r(x,y,s,t)|u(s, t)|1dt ds


+∫ a

0

∫ b

0r1(x,y,s,t)|u(s,t)|1dt ds+

∫ a

0

∫ b

0r2(x,y,s, t)|u(s,t)|1dt ds

� β exp(λ (x+ y))

+|u|S1

∫ a

0

∫ b

0[r(x,y,s,t)+ r1(x,y,s,t)+ r2(x,y,s,t)]exp(λ (s+ t))dt ds

� [β +Nα ]exp(λ (x+ y)). (1.6.11)

From (1.6.11), it follows that Tu ∈ S1. This proves that the operator T maps S1 into itself.

Let u(x,y), v(x,y) ∈ S1. From (1.6.9), (1.6.10) and using the hypotheses, we have

|(Tu)(x,y)− (Tv)(x,y)|1 �∫ a

0

∫ b

0|g(x,y,s, t,u(s,t),D1u(s,t),D2u(s,t))

−g(x,y,s, t,v(s, t),D1v(s,t),D2v(s,t))|dsdt

+∫ a

0

∫ b

0|D1g(x,y,s,t,u(s,t),D1u(s,t),D2u(s,t))

−D1g(x,y,s,t,v(s,t),D1v(s,t),D2v(s,t))|dsdt

+∫ a

0

∫ b


−D2g(x,y,s,t,v(s,t),D1v(s,t),D2v(s,t))|dsdt

�∫ a

0

∫ b

0r(x,y,s,t)|u(s,t)− v(s,t)|1dt ds

+∫ a

0

∫ b

0r1(x,y,s,t)|u(s,t)− v(s,t)|1dt ds+

∫ a

0

∫ b

0r2(x,y,s, t)|u(s,t)− v(s,t)|1dt ds

� |u− v|S1

∫ a

0

∫ b

0[r(x,y,s,t)+ r1(x,y,s,t)+ r2(x,y,s,t)]exp(λ (s+ t))dt ds

� |u− v|S1 α exp(λ (x+ y)). (1.6.12)

From (1.6.12), we obtain

|Tu−T v|S1 � α|u− v|S1 .

Since α < 1, it follows from Banach fixed point theorem (see [51, p. 372]) that T has a

unique fixed point in S1. The fixed point of T is however a solution of equation (1.6.1).

Using the inequality in Theorem 1.2.3, we present the following result which deals with

the uniqueness of solutions of equation (1.6.1) on E0 in R.


Theorem 1.6.2. Suppose that the function g in equation (1.6.1) and its derivatives

Dig for i = 1, 2 satisfy the conditions (1.6.5) and (1.6.6) with r(x, y,s,t) = c(x,y)h(s,t),

ri(x,y,s, t) = c(x,y)hi(s,t) for i = 1, 2, where c, h, hi ∈C(E0,R+) and

d1 =∫ a

0

∫ b

0[h(s,t)+h1(s,t)+h2(s,t)]c(s, t)dt ds < 1. (1.6.13)

Then the equation (1.6.1) has at most one solution on E0 in R.

Proof. Let u(x,y) and v(x,y) be two solutions of equation (1.6.1). Then from the hy-

potheses, we have

|u(x,y)− v(x,y)|1 �∫ a

0

∫ b

0|g(x,y,s,t,u(s, t),D1u(s, t),D2u(s,t))

−g(x,y,s, t,v(s, t),D1v(s,t),D2v(s,t))|dt ds

+∫ a

0

∫ b


−D1g(x,y,s,t,v(s,t),D1v(s,t),D2v(s,t))|dt ds

+∫ a

0

∫ b


−D2g(x,y,s,t,v(s,t),D1v(s,t),D2v(s,t))|dt ds

�∫ a

0

∫ b

0c(x,y)h(s,t)|u(s, t)− v(s,t)|1dt ds

+∫ a

0

∫ b

0c(x,y)h1(s, t)|u(s,t)− v(s,t)|1dt ds+

∫ a

0

∫ b

0c(x,y)h2(s,t)|u(s,t)− v(s, t)|1dt ds

= c(x,y)∫ a

0

∫ b

0[h(s,t)+h1(s,t)+h2(s,t)]|u(s,t)− v(s,t)|1dt ds. (1.6.14)

Now, an application of Theorem 1.2.3 (when p(x,y) = 0) to (1.6.14) yields

|u(x,y)− v(x,y)|1 � 0, and hence u(x,y) = v(x,y), which proves the uniqueness of solu-

tions of equation (1.6.1) on E0 in R.


Theorem 1.6.3. Suppose that the function g in equation (1.6.1) and its derivatives Dig for

i = 1, 2 satisfy the conditions

|g(x,y,s,t,u,v,w)| � k(x,y)e(s,t)[|u|+ |v|+ |w|], (1.6.15)

|Dig(x,y,s,t,u,v,w)| � k(x,y)ei(s,t)[|u|+ |v|+ |w|], (1.6.16)

where k, e, ei ∈C(E0,R) and

d2 =∫ a

0

∫ b

0[e(s,t)+ e1(s,t)+ e2(s, t)]k(s,t)dt ds < 1. (1.6.17)

Then for every solution u ∈C(E0,R) of equation (1.6.1), the estimate

|u(x,y)|1 � | f (x,y)|1 + k(x,y)

×{

11−d2

∫ a

0

∫ b

0[e(s,t)+ e1(s,t)+e2(s,t)]| f (s,t)|1dt ds

}

, (1.6.18)

holds for (x,y) ∈ E0.


Proof. Let u ∈C(E0,R) be a solution of equation (1.6.1). Then from the hypotheses, we

have

|u(x,y)|1 � | f (x,y)|+∫ a

0

∫ b

0|g(x,y,s,t,u(s,t),D1u(s,t),D2u(s,t))|dt ds

+|D1 f (x,y)|+∫ a

0

∫ b

0|D1g(x,y,s,t,u(s,t),D1u(s, t),D2u(s, t))|dt ds

+|D2 f (x,y)|+∫ a

0

∫ b

0|D2g(x,y,s,t,u(s,t),D1u(s, t),D2u(s, t))|dt ds

� | f (x,y)|1 +∫ a

0

∫ b

0k(x,y)e(s,t)|u(s, t)|1dt ds

+∫ a

0

∫ b

0k(x,y)e1(s,t)|u(s,t)|1dt ds+

∫ a

0

∫ b

0k(x,y)e2(s,t)|u(s,t)|1dt ds

= | f (x,y)|1 + k(x,y)∫ a

0

∫ b

0[e(s,t)+ e1(s,t)+ e2(s,t)]|u(s, t)|1dt ds. (1.6.19)

Now, an application of Theorem 1.2.3 to (1.6.19) gives the desired estimate in (1.6.18).

Remark 1.6.1. We note that the estimate obtained in (1.6.18) yields not only the bound

on the solution u(x,y) of equation (1.6.1), but also the bound on their derivatives Diu(x,y)

for i = 1, 2.

Next, we shall obtain the estimate on the solution of equation (1.6.1) assuming that the

function g and its derivatives Dig for i = 1, 2 satisfy Lipschitz type conditions.

Theorem 1.6.4. Suppose that the function g in equation (1.6.1) and its derivatives Dig

for i = 1, 2 satisfy the conditions in Theorem 1.6.2 and the condition (1.6.13) holds. If

u ∈C(E0,R) is any solution of equation (1.6.1) on E0, then

|u(x,y)− f (x,y)|1 � Q(x,y)+ c(x,y)

×{

11−d1

∫ a

0

∫ b

0[h(s,t)+h1(s,t)+h2(s,t)]Q(s,t)dt ds

}

, (1.6.20)

for (x,y) ∈ E0, where

Q(x,y) =∫ a

0

∫ b

0|g(x,y,σ ,τ, f (σ ,τ),D1 f (σ ,τ),D2 f (σ ,τ))|1dτ dσ , (1.6.21)

for (x,y) ∈ E0.


Proof. Since u(x,y) is a solution of equation (1.6.1), by using the hypotheses, we have

|u(x,y)− f (x,y)|1 �∫ a

0

∫ b

0|g(x,y,s,t,u(s,t),D1u(s,t),D2u(s,t))

−g(x,y,s,t, f (s,t),D1 f (s,t),D2 f (s,t))|dt ds

+∫ a

0

∫ b

0|g(x,y,s,t, f (s,t),D1 f (s,t),D2 f (s,t))|dt ds

+∫ a

0

∫ b


−D1g(x,y,s,t, f (s,t),D1 f (s,t),D2 f (s, t))|dt ds

+∫ a

0

∫ b

0|D1g(x,y,s, t, f (s, t),D1 f (s,t),D2 f (s,t))|dt ds

+∫ a

0

∫ b


−D2g(x,y,s,t, f (s,t),D1 f (s,t),D2 f (s, t))|dt ds

+∫ a

0

∫ b

0|D2g(x,y,s, t, f (s, t),D1 f (s,t),D2 f (s,t))|dt ds

� Q(x,y)+∫ a

0

∫ b

0c(x,y)h(s,t)|u(s,t)− f (s,t)|1dt ds

+∫ a

0

∫ b

0c(x,y)h1(s,t)|u(s,t)− f (s,t)|1dt ds

+∫ a

0

∫ b

0c(x,y)h2(s,t)|u(s,t)− f (s,t)|1dt ds

= Q(x,y)+ c(x,y)∫ a

0

∫ b

0[h(s,t)+h1(s,t)+h2(s, t)]|u(s, t)− f (s,t)|1dt ds. (1.6.22)


We next consider the equation (1.6.1) and the following Fredholm-type integral equation

z(x,y) = F(x,y)+∫ a

0

∫ b

0G(x,y,s,t,z(s,t),D1z(s,t),D2z(s, t))dt ds, (1.6.23)

for (x,y)∈ E0, where F ∈C(E0,R), G∈C(E20 ×R

3,R) and DiF ∈C(E0,R), DiG∈C(E20 ×

R3,R) for i = 1, 2.

The following theorem holds.


Theorem 1.6.5. Suppose that the function g in equation (1.6.1) and its derivatives Dig for

i = 1, 2 satisfy the conditions as in Theorem 1.6.2 and the condition (1.6.13) holds. Then

for every given solution z ∈ C(E0,R) of equation (1.6.23) and any solution u ∈ C(E0,R)

of equation (1.6.1), the estimate

|u(x,y)− z(x,y)|1 � [| f (x,y)−F(x,y)|1 +M(x,y)]+c(x,y)

×{

11−d1

∫ a

0

∫ b

0[h(s,t)+h1(s,t)+h2(s,t)][| f (s,t)−F(s,t)|1 +M(s,t)]dt ds

}

, (1.6.24)

holds for (x,y) ∈ E0, where

M(x,y) =∫ a

0

∫ b

0|g(x,y,σ ,τ,z(σ ,τ),D1z(σ ,τ),D2z(σ ,τ)).

−G(x,y,σ ,τ,z(σ ,τ),D1z(σ ,τ),D2z(σ ,τ))|1dτ dσ , (1.6.25)

for (x,y) ∈ E0.

Proof. From the hypotheses, we have

|u(x,y)− z(x,y)|1 � | f (x,y)−F(x,y)|1

+∫ a

0

∫ b

0|g(x,y,s,t,u(s, t),D1u(s, t),D2u(s, t))−g(x,y,s,t,z(s,t),D1z(s,t),D2z(s, t))|dt ds,

+∫ a

0

∫ b

0|g(x,y,s,t,z(s,t),D1z(s,t),D2z(s, t))−G(x,y,s,t,z(s, t),D1z(s,t),D2z(s,t))|dt ds

+∫ a

0

∫ b

0|D1g(x,y,s,t,u(s, t),D1u(s, t),D2u(s,t))

−D1g(x,y,s,t,z(s,t),D1z(s, t),D2z(s, t))|dt ds

+∫ a

0

∫ b

0|D1g(x,y,s,t,z(s, t),D1z(s,t),D2z(s,t))

−D1G(x,y,s,t,z(s,t),D1z(s,t),D2z(s, t))|dt ds

+∫ a

0

∫ b

0|D2g(x,y,s,t,u(s, t),D1u(s, t),D2u(s,t))

−D2g(x,y,s,t,z(s,t),D1z(s, t),D2z(s, t))|dt ds

+∫ a

0

∫ b

0|D2g(x,y,s,t,z(s, t),D1z(s,t),D2z(s,t))

−D2G(x,y,s,t,z(s,t),D1z(s,t),D2z(s, t))|dt ds

� [| f (x,y)−F(x,y)|1 +M(x,y)]+ c(x,y)

×∫ a

0

∫ b

0[h(s,t)+h1(s, t)+h2(s,t)]|u(x,y)− z(x,y)|1dt ds. (1.6.26)



Remark 1.6.2. We note that, one can formulate results on the continuous dependence of

solutions of equation (1.6.1) and its variants by closely looking at the corresponding results

given in Theorems 1.4.6 and 1.4.7. Furthermore, the idea used in this section can be very

easily extended to study the version of equation (1.6.1) involving functions of more than

two variables. Moreover, the results established in Theorems 1.6.1-1.6.5 can be extended

for equations of the form (1.6.1) when the function g is of the form

g(

x,y,s,t,u(s,t),∂ nu(s,t)

∂ sn ,∂ mu(s, t)

∂ sm

)

,

or

g(

x,y,s,t,u(s, t),∂ nu(s,t)

∂ sn ,∂ mu(s,t)

∂ sm ,∂ n+mu(s,t)

∂ sn∂ tm

)

,

under some suitable conditions. Naturally, these considerations will make the analysis

more complicated and we leave it to the reader to fill in where needed.

The generality of the equation (1.6.1) allow us to obtain results similar to the ones given

above, concerning the following equation

u(x,y) = f (x,y)+∫ a

0

∫ b

0K(x,y,s,t)h(s, t,u(s,t),D1u(s,t),D2u(s,t))dt ds, (1.6.27)

where f ∈ C(E0,R), K ∈ C(E20 ,R), h ∈ C(E0 ×R

3,R) and assume that Di f ∈ C(E0,R),

DiK ∈C(E20 ,R), for i = 1, 2. Below we present a result on the existence of a unique solu-

tion of equation (1.6.27) by using the Banach fixed point theorem coupled with maximum

norm. One can formulate other results given above in Theorems 1.6.2–1.6.5 for the equa-

tion (1.6.27).


(i) f , Di f ∈C(E0,R) for i = 1, 2,

(ii) h ∈C(E0 ×R3,R), h(s, t,0,0,0) = 0 and there is a constant L > 0 such that

|h(s,t,u,v,w)−h(s,t,u,v,w)| � L [|u−u|+ |v− v|+ |w−w|] , (1.6.28)

(iii) K, DiK ∈C(E20 ,R) for i = 1, 2 and

L∫ a

0

∫ b

0|K(x,y,s, t)|1dt ds � α < 1. (1.6.29)

Then the equation (1.6.27) has a unique solution u ∈C(E0,R).


Proof. Let B be a Banach space of bounded functions u ∈ C(E0,R), which are con-

tinuously differentiable with respect to x and y on E0 with maximum norm ‖ · ‖, where

‖u‖ = max(x,y)∈E0 |u(x,y)|1. Let u ∈ B and define the operator F by

(Fu)(x,y) = f (x,y)+∫ a

0

∫ b

0K(x,y,s,t)h(s,t,u(s,t),D1u(s,t),D2u(s,t))dt ds. (1.6.30)

Differentiating both sides of (1.6.30) partially with respect to x and y, we have

Di(Fu))(x,y) = Di f (x,y)

+∫ a

0

∫ b

0DiK(x,y,s,t)h(s,t,u(s, t),D1u(s,t),D2u(s,t))dt ds, (1.6.31)

for i = 1, 2. From the hypotheses, it follows that Fu, Di(Fu) (i = 1, 2) are continuous on

E0 and

|(Fu))(x,y)|1 � | f (x,y)|1 +∫ a

0

∫ b

0|K(x,y,s,t)|1

×|h(s,t,u(s,t),D1u(s, t),D2u(s, t))−h(s,t,0,0,0)|dt ds

� | f (x,y)|1 +L∫ a

0

∫ b

0|K(x,y,s, t)|1|u(s,t)|1dt ds

� | f (x,y)|1 +L‖u‖∫ a

0

∫ b

0|K(x,y,s,t)|1dt ds < ∞. (1.6.32)

Here, we have used the fact that | f (x,y)|1 is bounded, since f , Di f ∈ C(E0,R) and the

condition (1.6.29). This proves that the operator F maps B into itself.

Let u, v ∈ B. From (1.6.30), (1.6.31) and the hypotheses, we have

|(Fu)(x,y)− (Fv)(x,y)|1

�∫ a

0

∫ b

0|K(x,y,s,t)|1|h(s,t,u(s, t),D1u(s,t),D2u(s,t))

−h(s,t,v(s,t),D1v(s, t),D2v(s,t))|dt ds

� L∫ a

0

∫ b

0|K(x,y,s,t)|1|u(s,t)− v(s, t)|1dt ds

� L‖u− v‖∫ a

0

∫ b

0|K(x,y,s,t)|1dt ds

� α‖u− v‖. (1.6.33)

From (1.6.33), we have ‖Fu−Fv‖ � α‖u− v‖. Since α < 1, it follows that F is a con-

traction mapping, which proves that the equation (1.6.27) has a unique solution u in B

on E0.


Remark 1.6.3. We note that the equation (1.6.27) includes as a special case, the study of

the following important integral equation

u(x,y) = f (x,y)+∫ a

0

∫ b

0K(x,y,s, t)h(s,t,u(s,t))dt ds, (1.6.34)

which may be considered as a two independent variable generalization of the well-known

Hammerstein type integral equation studied by many authors in the literature.

1.7 Miscellanea

1.7.1 Hacia [42]

(e1) Let f and k be continuous functions in E and E2 respectively. If k is nonnegative, and

the continuous function u defined on E satisfies the inequality

u(x,y) � f (x,y)+∫ x

0

∫ y

0k(x,y,s,t)u(s,t)dsdt,

for (x,y) ∈ E, then

u(x,y) � f (x,y)+∫ x

0

∫ y

0r(x,y,s,t) f (s,t)dsdt,


r(x,y,s,t) =∞

∑n=1

kn(x,y,s,t),

is the resolvent kernel defined by formulas

k1(x,y,s, t) = k(x,y,s,t),

kn(x,y,s,t) =∫ x

0

∫ y

0k(x,y,ξ ,η)kn−1(ξ ,η,s,t)dξ dη,

for n = 2, 3, . . ..

Moreover, if f is nondecreasing with respect to every variable, then

u(x,y) � f (x,y)[

1+∫ x

0

∫ y

0r(x,y,s,t)dsdt

]

,

for (x,y) ∈ E.

(e2) Let a, b and f be nonnegative continuous functions in E . If the continuous function u

defined on E satisfies the inequality

u(x,y) � f (x,y)+a(x,y)∫ x

0

∫ y

0b(s, t)u(s,t)dsdt,

for (x,y) ∈ E, then

u(x,y) � f (x,y)+a(x,y)∫ x

0

∫ y

0b(s, t)exp

[∫ x

s

∫ y

ta(ξ ,η)b(ξ ,η)dξ dη

]

f (s,t)dsdt,

for (x,y) ∈ E.

Moreover, if f is nondecreasing with respect to every variable, then

u(x,y) � f (x,y)[

1+a(x,y)∫ x

0

∫ y

0b(s,t)exp

[∫ x

s

∫ y

ta(ξ ,η)b(ξ ,η)dξ dη

]

dsdt]

,

for (x,y) ∈ E.


1.7.2 Hacia [42]

Consider the system of integral equations

ui(x,y) = wi(x,y)+n

∑j=1

∫ x

0

∫ y

0ki j(x,y,s,t)ui(s,t)dsdt, (1.7.1)

for i = 1, 2, . . . ,n, where functions wi and ki j are continuous in E and E2 respectively.

(e3) If

n

∑i=1

max{|ki j(x,y,s,t)| : 1 � j � n

}� a(s,t),

where a ∈C(E,R+), then the estimate

n

∑i=1

|ui(x,y)| � sup

{n

∑i=1

|wi(s,t)| : 0 � s � x, 0 � t � y

}

exp[∫ x

0

∫ y

0a(s, t)dsdt

]

,

holds for (x,y) ∈ E . Moreover, if a ∈ L(R2+) and

sup

{n

∑i=1

|wi(s,t)| : 0 � s � x < ∞, 0 � t � y < ∞

}

< ∞,

then the solution {ui(x,y)}, i = 1, 2, . . . ,n, of (1.7.1) is bounded in E.

(e4) If

n

∑i=1

max{|ki j(x,y,s, t)| : 1 � j � n

}= b(x,y)a(s,t),

where a, b ∈C(E,R+) (b �= 0), then the estimate

n

∑i=1

|ui(x,y)| � b(x,y)sup

{n

∑i=1

|wi(s,t)|b(s,t)

: 0 � s � x < ∞, 0 � t � y < ∞

}

×exp[∫ x

0

∫ y

0a(s, t)b(s,t)dsdt

]

,

holds for (x,y) ∈ E . Moreover, if ab ∈ L(R2+) and

sup

{n

∑i=1

|wi(s,t)|b(s,t)

: 0 � s � x < ∞, 0 � t � y < ∞

}

< ∞,

then the solution {ui(x,y)} (i = 1, 2, . . . ,n), of (1.7.1) is bounded in E.


1.7.3 Pachpatte [76]

Consider the integral equation

z(x,y) = F(x,y,(T1z)(x,y),(T2z)(x,y),(T3z)(x,y),z(x,y)), (1.7.2)

for x ∈ Iα , y ∈ Iβ , where

(T1z)(x,y) =∫ x

0

∫ y

0f1(x,y,s,t,z(s,t))dsdt, (1.7.3)

(T2z)(x,y) =∫ x

0f2(x,y,s,z(s,y))ds, (1.7.4)

(T3z)(x,y) =∫ y

0f3(x,y,t,z(x,t))dt, (1.7.5)

f1, f2, f3, F are given functions and z is unknown function. Let (B,‖ · ‖) be a Banach

space, Δ = Iα × Iβ , Δ1 = {(x,y,s,t) : 0 � s � x � α , 0 � t � y � β}, Δ2 = {(x,y,s) :

0 � s � x � α , 0 � y � β}, Δ3 = {(x,y,t) : 0 � x � α , 0 � t � y � β}, fi ∈C(Δi ×B,B)

(i = 1, 2, 3), F ∈C(Δ×B4,B).

(H1) Suppose that

(i) there exist functions wi ∈C(Δi ×R+,R+) (i = 1, 2, 3), W ∈C(Δ×R4+,R+) such that

‖ f1(x,y,s,t,z)− f1(x,y,s,t,z)‖ � w1(x,y,s, t,‖z− z‖),

‖ f2(x,y,s,z)− f2(x,y,s,z)‖ � w2(x,y,s,‖z− z‖),

‖ f3(x,y, t,z)− f3(x,y,t,z)‖ � w3(x,y,t,‖z− z‖),

and

‖F(x,y,u1,u2,u3,u4)−F(x,y,u1,u2,u3,u4)‖

� W (x,y,‖u1 −u1‖,‖u2 −u2‖,‖u3 −u3‖,‖u4 −u4‖),

(ii) if u ∈C(Δ,R+) and

v(x,y) = W (x,y,(M1u)(x,y),(M2u)(x,y),(M3u)(x,y),u(x,y)),

for (x,y)∈Δ, where for i = 1, 2, 3, Miu are defined by the right hand sides in (1.7.3)–(1.7.5)

by replacing fi by wi and z by u, then v ∈C(Δ,R+),

(iii) If u, v ∈C(Δ,R+) and u � v, then

W (x,y,M1u,M2u,M3u,u) � W (x,y,M1v,M2v,M3v,v),

(iv) if un ∈C(Δ,R+), un+1 � un, n = 0, 1, 2, . . ., un → u, then

W (x,y,M1un,M2un,M3un,un) →W (x,y,M1u,M2u,M3u,u).


(H2) Suppose that

(i) there exists a solution u ∈C(Δ,R+) of the inequality

W (x,y,M1u,M2u,M3u,u)+h(x,y) � u,

where

h(x,y) = sup0�ξ�x0�η�y

‖F(ξ ,η ,(T10)(ξ ,η),(T20)(ξ ,η),(T30)(ξ ,η),0)‖,

(ii) in the class of functions satisfying the condition 0 � u(x,y) � u(x,y), the function

u(x,y) ≡ 0 is the only solution of the inequality

u(x,y) � W (x,y,(M1u)(x,y),(M2u)(x,y),(M3u)(x,y),u(x,y)).

Define the sequence {zn} by

z0(x,y) ≡ 0,

zn+1(x,y) = F(x,y,(T1zn)(x,y),(T2zn)(x,y),(T3zn)(x,y),zn(x,y)), (1.7.6)

for n = 0, 1, 2, . . .. Also define the sequence {un} by

u0(x,y) = u(x,y),

un+1(x,y) = W (x,y,(M1un)(x,y),(M2un)(x,y),(M3un)(x,y),un(x,y)), (1.7.7)

for n = 0, 1, 2, . . ., where u(x,y) is as in hypotheses (H2).

(e5) If hypothesis (H2) and the conditions (ii), (iii), (iv) of hypothesis (H1) are satisfied,

then

0 � un+1(x,y) � un(x,y) � u(x,y),

for n = 0, 1, 2, . . . and

limn→∞

un(x,y) = 0,

where the convergence is uniform in each bounded set.

(e6) If hypotheses (H1) and (H2) are satisfied, then there exists a continuous solution z of

equation (1.7.2) on Δ. The sequence {zn} defined by (1.7.6) converges uniformly on Δ to

z, and the following estimates

‖z(x,y)− zn(x,y)‖ � un(x,y), (1.7.8)

for (x,y) ∈ Δ, n = 0, 1, 2, . . ., and

‖z(x,y)‖ � u(x,y), (1.7.9)

for (x,y)∈ Δ hold true. The solution z of equation (1.7.2) is unique in the class of functions

satisfying the condition (1.7.9).

(e7) If hypothesis (H1) is satisfied and the function r(x,y) ≡ 0, (x,y) ∈ Δ is the only non-

negative continuous solution of the inequality

r(x,y) � W (x,y,(M1r)(x,y),(M2r)(x,y),(M3r)(x,y),r(x,y)),

for (x,y) ∈ Δ, then the equation (1.7.2) has at most one solution on Δ.



Consider the nonlinear functional integral equation

z(x,y) = F(x,y,(L1z)(x,y),(L2z)(x,y),(L3z)(x,y),z(g(x,y),h(x,y)),μ), (1.7.10)

for x ∈ Iα , y ∈ Iβ , where

(L1z)(x,y) =∫ a(x,y)

0

∫ b(x,y)

0f1(x,y,s, t,z(s,t))dt ds,

(L2z)(x,y) =∫ c(x,y)

0f2(x,y,s,z(s, p(x,y)))ds,

(L3z)(x,y) =∫ d(x,y)

0f3(x,y,t,z(q(x,y), t))dt,

f1, f2, f3, F, a, b, c, d, p, q, g, h are given functions, μ is a real parameter and z is the un-

known function. Let (B,‖ ·‖) be a Banach space and Δ, Δi (i = 1, 2, 3) be as defined below

the equation (1.7.2). The functions a, b, c, d, p, q, g, h are defined and continuous on Δ and

a, c, q, g ∈ Iα ; b, d, p, h ∈ Iβ , fi ∈C(Δi ×B,B) (i = 1, 2, 3), F ∈C(Δ×B4 ×R,B).

(H3) Suppose that

(i) there exist nonnegative constants Ki, Mi (i = 1, 2, 3) and M such that

‖ f1(x,y,s,t,z)− f1(x,y,s,t,z)‖ � M1‖z− z‖,

‖ f2(x,y,s,z)− f2(x,y,s,z)‖ � M2‖z− z‖,

‖ f3(x,y, t,z)− f3(x,y,t,z)‖ � M3‖z− z‖,

and∥∥F(x,y,z1,z2,z3,z4,μ)−F(x,y,z1,z2,z3,z4,μ)

∥∥

� K1‖z1 − z1‖+K2‖z2 − z2‖+K3‖z3 − z3‖+M‖z4 − z4‖,

(ii) for every fixed μ ∈ R, there exist constants λ > 0, Q � 0 such that∥∥F(x,y,(L10)(x,y),(L20)(x,y),(L30)(x,y),0,μ)

∥∥ � Qexp(λ (x + y)),

(iii) there exist constants N1, N2 such that 0 � N1 +N2 < 1 and

K1M1

λ 2 exp(λ [a(x,y)+b(x,y)])+K2M2

λexp(λ [c(x,y)+ p(x,y)])

+K3M3

λexp(λ [q(x,y)+d(x,y)]) � N1 exp(λ (x + y)),

M exp (λ [g(x,y)+h(x,y)]) � N2 exp(λ (x+ y)).


(H4) suppose that there exists a constant K � 0 and a function P ∈C(Δ,R+) such that

‖F(x,y,z1,z2,z3,z4,μ1)−F(x,y,z1,z2,z3,z4,μ2)‖ � P(x,y)|μ1 −μ2|,

and

max(x,y)∈Δ

[P(x,y)exp(−λ (x + y))] � K,

where λ > 0 is a constant.

(e8) Let S be the space defined as in Section 1.3. If hypothesis (H3) is satisfied, then for

every μ ∈ R, there exists exactly one solution z ∈ S of equation (1.7.10) on Δ.

(e9) Let S be the space defined as in Section 1.3. If hypotheses (H3) and (H4) are satisfied,

then the solution z(x,y,μ) of equation (1.7.10) belonging to S is continuous with respect to

the variables (x,y,μ) on Δ×R.


Consider the integrodifferential equation of the form

D2D1u(x,y) = f (x,y,u(x,y),(Ku)(x,y)), (1.7.11)

with the given initial boundary conditions

u(x,0) = σ(x), u(0,y) = τ(y), u(0,0) = 0, (1.7.12)


(Ku)(x,y) =∫ x

0

∫ y

0k(x,y,m,n,u(m,n))dndm,

k ∈C(E2 ×Rn,Rn), f ∈C(E ×R

n ×Rn,Rn), σ , τ ∈C(R+,Rn).

(e10) Suppose that the functions f , k in (1.7.11) satisfy the conditions

| f (x,y,u,v)− f (x,y,u,v)| � p(x,y) [|u−u|+ |v− v|] , (1.7.13)

|k(x,y,m,n,u)− k(x,y,m,n,u)| � r(x,y,m,n)|u−u|, (1.7.14)

where p ∈C(E,R+) and r, D1r, D2r, D2D1r ∈C(E2,R+). Let

c = sup(x,y)∈E

∣∣∣∣σ(x)+τ(y)+

∫ x

0

∫ y

0f (s, t,0,(K0)(s,t))dt ds

∣∣∣∣ < ∞,

where σ , τ are as in (1.7.12). If u(x,y) is any solution of problem (1.7.11)–(1.7.12) on E,

then

|u(x,y)| � c[

1+∫ x

0

∫ y

0p(s,t)exp

(∫ s

0

∫ t

0[p(m,n)+A(m,n)]dndm

)

dt ds]

,



A(x,y) = r(x,y,x,y)+∫ x

0D1r(x,y,ξ ,y)dξ

+∫ y

0D2r(x,y,x,η)dη +

∫ x

0

∫ y

0D2D1r(x,y,ξ ,η)dηdξ . (1.7.15)

(e11) Suppose that the functions f , k in (1.7.11) satisfy the conditions (1.7.13), (1.7.14)

respectively. Let ui(x,y) (i = 1, 2) be respectively εi-approximate solutions of (1.7.11) on

E, i.e.,

|D2D1ui(x,y)− f (x,y,ui(x,y),(Kui)(x,y))| � εi,

on E with

ui(x,0) = αi(x), ui(0,y) = βi(y), ui(0,0) = 0,

where αi, βi ∈C(R+,Rn) and assume that

|α1(x)−α2(x)+β1(y)−β1(y)| � δ ,

where δ � 0 is a constant. Then

|u1(x,y)−u2(x,y)| � e(x,y)[

1+∫ x

0

∫ y

0p(s, t)

×exp(∫ s

0

∫ t

0[p(m,n)+A(m,n)]dndm

)

dt ds]

,


e(x,y) = (ε1 + ε2)xy+δ ,

and A(x,y) is given by (1.7.15).

1.7.6 Brzychczy and Janus [20]

Consider the nonlinear integrodifferential equation of the form

zxy(x,y) = f (x,y,z(x,y),zx(x,y),zy(x,y),(Mz)(x,y)), (1.7.16)


z(x,0) = σ(x), z(0,y) = τ(y), σ(0) = τ(0) = c0, (1.7.17)

for x ∈ Ia, y ∈ Ib, where

(Mz)(x,y) =∫ y

0m(y− s)g(z(x,s))ds, (1.7.18)


f ∈ C(E0 ×R4,R), g ∈ C(R,R), m ∈ L2(Ib,R), σ ∈ C1(Ia,R), τ ∈ C1(Ib,R). Denote by

C1,∗(E0,R) the space of functions u∈C(E0,R) such that ux, uy, uxy exist and are continuous

on E0 with the norm

‖u‖ := max{

maxE0

|u(x,y)|,maxE0

|ux(x,y)|,maxE0

|uy(x,y)|}

.

The notation (u,ux,uy) � (v,vx,vy) (respectively (u,ux,uy) < (v,vx,vy)) means that

u(x,y) � v(x,y), ux(x,y) � vx(x,y), uy(x,y) � vy(x,y) (respectively u(x,y) < v(x,y),

ux(x,y) < vx(x,y), uy(x,y) < vy(x,y) for each (x,y) ∈ E0) for each (x,y) ∈ E0. A func-

tion u ∈C1,∗(E0,R) is called a lower function of problem (1.7.16)–(1.7.17) on E0 if

uxy(x,y) � f (x,y,u(x,y),ux(x,y),uy(x,y),(Mu)(x,y)),

ux(x,0) � σ ′(x),

uy(0,y) � τ ′(y),

u(0,0) � c0,

and an upper function of problem (1.7.16)–(1.7.17) on E0 if the reversed inequalities hold.

(e12) Assume that

(i) u, v ∈C1,∗(E0,R) and

uxy(x,y) < f (x,y,u(x,y),ux(x,y),uy(x,y),(Mu)(x,y)),

vxy(x,y) > f (x,y,v(x,y),vx(x,y),vy(x,y),(Mv)(x,y)),

for (x,y) ∈ E0,

ux(x,0) < vx(x,0), x ∈ Ia,

uy(0,y) < vy(0,y), y ∈ Ib,

u(0,0) < v(0,0);

(ii) f (x,y, p1, p2, p3, p4) is nondecreasing with respect to p1, p2, p3, p4 for all (x,y) ∈ E0;

(iii) g is nondecreasing;

(iv) m(ξ ) � 0 for ξ ∈ Ib.

Then (u,ux,uy) < (v,vx,vy) on E0.

(e13) Assume that

(i) u, v ∈C1,∗(E0,R) and

uxy(x,y) � f (x,y,u(x,y),ux(x,y),uy(x,y),(Mu)(x,y)),

vxy(x,y) � f (x,y,v(x,y),vx(x,y),vy(x,y),(Mv)(x,y)),


for (x,y) ∈ E0,

ux(x,0) � vx(x,0), x ∈ Ia,

uy(0,y) � vy(0,y), y ∈ Ib,

u(0,0) � v(0,0);

(ii) f (x,y, p1, p2, p3, p4) is nondecreasing with respect to p1, p2, p3, p4 for all (x,y)∈ E0;

(iii) f (x,y, p1, p2, p3, p4)− f (x,y, p1, p2, p3, p4) � L∑4i=1(pi − pi), whenever pi, pi ∈ R,

pi � pi (i = 1, 2, 3, 4) for (x,y) ∈ E0, where L > 0 is a constant;

(iv) 0 � g(ξ )−g(ξ ) � K(ξ −ξ ), whenever ξ � ξ for constant K > 0;

(v) m(ξ ) � 0 for ξ ∈ Ib.

Then (u,ux,uy) � (v,vx,vy) on E0.

(e14) let u, v be respectively, the lower and upper functions for problem (1.7.16)–(1.7.17)

and f , g, m as in (e13). Then for any solution z of problem (1.7.16)–(1.7.17), the inequalities

(u,ux,uy) � (z,zx,zy) � (v,vx,vy)

hold on E0.

1.7.7 Brzychczy and Janus [20]

Define the operator G : C1,∗(E0,R)→C1,∗(E0,R) as, for u ∈C1,∗(E0,R), let v = Gu be the

unique solution of the following linear problem (called (LP))

vxy(x,y) = F[u](x,y), (x,y) ∈ E0,

v(x,0) = σ(x), x ∈ Ia,

v(0,y) = τ(y), y ∈ Ib,

σ(0) = τ(0) = c0,

where

F [u](x,y) = f (x,y,u(x,y),ux(x,y),uy(x,y),(Mu)(x,y)),

the operator M is defined by (1.7.18). It is easy to see that the solution of problem (LP) is

given by the formula

v(x,y) =∫ x

0

∫ y

0F [u](ξ ,t)dξ dt +σ(x)+ τ(y)− c0.

(e15) Assume that

(i) there exist u0,v0, the lower and upper functions for problem (1.7.16)–(1.7.17) on E0

respectively;


(ii) f (x,y, p1, p2, p3, p4) is nondecreasing with respect to p1, p2, p3, p4 for all (x,y)∈ E0;

(iii) f (x,y, p1, p2, p3, p4) satisfies the Lipschitz condition

| f (x,y, p1, p2, p3, p4)− f (x,y, p1, p2, p3, p4)| � L4

∑i=1

|pi − pi|,

for (x,y) ∈ E0, pi, pi ∈ R, where L > 0 is a constant;

(iv) 0 � g(ξ )−g(ξ ) � K(ξ −ξ ), whenever ξ � ξ for a constant K > 0;

(v) m(ξ ) � 0 for ξ ∈ Ib.

Then, the sequences {un} and {vn} defined inductively by

u1 = Gu0, un = Gun−1, v1 = Gv0, vn = Gvn−1,

for n = 1, 2, . . ., satisfy the following conditions:

1o) un−1(x,y) � un(x,y),vn(x,y) � vn−1(x,y), for (x,y) ∈ E0;

2o) the functions un and vn for n = 1, 2, . . . are the lower and upper functions for problem

(1.7.16)–(1.7.17) on E0, respectively;

3o) the following inequalities hold

0 � vn(x,y)−un(x,y) � N0Cn(x+ y)n

n!,

0 � vnx(x,y)−un

x(x,y) � N0Cn(x+ y)n

n!,

0 � vny(x,y)−un

y(x,y) � N0Cn(x+ y)n

n!,

for n = 1, 2, . . . and (x,y) ∈ E0, where N0 := ‖v0−u0‖ and C is a some constant. Moreover,

a function z defined by

z(x,y) := limn→∞

un(x,y) = limn→∞

vn(x,y).

is the unique solution of problem (1.7.16)–(1.7.17) on E0.


Consider the Darboux problem for the hyperbolic partial integrodifferential equation of the

form

uxy(x,y) = f (x,y,u(x,y),ux(x,y),uy(x,y),(Ωu)(x,y),μ), (1.7.19)


u(x,0) = σ(x), u(0,y) = τ(y), u(0,0) = 0, (1.7.20)



(Ωu)(x,y) =∫ x

0

∫ y

0g(x,y,ξ ,η ,u(ξ ,η),uξ (ξ ,η),uη(ξ ,η))dξ dη ,

f ∈C(E ×Rn ×R

n ×Rn ×R

n ×R,Rn), g ∈C(E2 ×Rn ×R

n ×Rn,Rn), σ , τ ∈C(R+,Rn)

and μ is a parameter. Let S1 be the space defined as in Section 1.6.

(e16) Assume that

(i) the functions f , g in (1.7.19) satisfy the conditions

| f (x,y,u1,u2,u3,u4,μ)− f (x,y,u1,u2,u3,u4,μ)|

� L[|u1 −u1|+ |u2 −u2|+ |u3 −u3|+ |u4 −u4|

],

|g(x,y,s,t,u1,u2,u3)−g(x,y,s, t,u1,u2,u3)| � M[|u1 −u1|+ |u2 −u2|+ |u3 −u3|

],

where L � 0,M � 0 are constants,

(ii) for every μ ∈ R there exist constants Q j � 0 ( j = 1, 2, 3) such that

|σ(x)|+ |τ(y)|+∫ x

0

∫ y

0| f (s,t,0,0,0,(Ω0)(s,t),μ)|dsdt � Q1 exp(λ (x+ y)),

|σx(x)|+∫ y

0| f (x,t,0,0,0,(Ω0)(x,t),μ)|dt � Q2 exp(λ (x+ y)),

|τy(y)|+∫ x

0| f (s,y,0,0,0,(Ω0)(s,y),μ)|ds � Q3 exp(λ (x+ y)),

where λ > 0 is a constant and 0 is the zero element in Rn. If α =

L(

1+ Mλ 2

)(1

λ 2 + 2λ

)< 1, then there exists a unique solution u ∈ S1 of the problem

(1.7.19)–(1.7.20) on E.

(e17) Assume that

(i) the conditions in (e16) hold,

(ii) there exist constants Nj � 0 ( j = 1, 2, 3) and the function P ∈C(E,R+) such that

| f (x,y,u1,u2,u3,u4,μ1)− f (x,y,u1,u2,u3,u4,μ2)| � P(x,y)|μ1 −μ2|,

for μ1, μ2 ∈ R and∫ x

0

∫ y

0P(s, t)dsdt � N1 exp(λ (x+ y)),

∫ y

0P(x,t)dt � N2 exp(λ (x+ y)),

∫ x

0P(s,y)ds � N3 exp(λ (x+ y)).

Then the solution u(x,y,μ) of problem (1.7.19)–(1.7.20) belonging to S1 is continuous with

respect to the variables (x,y,μ) in E ×R.


1.8 Notes

The literature concerning the Volterra type equations and inequalities is particularly rich,

see [2,5,7,9,12,22,24,42,45] and the references given therein. Section 1.2 deals with some

basic integral inequalities with explicit estimates, needed in our subsequent discussion,

which are recently appeared in the literature. All the results are due to Pachpatte and are

taken from [24,27,40]. The results given in sections 1.3 and 1.4 deals with some important

qualitative properties of solutions of certain integral equations in two variables and adapted

from [40,26]. Sections 1.5 and 1.6 contains some basic qualitative properties concern-

ing certain partial integrodifferential and integral equations and are taken from [18,27,28].

Section 1.7 is written mostly to provide results on certain aspects related to some topics,

which we did not cover in our exposition. Moreover, many generalizations, extensions and

variants of the results are possible and progress is to be expected in the future.


Chapter 2

Integral inequalities and equations in two andthree variables

2.1 Introduction

The inequalities with explicit estimates are among the most powerful and widely used

analytic tools in the study of various dynamical systems. The extensive surveys of such

inequalities may be found in monographs [82,85,87,134]. It is easy to check that the

inequalities available in the literature are not directly applicable to study the qualitative

behavior of solutions of many dynamical systems arising in natural phenomena. For in-

stance, see the equations of the forms (11), (16), (19) to name a few. Motivated by the

needs of diverse applications and the desire to widen the scope of such inequalities, re-

cently in [108,,99,104,102,95,112] the present author investigated new explicit estimates

on some integral inequalities, which are equally important to achieve the diversity of de-

sired goals. In this chapter, we offer some such fundamental inequalities established in

[41,30,37,32,35,36], which can be used as tools for handling equations like (11), (16), (19)

and their variants. Some important qualitative aspects of the general forms of equations

(11), (19) are also studied in a simple and unified way. In what follows, we shall use the

notations given earlier without further mention.

2.2 Integral inequalities in two variables

In this section we present some basic integral inequalities with explicit estimates which can

be used in the study of qualitative properties of solutions of equations of the forms (11),

(16) and (19).

The inequalities established in [108] are embodied in the following theorems.

Theorem 2.2.1. Let u, r, n ∈C(E,R+) and c � 0 is a constant.

59


(a1) If

u(x,t) � c+∫ x

0

∫ s

0

∫ t

0r(σ ,τ)u(σ ,τ)dτ dσ ds, (2.2.1)

for (x, t) ∈ E, then

u(x,t) � cexp(∫ x

0

∫ s

0

∫ t

0r(σ ,τ)dτ dσ ds

)

, (2.2.2)

for (x, t) ∈ E

(a2) Let n(x,t) be nondecreasing in each variable x, t ∈ R+. If

u(x,t) � n(x,t)+∫ x

0

∫ s

0

∫ t

0r(σ ,τ)u(σ ,τ)dτ dσ ds, (2.2.3)


u(x,t) � n(x, t)exp(∫ x

0

∫ s

0

∫ t


)

, (2.2.4)

for (x, t) ∈ E.

Theorem 2.2.2. Let u, p, q, r ∈C(E,R+).

(a3) Let L ∈C(E ×R+,R+) be such that

0 � L(x,t,u)−L(x,t,v) � M(x,t,v)(u− v), (2.2.5)

for u � v � 0, where M ∈C(E ×R+,R+). If

u(x,t) � p(x,t)+q(x,t)∫ x

0

∫ s

0

∫ t

0L(σ ,τ,u(σ ,τ))dτ dσ ds, (2.2.6)


u(x,t) � p(x,t)+q(x,t)(∫ x

0

∫ s

0

∫ t

0L(σ ,τ, p(σ ,τ))dτ dσ ds

)

×exp(∫ x

0

∫ s

0

∫ t

0M(σ ,τ, p(σ ,τ))q(σ ,τ)dτ dσ ds

)

, (2.2.7)

for (x, t) ∈ E.

(a4) If

u(x,t) � p(x,t)+q(x,t)∫ x

0

∫ s

0

∫ t

0r(σ ,τ)u(σ ,τ)dτ dσ ds, (2.2.8)


u(x,t) � p(x, t)+q(x,t)(∫ x

0

∫ s

0

∫ t

0r(σ ,τ)p(σ ,τ)dτ dσ ds

)

×exp(∫ x

0

∫ s

0

∫ t

0r(σ ,τ)q(σ ,τ)dτ dσ ds

)

, (2.2.9)

for (x, t) ∈ E.

Integral inequalities and equations in two and three variables 61

Theorem 2.2.3. Let u, r ∈C(E,R+) and c � 0 is a constant.

(a5) If

u2(x,t) � c+∫ x

0

∫ s

0

∫ t

0r(σ ,τ)u(σ ,τ)dτ dσ ds, (2.2.10)


u(x,t) �√

c+12

∫ x

0

∫ s

0

∫ t

0r(σ ,τ)dτ dσ ds, (2.2.11)

for (x, t) ∈ E.

(a6) If u ∈C (E,R1), c � 1 and

u(x, t) � c+∫ x

0

∫ s

0

∫ t

0r(σ ,τ)u(σ ,τ) logu(σ ,τ)dτ dσ ds, (2.2.12)


u(x,t) � cA(x,t), (2.2.13)

for (x, t) ∈ E, where

A(x,t) = exp(∫ x

0

∫ s

0

∫ t


)

. (2.2.14)

In the following theorem we present the inequality proved in [104].

Theorem 2.2.4. Let IL = [0,L], IT = [0,T ] (L > 0, T > 0) are the given subsets of R and

D = IL × IT .

(a7) Let u, p, q, r ∈C(D,R+). If

u(x, t) � p(x,t)+q(x, t)∫ t

0

∫ s

0

∫ L

0r(y,τ)u(y,τ)dydτ ds, (2.2.15)

for (x, t) ∈ D, then

u(x,t) � p(x,t)+q(x,t)(∫ t

0

∫ s

0

∫ L

0r(y,τ)p(y,τ)dydτ ds

)

× exp(∫ t

0

∫ s

0

∫ L

0r(y,τ)q(y,τ)dydτ ds

)

, (2.2.16)

for (x, t) ∈ D.

Proofs of Theorems 2.2.1–2.2.4. To prove (a1) and (a5) it is sufficient to assume that

c > 0, since the standard limiting argument can be used to treat the remaining case, see [82,

p. 108].


(a1) Let c > 0 and define a function z(x,t) by the right hand side of (2.2.1), then z(x,0) =

z(0, t) = c, u(x, t) � z(x, t),

zt(x,t) =∫ x

0

∫ s

0r(σ ,τ)u(σ ,τ)dσ dτ,

zx(x,t) =∫ x

0

∫ t

0r(σ ,τ)u(σ ,τ)dτ dσ ,

zxx(x,t) =∫ t

0r(x,τ)u(x,τ)dτ,

and

zxxt(x,t) = r(x,t)u(x,t) � r(x,t)z(x,t). (2.2.17)

From (2.2.17) and using the facts that zxx(x,t) � 0, zt(x,t) � 0, z(x,t) > 0, we observe that

(see [22, p. 346])

zxxt(x,t)z(x,t)

� r(x,t)+zxx(x,t)zt(x,t)

z2(x,t),

i.e.,

∂∂ t

(zxx(x,t)z(x,t)

)

� r(x,t). (2.2.18)

By keeping x fixed in (2.2.18), we set t = τ and then integrating with respect to τ from 0 to

t and using the fact that zxx(x,0) = 0, we have

zxx(x,t)z(x, t)

�∫ t

0r(x,τ)dτ. (2.2.19)

Again as above, from (2.2.19) and the facts that zx(x,t) � 0,z(x,t) > 0, we observe that

∂∂x

(zx(x,t)z(x,t)

)

�∫ t

0r(x,τ)dτ. (2.2.20)

By taking t fixed in (2.2.20), set x = σ and then integrating with respect to σ from 0 to x

and using the fact that zx(0,t) = 0, we have

zx(x,t)z(x,t)

�∫ x

0

∫ t

0r(σ ,τ)dτ dσ . (2.2.21)

From (2.2.21), we get

z(x,t) � cexp(∫ x

0

∫ s

0

∫ t


)

. (2.2.22)

Using (2.2.22) in u(x,t) � z(x,t), we get (2.2.2).

(a2) First we assume that n(x,t) > 0 for (x,t) ∈ E. From (2.2.3), it is easy to observe that

u(x, t)n(x, t)

� 1+∫ x

0

∫ s

0

∫ t

0r(σ ,τ)

u(σ ,τ)n(σ ,τ)

dτ dσ ds. (2.2.23)


Now an application of the inequality in part (a1) to (2.2.23) yields (2.2.4). The proof of the

case when n(x,t) = 0 can be completed as in [82, p. 326].

(a3) Define a function z(x,t) by

z(x,t) =∫ x

0

∫ s

0

∫ t

0L(σ ,τ,u(σ ,τ))dτ dσ ds, (2.2.24)

then z(0,t) = z(x,0) = 0 and (2.2.6) can be restated as

u(x,t) � p(x,t)+q(x,t)z(x, t). (2.2.25)

From (2.2.24), (2.2.25) and (2.2.5), we observe that

z(x,t) �∫ x

0

∫ s

0

∫ t

0{L(σ ,τ, p(σ ,τ)+q(σ ,τ)z(σ ,τ))

−L(σ ,τ, p(σ ,τ))+L(σ ,τ, p(σ ,τ))}dτ dσ ds

�∫ x

0

∫ s

0

∫ t


+∫ x

0

∫ s

0

∫ t

0M(σ ,τ, p(σ ,τ))q(σ ,τ)z(σ ,τ)dτ dσ ds. (2.2.26)

Clearly, the first term on the right hand side in (2.2.26) is nonnegative and nondecreasing

in x, t ∈ R+. Now a suitable application of the inequality in part (a2) to (2.2.26) yields

z(x,t) �(∫ x

0

∫ s

0

∫ t


)

×exp(∫ x

0

∫ s

0

∫ t

0M(σ ,τ, p(σ ,τ))q(σ ,τ)dτ dσ ds

)

, (2.2.27)

for (x, t) ∈ E. Using (2.2.27) in (2.2.25), we get the required inequality in (2.2.7).

(a4) By taking L(σ ,τ,u(σ ,τ)) = r(σ ,τ)u(σ ,τ) in part (a3) for r,u ∈C(E,R+), we get the

required inequality.

(a5) Let c > 0 and define a function z(x, t) by the right hand side of (2.2.10), then z(x,0) =

z(0, t) = c, u(x, t) �√

z(x,t) and

zxxt(x,t) = r(x,t)u(x,t) � r(x,t)√

z(x,t). (2.2.28)

Now by following the same arguments as in the proof of part (a1) below (2.2.17) upto

(2.2.21) with suitable modifications (see [82, p. 528]), from (2.2.28), we get

zx(x,t)√z(x, t)

�∫ x

0

∫ t

0r(σ ,τ)dτ dσ . (2.2.29)


From (2.2.29), we obtain√

z(x,t) �√

c+12

∫ x

0

∫ s

0

∫ t

0r(σ ,τ)dτ dσ ds. (2.2.30)

Using (2.2.30) in u(x,t) �√

z(x, t), we get (2.2.11).

(a6) Define a function z(x, t) by the right hand side of (2.2.12), then z(x,0) = z(0, t) = c,

u(x,t) � z(x,t) and

zxxt(x,t) = r(x,t)u(x,t) logu(x,t) � r(x,t)z(x,t) logz(x, t). (2.2.31)

The remaining proof can be completed by following the proof of part (a1) and closely

looking at the proof of Theorem 5.10.1 given in [82, p. 544].

(a7) Introducing the notation

e(τ) =∫ L

0r(y,τ)u(y,τ)dy, (2.2.32)

in (2.2.15), we get

u(x,t) � p(x,t)+q(x, t)∫ t

0

∫ s

0e(τ)dτ ds, (2.2.33)

for (x, t) ∈ D. Define

z(t) =∫ t

0

∫ s

0e(τ)dτ ds, (2.2.34)

for t ∈ IT , then it is easy to see that z(0) = 0,z′(0) = 0 and

u(x,t) � p(x,t)+q(x,t)z(t). (2.2.35)

From (2.2.34), (2.2.32), (2.2.35), we observe that

z′′(t) = e(t) =∫ L

0r(y,t)u(y,t)dy �

∫ L

0r(y,t)[p(y,t)+q(y,t)z(t)]dy

=∫ L

0r(y,t)p(y,t)dy+ z(t)

∫ L

0r(y,t)q(y,t)dy. (2.2.36)

From (2.2.36) and the fact that z(t) is nondecreasing for t ∈ IT , it is easy to see that

z(t) �∫ t

0

∫ s

0

∫ L


+∫ t

0z(s)

{∫ s

0

∫ L

0r(y,τ)q(y,τ)dydτ

}

ds. (2.2.37)

Clearly, the first term on the right hand side in (2.2.37) is nonnegative and nondecreasing

in t ∈ IT . Now a suitable application of the inequality in Theorem 1.3.1 in [82] to (2.2.37)

yields

z(t) �(∫ t

0

∫ s

0

∫ L


)

× exp(∫ t

0

∫ s

0

∫ L

0r(y,τ)q(y,τ)dydτ ds

)

, (2.2.38)

for t ∈ IT . Using (2.2.38) in (2.2.35), we get the required inequality in (2.2.16).


2.3 Integral inequalities in three variables

Our main goal in this section is to present some integral inequalities with explicit estimates

in three variables established in [95,111,102], which can be used in certain new applications

for which the earlier inequalities do not apply directly. In what follows I = [a,b] (a < b)

denotes a given subset of R and G = R2+ × I.

First we shall give the following theorems which contains the inequalities investigated in

[95].

Theorem 2.3.1. Let u, p, q, f ∈C(G,R+) and k � 0 is a constant.

(b1) If

u(x,y,z) � k +∫ x

0

∫ y

0

∫ b

af (s, t,r)u(s,t,r)dr dt ds, (2.3.1)

for (x,y,z) ∈ G, then

u(x,y,z) � k exp(∫ x

0

∫ y

0

∫ b

af (s,t,r)dr dt ds

)

, (2.3.2)

for (x,y,z) ∈ G.

(b2) If

u(x,y,z) � p(x,y,z)+q(x,y,z)∫ x

0

∫ y

0

∫ b

af (s, t,r)u(s,t,r)dr dt ds, (2.3.3)


u(x,y,z) � p(x,y,z)+q(x,y,z)(∫ x

0

∫ y

0

∫ b

af (s,t,r)p(s,t,r)dr dt ds

)

×exp(∫ x

0

∫ y

0

∫ b

af (s,t,r)q(s,t,r)dr dt ds

)

, (2.3.4)

for (x,y,z) ∈ G.

Theorem 2.3.2. Let u, f ∈C(G,R+) and k � 0, c � 1 are constants.

(b3) If

u2(x,y,z) � k +∫ x

0

∫ y

0

∫ b

af (s,t,r)u(s,t,r)dr dt ds, (2.3.5)


u(x,y,z) �√

k +12

∫ x

0

∫ y

0

∫ b

af (s, t,r)dr dt ds, (2.3.6)

for (x,y,z) ∈ G.


(b4) If u(x,y,z) � 1 and

u(x,y,z) � c+∫ x

0

∫ y

0

∫ b

af (s,t,r)u(s,t,r) logu(s,t,r)dr dt ds, (2.3.7)


u(x,y,z) � cexp(∫ x

0∫ y

0∫ b

a f (s,t,r)dr dt ds), (2.3.8)

for (x,y,z) ∈ G.

In the following theorems we present the inequalities established in [102].

Theorem 2.3.3. Let u, p, q, f ∈C(G,R+).

(b5) If


0

∫ ∞

y

∫ b




0

∫ ∞

y

∫ b


)

×exp(∫ x

0

∫ ∞

y

∫ b

af (s,t,r)q(s, t,r)dr dt ds

)

, (2.3.10)

for (x,y,z) ∈ G.

(b6) If

u(x,y,z) � p(x,y,z)+q(x,y,z)∫ ∞

x

∫ ∞

y

∫ b



u(x,y,z) � p(x,y,z)+q(x,y,z)(∫ ∞

x

∫ ∞

y

∫ b


)

×exp(∫ ∞

x

∫ ∞

y

∫ b


)

, (2.3.12)

for (x,y,z) ∈ G.

Theorem 2.3.4. Let u, p,q,c, f , g ∈C(G,R+).

(b7) suppose that


0

∫ y

0

∫ b

af (s,t,r)u(s,t,r)dr dt ds

+ c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)u(s,t,r)dr dt ds, (2.3.13)


for (x,y,z) ∈ G. If

α1 =∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)B1(s,t,r)dr dt ds < 1, (2.3.14)

then

u(x,y,z) � A1(x,y,z)+D1B1(x,y,z), (2.3.15)

for (x,y,z) ∈ G, where

A1(x,y,z) = p(x,y,z)+q(x,y,z)(∫ x

0

∫ y

0

∫ b

af (s,t,r)p(s, t,r)dr dt ds

)

×exp(∫ x

0

∫ y

0

∫ b


)

(2.3.16)

B1(x,y,z) = c(x,y,z)+q(x,y,z)(∫ x

0

∫ y

0

∫ b

af (s,t,r)c(s, t,r)dr dt ds

)

×exp(∫ x

0

∫ y

0

∫ b


)

(2.3.17)

and

D1 =1

1−α1

∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)A1(s, t,r)dr dt ds. (2.3.18)

(b8) Suppose that

u(x,y,z) � p(x,y,z)+q(x,y,z)∫ ∞

x

∫ ∞

y

∫ b


+ c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)u(s,t,r)dr dt ds, (2.3.19)


α2 =∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)B2(s,t,r)dr dt ds < 1, (2.3.20)

then

u(x,y,z) � A2(x,y,z)+D2B2(x,y,z), (2.3.21)


A2(x,y,z) = p(x,y,z)+q(x,y,z)(∫ ∞

x

∫ ∞

y

∫ b

af (s, t,r)p(s,t,r)dr dt ds

)

×exp(∫ ∞

x

∫ ∞

y

∫ b


)

, (2.3.22)

B2(x,y,z) = c(x,y,z)+q(x,y,z)(∫ ∞

x

∫ ∞

y

∫ b

af (s, t,r)c(s,t,r)dr dt ds

)

×exp(∫ ∞

x

∫ ∞

y

∫ b


)

, (2.3.23)

and

D2 =1

1−α2

∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)A2(s, t,r)dr dt ds. (2.3.24)

The next theorem deals with a slight variant of the inequality proved in [111].


Theorem 2.3.5. Let u, p ∈C(G,R+), q ∈C(G× I,R+) and c � 0 is a constant. If

u(x,y,z) � c+∫ x

0

∫ y

0

[

p(s,t,z)u(s,t,z)+∫ b

aq(s,t,z,r)u(s,t,r)dr

]

dt ds, (2.3.25)


u(x,y,z) � cH(x,y,z)exp(∫ x

0

∫ y

0

∫ b

aq(s,t,z,r)H(s,t,r)dr dt ds

)

, (2.3.26)


H(x,y,z) = exp(∫ x

0

∫ y

0p(σ ,τ,z)dτ dσ

)

. (2.3.27)

Proofs of Theorems 2.3.1–2.3.5. We give the details of the proofs of (b2), (b3), (b5), (b8)

and the inequality in Theorem 2.3.5 only. The proofs of other inequalities can be completed

by following the proofs of these inequalities and closely looking at the proofs of the similar

results given in [82, 87].

(b2) Introducing the notation

E(s,t) =∫ b

af (s,t,r)u(s,t,r)dr, (2.3.28)

the inequality (2.3.3) can be restated as


0

∫ y

0E(s,t)dt ds. (2.3.29)

Define

w(x,y) =∫ x

0

∫ y

0E(s,t)dt ds, (2.3.30)

then w(x,0) = w(0,y) = 0 and

u(x,y,z) � p(x,y,z)+q(x,y,z)w(x,y). (2.3.31)

From (2.3.30), (2.3.28), (2.3.31), we observe that

wxy(x,y) = E(x,y) =∫ b

af (x,y,r)u(x,y,r)dr

�∫ b

af (x,y,r)[p(x,y,z)+q(x,y,z)w(x,y)]dr

= w(x,y)∫ b

af (x,y,r)q(x,y,r)dr +

∫ b

af (x,y,r)p(x,y,r)dr. (2.3.32)

Now, by following the similar arguments as in the proof of Theorem 4.3.1 given in [82,

p. 328] with suitable modifications, from (2.3.32), we obtain

w(x,y) �(∫ x

0

∫ y

0

∫ b


)


×exp(∫ x

0

∫ y

0

∫ b


)

. (2.3.33)

Now, using (2.3.33) in (2.3.31), we get the required inequality in (2.3.4).

(b3) Introducing the notation (2.3.28) in (2.3.5), we get

u2(x,y,z) � k +∫ x

0

∫ y

0E(s,t)dt ds. (2.3.34)

Let k > 0 and define

m(x,y) = k +∫ x

0

∫ y

0E(s,t)dt ds, (2.3.35)

then m(x,0) = m(0,y) = k and

u2(x,y,z) � m(x,y). (2.3.36)


mxy(x,y) = E(x,y) =∫ b

af (x,y,r)u(x,y,r)dr �

√m(x,y)

∫ b

af (x,y,r)dr. (2.3.37)

The inequality (2.3.37) implies (see [82, Theorem 5.8.1, p. 527])√

m(x,y) �√

k +12

∫ x

0

∫ y

0

∫ b

af (s,t,r)dr dt ds. (2.3.38)


If k � 0 we carry out the above procedure with k+ε instead of k, where ε > 0 is an arbitrary

small constant, and subsequently pass to the limit as ε → 0 to obtain (2.3.6).

(b5) Using the notation (2.3.28) in (2.3.9), we get


0

∫ ∞

yE(s,t)dt ds. (2.3.39)

Define

w(x,y) =∫ x

0

∫ ∞

yE(s,t)dt ds, (2.3.40)

then w(0,y) = 0 and

u(x,y,z) � p(x,y,z)+q(x,y,z)w(x,y). (2.3.41)


wx(x,y) =∫ ∞

yE(x,t)dt =

∫ ∞

y

{∫ b

af (x,t,r)u(x,t,r)dr

}

dt

�∫ ∞

y

{∫ b

af (x, t,r) [p(x,t,r)+q(x,t,r)w(x, t)]dr

}

dt


=∫ ∞

y

{∫ b

af (x,t,r)p(x,t,r)dr

}

dt +∫ ∞

yw(x,t)

{∫ b

af (x,t,r)q(x,t,r)dr

}

dt. (2.3.42)

By taking x = s in (2.3.42) and integrating both sides with respect to s from 0 to x, we get

w(x,y) � e1(x,y)+∫ x

0

∫ ∞

y

{∫ b

af (s,t,r)q(s,t,r)dr

}

w(s,t)dt ds, (2.3.43)

where

e1(x,y) =∫ x

0

∫ ∞

y

∫ b

af (s,t,r)p(s,t,r)dr dt ds. (2.3.44)

Clearly e1(x,y) is nonnegative, continuous, nondecreasing in x and nonincreasing in y for

x, y ∈ R+. Now a suitable application of the inequality in Theorem 1.2.4 given in [87,

p. 110] (see also [82, p. 440]) to (2.3.43) yields

w(x,y) � e1(x,y)exp(∫ x

0

∫ ∞

y

∫ b


)

. (2.3.45)

Using (2.3.45) in (2.3.41), we get (2.3.10).

(b8) Let E(s, t) be as in (2.3.28) and

λ =∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)u(s, t,r)dr dt ds. (2.3.46)

Then (2.3.19) can be restated as

u(x,y,z) � p(x,y,z)+q(x,y,z)∫ ∞

x

∫ ∞

yE(s, t)dt ds+ c(x,y,z)λ . (2.3.47)

Let

v(x,y) =∫ ∞

x

∫ ∞

yE(s, t)dt ds, (2.3.48)

then v(∞,y) = 0 and from (2.3.47), we have

u(x,y,z) � p(x,y,z)+q(x,y,z)v(x,y)+ c(x,y,z)λ . (2.3.49)

From (2.3.48), (2.3.28) and (2.3.49), we have

vx(x,y) = −∫ ∞

yE(x,t)dt = −

∫ ∞

y

{∫ b

af (x,t,r)u(x,t,r)dr

}

dt

� −∫ ∞

y

{∫ b

af (x,t,r)[p(x,t,r)+q(x,t,r)v(x,t)+ c(x, t,r)λ ]dr

}

dt

= −∫ ∞

y

{∫ b

af (x,t,r)p(x, t,r)dr

}

dt

−∫ ∞

y

{∫ b

af (x,t,r)[q(x,t,r)v(x, t)+ c(x, t,r)λ ]dr

}

dt. (2.3.50)


By taking x = s in (2.3.50) and integrating both sides with respect to s from x to ∞ for

x ∈ R+, we have

v(x,y) � e2(x,y)+∫ ∞

x

∫ ∞

y

{∫ b

af (s,t,r)q(s, t,r)dr

}

v(s,t)dt ds, (2.3.51)

where

e2(x,y) =∫ ∞

x

∫ ∞

y

∫ b

af (s, t,r)[p(s,t,r)+ c(s,t,r)λ ]dr dt ds. (2.3.52)

Clearly, e2(x,y) is nonnegative, continuous, nonincreasing in each variable x, y ∈R+. Now,

a suitable application of the inequality in Theorem 1.2.3 given in [87, p. 110] (see also [82,

p. 440]) to (2.3.51) yields

v(x,y) � e2(x,y)exp(∫ ∞

x

∫ ∞

y

∫ b


)

. (2.3.53)

Using (2.3.53), (2.3.52) in (2.3.49), we get

u(x,y,z) � p(x,y,z)+q(x,y,z)

[∫ ∞

x

∫ ∞

y

∫ b

af (s,t,r){p(s, t,r)+ c(s, t,r)λ}dr dt ds

]

×exp(∫ ∞

x

∫ ∞

y

∫ b


)

+ c(x,y,z)λ

= A2(x,y,z)+λB2(x,y,z). (2.3.54)

From (2.3.46) and (2.3.54), we have

λ =∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)u(s,t,r)dr dt ds

�∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)[A2(s,t,r)+λB2(s,t,r)]dr dt ds,

which implies

λ � D2. (2.3.55)

Using (2.3.55) in (2.3.54), we get (2.3.21).

To prove the inequality in Theorem 2.3.5, let

m(x,y,z) = c+∫ x

0

∫ y

0

∫ b

aq(s,t,z,r)u(s,t,r)dr dt ds, (2.3.56)


u(x,y,z) � m(x,y,z)+∫ x

0

∫ y

0p(s,t,z)u(s, t,z)dt ds. (2.3.57)


It is easy to observe that m(x,y,z) is nonnegative for (x,y,z) ∈ G and nondecreasing in each

variable x, y ∈ R+ and for z ∈ I. Treating (2.3.57) as two-dimensional integral inequal-

ity in x, y ∈ R+ for every z ∈ I and a suitable application of the inequality given in [82,

Theorem 4.2.2, p. 325] to (2.3.57) yields

u(x,y,z) � m(x,y,z)H(x,y,z). (2.3.58)

From (2.3.56) and (2.3.58), we observe that

m(x,y,z) � c+∫ x

0

∫ y

0

∫ b

aq(s,t,z,r)H(s,t,r)m(s,t,r)dr dt ds. (2.3.59)

Setting

e(s,t) =∫ b

aq(s, t,z,r)H(s, t,r)m(s,t,r)dr, (2.3.60)

for every z ∈ I, the inequality (2.3.59) can be restated as

m(x,y,z) � c+∫ x

0

∫ y

0e(s, t)dt ds. (2.3.61)

Let

n(x,y) = c+∫ x

0

∫ y

0e(s,t)dt ds, (2.3.62)

then n(x,0) = n(0,y) = c and

m(x,y,z) � n(x,y) (2.3.63)

for x, y ∈ R+ and for every z ∈ I. From (2.3.62), (2.3.60) and (2.3.63), we observe that

nxy(x,y) = e(x,y) =∫ b

aq(x,y,z,r)H(x,y,r)m(x,y,r)dr

� n(x,y)∫ b

aq(x,y,z,r)H(x,y,r)dr. (2.3.64)

The inequality (2.3.64) implies (see [82, p. 325])

n(x,y) � cexp(∫ x

0

∫ y

0

∫ b


)

, (2.3.65)

for x, y ∈R+ and for every z ∈ I. The required inequality in (5.3.26) follows from (2.3.65),

(2.3.63) and (2.3.58).


2.4 Integral equation in two variables


u(x,t) = f (x,t)+∫ t

0

∫ s

0

∫ L

0K(x,t,s,y,τ,u(y,τ))dydτ ds, (2.4.1)

for (x,t) ∈ D = J × I; J = [0,L], I = [0,T ], where L > 0, T > 0 are finite but can be

arbitrarily large constants and f ∈ C(D,R), K ∈ C(D × I × D×R,R). Such equations

arises, in the study of partial differential equations of the forms (13)–(15), see [4]. This

section is devoted to address some basic results related to the solution of equation (2.4.1)

given in [104]. Let U be the space of those functions φ ∈C(D,R) which fulfil the condition

|φ (x,t)| = O(exp(λ (x+ t))), (2.4.2)

where λ > 0 is a constant. In the space U , we define the norm

|φ |U = max(x,t)∈D

[|φ(x,t)|exp(−λ (x+ t))]. (2.4.3)

It is easy to see that U is a Banach space and

|φ |U � N, (2.4.4)


First, we formulate the following theorem which provide conditions for the existence of a

unique solution to equation (2.4.1).


(i) the function K in equation (2.4.1) satisfies the condition

|K(x, t,s,y,τ,u)−K(x,t,s,y,τ,v)| � h(x,t,s,y,τ)|u− v|, (2.4.5)

where h ∈C (D× I ×D,R+),


( j1) there exists a nonnegative constant α < 1 such that∫ t

0

∫ s

0

∫ L

0h(x,t,s,y,τ)exp(λ (y+ τ))dydτ ds � α exp(λ(x+ t)), (2.4.6)

( j2) there exists a nonnegative constant β such that

| f (x,t)|+∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,0)|dydτ ds � β exp(λ (x + t)), (2.4.7)

where f , K are as defined in equation (2.4.1).

Then the equation (2.4.1) has a unique solution u(x,t) in u on D.


Proof. Let u ∈U and define the operator F by

(Fu)(x,t) = f (x, t)+∫ t

0

∫ s

0

∫ L

0K(x,t,s,y,τ,u(y,τ))dydτ ds, (2.4.8)

for (x,t) ∈ D. The proof that F maps U into itself and is a contraction map can be com-

pleted by closely looking at the proof of Theorem 1.3.1 given in Chapter 1 with suitable

modifications. We leave the details to the reader.

The next theorem deals with the uniqueness of solutions of equation (2.4.1) on D.

Theorem 2.4.2. Suppose that the function K in equation (2.4.1) satisfies the condition

|K(x,t,s,y,τ,u)−K(x,t,s,y,τ,v)| � q(x, t)g(y,τ)|u− v|, (2.4.9)

where q, g ∈C(D,R+). Then the equation (2.4.1) has at most one solution on D.

Proof. Let u(x,t) and v(x,t) be two solutions of equation (2.4.1) on D. Using these facts

and hypotheses, we have

|u(x,t)− v(x,t)| �∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,u(y,τ))−K(x, t,s,y,τ,v(y,τ))|dydτ ds

� q(x,t)∫ t

0

∫ s

0

∫ L

0g(y,τ)|u(y,τ)− v(y,τ)|dydτ ds. (2.4.10)

Now a suitable application of Theorem 2.2.4 part (a7) (with p(x,t) = 0) to (2.4.10) yields

|u(x,t)− v(x,t)| � 0, which implies u(x,t) = v(x,t). Thus there is at most one solution to

equation (2.4.1) on D.

The following theorems provide estimates on the solution of equation (2.4.1).


|K(x, t,s,y,τ,u)| � q(x,t)g(y,τ)|u|, (2.4.11)

where q, g ∈C(D,R+). If u(x,t) is any solution of equation (2.4.1) on D, then

|u(x,t)| � | f (x,t)|+q(x,t)(∫ t

0

∫ s

0

∫ L

0g(y,τ)| f (y,τ)|dydτ ds

)

×exp(∫ t

0

∫ s

0

∫ L

0g(y,τ)q(y,τ)dydτ ds

)

, (2.4.12)

for (x, t) ∈ D.

Proof. Using the fact that u(x,t) is a solution of equation (2.4.1) and hypotheses, we have

|u(x,t)| � | f (x, t)|+∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,u(y,τ))|dydτ ds

� | f (x, t)|+q(x,t)∫ t

0

∫ s

0

∫ L

0g(y,τ)|u(y,τ)|dydτ ds. (2.4.13)

Now an application of Theorem 2.2.4 part (a7) to (2.4.13) yields (2.4.12).



(2.4.9). If u(x,t) is any solution of equation (2.4.1) on D, then

|u(x,t)− f (x,t)| � Q(x,t)+q(x,t)(∫ t

0

∫ s

0

∫ L

0g(y,τ)Q(y,τ)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫ L


)

, (2.4.14)

for (x, t) ∈ D, where

Q(x, t) =∫ t

0

∫ s

0

∫ L

0|K (x,t,s,y,τ, f (y,τ))|dydτ ds, (2.4.15)

for (x, t) ∈ D.

Proof. From the fact that u(x, t) is a solution of equation (2.4.1) and the condition (2.4.9),

we have

|u(x,t)− f (x,t)| �∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,u(y,τ))−K (x,t,s,y,τ, f (y,τ))|dydτ ds

+∫ t

0

∫ s

0

∫ L

0|K (x,t,s,y,τ, f (y,τ))|dydτ ds

� Q(x, t)+q(x,t)∫ t

0

∫ s

0

∫ L

0g(y,τ) |u(y,τ)− f (y,τ)|dydτ ds. (2.4.16)

An application of Theorem 2.2.4 part (a7) to (2.4.16) yields (2.4.14).

We call the function u ∈ C(D,R) an ε-approximate solution of equation (2.4.1), if there

exists a constant ε � 0 such that∣∣∣∣u(x,t)− f (x,t)−

∫ t

0

∫ s

0

∫ L

0K(x, t,s,y,τ,u(y,τ))dydτ ds

∣∣∣∣ � ε,

for (x, t) ∈ D.

The relation between an ε-approximate solution and a solution of equation (2.4.1) is shown

in the following theorem.


(i) the function K in equation (2.4.1) satisfies the condition (2.4.9),

(ii) the functions uε (x,t),u(x,t) ∈C(D,R) are respectively, an ε-approximate solution and

any solution of equation (2.4.1).

Then

|uε (x,t)−u(x,t)| � ε[

1+q(x,t)(∫ t

0

∫ s

0

∫ L

0g(y,τ)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫ L


)]

, (2.4.17)

for (x, t) ∈ D.


Proof. Let z(x,t) = |uε (x,t)−u(x, t)|, (x,t) ∈ D. From the hypotheses, we observe that

z(x, t) =∣∣∣∣uε (x,t)− f (x,t)−

∫ t

0

∫ s

0

∫ L

0K(x,t,s,y,τ,uε(y,τ))dydτ ds

+∫ t

0

∫ s

0

∫ L

0{K(x,t,s,y,τ,uε(y,τ))−K(x,t,s,y,τ,u(y,τ))}dydτ ds

∣∣∣∣

�∣∣∣∣uε (x,t)− f (x,t)−

∫ t

0

∫ s

0

∫ L

0K(x,t,s,y,τ,uε(y,τ))dydτ ds

∣∣∣∣

+∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,uε(y,τ))−K(x,t,s,y,τ,u(y,τ))|dydτ ds

� ε +q(x, t)∫ t

0

∫ s

0

∫ L

0g(y,τ)z(y,τ)dydτ ds. (2.4.18)

Now an application of Theorem 2.2.4 part (a7) to (2.4.18) yields (2.4.17).

In order to establish the dependency of solutions on parameters, we consider the equations

u(x,t) = fi(x,t)+∫ t

0

∫ s

0

∫ L

0K(x, t,s,y,τ,u(y,τ))dydτ ds, (2.4.19)

for (x, t) ∈ D, i = 1, 2; where fi ∈C(D,R), K ∈C(D× I ×D×R,R).

In the following theorem we formulate conditions for continuous dependence of solutions

of equations (2.4.19) on the functions involved therein.


(i) the function K in (2.4.19) satisfies the condition (2.4.9),

(ii) for i = 1, 2 the functions ui ∈C(D,R) are respectively the εi-approximate solutions of

(2.4.19) and let f (x,t) = | f1(x,t)− f2(x, t)|+ ε1 + ε2.

Then

|u1(x,t)−u2(x, t)| � f (x, t)+q(x,t)(∫ t

0

∫ s

0

∫ L

0g(y,τ) f (y,τ)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫ L


)

, (2.4.20)

for (x, t) ∈ D.


Proof. Let z(x,t) = |u1(x,t)−u2(x,t)|, (x,t) ∈ D. Following the proof of Theorem 1.3.5

in Chapter 1 and using the hypotheses, we obtain

z(x, t) � | f1(x,t)− f2(x,t)|+ ε1 + ε2

+∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,u1(y,τ))−K(x,t,s,y,τ ,u2(y,τ))|dydτ ds

� f (x,t)+q(x, t)∫ t

0

∫ s

0

∫ L

0g(y,τ)z(y,τ)dydτ ds. (2.4.21)

Applying Theorem 2.2.4 part (a7) to (2.4.21) yields (2.4.20), which shows that the solutions

of (2.4.19) depends continuously on functions on the right hand side of (2.4.19).

We next consider the equation (2.4.1) and the integral equation

w(x,t) = f (x,t)+∫ t

0

∫ s

0

∫ L

0K(x,t,s,y,τ,w(y,τ))dydτ ds, (2.4.22)

for (x, t) ∈ D, where f ∈C(D,R), K ∈C(D× I×D×R,R).

The following theorem holds.


(2.4.9). Then for every given solution w ∈ C(D,R) of equation (2.4.22) and any solution

u ∈C(D,R) of equation (2.4.1), the estimation

|u(x,t)−w(x, t)| � h(x,t)+q(x, t)(∫ t

0

∫ s

0

∫ L

0g(y,τ)h(y,τ)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫ L


)

(2.4.23)

holds for (x,t) ∈ D, where

h(x,t) = | f (x, t)− f (x,t)|

+∫ t

0

∫ η

0

∫ L

0|K(x,t,η ,z,σ ,w(z,σ))−K(x,t,η ,z,σ ,w(z,σ))|dzdσ dη , (2.4.24)

for (x, t) ∈ D.

Proof. Using the facts that u(x,t) and w(x, t) are respectively the solutions of equations

(2.4.1) and (2.4.22) and hypotheses, we have

|u(x, t)−w(x, t)| � | f (x,t)− f (x,t)|

+∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,u(y,τ))−K(x,t,s,y,τ,w(y,τ))|dydτ ds

� | f (x,t)− f (x,t)|


+∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,u(y,τ))−K(x,t,s,y,τ,w(y,τ))|dydτ ds

+∫ t

0

∫ s

0

∫ L

0|K(x,t,s,y,τ,w(y,τ))−K(x,t,s,y,τ,w(y,τ))|dydτ ds

� h(x,t)+q(x,t)∫ t

0

∫ s

0

∫ L

0g(y,τ)|u(y,τ)−w(y,τ)|dydτ ds. (2.4.25)

Now applying Theorem 2.2.4 part (a7) to (2.4.25) yields (2.4.23).

We note that, one can use the approach here to establish the results similar to those given

above to study the equation of the form

u(x,t) = h(x,t)+∫ x

0

∫ s

0

∫ t

0F(x,t,σ ,τ,u(σ ,τ))dτ dσ ds, (2.4.26)

under some suitable conditions on the functions involved in (2.4.26) and using Theo-

rem 2.2.2 part (a4). Here, we do not discuss the details.

2.5 Integral equation in three variables


u(x,y,z) = e(x,y,z)+(Lu)(x,y,z)+(Mu)(x,y,z), (2.5.1)

for x, y ∈ R+, z ∈ I = [a,b] (a < b), where

(Lu)(x,y,z) =∫ x

0

∫ y

0

∫ b

aF(x,y,z,s,t,r,u(s,t,r))dr dt ds, (2.5.2)

(Mu)(x,y,z) =∫ ∞

0

∫ ∞

0

∫ b

aH(x,y,z,s,t,r,u(s,t,r))dr dt ds, (2.5.3)

e, F, H are the given functions and u is the unknown function. The roots of the special

version of equation (2.5.1) can be found in the work of Lovelady [64] in the field of partial

differential equations, see also [5,18,132]. The main aim of the present section is to offer

some basic properties of solutions of equation (2.5.1), recently studied in [112]. Let G be

as defined in Section 2.3 and E3 = {(x,y,z,s,t,r) ∈ G2 : 0 � s � x < ∞, 0 � t � y < ∞,

z, r ∈ I}. Throughout, we assume that e ∈C(G,R), F ∈C(E3 ×R,R), H ∈C(G2 ×R,R).

Let Ω be the space of functions φ ∈C(G,R) which fulfil the condition

|φ(x,y,z)| = O(exp(λ (x+ y+ |z|))), (2.5.4)

for (x,y,z) ∈ G, where λ > 0 is a constant. In the space Ω we define the norm

|φ |Ω = sup(x,y,z)∈G

[|φ(x,y,z)|exp(−λ (x+ y+ |z|))]. (2.5.5)


It is easy to see that Ω with the norm defined in (2.5.5) is a Banach space and

|φ |Ω � N, (2.5.6)


First, we formulate the following theorem concerning the existence of a unique solution of

equation (2.5.1).


(i) the functions F, H in equation (2.5.1) satisfy the conditions

|F(x,y,z,s,t,r,u)−F(x,y,z,s,t,r,u)| � p(x,y,z,s, t,r)|u−u|, (2.5.7)

|H(x,y,z,s,t,r,u)−H(x,y,z,s,t,r,u)| � q(x,y,z,s, t,r)|u−u|, (2.5.8)

where p ∈C(E3,R+), q ∈C(G2,R+),


(j1) there exists a nonnegative constant α such that α < 1 and∫ x

0

∫ y

0

∫ b

ap(x,y,z,s,t,r)exp(λ (s+ t + |r|))dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

aq(x,y,z,s,t,r)exp(λ (s+ t + |r|))dr dt ds

� α exp(λ (x+ y+ |z|)), (2.5.9)

(j2) there exists a nonnegative constant β such that

|e(x,y,z)+(L0)(x,y,z)+(M0)(x,y,z)| � β exp(λ (x+ y+ |z|)), (2.5.10)

where e, L, M are as in equation (2.5.1).

Then the equation (2.5.1) has a unique solution u(x,y,z) on G in Ω.

The proof is analogous to the proof of Theorem 1.3.1. Here, we omit the details.

Next, we shall give the following theorem concerning the uniqueness of solutions of equa-

tion (2.5.1).

Theorem 2.5.2. Suppose that the functions F, H in equation (2.5.1) satisfy the conditions

|F(x,y,z,s,t,r,u)−F(x,y,z,s,t,r,u)| � q(x,y,z) f (s,t,r)|u−u|, (2.5.11)

|H(x,y,z,s, t,r,u)−H(x,y,z,s,t,r,u)| � c(x,y,z)g(s,t,r)|u−u|, (2.5.12)

where q, f , c, g ∈C(G,R+). Let α1, D1, A1, B1 be as in Theorem 2.3.4 part (b7). Then the

equation (2.5.1) has at most one solution on G.


Proof. Let u1(x,y,z) and u2(x,y,z) be two solutions of equation (2.5.1). Then by using

the hypotheses, we have

|u1(x,y,z)−u2(x,y,z)|

�∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s, t,r,u1(s, t,r))−F(x,y,z,s,t,r,u2(s,t,r))|dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

a|H(x,y,z,s,t,r,u1(s,t,r))−H(x,y,z,s,t,r,u2(s,t,r))|dr dt ds

� q(x,y,z)∫ x

0

∫ y

0

∫ b

af (s,t,r)|u1(s,t,r)−u2(s,t,r)|dr dt ds

+c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)|u1(s,t,r)−u2(s, t,r)|dr dt ds. (2.5.13)

Here, it is easy to see that A1(x,y,z) and D1 defined in (2.3.16) and (2.3.18) reduces to

A1(x,y,z) = 0 and D1 = 0. Now a suitable application of Theorem 2.3.4 part (b7) to (2.5.13)

yields |u1(x,y,z)−u2(x,y,z)| � 0, and hence u1(x,y,z) = u2(x,y,z). Thus, there is at most

one solution to equation (2.5.1) on G.

The following theorems deal with the estimates on the solution of equation (2.5.1).

Theorem 2.5.3. Suppose that the functions F, H in equation (2.5.1) satisfy the conditions

|F(x,y,z,s,t,r,u)| � q(x,y,z) f (s,t,r)|u|, (2.5.14)

|H(x,y,z,s,t,r,u)| � c(x,y,z)g(s,t,r)|u|, (2.5.15)

where q, f , c, g ∈C(G,R+). Let α1, B1 be as in Theorem 2.3.4 part (b7) and

D2 =1

1−α1

∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)A2(s, t,r)dr dt ds, (2.5.16)

where A2(x,y,z) is defined by the right hand side of (2.3.16) by replacing p(x,y,z) by

|e(x,y,z)|. If u(x,y,z) is any solution of equation (2.5.1) on G, then

|u(x,y,z)| � A2(x,y,z)+D2B1(x,y,z), (2.5.17)

for (x,y,z) ∈ G.

Proof. Using the fact that u(x,y,z) is a solution of equation (2.5.1) and hypotheses, we

have

|u(x,y,z)| � |e(x,y,z)|+∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s,t,r,u(s, t,r))|dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

a|H(x,y,z,s,t,r,u(s,t,r))|dr dt ds

� |e(x,y,z)|+q(x,y,z)∫ x

0

∫ y

0

∫ b

af (s,t,r)|u(s,t,r)|dr dt ds

+c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)|u(s,t,r)|dr dt ds. (2.5.18)

Now, an application of Theorem 2.3.4 part (b7) to (2.5.18) yields (2.5.17).


Theorem 2.5.4. Suppose that the functions F,H in equation (2.5.1) satisfy the conditions

(2.5.11), (2.5.12). Let α1, B1 be as in Theorem 2.3.4 part (b7) and

e0(x,y,z) =∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s,t,r,e(s,t,r))|dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

a|H(x,y,z,s,t,r,e(s, t,r))|dr dt ds, (2.5.19)

D3 =1

1−α1

∫ ∞

0

∫ ∞

0

∫ b



e0(x,y,z). If u(x,y,z) is any solution of equation (2.5.1) on G, then

|u(x,y,z)− e(x,y,z)| � A3(x,y,z)+D3B1(x,y,z), (2.5.21)

for (x,y,z) ∈ G.

Proof. Using the fact that u(x,y,z) is a solution of equation (2.5.1) and the hypotheses,

we have

|u(x,y,z)− e(x,y,z)|

�∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s,t,r,u(s, t,r))−F(x,y,z,s,t,r,e(s,t,r))|dr dt ds

+∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s,t,r,e(s,t,r))|dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

a|H(x,y,z,s,t,r,u(s,t,r))−H(x,y,z,s,t,r,e(s,t,r))|dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

a|H(x,y,z,s,t,r,e(s,t,r))|dr dt ds

� e0(x,y,z)+q(x,y,z)∫ x

0

∫ y

0

∫ b

af (s,t,r)|u(s,t,r)− e(s,t,r)|dr dt ds

+c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)|u(s,t,r)− e(s,t,r)|dr dt ds. (2.5.22)

Applying Theorem 2.3.4 part (b7) to (2.5.22), we get (2.5.21).

Now, we present the following theorem which deals with the estimate on the difference

between the solutions of equation (2.5.1) and the equation of the form

v(x,y,z) = e(x,y,z)+∫ x

0

∫ y

0

∫ b

aF(x,y,z,s, t,r,v(s,t,r))dr dt ds, (2.5.23)

for (x,y,z) ∈ G, where the functions e,F are as in equation (2.5.1).


Theorem 2.5.5. Suppose that the functions F, H in equations (2.5.1), (2.5.23) satisfy

the conditions (2.5.11), (2.5.12) and H(x,y,z,s,t,r,0) = 0. Let v(x,y,z) be a solution of

equation (2.5.23) on G such that |v(x,y,z)| � Q, where Q � 0 is a constant. Let α1, B1 be

as in Theorem 2.3.4 part (b7) and

p(x,y,z) = Qc(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)dr dt ds, (2.5.24)

D4 =1

1−α1

∫ ∞

0

∫ ∞

0

∫ b



p(x,y,z). If u(x,y,z) is a solution of equation (2.5.1) on G, then

|u(x,y,z)− v(x,y,z)| � A4(x,y,z)+D4B1(x,y,z), (2.5.26)

for (x,y,z) ∈ G.

Proof. Using the facts that u(x,y,z) and v(x,y,z) are the solutions of equations (2.5.1) and

(2.5.23) and hypotheses, we observe that

|u(x,y,z)− v(x,y,z)|

�∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s,t,r,u(s, t,r))−F(x,y,z,s,t,r,v(s,t,r))|dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

a|H(x,y,z,s,t,r,u(s,t,r))−H(x,y,z,s,t,r,v(s,t,r))|dr dt ds

+∫ ∞

0

∫ ∞

0

∫ b

a|H(x,y,z,s, t,r,v(s,t,r))−H(x,y,z,s,t,r,0)|dr dt ds

� q(x,y,z)∫ x

0

∫ y

0

∫ b

af (s,t,r)|u(s,t,r)− v(s,t,r)|dr dt ds

+c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)|u(s,t,r)− v(s,t,r)|dr dt ds

+c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r) |v(s,t,r)| dr dt ds

� p(x,y,z)+q(x,y,z)∫ x

0

∫ y

0

∫ b

af (s,t,r)|u(s,t,r)− v(s,t,r)|dr dt ds

+c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)|u(s,t,r)− v(s,t,r)|dr dt ds. (2.5.27)

Applying Theorem 2.3.4 part (b7) to (2.5.27), we get (2.5.26).


In concluding we note that the idea used in this section can be used to formulate the re-

sults similar to those given above for the equations of the form (2.5.1) when the operator

(Lu)(x,y,z) defined in (2.5.2) is replaced by∫ ∞

x

∫ ∞

y

∫ b

aF(x,y,z,s,t,r,u(s,t,r))dr dt ds,

or∫ x

0

∫ ∞

y

∫ b

aF(x,y,z,s,t,r,u(s,t,r))dr dt ds.

Furthermore, one can formulate results on the continuous dependence of solutions and ε-

approximate solutions of equations of the form (2.5.1) by making use of suitable inequali-

ties given in Section 2.3 or their variants. We leave the details of such results to the reader

to fill in where needed.

2.6 Hyperbolic-type Fredholm integrodifferential equation

In this section we shall be concerned with the hyperbolic type Fredholm integrodifferential

equation

D2D2u(x,y,z) = F(x,y,z,u(x,y,z),D1u(x,y,z),D2u(x,y,z),(Hu)(x,y,z)), (2.6.1)

with the given data

u(x,0,z) = σ(x,z), u(0,y,z) = τ(y,z), (2.6.2)

for x, y ∈ R+, z ∈ I = [a,b] ⊂ R (a < b), where

(Hu)(x,y,z) =∫ b

aK(x,y,z,r,u(x,y,r),D1u(x,y,r),D2u(x,y,r))dr,

F, K are the given functions and u is the unknown function. Obviously, (H0)(x,y,z)

=∫ b

a K(x,y,z,r,0,0,0)dr. The origin of problem (2.6.1)–(2.6.2) can be traced back in

the work of Lovelady [64], who studied the existence and uniqueness of solutions of

special form of equation (2.6.1) with the given data in (2.6.2). Inspired by the results

in [64], recently in [111] the present author has studied some basic aspects of solu-

tions of problem (2.6.1)–(2.6.2). Our main goal here is to present the results given in

[111], which deals with some important qualitative properties of solutions of problem

(2.6.1)–(2.6.2) and its special version. Throughout, we assume that F ∈ C(G ×R4,R),

K ∈ C(G × I × R3,R), σ ,σx,τ,τy ∈ C(R+ × I,R), in which G is defined as in Sec-

tion 1.3. By a solution of problem (2.6.1)–(2.6.2) we mean a function u ∈ C(G,R)

which satisfy the equations (2.6.1), (2.6.2). For u, D1u, D2u ∈ C(G,R), we denote by


|u(x,y,z)|0 = |u(x,y,z)|+ |D1u(x,y,z)|+ |D2u(x,y,z)|. Let V be the space of functions

u, D1u, D2u ∈C(G,R) which fulfil the condition

|u(x,y,z)|0 = O(

exp(λ (x+ y+ |z|))), (2.6.3)

for (x,y,z) ∈ G, where λ > 0 is a constant. In the space V we define the norm

|u|V = sup(x,y,z)∈G

[|u(x,y,z)|0 exp(−λ (x + y+ |z|))]. (2.6.4)

It is easy to see that V with norm defined in (2.6.4) is a Banach space and

|u|V � N, (2.6.5)


The following theorem ensures the existence of a unique solution to problem (2.6.1)–

(2.6.2).


(i) the functions F, K in (2.6.1) satisfy the conditions

|F(x,y,z,u1,u2,u3,u4)−F(x,y,z,u1,u2,u3,u4)|

� L(x,y,z) [|u1 −u1|+ |u2 −u2|+ |u3 −u3|+ |u4 −u4|] , (2.6.6)

|K(x,y,z,r,u1,u2,u3)−K(x,y,z,r,u1,u2,u3)|

� M(x,y,z,r)[|u1 −u1|+ |u2 −u2|+ |u3 −u3|], (2.6.7)

where L ∈C(G,R+),M ∈C(G× I,R+),


( j1) there exist nonnegative constants αi (i = 1, 2, 3) such that∫ x

0

∫ y

0P(s,t,z)dt ds � α1 exp(λ (x+ y+ |z|)), (2.6.8)

∫ y

0P(x,t,z)dt � α2 exp(λ (x+ y+ |z|)), (2.6.9)

∫ x

0P(s,y,z)ds � α3 exp(λ (x+ y+ |z|)), (2.6.10)

where

P(x,y,z) = L(x,y,z)

[

exp(λ (x + y+ |z|))+∫ b

aM(x,y,z,r)exp(λ (x + y+ |r|))dr

]

,


(j2) there exist nonnegative constants βi (i = 1, 2, 3) such that

|σ(x,z)+ τ(y,z)−σ(0,z)|+∫ x

0

∫ y

0|F(s,t,z,0,0,0,(H0)(s,t,z))|dt ds

� β1 exp(λ (x + y+ |z|)), (2.6.11)

|σx(x,z)|+∫ y

0|F(x,t,z,0,0,0,(H0)(x,t,z))|dt � β2 exp(λ(x + y+ |z|)), (2.6.12)

|τy(y,z)|+∫ x

0|F(s,y,z,0,0,0,(H0)(s,y,z))|ds � β3 exp(λ(x + y+ |z|)), (2.6.13)

where σ , τ are as in (2.6.2).

If α = α1 + α2 + α3 < 1, then the problem (2.6.1)–(2.6.2) has a unique solution u(x,y,z)

on G in V .

Proof. Let u ∈V and define the operator T by

(Tu)(x,y,z) = σ(x,z)+ τ(y,z)−σ(0,z)

+∫ x

0

∫ y

0F(s,t,z,u(s,t,z),D1u(s,t,z),D2u(s, t,z),(Hu)(s,t,z))dt ds. (2.6.14)

First, we shall show that Tu maps V into itself. Evidently, Tu is continuous on G and

Tu ∈ R. We verify that (2.6.3) is fulfilled. From (2.6.14), hypotheses and (2.6.5), we have

|(Tu)(x,y,z)| � |σ(x,z)+ τ(y,z)−σ(0,z)|

+∫ x

0

∫ y

0|F(s,t,z,u(s,t,z),D1u(s,t,z),D2u(s,t,z),(Hu)(s,t,z))

−F(s,t,z,0,0,0,(H0)(s,t,z))|dt ds

+∫ x

0

∫ y

0|F(s,t,z,0,0,0,(H0)(s,t,z))|dt ds

� β1 exp(λ (x+ y+ |z|))

+∫ x

0

∫ y

0L(s,t,z)

[

|u(s, t,z)|0 +∫ b

aM(s, t,z,r)|u(s,t,r)|0dr

]

dt ds

� β1 exp(λ (x + y+ |z|))+ |u|V∫ x

0

∫ y

0P(s, t,z)dt ds

� [β1 +Nα1]exp(λ (x+ y+ |z|)). (2.6.15)

Differentiating both sides of (2.6.14) with respect to x, using hypotheses and (2.6.5), we

have

|D1(Tu)(x,y,z)| � |σx(x,z)|+∫ y

0|F(x,t,z,u(x, t,z),D1u(x, t,z),D2u(x,t,z),(Hu)(x, t,z))


−F(x,t,z,0,0,0,(H0)(x,t,z))|dt +∫ y

0|F(x,t,z,0,0,0,(H0)(x,t,z))|dt

� β2 exp(λ (x+ y+ |z|))+ |u|V∫ y

0P(x, t,z)dt

� [β2 +Nα2]exp(λ (x+ y+ |z|)). (2.6.16)

Similarly, we obtain

|D2(Tu)(x,y,z)| � [β3 +Nα3]exp(λ (x+ y+ |z|)). (2.6.17)

From (2.6.15)–(2.6.17), we observe that

|Tu|V � [β1 +β1 +β1 +Nα].

This shows that T maps V into itself.

Next, we verify that the operator T is a contraction map. Let u, u ∈ V . From (2.6.14) and

using the hypotheses, we have

|(Tu)(x,y,z)− (T u)(x,y,z)|

�∫ x

0

∫ y

0|F(s,t,z,u(s,t,z),D1u(s, t,z),D2u(s,t,z),(Hu)(s,t,z))

−F(s,t,z,u(s,t,z),D1u(s, t,z),D2u(s,t,z),(Hu)(s,t,z))|dt ds

� |u−u|V∫ x

0

∫ y

0P(s,t,z)dt ds

� |u−u|V α1 exp(λ(x+ y+ |z|)). (2.6.18)

Similarly, differentiating both sides of (2.6.14), with respect to x and with respect to y and

using hypotheses, we obtain∣∣D1(Tu)(x,y,z)−D1(T u)(x,y,z)

∣∣ � |u−u|V α2 exp(λ (x+ y+ |z|)) (2.6.19)

and

|D2(Tu)(x,y,z)−D2(T u)(x,y,z)| � |u−u|V α3 exp(λ (x+ y+ |z|)). (2.6.20)

From (2.6.18)–(2.6.20), we obtain

|Tu−T u|V � α|u−u|V . (2.6.21)

Since α < 1, from (2.6.21), it follows from Banach fixed point theorem (see [51, p. 372])

that T has a unique fixed point in V . The fixed point of T is however a solution of problem

(2.6.1)–(2.6.2). The proof is complete.


Remark 2.6.1. We note that in [64], the existence and uniqueness of solutions to problem

(2.6.1)–(2.6.2) have been analyzed as an application of the fixed point theorem established

therein to study perturbed differential equations. The result in Theorem 2.6.1 ensure the

existence of a unique solution to more general problem (2.6.1)–(2.6.2) under different con-

ditions from those used in [64].

Below, we study some fundamental qualitative properties of solutions of a hyperbolic type

Fredholm integrodifferential equation of the form (see [111])

D2D1u(x,y,z) = f (x,y,z,u(x,y,z),(hu)(x,y,z)), (2.6.22)

with the given data (2.6.2) for x, y ∈ R+,z ∈ I, where

(hu)(x,y,z) =∫ b

ak(x,y,z,r,u(x,y,r))dr,

f ∈C(G×R2,R), k ∈C(G× I ×R,R).

First, we shall give the following theorem which deals with the uniqueness of solutions of

problem (2.6.22)–(2.6.2) on G in R.

Theorem 2.6.2. Assume that the functions f , k in (2.6.22) satisfy the conditions

| f (x,y,z,u,v)− f (x,y,z,u,v)| � p(x,y,z)|u−u|+ |v− v|, (2.6.23)

|k(x,y,z,r,u)− k(x,y,z,r,u)| � q(x,y,z,r)|u−u|, (2.6.24)

where p ∈C(G,R+), q ∈C(G× I,R+) and∫ ∞

0

∫ ∞

0p(s,t,z)dt ds < ∞,

∫ ∞

0

∫ ∞

0

∫ b

aq(s,t,z,r)H(s,t,r)dr dt ds < ∞, (2.6.25)

for z ∈ I, where H(x,y,z) is given by (2.3.27).

Then the problem (2.6.22)–(2.6.2) has at most one solution on G in R.

Proof. Let u1(x,y,z) and u2(x,y,z) be two solutions of problem (2.6.22)–(2.6.2) on G.

Using these facts and the hypotheses, we have

|u1(x,y,z)−u2(x,y,z)| �∫ x

0

∫ y

0

∣∣ f (s,t,z,u1(s,t,z),(hu1)(s,t,z))

− f (s,t,z,u2(s,t,z),(hu2)(s,t,z))∣∣dt ds

�∫ x

0

∫ y

0

[p(s,t,z)|u1(s,t,z)−u2(s,t,z)|+ |(hu1)(s,t,z)− (hu2)(s,t,z|

]dt ds


�∫ x

0

∫ y

0

[

p(s,t,z)|u1(s, t,z)−u2(s,t,z)|+∫ b

aq(s,t,z,r)|u1(s,t,r)−u2(s, t,r)|dr

]

dt ds.

(2.6.26)

Now an application of Theorem 2.3.5 (when c = 0) to (2.6.26) yields |u1(x,y,z) −u2(x,y,z)| � 0, which implies u1(x,y,z) = u2(x,y,z). Thus there is at most one solution

to the problem (2.6.22)–(2.6.2) on G in R.

Next, we shall give the following theorem which deals with the bound on the solution of

problem (2.6.22)–(2.6.2).

Theorem 2.6.3. Suppose That the functions f , k, σ , τ in (2.6.22)–(2.6.2) satisfy the con-

ditions

| f (x,y,z,u,v)| � p(x,y,z)|u|+ |v|, (2.6.27)

|k(x,y,z,r,u)| � q(x,y,z,r)|u| (2.6.28)

|σ(x,z)+ τ(y,z)−σ(0,z)| � c, (2.6.29)

where p ∈C(G,R+), q ∈C(G× I,R+), c � 0 is a constant. If u(x,y,z) is any solution of

problem (2.6.22)–(2.6.2) on G, then

|u(x,y,z)| � cH(x,y,z)exp(∫ x

0

∫ y

0

∫ b

aq(s, t,z,r)H(s,t,r)dr dt ds

)

, (2.6.30)

for (x,y,z) ∈ G, where H(x,y,z) is given by (2.3.27).

Proof. Using the fact that u(x,y,z) is a solution of problem (2.6.22)–(2.6.2) and hypothe-

ses, we have

|u(x,y,z)| � |σ(x,z)+ τ(y,z)−σ(0,z)|+∫ x

0

∫ y

0| f (s,t,z,u(s,t,z),(hu)(s,t,z))|dt ds

� c+∫ x

0

∫ y

0

[

p(s,t,z)|u(s,t,z)|+∫ b

aq(s,t,z,r)|u(s,t,r)|dr

]

dt ds. (2.6.31)

Applying Theorem 2.3.5 to (2.6.31) yields (2.6.30).

The following theorem deals with the dependency of solutions of equation (2.6.22) on given

data.

Theorem 2.6.4. Suppose that the functions f , k in (2.6.22) satisfy the conditions (2.6.23),

(2.6.24). Let u(x,y,z) and v(x,y,z) be the solutions of equation (2.6.22) with the given data

(2.6.2) and

v(x,0,z) = σ(x,z), v(0,y,z) = τ(x,z), (2.6.32)


respectively, where σ , τ, σ x, τy ∈C(R+ × I,R) and∣∣σ(x,z)+ τ(y,z)−σ(0,z)−{σ(x,z)+ τ(y,z)−σ(0,z)}

∣∣ � d, (2.6.33)

where d � 0 is a constant. Then

|u(x,y,z)− v(x,y,z)| � dH(x,y,z)exp(∫ x

0

∫ y

0

∫ b


)

, (2.6.34)

for (x,y,z) ∈ G, where H(x,y,z) is given by (2.3.27).

Proof. Using the facts that u(x,y,z) and v(x,y,z) are the solutions of problems (2.6.22)–

(2.6.2) and (2.6.22)–(2.6.32) and the hypotheses, we have

|u(x,y,z)− v(x,y,z)| � |σ(x,z)+ τ(y,z)−σ(0,z)−{σ(x,z)+ τ(y,z)−σ(0,z)}|

+∫ x

0

∫ y

0| f (s,t,z,u(s,t,z),(hu)(s,t,z))− f (s,t,z,v(s,t,z),(hv)(s,t,z))|dt ds

� d +∫ x

0

∫ y

0

[

p(s, t,z)|u(s,t,z)− v(s,t,z)|

+∫ b

aq(s,t,z,r)|u(s,t,r)− v(s,t,r)|dr

]

dt ds. (2.6.35)

Now an application of Theorem 2.3.5 to (2.6.35) yields the estimate (2.6.34), which shows

the dependency of solutions of equation (2.6.22) on given data.

Remark 2.6.2. In general one may expect that the analysis used to study the properties of

solutions of problem (2.6.22)–(2.6.2) in Theorems 2.6.2–2.6.4 will also be equally useful

in the study of general problem (2.6.1)–(2.6.2). In fact, it involves the task of designing a

new inequality, similar to the one given in Theorem 2.3.5, which will allow applications in

the discussion of problem (2.6.1)–(2.6.2). Indeed, it is not an easy matter and any attempt

to extend the analysis to general problem (2.6.1)–(2.6.2) is of great interest and importance

and its detailed treatment is desired.

2.7 Miscellanea

2.7.1 Hacia and Kaczmarek [41]

(h1) Let u, w, a, b ∈C(E,R+) and b(x,y) > 0. If u(x,y) satisfies

u(x,y) � w(x,y)+b(x,y)∫ x

0

∫ y

0a(s,t)u(s,t)dsdt,

then

u(x,y) � b(x,y)h(x,y)exp(∫ x

0

∫ y

0a(s, t)b(s,t)dsdt

)

,


where

h(x,y) = sup{

w(s,t)b(s,t)

: 0 � s � x, 0 � t � y}

.

(h2) Let u, w ∈C(E,R+), K ∈C(E2,R+) and

p(x,y) = sup0�s�x0�t�y

K(x,y,s,t) > 0.

If u(x,y) satisfies

u(x,y) � w(x,y)+∫ x

0

∫ y

0K(x,y,s,t)u(s, t)dsdt,

then

u(x,y) � p(x,y)sup{

w(s, t)p(s,t)

: 0 � s � x, 0 � t � y}

exp(∫ x

0

∫ y

0p(s,t)dsdt

)

.

2.7.2 Hacia [40]

(h3) Let f be a continuous function in D = {(x, t) : a � x � b, t � 0} and K be nonnegative

and continuous in Ω = {(x,t,y,s) : a � x, y � b, 0 � s � t < ∞}. If the continuous function u

satisfies the inequality

u(x, t) � f (x,t)+∫ t

0

∫ b

aK(x,t,y,s)u(y,s)dyds,


u(x,t) � f (x,t)+∫ t

0

∫ b

ar(x,t,y,s) f (y,s)dyds,

where

r(x,t,y,s) =∞

∑n=0

Kn(x,t,y,s),

is the resolvent kernel defined by formulas

K0(x, t,y,s) = K(x,t,y,s),

Kn(x,t,y,s) =∫ t

s

∫ b

aK(x,t, p,q)Kn−1(p,q,y,s)d pdq,

for n = 1, 2, . . ..

(h4) Let u, f , A, B be continuous in D and A ·B is nonnegative. If u satisfies the inequality

u(x,t) � f (x,t)+A(x,t)∫ t

0

∫ b

aB(y,s)u(y,s)dyds,


u(x,t) � f (x, t)+A(x,t)∫ t

0

∫ b

aB(y,s)exp

[∫ t

s

∫ b

aA(z,τ)B(z,τ)dzdτ

]

f (y,s)dyds,

for (x, t) ∈ D.


2.7.3 Hacia [40]

Consider the following nonlinear integral equation of the Volterra-Fredholm-type

u(x,t) = f (x, t)+∫ t

0

∫ b

aK(x, t,y,s,u(y,s))dyds, (2.7.1)

with assumptions:

(γ1) f and K are continuous in D and Ω×R respectively,

(γ2) |K(x,t,y,s,u)| � B(y,s)|u|,(γ3) |K(x,t,y,s,u)−K(x,t,y,s,u)| � B(y,s)|u−u|,where B is continuous and integrable in D.

(h5) If assumptions (γ1) and (γ2) are satisfied, then a solution u(x,t) of equation (2.7.1) is

bounded in D and

|u(x,t)| � ψ(t)exp(∫ t

0

∫ b

aB(y,s)dyds

)

,

where ψ(t) = sup{| f (x,t)| : a � x � b, 0 � s � t}.

(h6) If assumptions (γ1) and (γ3) are satisfied, then the equation (2.7.1) has at most one

solution, which is stable.



u(x,t) = h(x,t)+∫ x

0

∫ s

0

∫ t

0F(x,t,σ ,τ,u(σ ,τ))dτ dσ ds, (2.7.2)

where h ∈C(R2+,R), F ∈C(R4

+ ×R,R) and u is the unknown function.

(h7) Suppose that the functions F, h in equation (2.7.2) satisfy the conditions

|F(x,t,σ ,τ,u)−F(x,t,σ ,τ,v)| � q(x,t)r(σ ,τ)|u− v|, (2.7.3)∣∣∣∣h(x,t)+

∫ x

0

∫ s

0

∫ t

0F(x,t,σ ,τ,0)dτ dσ ds

∣∣∣∣ � p(x,t),

where p, q, r ∈ C(R2+,R+). If u(x,t) is any solution of equation (2.7.2) for (x,t) ∈ R

2+,

then

|u(x,t)| � p(x,t)+q(x,t)(∫ x

0

∫ s

0

∫ t

0r(σ ,τ)p(σ ,τ)dτ dσ ds

)

×exp(∫ x

0

∫ s

0

∫ t


)

,

for (x, t) ∈ R2+.


(h8) Let ui(x,t) (i = 1, 2) be respectively εi-approximate solutions of equation (2.7.2) for

(x, t) ∈ R2+ i.e.,∣∣∣∣ui(x, t)−

{

h(x, t)+∫ x

0

∫ s

0

∫ t

0F(x,t,σ ,τ,ui(σ ,τ))dτ dσ ds

}∣∣∣∣ � εi,

for (x,t) ∈ R2+. Suppose that the function F in equation (2.7.2) satisfies the condition

(2.7.3). Then

|u1(x,t)−u2(x,t)| � (ε1 + ε2)[

1+q(x,t)(∫ x

0

∫ s

0

∫ t


)

×exp(∫ x

0

∫ s

0

∫ t


)]

,

for (x, t) ∈ R2+.



u(x,y,z) = h(x,y,z)+∫ x

0

∫ y

0

∫ b

aF(x,y,z,s,t,r,u(s, t,r))dr dt ds, (2.7.4)

for (x,y,z) ∈ G, where h ∈C(G,R), F ∈C(G2 ×R,R) and G is as defined in section 2.3.

(h9) Suppose that the function F in equation (2.7.4) satisfies the condition

|F(x,y,z,s,t,r,u)−F(x,y,z,s, t,r,v)| � q(x,y,z) f (s,t,r)|u− v|, (2.7.5)

where q, f ∈C(G,R+). If u(x,y,z) is any solution of equation (2.7.4) on G, then

|u(x,y,z)−h(x,y,z)| � d(x,y,z)+q(x,y,z)(∫ x

0

∫ y

0

∫ b

af (s,t,r)d(s,t,r)dr dt ds

)

×exp(∫ x

0

∫ y

0

∫ b


)

,


d(x,y,z) =∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s,t,r,h(s,t,r))|dr dt ds.

(h10) Suppose that the function F in equation (2.7.4) satisfies the condition (2.7.5). Then

for every given solution v ∈C(G,R) of equation

v(x,y,z) = g(x,y,z)+∫ x

0

∫ y

0

∫ b

aL(x,y,z,s,t,r,v(s,t,r))dr dt ds,

where g ∈C(G,R), L ∈C(G2×R,R) and any solution u ∈C(G,R) of equation (2.7.4), the

estimation

|u(x,y,z)− v(x,y,z)| � [h(x,y,z)+d(x,y,z)]


+q(x,y,z)(∫ x

0

∫ y

0

∫ b

af (s,t,r)[h(s,t,r)+d(s, t,r)]dr dt ds

)

×exp(∫ x

0

∫ y

0

∫ b


)

,

holds for (x,y,z) ∈ G, in which

h(x,y,z) = |h(x,y,z)−g(x,y,z)|,

d(x,y,z) =∫ x

0

∫ y

0

∫ b

a|F(x,y,z,s,t,r,v(s,t,r))−L(x,y,z,s,t,r,v(s,t,r))|dr dt ds,

for (x,y,z) ∈ G.


(h11) Let u, p,q ∈C(G,R+) and L ∈C (G×R+,R+) be such that

0 � L(x,y,z,u)−L(x,y,z,v) � M(x,y,z,v)(u− v), (2.7.6)

for u � v � 0, where M ∈C(G×R+,R+) and G is as defined in section 2.3. If


0

∫ y

0

∫ b

aL(s, t,r,u(s,t,r))dr dt ds,



0

∫ y

0

∫ b

aL(s,t,r, p(s, t,r))dr dt ds

)

×exp(∫ x

0

∫ y

0

∫ b

aM(s, t,r, p(s,t,r))q(s,t,r)dr dt ds

)

,

for (x,y,z) ∈ G.

(h12) Let u, f , g∈C(G,R+) and L is as defined in (h11), which verifies the condition (2.7.6)

and k � 0 is a real constant. If

u2(x,y,z) � k2 +2x∫

0

y∫

0

b∫

a

[ f (s,t,r)u(s,t,r)L(s,t,r,u(s,t,r))+g(s, t,r)u(s,t,r)]dr dt ds,


u(x,y,z) � n(x,y)+(∫ x

0

∫ y

0

∫ b

af (s, t,r)L(s,t,r,n(s,t))dr dt ds

)

×exp(∫ x

0

∫ y

0

∫ b

af (s,t,r)M(s,t,r,n(s, t))dr dt ds

)

,

for (x,y,z) ∈ G, where M ∈C(G×R+,R+) and

n(x,y) = k +∫ x

0

∫ y

0

∫ b

ag(σ ,τ,η)dηdτ dσ ,

for x, y ∈ R+.



(h13) Let u, p,q,c, f , g ∈C(G,R+) and suppose that


0

∫ ∞

y

∫ b


+c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s, t,r)u(s,t,r)dr dt ds,

for (x,y,z) ∈ G, where G is as defined in section 2.3. If

α3 =∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)B3(s,t,r)dr dt ds < 1,

then

u(x,y,z) � A3(x,y,z)+D3B3(x,y,z),


A3(x,y,z) = p(x,y,z)+q(x,y,z)(∫ x

0

∫ ∞

y

∫ b

af (s,t,r)p(s, t,r)dr dt ds

)

×exp(∫ x

0

∫ ∞

y

∫ b


)

,

B3(x,y,z) = c(x,y,z)+q(x,y,z)(∫ x

0

∫ ∞

y

∫ b

af (s,t,r)c(s, t,r)dr dt ds

)

×exp(∫ x

0

∫ ∞

y

∫ b


)

,

and

D3 =1

1−α3

∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)A3(s, t,r)dr dt ds.

(h14) Let u, p, c, g ∈C(G,R+) and suppose that

u(x,y,z) � p(x,y,z)+ c(x,y,z)∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)u(s,t,r)dr dt ds,


α0 =∫ ∞

0

∫ ∞

0

∫ b

ag(s,t,r)c(s,t,r)dr dt ds < 1,

then

u(x,y,z) � p(x,y,z)+ c(x,y,z){

11−α0

∫ ∞

0

∫ ∞

0

∫ b

ag(s, t,r)p(s,t,r)dr dt ds

}

,

for (x,y,z) ∈ G.



Consider the following integral equation

u(x,y,z) = h(x,y,z)+∫ x

0

∫ ∞

y

∫ b

aF(x,y,z,s,t,r,u(s, t,r))dr dt ds, (2.7.7)

with assumptions

(δ1) h ∈C(G,R), F ∈C(G2 ×R,R),

(δ2) |F(x,y,z,s,t,r,u)| � q(x,y,z) f (s,t,r)|u|,(δ3) |F(x,y,z,s,t,r,u)−F(x,y,z,s,t,r,v)| � q(x,y,z) f (s, t,r)|u− v|,where q, f ∈C(G,R+) and G is as defined in section 2.3.

(h15) Suppose that the assumptions (δ1) and (δ2) are satisfied. If u(x,y,z) is any solution

of equation (2.7.7) on G, then

|u(x,y,z)| � |h(x,y,z)|+q(x,y,z)(∫ x

0

∫ ∞

y

∫ b

af (s,t,r)|h(s,t,r)|dr dt ds

)

×exp(∫ x

0

∫ ∞

y

∫ b


)

,

for (x,y,z) ∈ G.

(h16) If assumptions (δ1) and (δ3) are satisfied, then the equation (2.7.7) has at most one

solution on G.

2.8 Notes

Various approaches are developed by different researchers for studying multidimensional

integral equations. Sections 2.2 and 2.3 deals with some basic integral inequalities

with explicit estimates in two and three variables, recently established by Pachpatte

[108,98,104,95,111,102] which will be equally important in handling the dynamic equa-

tions of various forms, when the earlier inequalities in the literature do not apply directly.

The material in sections 2.4 and 2.5 contains some fundamental qualitative properties of

solutions of various types of integral equations in two and three variables and is adapted

from Pachpatte [104,112]. The results included in section 2.6 are recently obtained by

Pachpatte in [111], which are motivated by the work of Lovelady in [64]. A detailed ac-

count including a comprehensive list of references related to such equations can be found

in books by Walter [134] and Appell, Kalitvin and Zabrejko [5]. Section 2.7 is devoted to

miscellanea related to the results given in earlier sections, which we hope will stimulate the

reader’s interest.


Chapter 3

Mixed integral equations and inequalities

3.1 Introduction

In the study of many basic models in parabolic differential equations which describe diffu-

sion or heat transfer phenomena and epidemiology, the integral equation of the form

u(t,x) = f (t,x)+∫ t

0

∫

Bk(t,x, s,y)g(u(s,y))dyds, (3.1.1)

where B is a bounded domain in Rn, t ∈ R+; f , k, g are given functions and u is the un-

known function, occur in a natural way, see [5,8,28,31,32,37-39,69,134 ]. The integral

equation (3.1.1) appears to be Volterra-type in t, and of Fredholm-type with respect to x

and hence it can be viewed as a mixed Volterra-Fredholm-type integral equation. In the

general case solving integral equation (3.1.1) is highly nontrivial problem and handling the

study of its qualitative properties need a fresh outlook. The method of integral inequali-

ties with explicit estimates serve as an important tool which provides valuable information

of various dynamic equations, without the need to know in advance the solutions explic-

itly. Recently in [99,103,105,107,116,119] explicit estimates on a number of new integral

inequalities are considered and used in various applications. In the present chapter, we

offer some fundamental mixed integral inequalities with explicit estimates established in

the above noted papers and also focus our attention on some basic qualitative aspects of

solutions of equations of the form (3.1.1). A particular feature of our approach here is that

it is elementary and provide some basic results for future advanced studies in the field.

3.2 Volterra-Fredholm-type integral inequalities I

In this section we present some basic integral inequalities with explicit estimates in-

vestigated in [99,103,105,116], which can be used as tools for handling the equations

97


like (3.1.1). In what follows, we denote by B a bounded domain in Rn defined by

B =n

∏i=1

[ai,bi] (ai < bi),

x = (x1, . . . ,xn) (xi ∈ R) is a variable point in B and dx = dx1 · · ·dxn.

For any continuous function z : B → R, we denote by∫

B z(x)dx the n-fold integral∫ bn

an

· · ·∫ b1

a1

z(x1, . . . ,xn)dx1 · · ·dxn.

Let D0 = R+ ×B, Ω = {(t,x,s,y) : 0 � s � t < ∞; x, y ∈ B} and denote by D1h(t,x,s,y)

the partial derivative of a function h(t,x,s,y) defined on Ω with respect to the first variable.

The first theorem deals with the inequalities given in [99].

Theorem 3.2.1. Let u, p, q, f ∈C (D0,R+).

(c1) Let L ∈C (D0 ×R+,R+) be such that

0 � L(t,x,u)−L(t,x,v) � M (t,x,v)(u− v), (3.2.1)

for u � v � 0, where M ∈C (D0 ×R+,R+). If

u(t,x) � p(t,x)+q(t,x)∫ t

0

∫

BL(s,y,u(s,y))dyds, (3.2.2)

for (t,x) ∈ D0, then


0

∫

BL(s,y, p(s,y))

× exp(∫ t

s

∫

BM(τ,z, p(τ,z))q(τ,z)dzdτ

)

dyds, (3.2.3)

for (t,x) ∈ D0.

(c2) If


0

∫

Bf (s,y)u(s,y)dyds, (3.2.4)



0

∫

Bf (s,y)p(s,y)

× exp(∫ t

s

∫

Bf (τ,z)q(τ,z)dzdτ

)

dyds, (3.2.5)

for (t,x) ∈ D0.

The inequalities established in [105] are given in the following theorems.

Mixed integral equations and inequalities 99

Theorem 3.2.2. Let u, p, q, f , g ∈C(D0,R+).

(c3) If


0

∫

B[ f (s,y)u(s,y)+g(s,y)]dyds, (3.2.6)



0

∫

B[ f (s,y)p(s,y)+g(s,y)]

× exp(∫ t

s

∫


)

dyds, (3.2.7)

for (t,x) ∈ D0.

(c4) Let c � 0 and 0 < α < 1 be real constants. If

u(t,x) � c+∫ t

0

∫

B[ f (s,y)u(s,y)+g(s,y)uα(s,y)]dyds, (3.2.8)


u(t,x) � exp(∫ t

0

∫

Bf (τ,z)dzdτ

)[

c1−α +(1−α)∫ t

0

∫

Bg(s,y)

×exp(

−(1−α)∫ s

0

∫

Bf (τ,z)dzdτ

)

dyds] 1

1−α, (3.2.9)

for (t,x) ∈ D0.

Remark 3.2.1. If we take g = 0 in (3.2.6), then the bound obtained in (3.2.7) reduces to

(3.2.5). In this case, we observe that the obtained result is a new variant of the inequality

in Corollary 4.3.1 given in [82, p. 329]. We note that the bound on the unknown function

u(t,x) involved in (3.2.8) when α �= 1, 1 < α < ∞ can be obtained by closely looking at

the proof of the inequality given in [82, Theorem 2.7.4, p. 153]. By taking (i) g = 0 and

(ii) f = 0 in (3.2.8), it is easy to see that the bound obtained in (3.2.9) reduces respectively

to

u(t,x) � cexp(∫ t

0

∫

Bf (τ,z)dzdτ

)

, (3.2.10)

and

u(t,x) �[

c1−α +(1−α)∫ t

0

∫

Bg(s,y)dyds

] 11−α

, (3.2.11)

for (t,x) ∈ D0.


Theorem 3.2.3. Let u, f , g ∈C(D0,R+) and k � 0, c � 1, β > 1 be real constants.

(c5) If

uβ (t,x) � kβ +β∫ t

0

∫

B

[f (s,y)u(s,y)+g(s,y)uβ (s,y)

]dyds, (3.2.12)


u(t,x) � exp(∫ t

0

∫

Bg(τ,z)dzdτ

)[

kβ−1 +(β −1)∫ t

0

∫

Bf (s,y)

×exp(

−(β −1)∫ s

0

∫

Bg(τ,z)dzdτ

)

dyds] 1

β−1, (3.2.13)

for (t,x) ∈ D0.

(c6) If u ∈C(D0,R1) and

u(t,x) � c+∫ t

0

∫

Bf (s,y)u(s,y) logu(s,y)dyds, (3.2.14)



0∫

B f (s,y)dyds), (3.2.15)

for (t,x) ∈ D0.

Remark 3.2.2. If we take g = 0 in (3.2.12), then the bound obtained in (3.2.13) reduces

to

u(t,x) �[

kβ−1 +(β −1)∫ t

0

∫

Bf (s,y)

] 1β−1

, (3.2.16)

for (t,x)∈D0. By taking g = 0 and β = 2 in part (c5), we get a new variant of the inequality

given in Theorem 5.8.1 in [82, p. 527].

The inequalities in the following theorem are established in [99].

Theorem 3.2.4. Let u, p, q, r, f , g ∈C(D0,R+).

(c7) Suppose that


0

∫

Bf (s,y)u(s,y)dyds

+ r(t,x)∫ ∞

0

∫

Bg(s,y)u(s,y)dyds, (3.2.17)

for (t,x) ∈ D0. If

d =∫ ∞

0

∫

Bg(s,y)K2(s,y)dyds < 1, (3.2.18)

then

u(t,x) � K1(t,x)+DK2(t,x), (3.2.19)


for (t,x) ∈ D0, where

K1(t,x) = p(t,x)+q(t,x)∫ t

0

∫

Bf (s,y)p(s,y)

×exp(∫ t

s

∫


)

dyds, (3.2.20)

K2(t,x) = r(t,x)+q(t,x)∫ t

0

∫

Bf (s,y)r(s,y)

×exp(∫ t

s

∫


)

dyds, (3.2.21)

and

D =1

1−d

∫ ∞

0

∫

Bg(s,y)K1(s,y)dyds. (3.2.22)

(c8) Suppose that

u(t,x) � p(t,x)+ r(t,x)∫ ∞

0

∫


for (t,x) ∈ D0. If

d0 =∫ ∞

0

∫

Bg(s,y)r(s,y)dyds < 1, (3.2.24)

then

u(t,x) � p(t,x)+ r(t,x){

11−d0

∫ ∞

0

∫

Bg(s,y)p(s,y)dyds

}

, (3.2.25)

for (t,x) ∈ D0.

Remark 3.2.3. By taking g = 0 in Theorem 3.2.4 part (c7), we get the inequality given in

Theorem 3.2.1 part (c2).

The next two theorems deal with the integral inequalities established in [116].

Theorem 3.2.5. Let u ∈C(D0,R+), k, D1k ∈C(Ω,R+) and c � 0 is a real constant.

(c9) If

u(t,x) � c+∫ t

0

∫

Bk(t,x,s,y)u(s,y)dyds, (3.2.26)



0A(σ ,x)dσ

)

, (3.2.27)


A(t,x) =∫

Bk(t,x,t,y)dy+

∫ t

0

∫

BD1k(t,x,s,y)dyds, (3.2.28)


for (t,x) ∈ D0.

(c10) Let g ∈C(R+,R+) be a nondecreasing function, g(u) > 0 on (0,∞). If

u(t,x) � c+∫ t

0

∫

Bk(t,x,s,y)g(u(s,y))dyds, (3.2.29)

for (t,x) ∈ D0, then for 0 � t � t1; t, t1 ∈ R+, x ∈ B,

u(t,x) � W−1[

W (c)+∫ t

0A(σ ,x)dσ

]

, (3.2.30)

where

W (r) =∫ r

r0

dsg(s)

, r > 0, (3.2.31)

r0 > 0 is arbitrary and W−1 is the inverse of W and A(t,x) is given by (3.2.28) and t1 ∈ R+

is chosen so that

W (c)+∫ t

0A(σ ,x)dσ ∈ Dom

(W−1) ,

for all t ∈ R+ lying in the interval 0 � t � t1 and x ∈ B.

Theorem 3.2.6. Let u ∈C(D0,R+); k, D1k ∈C(Ω,R+) and c � 0 is a real constant.

(c11) If

u2(t,x) � c+∫ t

0

∫



u(t,x) �√

c+12

∫ t

0A(σ ,x)dσ , (3.2.33)

for (t,x) ∈ D0, where A(t,x) is given by (3.2.28).

(c12) Let g(u) be as in part (c10). If

u2(t,x) � c+∫ t

0

∫

Bk(t,x,s,y)u(s,y)g(u(s,y))dyds, (3.2.34)


u(t,x) � W−1[

W(√

c)+

12

∫ t

0A(σ ,x)dσ

]

, (3.2.35)

where W, W−1, A(t,x) are as in part (c10) and t2 ∈ R+ is chosen so that

W(√

c)+

12

∫ t


(W−1),


The integral inequality established in [103] and its variant are given in the following theo-

rem.


Theorem 3.2.7. Let u, p, q, f , g ∈C(D0,R+).

(c13) If

u(t,x) � p(t,x)

+q(t,x)∫ t

0

∫

Bf (s,y)

[

u(s,y)+q(s,y)∫ s

0

∫

Bg(τ,z)u(τ,z)dzdτ

]

dyds, (3.2.36)



0

∫

Bp(s,y) [ f (s,y)+g(s,y)]

×exp(∫ t

s

∫

Bq(τ,z)[ f (τ,z)+g(τ,z)]dzdτ

)

dyds, (3.2.37)

for (t,x) ∈ D0.

(c14) If

u(t,x) � p(t,x)

+q(t,x)∫ t

0

∫ s

0

∫

Bf (τ,y)

[

u(τ,y)+q(τ,y)∫ τ

0

∫ σ

0

∫

Bg(ξ ,z)u(ξ ,z)dzdξ

]

dydτ, (3.2.38)


u(t,x) � p(t,x)+q(t,x)(∫ t

0

∫ s

0

∫

Bp(τ,y)[ f (τ,y)+g(τ,y)]dydτ

)

×exp(∫ t

0

∫ s

0

∫

Bq(τ,y)[ f (τ,y)+g(τ,y)]dydτ

)

, (3.2.39)

for (t,x) ∈ D0.

Proofs of Theorems 3.2.1–3.2.7. The proofs resemble one another, we give the details for

(c1), (c3), (c5), (c7), (c9), (c11), (c13) only; the proofs of other inequalities can be completed

by following the proofs of the above inequalities, see also [82,87].

(c1) Setting

e(s) =∫

BL(s,y,u(s,y))dy, (3.2.40)



0e(s)ds. (3.2.41)

Define

m(t) =∫ t

0e(s)ds, (3.2.42)


then m(0) = 0 and

u(t,x) � p(t,x)+q(t,x)m(t). (3.2.43)

From (3.2.42), (3.2.40), (3.2.43) and (3.2.1), we observe that

m′(t) = e(t) =∫

BL(t,y,u(t,y))dy �

∫

BL(t,y, p(t,y)+q(t,y)m(t))dy

=∫

B{L(t,y, p(t,y)+q(t,y)m(t))−L(t,y, p(t,y))+L(t,y, p(t,y))}dy

�∫

BM(t,y, p(t,y))q(t,y)m(t)dy+

∫

BL(t,y, p(t,y))dy

= m(t)∫

BM(t,y, p(t,y))q(t,y)dy+

∫

BL(t,y, p(t,y))dy. (3.2.44)

The inequality (3.2.44) implies (see [82, Theorem 1.3.2])

m(t) �∫ t

0

∫

BL(s,y, p(s,y))exp

(∫ t

s

∫

BM(τ,z, p(τ,z))q(τ,z)dzdτ

)

dyds, (3.2.45)

for (t,x) ∈ D0. Using (3.2.45) in (3.2.43), we get the required inequality in (3.2.3).

(c3) Setting

e1(s) =∫

B[ f (s,y)u(s,y)+g(s,y)]dy,

and following the proof of part (c1) with suitable changes we get the required inequality in

(3.2.7).

(c5) Introducing the notation

e2(s) =∫

B

[f (s,y)u(s,y)+g(s,y)uβ (s,y)

]dy, (3.2.46)

in (3.2.12), we get

uβ (t,x) � kβ +β∫ t

0e2(s)ds.

Let k > 0 and define

z(t) = kβ +β∫ t

0e2(s), (3.2.47)

then z(0) = kβ and

uβ (t,x) � z(t), (3.2.48)

for (t,x) ∈ D0. From (3.2.47), (3.2.46), (3.2.48), we observe that

z′(t) = βe2(t)

= β∫

B

[f (t,y)u(t,y)+g(t,y)uβ (t,y)

]dy

� β∫

B

[f (t,y)(z(t))

1β +g(t,y)z(t)

]dy

= β[

z(t)∫

Bg(t,y)dy+(z(t))

1β

∫

Bf (t,y)dy

]

. (3.2.49)



z(t) � exp(

β∫ t

0

∫

Bg(τ,z)dzdτ

)[

kβ−1 +(β −1)∫ t

0

∫

Bf (s,y)

×exp(

−(β −1)∫ s

0

∫

Bg(τ,z)dzdτ

)

dyds] β

β−1. (3.2.50)

Using (3.2.50) in (3.2.48), we get the required inequality in (3.2.13). If k � 0, we carry out

the above procedure with k+ε instead of k, where ε > 0 is an arbitrary small constant, and

then pass to the limit as ε → 0 to obtain (3.2.13).

(c7) Let

w(t) =∫ t

0

∫

Bf (s,y)u(s,y)dyds, (3.2.51)

λ =∫ ∞

0

∫



u(t,x) � p(t,x)+q(t,x)w(t)+ r(t,x)λ . (3.2.53)

Introducing the notation

e3(s) =∫

Bf (s,y)u(s,y)dy, (3.2.54)

in (3.2.51), we get

w(t) =∫ t

0e3(s)ds. (3.2.55)

From (3.2.55), (3.2.54) and (3.2.53), we have

w′(t) = e3(t) =∫

Bf (t,y)u(t,y)dy �

∫

Bf (t,y)[p(t,y)+q(t,y)w(t)+ r(t,y)λ ]dy

= w(t)∫

Bf (t,y)q(t,y)dy+

∫

Bf (t,y) [p(t,y)+ r(t,y)λ ]dy. (3.2.56)


w(t) �∫ t

0

∫

Bf (s,y)[p(s,y)+ r(s,y)λ ]exp

(∫ t

s

∫


)

dyds

=∫ t

0

∫

Bf (s,y)p(s,y)exp

(∫ t

s

∫


)

dyds

+λ∫ t

0

∫

Bf (s,y)r(s,y)exp

(∫ t

s

∫


)

dyds. (3.2.57)

From (3.2.53) and (3.2.57), we get

u(t,x) � p(t,x)+q(t,x)


×{∫ t

0

∫

Bf (s,y)p(s,y)exp

(∫ t

s

∫


)

dyds

+λ∫ t

0

∫

Bf (s,y)r(s,y)exp

(∫ t

s

∫


)

dyds}

+ r(t,x)λ

= K1(t,x)+λK2(t,x). (3.2.58)


λ �∫ ∞

0

∫

Bg(s,y)[K1(s,y)+λK2(s,y)]dyds,

which implies

λ � D. (3.2.59)

Using (3.2.59) in (3.2.58), we get (3.2.19).

(c9) Setting

E(t,s) =∫

Bk(t,x,s,y)u(s,y)dy, (3.2.60)

for every x ∈ B, the inequality (3.2.26) can be restated as

u(t,x) � c+∫ t

0E(t,s)ds. (3.2.61)

Define

z(t) = c+∫ t

0E(t,s)ds, (3.2.62)

then z(0) = c and

u(t,x) � z(t). (3.2.63)

From (3.2.62), (3.2.60), (3.2.63) and the fact that z(t) is nondecreasing in t ∈ R+, we

observe that

z′(t) = E (t,t)+∫ t

0D1E (t,s) ds

=∫

Bk(t,x,t,y)u(t,y)dy+

∫ t

0D1

{∫

Bk(t,x,s,y)u(s,y)dy

}

ds

�∫

Bk(t,x,t,y)z(t)dy+

∫ t

0

∫

BD1k(t,x,s,y)z(s)dyds

� A(t,x)z(t). (3.2.64)

The inequality (3.2.64) implies

z(t) � cexp(∫ t

0A(σ ,x)dσ

)

. (3.2.65)



(c11) Let E(t,s) be given by (3.2.60). Then (3.2.32) can be restated as

u2(t,x) � c+∫ t

0E (t,s) ds. (3.2.66)

Let c > 0 and define by z(t) the right hand side of (3.2.66), then z(0) = c and u(t,x) �√

z(t). Following the proof of part (c9), we get

z′(t) � A(t,x)√

z(t). (3.2.67)

The inequality (3.2.67) implies√

z(t) �√

c+12

∫ t

0A(σ ,x)dσ . (3.2.68)

The required inequality in (3.2.33) follows by using (3.2.68) in u(t,x) �√

z(t). The proof

of the case when c � 0 can be completed as mentioned in the proof of part (c5).

(c13) Introducing the notation

E0(s) =∫

Bf (s,y)

[

u(s,y)+q(s,y)∫ s

0

∫


]

dy, (3.2.69)



0E0(s)ds. (3.2.70)

Define

m(t) =∫ t

0E0(s)ds, (3.2.71)

for t ∈ R+, then m(0) = 0 and from (3.2.70), we get

u(t,x) � p(t,x)+q(t,x)m(t), (3.2.72)

for (t,x) ∈ D0. From (3.2.71), (3.2.69) and (3.2.72), we observe that

m′(t) = E0(t) =∫

Bf (t,y)

[

u(t,y)+q(t,y)∫ t

0

∫


]

dy

�∫

Bf (t,y)

[

p(t,y)+q(t,y)m(t)

+q(t,y)∫ t

0

∫

Bg(τ,z)[p(τ,z)+q(τ,z)m(τ)]dzdτ

]

dy, (3.2.73)

for t ∈ R+ Introducing the notation

r(τ) =∫

Bg(τ,z) [p(τ,z)+q(τ,z)m(τ)]dz, (3.2.74)


the inequality (3.2.73) can be written as

m′(t) �∫

Bf (t,y)

[

p(t,y)+q(t,y){

m(t)+∫ t

0r(τ)dτ

}]

dy, (3.2.75)

for t ∈ R+. Define

v(t) = m(t)+∫ t

0r(τ)dτ, (3.2.76)

then m(t) � v(t), v(0) = m(0) = 0 and

m′(t) �∫

Bf (t,y) [p(t,y)+q(t,y)v(t)]dy. (3.2.77)

From (3.2.76), (3.2.77), (3.2.74) and the fact that m(t) � v(t), t ∈ R+, we observe that

v′(t) = m′(t)+ r(t)

�∫

Bf (t,y)[p(t,y)+q(t,y)v(t)]dy+

∫

Bg(t,z)[p(t,z)+q(t,z)m(t)]dz

� v(t)∫

Bq(t,y)[ f (t,y)+g(t,y)]dy +

∫

Bp(t,y)[ f (t,y)+g(t,y)]dy. (3.2.78)

The inequality (3.2.78) implies

v(t) �∫ t

0

∫

Bp(s,y)[ f (s,y)+g(s,y)]

×exp(∫ t

s

∫


)

dyds. (3.2.79)

Using the fact that m(t) � v(t), t ∈ R+ in (3.2.79) and then using the bound on m(t) in

(3.2.72), we get the required inequality in (3.2.37).

3.3 Volterra-Fredholm-type integral inequalities II

The main objective of this section is to present some more mixed Volterra-Fredholm-type

integral inequalities established in [107,119 ] which can be used as tools in certain new

situations. In what follows we shall use the notation as given in section 3.2. Furthermore,

let x = (x1, . . . ,xn) be any point in Rn+, Di = ∂

∂ xi, Dn · · ·D1 = ∂

∂xn· · · ∂

∂x1for 1 � i � n,

B0,x = {x ∈ Rn+ : 0 < x < ∞}, Ha,b = ∏m

i=1[ai,bi] ⊂ Rm (ai < bi) and H = R

n+ ×Ha,b. For

the functions u(s) and v(t) defined on B0,x and Ha,b we denote by∫

B0,xu(s)ds,

∫Ha,b

v(t)dt

the n-fold and m-fold integrals∫ x1

0· · ·

∫ xn

0u(s1, . . . ,sn)dsn · · ·ds1,

∫ b1

a1

· · ·∫ bm

am

v(t1, . . . , tm)dtm · · ·dt1

respectively.

The mixed Volterra-Fredholm-type integral inequalities established in [107] are given in

the following theorems.


Theorem 3.3.1. Let u, p, q, f ∈C(D0,R+) and c � 0 be a constant.

(d1) Let L ∈C(D0 ×R+,R+) satisfies the condition (3.2.1) in Theorem 3.2.1 part (c1). If


0

∫ s

0

∫

BL(τ,y,u(τ,y))dydτ ds, (3.3.1)


u(t,x) � p(t,x)+q(t,x)(∫ t

0

∫ s

0

∫

BL(τ,y, p(τ,y))dydτ ds

)

×exp(∫ t

0

∫ s

0

∫

BM(τ,y, p(τ,y))q(τ,y)dydτ ds

)

, (3.3.2)

for (t,x) ∈ D0.

(d2) Let g ∈C(R+,R+) be a nondecreasing function, g(u) > 0 on (0,∞). If

u(t,x) � c+∫ t

0

∫ s

0

∫

Bf (τ,y)g(u(τ,y))dydτ ds, (3.3.3)


u(t,x) � W−1[

W (c)+∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds

]

, (3.3.4)

where W, W−1 are as in Theorem 3.2.5 part (c10) and t1 ∈ R+ is chosen so that

W (c)+∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds ∈ Dom

(W−1), (3.3.5)

for all t ∈ R+ lying in 0 � t � t1 and x ∈ B.

Theorem 3.3.2. Let u, f ∈C(D0,R+) and c � 0 be a constant.

(d3) If

u2(t,x) � c+∫ t

0

∫ s

0

∫

Bf (τ,y)u(τ,y)dydτ ds, (3.3.6)


u(t,x) �√

c+12

∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds, (3.3.7)

for (t,x) ∈ D0.

(d4) Let g(u) be as in part (d2). If

u2(t,x) � c+∫ t

0

∫ s

0

∫

Bf (τ,y)u(τ,y)g(u(τ,y))dydτ ds, (3.3.8)


u(t,x) � W−1[

W(√

c)+

12

∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds

]

, (3.3.9)

where W, W−1 are as in part (d2) and t2 ∈ R+ is chosen so that

W(√

c)+

12

∫ t

0

∫ s

0

∫


(W−1),

for all t ∈ R+ lying in 0 � t � t2 and x ∈ B.


Theorem 3.3.3. Let u ∈C(D0,R1), f ∈C(D0,R+) and c � 1 be a constant.

(d5) If

u(t,x) � c+∫ t

0

∫ s

0

∫

Bf (τ,y)u(τ,y) logu(τ,y)dydτ ds, (3.3.10)



0∫ s

0∫

B f (τ,y)dydτ ds), (3.3.11)

for (t,x) ∈ D0.

(d6) Let g(u) be as in part (d2). If

u(t,x) � c+∫ t

0

∫ s

0

∫

Bf (τ,y)u(τ,y)g(logu(τ,y))dydτ ds, (3.3.12)


u(t,x) � exp(

W−1[

W (logc)+∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds

])

, (3.3.13)

where W, W−1 are as in part (d2) and t3 ∈ R+ be chosen so that

W (logc)+∫ t

0

∫ s

0

∫


(W−1),


The following theorems contain the inequalities given in [119].

Theorem 3.3.4. Let u, f ∈C(H,R+) and c � 0 is a constant.

(d7) If

u(x,y) � c+∫

B0,x

∫

Ha,b

f (s,t)u(s,t)dt ds, (3.3.14)

for (x,y) ∈ H, then

u(x,y) � cexp(∫

B0,x

∫

Ha,b

f (s, t)dt ds)

, (3.3.15)

for (x,y) ∈ H.

(d8) If c � 1, u � 1 and

u(x,y) � c+∫

B0,x

∫

Ha,b

f (s,t)u(s,t) logu(s, t)dt ds, (3.3.16)


u(x,y) � cexp

(∫B0,x

∫Ha,b

f (s,t)dt ds)

, (3.3.17)

for (x,y) ∈ H.


Theorem 3.3.5. (d9) Let u, f ∈C(H,R+); c, p,q be positive constants and suppose that

up(x,y) � c+∫

B0,x

∫

Ha,b

f (s,t)uq(s,t)dt ds, (3.3.18)

for (x,y) ∈ H. If 0 < q < p, then

u(x,y) �[

(c)p−q

p +p−q

p

∫

B0,x

∫

Ha,b

f (s,t)dt ds] 1

p−q

, (3.3.19)

for (x,y) ∈ H.

(d10) Let u, p, q, f ∈C(H,R+) and L ∈C(H ×R+,R+) be such that

0 � L(x,y,u)−L(x,y,v) � M(x,y,v)(u− v), (3.3.20)

for u � v � 0, where M ∈C(H ×R+,R+). If

u(x,y) � p(x,y)+q(x,y)∫

B0,x

∫

Ha,b

L(s, t,u(s,t))dt ds, (3.3.21)


u(x,y) � p(x,y)+q(x,y)(∫

B0,x

∫

Ha,b

L(s,t, p(s,t)) dt ds)

×exp(∫

B0,x

∫

Ha,b

M(s,t, p(s,t))q(s,t)dt ds)

, (3.3.22)

for (x,y) ∈ H.

Proofs of Theorems 3.3.1–3.3.5. To prove (d2)–(d4), (d7), it is sufficient to assume that

c > 0, since the standard limiting argument can be used to treat the remaining case, see [82,

p. 108].

(d1) Setting

r(τ) =∫

BL(τ,y,u(τ,y))dy, (3.3.23)



0

∫ s

0r(τ)dτ ds. (3.3.24)

Define

z(t) =∫ t

0

∫ s

0r(τ)dτ ds, (3.3.25)

then, it is easy to see that z(0) = 0, z′(0) = 0 and

u(t,x) � p(t,x)+q(t,x)z(t). (3.3.26)

From (3.3.25), (3.3.23), (3.3.26) and (3.2.1), we observe that

z′′(t) = r(t) =∫

BL(t,y,u(t,y))dy


�∫

B

{L(t,y, p(t,y)+q(t,y)z(t))−L(t,y, p(t,y))

}dy+

∫

BL(t,y, p(t,y))dy

� z(t)∫

BM(t,y, p(t,y))q(t,y)dy+

∫

BL(t,y, p(t,y))dy. (3.3.27)

From (3.3.27) and using the fact that z(t) is nondecreasing in t ∈ R+, it is easy to see that

z(t) �∫ t

0

∫ s

0

∫


+∫ t

0z(s)

{∫ s

0

∫

BM(τ,y, p(τ,y))q(τ,y)dydτ

}

ds. (3.3.28)

Clearly, the first term on the right hand side of (3.3.28) is continuous, nonnegative and non-

decreasing in t ∈ R+. Now a suitable application of the inequality in [82, Theorem 1.3.1]

to (3.3.28) yields

z(t) �(∫ t

0

∫ s

0

∫


)

×exp(∫ t

0

∫ s

0

∫

BM(τ,y, p(τ,y))q(τ,y)dydτ ds

)

. (3.3.29)


(d2) Setting

r1(τ) =∫

Bf (τ,y)g(u(τ,y))dy, (3.3.30)


u(t,x) � c+∫ t

0

∫ s

0r1(τ)dτ ds. (3.3.31)

Let c > 0 and define by z(t) the right hand side of (3.3.31). Following the proof of part (d1)

given above, we get

z′′(t) = r1(t) =∫

Bf (t,y)g(u(t,y))dy � g(z(t))

∫

Bf (t,y)dy. (3.3.32)

From (3.3.32) and using the fact that z(t) is nondecreasing in t ∈ R+, it is easy to see that

z(t) � c +∫ t

0g(z(s))

{∫ s

0

∫

Bf (τ,y)dydτ

}

ds. (3.3.33)

Now a suitable application of the inequality in [82, Theorem 2.3.1] to (3.3.33) yields

z(t) � W−1[

W (c)+∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds

]

. (3.3.34)

Using (3.3.34) in u(t,x) � z(t) we get the required inequality in (3.3.4). The proof of the

case when c � 0 can be completed as mentioned in the proof of part (c5). The subinterval

0 � t � t1 is obvious.


(d3) setting

r2(τ) =∫

Bf (τ,y)u(τ,y)dy, (3.3.35)


u2(t,x) � c+∫ t

0

∫ s

0r2(τ)dτ ds. (3.3.36)

Let c > 0 and define by z(t) the right hand side of (3.3.36), then z(0) = c, z′(0) = 0 and

u(t,x) �√

z(t). Following the proof of part (d1), we get

z′′(t) = r2(t) =∫

Bf (t,y)u(t,y)dy �

√z(t)

∫

Bf (t,y)dy. (3.3.37)

By taking t = τ in (3.3.37) and integrating it over τ from 0 to t and using the fact that z(t)

is nondecreasing in t ∈ R+, we get

z′(t) �√

z(t)∫ t

0

∫

Bf (τ,y)dydτ. (3.3.38)

The inequality (3.3.38) implies (see [82, p. 233])√

z(t) �√

c+12

∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds. (3.3.39)

The required inequality in (3.3.7) follows by using (3.3.39) in u(t,x) �√

z(t).

(d4) Setting

r3(τ) =∫

Bf (τ,y)u(τ,y)g(u(τ,y))dy, (3.3.40)


u2(t,x) � c+∫ t

0

∫ s

0r3(τ)dτ ds. (3.3.41)

Let c > 0 and define by z(t) the right hand side of (3.3.41). Following the proof of part

(d1), we get

z′′(t) = r3(t) �√

z(t)g(√

z(t))∫

Bf (t,y)dy. (3.3.42)

From (3.3.42) and using the fact that z(t) is nondecreasing in t ∈ R+, it is easy to observe

thatz′(t)

√z(t)

� g(√

z(t))∫ t

0

∫

Bf (τ,y)dydτ. (3.3.43)

From (3.3.43), we get√

z(t) �√

c+12

∫ t

0g(√

z(s)){∫ s

0

∫

Bf (τ,y)dydτ

}

ds. (3.3.44)

Now a suitable application of the inequality in [82, Theorem 2.3.1] to (3.3.44) yields√

z(t) � W−1[

W(√

c)+

12

∫ t

0

∫ s

0

∫

Bf (τ,y)dydτ ds

]

. (3.3.45)


Using (3.3.45) in u(t,x) �√

z(t), we get (3.3.9).

(d5) Setting

r4(τ) =∫

Bf (τ,y)u(τ,y) logu(τ,y)dy, (3.3.46)


u(t,x) � c+∫ t

0

∫ s

0r4(τ)dτ ds. (3.3.47)

Let c > 0 and define by z(t) the right hand side of (3.3.47). Following the proof of (d1) and

using the fact that u(t,x) � z(t), we have

z′′(t) = r4(t) =∫

Bf (t,y)u(t,y) logu(t,y)dy � z(t) log z(t)

∫

Bf (t,y)dy. (3.3.48)

From (3.3.48) and using the fact that z(t) is nondecreasing in t ∈ R+, it is easy to observe

that

z′(t) � z(t) logz(t)∫ t

0

∫

Bf (τ,y)dydτ, (3.3.49)

which implies

log z(t) � logc+∫ t

0log z(s)

{∫ s

0

∫

Bf (τ,y)dydτ

}

ds. (3.3.50)

Now by following the proof of Theorem 3.8.2 given in [82, p. 269] we get

z(t) � cexp(∫ t

0∫ s

0∫

B f (τ,y)dydτ ds). (3.3.51)

Using (3.3.51) in u(t,x) � z(t), we get the required inequality in (3.3.11).

(d6) The proof can be completed by following the proof of (d5) and closely looking at the

proof of Theorem 3.9.1 given in [82, p. 270]. Here, we omit the details.

Next, we will give the proofs of (d7) and (d9); the proofs of (d8) and (d10) can be completed

by following the proofs of (d7) and (d9) and closely looking at the proofs of (d5) and (d1).

(d7) Introducing the notation

R1(s) =∫

Ha,b

f (s, t)u(s,t)dt, (3.3.52)

in (3.3.14), we get

u(x,y) � c+∫

B0,x

R1(s)ds, (3.3.53)

for (x,y) ∈ H. Let c > 0 and define

z(x) = c+∫

B0,x

R1(s)ds, (3.3.54)


for x ∈ Rn+, then z(0) = c and

u(x,y) � z(x). (3.3.55)

From (3.3.54), (3.3.52), (3.3.55), we observe that (see [82, Theorem 4.9.1])

Dn · · ·D1z(x) = R1(x) =∫

Ha,b

f (x,t)u(x, t)dt � z(x)∫

Ha,b

f (x,t)dt. (3.3.56)

From (3.3.56), it is easy to observe that

z(x) � c+∫

B0,x

z(s)∫

Ha,b

f (s,t)dt ds. (3.3.57)

Now a suitable application of Theorem 4.9.1 part (i) in [82, p. 397] to (3.3.57) yields

z(x) � cexp(∫

B0,x

∫

Ha,b

f (s, t)dt ds)

. (3.3.58)

Using (3.3.58) in (3.3.55) yields (3.3.15).

(d9) Introducing the notation

R2(s) =∫

Ha,b

f (s, t)uq(s,t)dt, (3.3.59)


up(x,y) � c+∫

B0,x

R2(s)ds. (3.3.60)

Define

z(x) = c+∫

B0,x

R2(s)ds, (3.3.61)

for x ∈ Rn+, then z(0) = c and

up(x,y) � z(x), (3.3.62)

for (x,y) ∈ H. From (3.3.61), (3.3.59) and (3.3.62), we observe that

Dn · · ·D1z(x) = R2(x) =∫

Ha,b

f (x,t)uq(x,t)dt � (z(x))qp

∫

Ha,b

f (x,t)dt.

i.e.,Dn · · ·D1z(x)

(z(x))qp

�∫

Ha,b

f (x,t)dt. (3.3.63)

Now by following the similar arguments as in the proofs of Theorems 4.9.1 and 5.9.1 given

in [82] with suitable modifications, from (3.3.63), we obtain

(z(x))p−q

p − (c)p−q

p � p−qp

∫

B0,x

∫

Ha,b

f (s,t)dt ds,

for x ∈ Rn+, which implies

z(x) �[

(c)p−q

p +p−q

p

∫

B0,x

∫

Ha,b

f (s,t)dt ds] p

p−q

. (3.3.64)

The assertion (3.3.19) follows by using (3.3.64) in (3.3.62).


3.4 Integral equation of Volterra-Fredholm-type

The mixed Volterra-Fredholm integral equations of the form

u(t,x) = f (t,x)+∫ t

0

∫

BF(t,x,s,y,u(s,y))dyds, (3.4.1)

often arise from mathematical modeling of many physical and biological phenomena, see

[5,16–18,28,31,32] and the related references given therein. In (3.4.1) f , F are given func-

tions, B is as defined in section 3.2 and u is the unknown function. We assume that

f ∈ C(D0,Rn), F ∈ C(Ω×R

n,Rn), where D0,Ω are as defined in section 3.2. Let Z be

the space of functions φ ∈C(D0,Rn) which fulfil the condition

|φ(t,x)| = O(exp(λ (t + |x|))), (3.4.2)

where λ > 0 is a constant. In the space Z we define the norm

|φ |Z = sup(t,x)∈D0

[|φ(t,x)|exp(−λ (t + |x|))]. (3.4.3)

It is easy to see that Z with norm defined in (3.4.3) is a Banach space and

|φ |Z � M, (3.4.4)

where M � 0 is a constant. The main objective of this section is to present some qualitative

aspects of solutions of equation (3.4.1) developed in [79,99,105,116].

The following result guarantees the existence and uniqueness of solutions of equation

(3.4.1).


(i) the function F in equation (3.4.1) satisfies the condition

|F(t,x,s,y,u)−F(t,x,s,y,v)| � h(t,x,s,y)|u− v|, (3.4.5)

where h ∈C(D20,R+),


(a1) there exists a nonnegative constant α < 1 such that∫ t

0

∫

Bh(t,x,s,y)exp(λ (s+ |y|))dyds � α exp(λ (t + |x|)), (3.4.6)

(a2) there exists a nonnegative constant β such that

| f (t,x)|+∫ t

0

∫

B|F(t,x,s,y,0)|dyds � β exp(λ (t + |x|)), (3.4.7)

where f , F are as defined in equation (3.4.1).

Then the equation (3.4.1) has a unique solution u(t,x) in Z on D0.


Proof. Let u ∈ Z and define the operator T by

(Tu)(t,x) = f (t,x)+∫ t

0

∫


for (t,x) ∈ D0. From (3.4.8) and using the hypotheses, we have

|(Tu)(t,x)| � | f (t,x)|+∫ t

0

∫

B|F(t,x,s,y,u(s,y))−F(t,x,s,y,0)|dyds

+∫ t

0

∫

B|F(t,x,s,y,0)|dyds

� β exp(λ (t + |x|))+ |u|Z∫ t

0

∫

Bh(t,x,s,y)exp(λ(s+ |y|))dyds

� [Mα +β ]exp(λ (t + |x|)). (3.4.9)

From (3.4.9), it follows that Tu ∈ Z.

Let u, v ∈ Z. From (3.4.8) and using the hypotheses, we have

|(Tu)(t,x)− (Tv)(t,x)| �∫ t

0

∫

B|F(t,x,s,y,u(s,y))−F(t,x,s,y,v(s,y))|dyds

� |u− v|Z∫ t

0

∫

Bh(t,x,s,y)exp(λ (s+ |y|))dyds

� α|u− v|Z exp(λ (t + |x|)). (3.4.10)

From (3.4.10), it follows that

|Tu−T v|Z � α|u− v|Z.

Since α < 1, it follows from Banach fixed point theorem (see [28] and [51]) that T has a

unique fixed point in Z. The fixed point of T is a solution of equation (3.4.1).

Remark 3.4.1. We note that one can formulate existence and uniqueness result similar

to that of Theorem 1.3.2 for the solution u ∈C (D0,Rn) of equation (3.4.1). Furthermore,

Theorem 3.4.1 can also be extended to more general mixed Volterra-Fredholm integral

equation of the form

u(t,x) = f (t,x)+∫ t

0G(t,x,s,u(s,x))ds

+∫

BH(t,x,y,u(t,y))dy+

∫ t

0

∫


under some suitable conditions on the functions involved in (3.4.11). We leave the precise

formulations of such results to the reader.

The next result concerning the estimate on the solution of equation (3.4.1) holds.


Theorem 3.4.2. Suppose that the function F in equation (3.4.1) satisfies the condition

|F(t,x,s,y,u)−F(t,x,s,y,v)| � k(t,x,s,y)|u− v|, (3.4.12)

where k, D1k ∈C(Ω,R+). Let

c = sup(t,x)∈D0

∣∣∣∣ f (t,x)+

∫ t

0

∫

BF(t,x,s,y,0)dyds

∣∣∣∣ < ∞, (3.4.13)

where f , F are the functions in equation (3.4.1). If u(t,x) is any solution of equation (3.4.1)

on D0, then

|u(t,x)| � cexp(∫ t

0A(σ ,x)dσ

)

, (3.4.14)


Proof. Using the fact that u(t,x) is a solution of equation (3.4.1) on D0 and hypotheses,

we have

|u(t,x)| �∣∣∣∣ f (t,x)+

∫ t

0

∫

BF(t,x,s,y,0)dyds

∣∣∣∣

+∫ t

0

∫

B|F(t,x,s,y,u(s,y))−F(t,x,s,y,0)|dyds

� c+∫ t

0

∫

Bk(t,x,s,y)|u(s,y)|dyds. (3.4.15)

Now an application of the inequality in Theorem 3.2.5 part (c9) to (3.4.15) yields (3.4.14).

A variant of Theorem 3.4.2 is given in the following theorem.


(3.4.12). Let

d = sup(t,x)∈D0

∫ t

0

∫

B|F(t,x,s,y, f (s,y))|dyds < ∞, (3.4.16)

where f , F are the functions in equation (3.4.1). If u(t,x) is any solution of equation (3.4.1)

on D0, then

|u(t,x)− f (t,x)| � d exp(∫ t

0A(σ ,x)dσ

)

, (3.4.17)



Proof. Let z(t,x) = |u(t,x)− f (t,x)| for (t,x)∈D0. Using the fact that u(t,x) is a solution

of equation (3.4.1) and hypotheses, we have

z(t,x) �∫ t

0

∫

B|F(t,x,s,y,u(s,y))−F(t,x,s,y, f (s,y))|dyds

+∫ t

0

∫

B|F(t,x,s,y, f (s,y))|dyds

� d +∫ t

0

∫

Bk(t,x,s,y)z(s,y)dyds. (3.4.18)


We call the function u ∈ C(D0,Rn) an ε-approximate solution of equation (3.4.1) if there

exists a constant ε � 0 such that∣∣∣∣u(t,x)−

{

f (t,x)+∫ t

0

∫

BF(t,x,s,y,u(s,y))dyds

}∣∣∣∣ � ε,

for (t,x) ∈ D0.

The next theorem deals with the estimate on the difference between the two approximate

solutions of equation (3.4.1).

Theorem 3.4.4. Let u1(t,x) and u2(t,x) be respectively ε1- and ε2-approximate solutions

of equation (3.4.1) on D0. Suppose that the function F in equation (3.4.1) satisfies the

condition (3.4.12). Then

|u1(t,x)−u2(t,x)| � (ε1 + ε2)exp(∫ t

0A(σ ,x)dσ

)

, (3.4.19)


Proof. Since u1(t,x) and u2(t,x) are respectively, ε1- and ε2-approximate solutions of

equation (3.4.1) on D0, we have∣∣∣∣ui(t,x)−

{

f (t,x)+∫ t

0

∫

BF(t,x,s,y,ui(s,y))dyds

}∣∣∣∣ � εi, (3.4.20)

for i = 1, 2. From (3.4.20) and using the elementary inequalities in (1.3.25), we observe

that

ε2 + ε2 �∣∣∣∣u1(t,x)−

{

f (t,x)+∫ t

0

∫

BF(t,x,s,y,u1(s,y))dyds

}∣∣∣∣

+∣∣∣∣u2(t,x)−

{

f (t,x)+∫ t

0

∫


}∣∣∣∣

�∣∣∣∣[u1(t,x)−u2(t,x)]−

[{

f (t,x)+∫ t

0

∫


}


−{

f (t,x)+∫ t

0

∫


}]∣∣∣∣

� |u1(t,x)−u2(t,x)|−∣∣∣∣

∫ t

0

∫

B{F(t,x,s,y,u1(s,y))−F(t,x,s,y,u2(s,y))}dyds

∣∣∣∣. (3.4.21)

Let w(t,x) = |u1(t,x)−u2(t,x)| for (t,x) ∈ D0. From (3.4.21) and using (3.4.12), we have

w(t,x) � (ε2 + ε2)+∫ t

0

∫

B|F(t,x,s,y,u1(s,y))−F(t,x,s,y,u2(s,y))|dyds

� (ε2 + ε2)+∫ t

0

∫

Bk(t,x,s,y)w(s,y)dyds. (3.4.22)


Remark 3.4.2. In case u1(t,x) is a solution of equation (3.4.1), then we have ε1 = 0 and

from (3.4.19), we see that u2(t,x) → u1(t,x) as ε2 → 0. Moreover, from (3.4.19) it follows

that if ε1 = ε2 = 0, then the uniqueness of solutions of equation (3.4.1) is established.

Consider the equation (3.4.1) and the following Volterra-Fredholm integral equation

v(t,x) = f (t,x)+∫ t

0

∫

BF(t,x,s,y,v(s,y))dyds, (3.4.23)

where f ∈C(D0,Rn), F ∈C(Ω×R

n,Rn).

The following result that relates the solutions of equations (3.4.1) and (3.4.23) holds.


(3.4.12) and there exist constants δi � 0 (i = 1, 2) such that

| f (t,x)− f (t,x)| � δ1, (3.4.24)

∫ t

0

∫

B|F(t,x,s,y, p)−F(t,x,s,y, p)|dyds � δ2, (3.4.25)

where f , F and f , F are as given in equations (3.4.1) and (3.4.23). Let u(t,x) and v(t,x)

be respectively, solutions of (3.4.1) and (3.4.23) on D0, then

|u(t,x)− v(t,x)| � (δ1 +δ2)exp(∫ t

0A(σ ,x)dσ

)

, (3.4.26)



Proof. Let r(t,x) = |u(t,x)− v(t,x)|, (t,x) ∈ D0. Using the facts that u(t,x), v(t,x) are

respectively the solutions of equations (3.4.1), (3.4.23) on D0 and hypotheses, we have

r(t,x) � | f (t,x)− f (t,x)|

+∫ t

0

∫


+∫ t

0

∫

B|F(t,x,s,y,v(s,y))−F(t,x,s,y,v(s,y))|dyds

� (δ1 +δ2)+∫ t

0

∫

Bk(t,x,s,y)r(s,y)dyds. (3.4.27)


We now consider the system of nonlinear Volterra-Fredholm integral equations of the form

u(t,x) = f (t,x)+∫ t

0

∫

B[k(t,x,s,y)u(s,y)+F(t,x,s,y,u(s,y))]dyds, (3.4.28)

as a perturbation of the linear system of Volterra-Fredholm integral equations

v(t,x) = f (t,x)+∫ t

0

∫

Bk(t,x,s,y)v(s,y)dyds, (3.4.29)

where f , F are as given in equation (3.4.1) and k ∈C(Ω,R).

The following theorem deals with the estimate on the difference between the solutions of

equations (3.4.28) and (3.4.29).

Theorem 3.4.6. Suppose that the functions F and k in equation (3.4.28) satisfy respec-

tively the conditions

|F(t,x,s,y,u1)−F(t,x,s,y,u2)| � q(t,x)h(s,y)|u1 −u2|, (3.4.30)

F(t,x,s,y,0) = 0 and

|k(t,x,s,y)| � q(t,x)g(s,y), (3.4.31)

where q, h, g ∈ C(D0,R+). Let v(t,x) be a solution of equation (3.4.29) on D0 such that

|v(t,x)| � M, where M � 0 is a constant and

p(t,x) = Mq(t,x)∫ t

0

∫

Bh(s,y)dyds. (3.4.32)

If u(t,x) is any solution of equation (3.4.28) on D0, then

|u(t,x)− v(t,x)| � p(t,x)+q(t,x)∫ t

0

∫

B[h(s,y)+g(s,y)]p(s,y)

×exp(∫ t

s

∫

B[h(τ,z)+g(τ,z)]q(τ,z)dzdτ

)

dyds, (3.4.33)

for (t,x) ∈ D0.


Proof. From the hypotheses, we observe that

|u(t,x)− v(t,x)| �∫ t

0

∫

B|k(t,x,s,y)| |u(s,y)− v(s,y)|dyds

+∫ t

0

∫


+∫ t

0

∫

B|F(t,x,s,y,v(s,y))−F(t,x,s,y,0)|dyds

� q(t,x)∫ t

0

∫

Bg(s,y)|u(s,y)− v(s,y)|dyds

+q(t,x)∫ t

0

∫

Bh(s,y)|u(s,y)− v(s,y)|dyds+q(t,x)

∫ t

0

∫

Bh(s,y)|v(s,y)|dyds

� p(t,x)+q(t,x)∫ t

0

∫

B[h(s,y)+g(s,y)]|u(s,y)− v(s,y)|dyds. (3.4.34)


Remark 3.4.3. We note that the inequality in Theorem 3.2.5 part (c9) can be used to

formulate results on the uniqueness and continuous dependence of solutions of equation

(3.4.1) by following the corresponding results given in section 1.4.

3.5 Volterra-Fredholm-type integral equations

Inspired by the equations like (16) and (19) (see [4,64]), we consider the Volterra-

Fredholm-type integral equations of the forms

u(t,x) = h(t,x)+∫ t

0

∫ s

0

∫

BF(t,x,τ,y,u(τ,y))dydτ ds, (3.5.1)

and

u(x,y) = f (x,y)+∫

B0,x

∫

Ha,b

K(x,y,s,t,u(s,t))dt ds, (3.5.2)

where h, F and f , K are given functions and u is the unknown function. In this section we

will make use of the notations and definitions given in sections 3.2 and 3.4 and the concept

of the ε-approximate solution extended in a natural way to equations (3.5.1) and (3.5.2).

We assume that h ∈ C(D0,Rn), F ∈ C(Ω×R

n,Rn), f ∈ C(H,Rn), K ∈ C(H2 ×Rn,Rn).

Owing to the importance of equations (3.5.1), (3.5.2), we believe that the qualitative theory

of such equations needs to be developed in various directions. The problems of existence

of solutions of equations (3.5.1) and (3.5.2) can be dealt with the method employed in

section 3.4. In this section, we present conditions under which we can offer simple, unified

and concise proofs of some of the important qualitative properties of solutions of equations

(3.5.1) and (3.5.2).

In our discussion, we use the following special forms of the inequalities given in (d1) and

(d10).


Lemma 3.5.1. Let u, p,q,g ∈C (D0,R+) . If


0

∫ s

0

∫

Bg(τ,y)u(τ,y)dydτ ds,


u(t,x) � p(t,x)+q(t,x)(∫ t

0

∫ s

0

∫

Bg(τ,y)p(τ,y)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫

Bg(τ,y)q(τ,y)dydτ ds

)

,

for (t,x) ∈ D0.

Lemma 3.5.2. Let u, p, q, g ∈C(H,R+). If

u(x,y) � p(x,y)+q(x,y)∫

B0,x

∫

Ha,b

g(s,t)u(s,t)dt ds,


u(x,y) � p(x,y)+q(x,y)(∫

B0,x

∫

Ha,b

g(s,t)p(s,t)dt ds)

×exp(∫

B0,x

∫

Ha,b

g(s,t)q(s, t)dt ds)

,

for (x,y) ∈ H.

We start with the following theorem which deals with the estimate on the difference be-

tween the two approximate solutions of equation (3.5.1).

Theorem 3.5.1. Let ui(t,x) (i = 1, 2) be respectively εi-approximate solutions of equation

(3.5.1) on D0. Suppose that the function F in equation (3.5.1) satisfies the condition

|F(t,x,τ,y,u)−F(t,x,τ,y,v)| � q(t,x)g(τ,y)|u− v|, (3.5.3)

where q, g ∈C(D0,R+). Then

|u1(t,x)−u2(t,x)| � (ε1 + ε2)[

1+q(t,x)(∫ t

0

∫ s

0

∫

Bg(τ,y)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫


)]

, (3.5.4)

for (t,x) ∈ D0.


Proof. Since ui(t,x)(i = 1, 2) are respectively εi-approximate solutions of equation

(3.5.1), we have∣∣∣∣ui(t,x)−

{

h(t,x)+∫ t

0

∫ s

0

∫

BF(t,x,τ,y,ui(τ,y))dydτ ds

}∣∣∣∣ � εi.

From the above inequality and using the elementary inequalities in (1.3.25) and following

the proof of Theorem 3.4.4, we get

|u1(t,x)−u2(t,x)| � (ε1 + ε2)

+∫ t

0

∫ s

0

∫

B|F(t,x,τ,y,u1(τ,y))−F(t,x,τ,y,u2(τ,y))|dydτ ds. (3.5.5)

Let w(t,x) = |u1(t,x)−u2(t,x)|, (t,x) ∈ D0. From (3.5.5) and using (3.5.3), we have

w(t,x) � (ε1 + ε2)+q(t,x)∫ t

0

∫ s

0

∫

Bg(τ,y)w(τ,y)dydτ ds. (3.5.6)

Now an application of Lemma 3.5.1 to (3.5.6) yields (3.5.4).

Remark 3.5.1. When u1(t,x) is a solution of equation (3.5.1), we have ε1 = 0 and from

(3.5.4), we see that u2(t,x) → u1(t,x) as ε2 → 0. Furthermore, if we put ε1 = ε2 = 0 in

(3.5.4), then the uniqueness of solutions of equation (3.5.1) is established.

Consider the equation (3.5.1) together with the equation

v(t,x) = h(t,x)+∫ t

0

∫ s

0

∫

BF(t,x,τ,y,v(τ,y))dydτ ds, (3.5.7)

where h ∈C(D0,Rn), F ∈C(Ω×R

n,Rn).

In the next theorem we provide conditions concerning the closeness of solutions of equa-

tions (3.5.1) and (3.5.7).

Theorem 3.5.2. Suppose that the function F in equation (3.5.1) satisfies (3.5.3) and there

exist constants δi � 0 (i = 1, 2) such that

|h(t,x)−h(t,x)| � δ1, (3.5.8)∫ t

0

∫ s

0

∫

B|F(t,x,τ,y,z)−F(t,x,τ,y,z)|dydτ ds � δ2, (3.5.9)

where h, F and h, F are given as in (3.5.1) and (3.5.7). Let u(t,x) and v(t,x) for (t,x)∈D0,

be solutions of equations (3.5.1) and (3.5.7) respectively. Then

|u(t,x)− v(t,x)| � (δ1 +δ2)[

1+q(t,x)(∫ t

0

∫ s

0

∫

Bg(τ,y)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫


)]

, (3.5.10)

for (t,x) ∈ D0.


Proof. Let r(t,x) = |u(t,x)− v(t,x)|, (t,x) ∈ D0. Using the facts that u(t,x), v(t,x) are

solutions of equations (3.5.1), (3.5.7) and the hypotheses, we have

r(t,x) � |h(t,x)−h(t,x)|

+∫ t

0

∫ s

0

∫

B|F(t,x,τ,y,u(τ,y))−F(t,x,τ,y,v(τ,y))|dydτ ds

+∫ t

0

∫ s

0

∫

B|F(t,x,τ,y,v(τ,y))−F(t,x,τ,y,v(τ,y))|dydτ ds

� (δ1 +δ2)+q(t,x)∫ t

0

∫ s

0

∫

Bg(τ,y)r(τ,y)dydτ ds. (3.5.11)

Now, an application of Lemma 3.5.1 to (3.5.11) yields (3.5.10).

Remark 3.5.2. The result given in Theorem 3.5.2 relates the solutions of equations

(3.5.1) and (3.5.7) in the sense that if F is close to F and h is close to h, then the solu-

tions of equations (3.5.1) and (3.5.7) are also close to each other.

A slight variation of Theorem 3.5.2 is embodied in the following theorem.

Theorem 3.5.3. Suppose that the functions F and F in equations (3.5.1) and (3.5.7) sat-

isfies the condition

|F(t,x,τ,y,u)−F(t,x,τ,y,v)| � q(t,x)g(τ,y)|u− v|, (3.5.12)

where q, g ∈C(D0,R+) and (3.5.8) holds. Let u(t,x) and v(t,x) be solutions of equations

(3.5.1) and (3.5.7), respectively, on D0. Then

|u(t,x)− v(t,x)| � δ1

[

1+q(t,x)(∫ t

0

∫ s

0

∫

Bg(τ,y)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫


)]

, (3.5.13)

for (t,x) ∈ D0.

The proof of the above theorem is similar to that of Theorem 3.5.2, with suitable modifica-

tions, and hence we omit the details.

We next consider the equations

u(t,x) = h(t,x)+∫ t

0

∫ s

0

∫

BF(t,x,τ,y,u(τ,y),μ)dydτ ds, (3.5.14)

u(t,x) = h(t,x)+∫ t

0

∫ s

0

∫

BF(t,x,τ,y,u(τ,y),μ0)dydτ ds, (3.5.15)

for (t,x) ∈ D0, where h ∈C(D0,Rn), F ∈C(Ω×R

n ×R,Rn) and μ , μ0 are parameters.

The following theorem shows the dependency of solutions of equations (3.5.14) and

(3.5.15) on parameters.


Theorem 3.5.4. Suppose that the function F in (3.5.14), (3.5.15) satisfy the conditions

|F(t,x,τ,y,u,μ)−F(t,x,τ,y,u,μ)| � q0(t,x)g0(τ,y)|u−u|, (3.5.16)

|F(t,x,τ ,y,u,μ)−F(t,x,τ,y,u,μ0)| � N|μ −μ0|, (3.5.17)

where q0, g0 ∈C(D0,R+) and N � 0 is a constant. Let u1(t,x) and u2(t,x) be the solutions

of equations (3.5.14) and (3.5.15) respectively. Then

|u1(t,x)−u2(t,x)| � N|μ −μ0|[

1+q0(t,x)(∫ t

0

∫ s

0

∫

Bg0(τ,y)dydτ ds

)

×exp(∫ t

0

∫ s

0

∫

Bg0(τ,y)q0(τ,y)dydτ ds

)]

, (3.5.18)

for (t,x) ∈ D0.

Proof. Let z(t,x) = |u1(t,x)−u2(t,x)|, (t,x)∈D0. Using the facts that u1(t,x) and u2(t,x)

are respectively the solutions of equations (3.5.14) and (3.5.15) and the hypotheses, we

have

z(t,x) �∫ t

0

∫ s

0

∫

B|F(t,x,τ,y,u1(τ,y),μ)−F(t,x,τ,y,u2(τ,y),μ)|dydτ ds

+∫ t

0

∫ s

0

∫

B|F(t,x,τ,y,u2(τ,y),μ)−F(t,x,τ,y,u2(τ,y),μ0)|dydτ ds

� N|μ −μ0|+q0(t,x)∫ t

0

∫ s

0

∫

Bg0(τ,y)z(τ,y)dydτ ds. (3.5.19)

Now an application of Lemma 3.5.1 to (3.5.19) yields (3.5.18), which shows the depen-


Below, we apply the inequality in Lemma 3.5.2 to obtain the uniqueness and explicit esti-

mates on the solutions of equation (3.5.2). One can formulate results similar to those given

in Theorems 3.5.1–3.5.4 for the equation (3.5.2) by using Lemma 3.5.2.

The following theorem deals with the uniqueness of solutions of equation (3.5.2).


|K(x,y,s,t,u)−K(x,y,s,t,v)| � q(x,y)g(s,t)|u− v|, (3.5.20)

where q, g ∈C(H,R+).Then the equation (3.5.2) has at most one solution on H.


Proof. Let u(x,y) and v(x,y) be two solutions of equation (3.5.2) on H. Using these facts

and the hypotheses (3.5.20), we have

|u(x,y)− v(x,y)| �∫

B0,x

∫

Ha,b

|K(x,y,s,t,u(s,t))−K(x,y,s, t,v(s,t))|dt ds

� q(x,y)∫

B0,x

∫

Ha,b

g(s,t)|u(s,t)− v(s, t)|dt ds. (3.5.21)

Now a suitable application of Lemma 3.5.2 (when p(x,y) = 0) to (3.5.21) yields

|u(x,y)− v(x,y)| � 0, which implies u(x,y) = v(x,y). Thus there is at most one solution

to equation (3.5.2) on H.

The next theorem deals with the estimate on the solution of equation (3.5.2).


|K(x,y,s,t,u)| � q(x,y)g(s,t)|u|, (3.5.22)

where q, g ∈C(H,R+). If u(x,y) is any solution of equation (3.5.2) on H, then

|u(x,y)| � | f (x,y)|+q(x,y)(∫

B0,x

∫

Ha,b

g(s,t)| f (s,t)|dt ds)

× exp(∫

B0,x

∫

Ha,b

g(s,t)q(s,t)dt ds)

, (3.5.23)

for (x,y) ∈ H.

Proof. Using the fact that u(x,y) is a solution of equation (3.5.2) and hypotheses, we have

|u(x,y)| � | f (x,y)|+∫

B0,x

∫

Ha,b

|K(x,y,s,t,u(s, t))|dt ds

� | f (x,y)|+q(x,y)∫

B0,x

∫

Ha,b

g(s,t)|u(s,t)|dt ds. (3.5.24)


A slight variation of Theorem 3.5.6 is given in the following theorem.


(3.5.20). If u(x,y) is any solution of equation (3.5.2) on H, then

|u(x,y)− f (x,y)| � Q(x,y)+q(x,y)(∫

B0,x

∫

Ha,b

g(s,t)Q(s,t)dt ds)

× exp(∫

B0,x

∫

Ha,b

g(s,t)q(s,t)dt ds)

, (3.5.25)

for (x,y) ∈ H, where

Q(x,y) =∫

B0,x

∫

Ha,b

|K(x,y,σ ,τ, f (σ ,τ))|dτ dσ , (3.5.26)

for (x,y) ∈ H.


Proof. Using the fact that u(x,y) is a solution of equation (3.5.2) and hypotheses, we

observe that

|u(x,y)− f (x,y)| �∫

B0,x

∫

Ha,b

|K(x,y,s,t, f (s, t))|dt ds

+∫

B0,x

∫

Ha,b

|K(x,y,s,t,u(s,t))−K(x,y,s, t, f (s,t))|dt ds

� Q(x,y)+q(x,y)∫

B0,x

∫

Ha,b

g(s, t)|u(s,t)− f (s,t)|dt ds, (3.5.27)

for (x,y) ∈ H. Now an application of Lemma 3.5.2 to (3.5.27) gives the required estimate

in (3.5.25).

Remark 3.5.3. We note that the equation (3.5.1) contains as a special case, the study of

integral equation (16). Moreover, the generality of the equation (3.5.2) allow us to include

in the special cases (i) n = 1, m = 1, (ii) n = 2, m = 1, (iii) n = 2, m = 2; respectively the

study of integral equations of the forms

u(x1,y1) = f (x1,y1)+∫ x1

0

∫ b1

a1

K(x1,s1,y1,u(s1,y1))dy1ds1, (3.5.28)

u(x1,x2,y1) = f (x1,x2,y1)

+∫ x1

0

∫ x2

0

∫ b1

a1

K(x1,x2,s1,s2,y1,u(s1,s2,y1))dy1ds2ds1, (3.5.29)

u(x1,x2,y1,y2) = f (x1,x2,y1,y2)

+∫ x1

0

∫ x2

0

∫ b1

a1

∫ b2

a2

K(x1,x2,s1,s2,y1,y2,u(s1,s2,y1,y2))dy2dy1ds2ds1. (3.5.30)

3.6 General Volterra-Fredholm-type integral equations

Consider the general nonlinear Volterra-Fredholm-type integral equations of the forms:

u(t,x) = h(t,x)+∫ t

0

∫

BF(t,x,s,y,u(s,y),(Tu)(s,y))dyds, (3.6.1)

and

u(t,x) = e(t,x)+∫ t

0

∫

BG(t,x,s,y,u(s,y))dyds

+∫ ∞

0

∫

BL(t,x,s,y,u(s,y))dyds, (3.6.2)

where

(Tu)(t,x) =∫ t

0

∫

BK(t,x,τ,z,u(τ,z))dzdτ, (3.6.3)


h, F, K and e, G, L are given functions and u is the unknown function. In this section

we shall use the notations and definitions given in sections 3.2 and 3.4 without further

mention. We assume that h, e ∈C(D0,Rn), K ∈C(Ω×R

n,Rn), F ∈C(Ω×Rn ×R

n,Rn),

G ∈ C(Ω×Rn,Rn), L ∈ C(D2

0 ×Rn,Rn). The present section is devoted to study some

fundamental qualitative properties of solutions of equations (3.6.1) and (3.6.2), which we

hope will serve as a source for further investigations.

First, we formulate the following theorem concerning the existence of a unique solution of

equation (3.6.1).


(i) the functions F and K in equation (3.6.1) satisfy the conditions

|F(t,x,s,y,u,v)−F(t,x,s,y,u,v)| � r(t,x,s,y) [|u−u|+ |v− v|] , (3.6.4)

and

|K(t,x,τ,z,u)−K(t,x,τ ,z,u)| � m(t,x,τ,z)|u−u|, (3.6.5)

where r, m ∈C(Ω,R+),


(b1) there exists a nonnegative constant α < 1 such that

∫ t

0

∫

Br(t,x,s,y)

[

exp(λ (s+ |y|))+∫ t

0

∫

Bm(s,y,τ,z)exp(λ (τ + |z|))dzdτ

]

dyds

� α exp(λ (t + |x|)), (3.6.6)

(b2) there exists a nonnegative constant β such that

|h(t,x)|+∫ t

0

∫

B|F(t,x,s,y,0,(T 0)(s,y))|dyds � β exp(λ (t + |x|)), (3.6.7)

where h, F are as defined in equation (3.6.1).

Under the assumptions (i) and (ii) the equation (3.6.1) has a unique solution u(t,x) on D0

in Z.

The proof follows by the similar arguments as in the proof of Theorem 3.4.1. We omit the

details.



Theorem 3.6.2. Suppose that the functions F, K in equation (3.6.1) satisfy the conditions

|F(t,x,s,y,u,v)−F(t,x,s,y,u,v)| � q(t,x) f (s,y) [|u−u|+ |v− v|] , (3.6.8)

|K(t,x,τ,z,u)−K(t,x,τ,z,u)| � q(t,x)g(τ,z)|u−u|, (3.6.9)

where q, f , g ∈C(D0,R+). Let

c = sup(t,x)∈D0

∣∣∣∣h(t,x)+

∫ t

0

∫

BF(t,x,s,y,0,(T 0)(s,y))dyds

∣∣∣∣ < ∞, (3.6.10)

where h, F are the functions in equation (3.6.1). If u(t,x) is any solution of equation (3.6.1)

on D0, then

|u(t,x)| � c[

1+q(t,x)∫ t

0

∫

B[ f (s,y)+g(s,y)]

×exp(∫ t

s

∫


)

dyds]

, (3.6.11)

for (t,x) ∈ D0.

Proof. Using the fact that u(t,x) is a solution of equation (3.6.1) and hypotheses, we have

|u(t,x)| �∣∣∣∣h(t,x)+

∫ t

0

∫

BF(t,x,s,y,0,(T 0)(s,y))dyds

∣∣∣∣

+∫ t

0

∫

B|F(t,x,s,y,u(s,y),(Tu)(s,y))−F(t,x,s,y,0,(T 0)(s,y))|dyds

� c+q(t,x)∫ t

0

∫

Bf (s,y)

×[

|u(s,y)|+q(s,y)∫ s

0

∫

Bg(τ,z)|u(τ,z)|dzdτ

]

dyds. (3.6.12)

Now applying Theorem 3.2.7 part (c13) to (3.6.12) yields (3.6.11).

In the next theorem we will employ Theorem 3.2.7 part (c13) to obtain the uniqueness of



(3.6.8), (3.6.9) respectively. Then the equation (3.6.1) has at most one solution on D0.


Proof. Let u1(t,x) and u2(t,x) be two solutions of equation (3.6.1) on D0 and w(t,x) =

|u1(t,x)−u2(t,x)|, (t,x) ∈ D0. From the hypotheses, we have

w(t,x) �∫ t

0

∫

B|F(t,x,s,y,u1(s,y),(Tu1)(s,y))−F(t,x,s,y,u2(s,y),(Tu2)(s,y))|dyds

� q(t,x)∫ t

0

∫

Bf (s,y)

[

w(s,y)+q(s,y)∫ s

0

∫

Bg(τ,z)w(τ,z)dzdτ

]

dyds. (3.6.13)

Now applying Theorem 3.2.7 part (c13) (with p(t,x) = 0) to (3.6.13) yields |u1(t,x)−u2(t,x)|� 0, which implies u1(t,x) = u2(t,x). Thus there is at most one solution to equation

(3.6.1) on D0.

Now, we present a result on the continuous dependence of solution of equation (3.6.1)

on the functions involved therein. Consider the equation (3.6.1) and the corresponding

equation

v(t,x) = h(t · x)+∫ t

0

∫

BF(t,x,s,y,v(s,y),(Tv)(s,y))dyds, (3.6.14)


(T v)(t,x) =∫ t

0

∫

BK(t,x,τ,z,v(τ,z))dzdτ, (3.6.15)

h ∈C(D0,Rn), K ∈C(Ω×R

n,Rn), F ∈C(Ω×Rn ×R

n,Rn).


(3.6.8), (3.6.9) respectively. Furthermore, suppose that v(t,x) is a given solution of equation

(3.6.14) on D0 and

|h(t,x)−h(t,x)|+∫ t

0

∫

B|F(t,x,s,y,v(s,y),(Tv)(s,y))

−F(t,x,s,y,v(s,y),(T v)(s,y))|dyds � ε, (3.6.16)

where h, F, Tu and h, F, T v are as in equations (3.6.1) and (3.6.14) respectively and ε > 0

is an arbitrary small constant. Then the solution u(t,x) of equation (3.6.1) on D0 depends

continuously on the functions involved in equation (3.6.1).

Proof. Let w(t,x) = |u(t,x)− v(t,x)|, for (t,x) ∈ D0. Using the hypotheses, we have

w(t,x) � |h(t,x)−h(t,x)|

+∫ t

0

∫

B|F(t,x,s,y,u(s,y),(Tu)(s,y))−F(t,x,s,y,v(s,y),(Tv)(s,y))|dyds

+∫ t

0

∫

B|F(t,x,s,y,v(s,y),(Tv)(s,y))−F(t,x,s,y,v(s,y),(T v)(s,y))|dyds


� ε +q(t,x)∫ t

0

∫

Bf (s,y)

[

w(s,y)+q(s,y)∫ s

0

∫

Bg(τ,z)w(τ,z)dzdτ

]

dyds. (3.6.17)

Now using Theorem 3.2.7 part (c13) to (3.6.17) yields

|u(t,x)− v(t,x)| � ε[

1+q(t,x)∫ t

0

∫

B[ f (s,y)+g(s,y)]

×exp(∫ t

s

∫


)

dyds]

, (3.6.18)

for (t,x) ∈ D0. From (3.6.18) it follows that the solution u(t,x) of equation (3.6.1) depends

continuously on the functions involved therein.

We now turn our attention to some basic qualitative properties of solutions of equation

(3.6.2). Here, we note that one can obtain the existence and uniqueness result for the

solution of equation (3.6.2) by using the idea employed in Theorem 3.4.1.

Now we formulate the following theorem which estimates the difference between the two

approximate solutions of equation (3.6.2).


(i) The functions G,L in equation (3.6.2) satisfy the conditions

|G(t,x,s,y,u)−G(t,x,s,y,v)| � q(t,x) f (s,y)|u− v|, (3.6.19)

|L(t,x,s,y,u)−L(t,x,s,y,v)| � r(t,x)g(s,y)|u− v|, (3.6.20)

where q, f , r, g ∈C(D0,R+),

(ii) the functions ui(t,x) ∈C(D0,Rn) (i = 1, 2) be respectively εi-approximate solutions of

equation (3.6.2) on D0,

(iii) let d, K2(t,x) be as in (3.2.18), (3.2.21) respectively and

D1 =1

1−d

∫ ∞

0

∫

Bg(s,y)A1(s,y)dyds, (3.6.21)

where A1(t,x) is defined by the right hand side of (3.2.20) by replacing p(t,x) by ε1 + ε2.

Then

|u1(t,x)−u2(t,x)| � A1(t,x)+D1K2(t,x), (3.6.22)

for (t,x) ∈ D0.


Proof. Since ui(t,x) (i = 1, 2) are respectively εi-approximate solutions of equation

(3.6.2), we have∣∣∣∣ui(t,x)−

{

e(t,x)+∫ t

0

∫

BG(t,x,s,y,ui(s,y))dyds

+∫ ∞

0

∫

BL(t,x,s,y,ui(s,y))dyds

}∣∣∣∣ � εi. (3.6.23)

From (3.6.23) and following the proof of Theorem 3.4.4, we get

|u1(t,x)−u2(t,x)| � ε1 + ε2

+∫ t

0

∫

B|g(t,x,s,y,u1(s,y))−g(t,x,s,y,u2(s,y))|dyds

+∫ ∞

0

∫

B|L(t,x,s,y,u1(s,y))−L(t,x,s,y,u2(s,y))|dyds. (3.6.24)

Let w(t,x) = |u1(t,x)−u2(t,x)|, (t,x) ∈ D0. From (3.6.24) and using (3.6.19), (3.6.20), we

have

w(t,x) � (ε1 + ε2)+q(t,x)∫ t

0

∫

Bf (s,y)w(s,y)dyds

+r(t,x)∫ ∞

0

∫

Bg(s,y)w(s,y)dyds. (3.6.25)

Now an application of Theorem 3.2.4 part (c7) to (3.6.25) yields (3.6.22).

The next result is a consequence of Theorem 3.6.5.


(i) the functions G, L in equation (3.6.2) satisfy the conditions (3.6.19), (3.6.20) respec-

tively,

(ii) the functions uε(t,x) and u(t,x) be respectively an ε-approximate solution and any

solution of equation (3.6.2) on D0,

(iii) let d,K2(t,x) be as in (3.2.18), (3.2.21) respectively and

D2 =1

1−d

∫ ∞

0

∫

Bg(s,y)A2(s,y)dyds, (3.6.26)

where A2(t,x) is defined by the right hand side of (3.2.20) by replacing p(t,x) by ε . Then

|uε(t,x)−u(t,x)| � A2(t,x)+D2K2(t,x), (3.6.27)

for (t,x) ∈ D0.


Proof. Let w(t,x) = |uε (t,x)−u(t,x)|, (t,x) ∈ D0. From the hypotheses, we observe that

w(t,x) =∣∣∣∣uε(t,x)− e(t,x)−

∫ t

0

∫

BG(t,x,s,y,uε(s,y))dyds

−∫ ∞

0

∫

BL(t,x,s,y,uε (s,y))dyds

∣∣∣∣

+∫ t

0

∫

B|G(t,x,s,y,uε (s,y))−G(t,x,s,y,u(s,y))|dyds

+∫ ∞

0

∫

B|L(t,x,s,y,uε(s,y))−L(t,x,s,y,u(s,y))|dyds

� ε +q(t,x)∫ t

0

∫

Bf (s,y)w(s,y)dyds+ r(t,x)

∫ ∞

0

∫

Bg(s,y)w(s,y)dyds. (3.6.28)

Now an application of Theorem 3.2.4 part (c7) to (3.6.28) yields (3.6.27).

In the following theorem we present conditions for continuous dependence on parameters

of solutions of equations

u(t,x) = ei(t,x)+∫ t

0

∫

BG(t,x,s,y,u(s,y))dyds+

∫ ∞

0

∫

BL(t,x,s,y,u(s,y))dyds, (3.6.29)

for (t,x) ∈ D0, i = 1, 2; where ei ∈C(D0,Rn) and G, L are as in equation (3.6.2).


(i) the functions G, L in (3.6.29) satisfy the conditions (3,6,19), (3.6.20) respectively,

(ii) the functions ui(t,x)(i = 1, 2) are respectively the εi-approximate solutions of equations

in (3.6.29) corresponding to i = 1, 2,

(iii) let d, K2(t,x) be as in (3.2.18), (3.2.21) respectively and

D3 =1

1−d

∫ ∞

0

∫

Bg(s,y)A3(s,y)dyds, (3.6.30)

where A3(t,x) is defined by the right hand side of (3.2.20) by replacing p(t,x) by ε1 +ε2 +

|e1(t,x)− e2(t,x)|.Then

|u1(t,x)−u2(t,x)| � A3(t,x)+D3K2(t,x), (3.6.31)

for (t,x) ∈ D0.

The proof follows by the similar arguments as in the proof of Theorem 3.6.6 with suitable

modifications. Here, we omit the details.


3.7 Miscellanea

3.7.1 Bacotiu [7]

Let (X ,‖ ‖X ) be a Banach space. Consider the nonlinear integral equation of Volterra-

Fredholm-type

u(t,x) = g(t,x,u(t,x))+∫ t

0H(t,x,s,u(s,x))ds

+∫ t

0

∫ b

aK(t,x,s,y,u(s,y),u(φ1(s,y),φ2(s,y)))dyds, (3.7.1)

for all (t,x) ∈ [0,T ]× [a,b] := D; u ∈C(D,X), where b > a > 0 and T > 0. Assume that

(i) g ∈ C(D×X ,X), H ∈ C(D× [0,T ]×X ,X), K ∈ C(D2 ×X2,X), φ1 ∈ C(D, [0,T ]) and

φ2 ∈C(D, [a,b]);

(ii) there exists Lg > 0 such that

‖g(t,x,u)−g(t,x,v)‖X � Lg‖u− v‖X ,

for all (t,x) ∈ D and u, v ∈ X ;

(iii) there exists LH > 0 such that

‖H(t,x,s,u)−H(t,x,s,v)‖X � LH‖u− v‖X ,

for all (t,x,s) ∈ D× [0,T ] and u, v ∈ X ;

(iv) there exists LK > 0 such that

‖K(t,x,s,y,u,u)−K(t,x,s,y,v,v)‖X � LK [‖u− v‖X +‖u− v‖X ] ,

for all (t,x,s,y) ∈ D2 and u, v, u, v ∈ X ;

(v) Lg < 1;

(vi) there exists τ > 0 such that

α = Lg +1τ

LH +b−a

τLK +max

{∫ t

0

∫ b

aexp(τ[φ1(s,y)− t])dyds : t ∈ [0,T ]

}

LK < 1.

Then (3.7.1) has a unique solution u ∈C(D,X).

3.7.2 Kauthen [50]

Consider the linear Volterra-Fredholm integral equation

u(t,x) = f (t,x)+∫ t

0

∫

ΩK(t,s,x,ξ )u(s,ξ )dξ ds, (3.7.2)

where Ω is a closed subset of Rn (with n = 1, 2, 3) with assumptions

(i) f ∈C(I ×Ω), where I = [0,T ],

(ii) K ∈C(D×Ω2), where D = {(t,s) : 0 � s � t � T} and Ω2 = Ω×Ω.


Then the equation (3.7.2) has a unique solution u ∈C(I ×Ω) and this solution is given by

u(t,x) = f (t,x)+∫ t

0

∫

ΩR(t,s,x,ξ ) f (s,ξ )dξ ds,

for (t,x) ∈ I ×Ω, where the resolvent kernel R ∈ C(D×Ω2) associated with the kernel

K(t,s,x,ξ ) is the limit of the Neumann series for K and solves the resolvent equation

R(t,s,x,ξ ) = K(t,s,x,ξ )+∫ t

0

∫

ΩK(t,v,x,z)R(v,s,z,ξ )dzdv,

on D×Ω2.

3.7.3 Diekmann [31]

Consider the nonlinear integral equation

u(t,x) =∫ t

0

∫

Ωg(u(t − τ,ξ ))S0(ξ )A(τ,x,ξ )dξ dτ + f (t,x), (3.7.3)

where Ω is a closed subset of Rn. Let BC(Ω) denote the Banach space of the bounded

continuous functions on Ω equipped with supremum norm and CT = C([0,T ],BC(Ω)) be

the Banach space of continuous functions on [0,T ] with values in BC(Ω), equipped with

the norm

‖z‖CT = sup0�t�T

‖z[t]‖BC(Ω).

When looking at u as an element of CT , write u[t](x) instead of u(t,x). With this convention

(3.7.3) can be written as

u[t]+Qu[t]+ f [t], (3.7.4)

where Q is defined by

Qu[t](x) =∫ t

0

∫

Ωg(u[t − τ](ξ ))S0(ξ )A(τ,x,ξ )dξ dτ.

Assume that

(i) S0 ∈ L∞(Ω);S0 is nonnegative,

(ii) g : R → R is continuous; g(0) = 0,

(iii) A(·, ·, ·) is defined and nonnegative on [0,∞)×Ω×Ω; for every x ∈ Ω and every T > 0,

A(·,x, ·) ∈ L1([0,T ]×Ω),

(iv) let

η(t,x) =∫ t

0

∫

ΩA(τ,x,ξ )dξ dτ

and T > 0 be arbitrary, then the family of functions on [0,T ], {η(·,x) : x ∈ Ω} is uniformly

bounded and equicontinuous.


(v) For every T > 0 and every ε > 0 there exists δ = δ (ε,T ) > 0 such that if x1, x2 ∈ Ωand |x1 − x2| < δ , then

∫ T

0

∫

Ω|A(τ,x1,ξ )−A(τ,x2,ξ )|dξ dτ < ε,

(vi) f : [0,∞) → BC(Ω) is continuous.

Then

1. For every T > 0 and every u ∈CT , Qu ∈CT .

2. Suppose g is locally Lipschitz continuous, then there exists a T > 0 such that (3.7.3) has

a unique solution u in CT and the mapping f → u is continuous from CT into CT .

3. Suppose g is uniformly Lipschitz continuous, then (3.7.3) has a unique continuous

solution u : [0,∞) → BC(Ω).

3.7.4 Diekmann [31]

Suppose that the functions g, f in (3.7.3) satisfy

1. g(y) > 0 for y > 0 and f [t] � 0 for all t � 0, then u[t] � 0 on the domain of definition

of u.

2. In addition, g is monotone nondecreasing and f [t +h] � f [t] for all h � 0, then u[t +h] �u[t] for all h � 0 and t � 0 such that t +h is in the domain of definition of u.

3. In addition to the assumptions 1, 2, suppose that g is bounded and uniformly Lipschitz

continuous on [0,∞), that the subset { f [t] : t � 0} of BC(Ω) is uniformly bounded and

equicontinuous and that A satisfies

(vii) for each x ∈ Ω∫ t

0A(τ,x, ·)dτ →

∫ ∞

0A(τ,x, ·)dτ,

in L1(Ω) as t → ∞, and for some C > 0,

supx∈Ω

∫

Ω

∫ ∞

0A(τ,x,ξ )dτ dξ < C.

(viii) For each ε > 0 there exists δ = δ (ε) > 0 such that if x1, x2 ∈ Ω and |x1 − x2| < δ ,

then∫

Ω

∫ ∞

0|A(τ,x1,ξ )−A(τ,x2,ξ )|dτ dξ < ε.

Then the solution u of (3.7.3) is defined on [0,∞) and there exists u[∞] ∈ BC(Ω) such that,

as t → ∞, u[t] → u[∞] in BC(Ω) if Ω is compact, and uniformly on compact subsets of Ω if

Ω is not compact. Moreover, u[∞] satisfies the limit equation

u[∞](x) =∫

Ωg(u[∞](ξ ))S0(ξ )

∫ ∞

0A(τ,x,ξ )dτ dξ + f [∞](x).



Under the notations as in section 3.2, let u ∈C(D0,R1), k, D1k ∈C(Ω,R+) and c � 1 is a

constant.

(f1) If

u(t,x) � c+∫ t

0

∫

Bk(t,x,s,y)u(s,y) logu(s,y)dyds,



0 A(σ ,x)dσ),


(f2) Let g(u) be as in Theorem 3.2.5 part (c10). If

u(t,x) � c+∫ t

0

∫

Bk(t,x,s,y)u(s,y)g(logu(s,y))dyds,


u(t,x) � W−1[

W (logc)+∫ t

0A(σ ,x)dσ

]

,

where W, W−1, A(t,x) are as in Theorem 3.2.5 and t1 ∈ R+ is chosen so that

W (logc)+∫ t


(W−1) ,

for all t ∈ R+ lying in the interval 0 � t � t1, x ∈ B.


Under the notations as in section 3.3, let u, p, q, f ∈C(H,R+) and r > 1 be a real constant.

(f3) Let L ∈C(H ×R+,R+) satisfies the condition (3.3.20) in Theorem 3.3.5. If

ur(x,y) � p(x,y)+q(x,y)∫

B0,x

∫

Ha,b

L(s,t,u(s,t))dt ds,


u(x,y) �[

p(x,y)+q(x,y)(∫

B0,x

∫

Ha,b

L(

s,t,[

r−1r

+p(s,t)

r

])

dt ds)

×exp(∫

B0,x

∫

Ha,b

M(

s,t,[

r−1r

+p(s, t)

r

])q(s,t)

rdt ds

)] 1r

,

for (x,y) ∈ H, where M is the function given in Theorem 3.3.5.

(f4) If

ur(x,y) � p(x,y)+q(x,y)∫

B0,x

∫

Ha,b

f (s, t)u(s,t)dt ds,


u(x,y) �[

p(x,y)+q(x,y)(∫

B0,x

∫

Ha,b

f (s,t)[

r−1r

+p(s,t)

r

]

dt ds)

×exp(∫

B0,x

∫

Ha,b

f (s, t)q(s,t)

rdt ds

)] 1r

,

for (x,y) ∈ H.



Under the notations as in section 3.2, consider the Volterra-Fredholm-type integral equa-

tions

v(t,x) = h1(t,x)+∫ t

0

∫

BL(t,x,s,y,v(s,y))dyds, (3.7.5)

w(t,x) = h2(t,x)+∫ t

0

∫

BM(t,x,s,y,w(s,y))dyds, (3.7.6)

with assumptions

(i) h1, h2 ∈C(D0,R), L, M ∈C(Ω×R,R),

(ii) the function L in equation (3.7.5) satisfies the condition

|L(t,x,s,y,v)−L(t,x,s,y,w)| � q(t,x) f (s,y)|v−w|,

where q, f ∈C(D0,R+).

Then for every given solution w ∈C(D0,R) of equation (3.7.6) and v ∈C(D0,R) a solution

of equation (3.7.5), the estimate

|v(t,x)−w(t,x)| � [h(t,x)+ r(t,x)]

+q(t,x)∫ t

0

∫

Bf (s,y)[h(s,y)+ r(s,y)]exp

(∫ t

s

∫


)

dyds,

holds for (t,x) ∈ D0, in which

h(t,x) = |h1(t,x)−h2(t,x)|,

r(t,x) =∫ t

0

∫

B|L(t,x,s,y,w(s,y))−M(t,x,s,y,w(s,y))|dyds,

for (t,x) ∈ D0.


Consider the Volterra-Fredholm integral equation of the form

u(t,x) =∫ t

0

∫

ΩG(t,x;s,y)F(s,y,u(s,y))dyds

+∫ t

0

∫

∂ ΩG(t,x;s,y)ψ(s,y)dyds+

∫

ΩG(t,x;0,y)u0(y)dy, (3.7.7)

arising in the study of nonlinear parabolic system

∂u∂ t

−Lu = F(t,x,u), t ∈ (0,T ], x ∈ Ω. (3.7.8)


with the given boundary and initial conditions

∂ u∂ν

+b(x)u = c0(x), t ∈ (0,T ], x ∈ ∂Ω, (3.7.9)

u(0,x) = u0(x), x ∈ Ω, (3.7.10)

where Ω is a bounded domain in Rn and ∂ Ω is a smooth boundary of Ω and L is the

uniformly elliptic operator of the form

L =n

∑i, j=1

ai j(x)∂ 2

∂xi∂ x j+

n

∑i=1

ai(x)∂

∂ xi.

For detailed derivation of (3.7.7) and the assumptions on the functions involed in (3.7.8)–

(3.7.10), see [37]. Let D = (0,T ]×Ω and D is the closure of D and in (3.7.7), G is the

fundamental solution of ∂w∂ t − Lw = 0. The second integral in (3.7.7) is a single layer

potential with the density ψ(t,x) which can be determined from the following Volterra-

type integral equation

ψ(t,x) =∫ t

0

∫

∂ ΩR(t,x;s,y)ψ(s,y)dyds+H(t,x,u(t,x)),

where

R(t,x;s,y) = 2[

∂∂ν

G(t,x;s,y)+b(x)G(t,x;s,y)]

,

H(t,x,u(t,x)) =∫

ΩR(t,x;0,y)u0(y)dy+

∫ t

0

∫

ΩR(t,x;s,y)F(s,y,u(s,y))dyds− c0(x).

In fact ψ is given by (see [37, p. 145])

ψ(t,x) = H(t,x,u(t,x))+2∫ t

0

∫

∂ΩR(t,x;s,y)H(s,y,u(s,y))dyds.

Consider the equation (3.7.7), under the following assumptions:

(H1) the function F satisfies the condition

|F(t,x,u)−F(t,x,v)| � g(t,x)|u− v|,

where g ∈C(D,R+),

(H2) there exist nonnegative constants Qi (i = 1, 2, 3) and a constant μ > 0 such that∫ t

0

∫

Ω|G(t,x;s,y)|g(s,y)exp(μ(s+ |y|))dyds � Q1 exp(μ(t + |x|)),

t∫

0

∫

∂Ω

|G(t,x;s,y)|[

exp(μ(s+ |y|))+2s∫

0

∫

∂Ω

|R(s,y;τ,z)|exp(μ(τ + |z|))dzdτ

]

dyds

� Q2 exp(μ(t + |x|)),


∫ t

0

∫

Ω|R(t,x;s,y)|g(s,y)exp(μ(s+ |y|))dyds � Q3 exp(μ(t + |x|)),

(H3) there exist nonnegative constants N1, N2 such that

∫

Ω

|G(t,x;0,y)||u0(y)|dy+t∫

0

∫

Ω

|G(t,x;s,y)||F(s,y,0)|dyds � N1 exp(μ(t + |x|)),

∫

Ω

|R(t,x;0,y)||u0(y)|dy+t∫

0

∫

Ω

|R(t,x;s,y)||F(s,y,0)|dyds+ |c0(x)| � N2 exp(μ(t + |x|)).

(f5) Suppose that the assumptions (H1)–(H3) hold and assume that 0 � Q1 + Q2Q3 < 1.

Then equation (3.7.7) has a unique solution u(t,x) in Z on D, where Z is the space of

functions as defined in section 3.4.

3.8 Notes

Mixed Volterra-Fredholm-type integral equations occur in a natural way while studying

parabolic equations which describe diffusion or heat transfer phenomena and other areas

of science and technology. Most of the basic problems for various types of such equa-

tions are still in a very early stage of development. The material included in sections 3.2

and 3.3 is adapted from the recent work of Pachpatte [99,103,105,107,116,119], which

in fact is motivated by the desire to widen the scope of the integral inequalities with ex-

plicit estimates. Section 3.4 contains some basic results on the theory of nonlinear mixed

Volterra-Fredholm-type integral equation and are adapted from Pachpatte [79,116]. The

results in section 3.5 deals with the qualitative properties of solutions of certain general

Volterra-Fredholm-type integral equations, which are inspired by the integral equations

arising while studying certain partial differential and integrodifferential equations and they

are adapted from Pachpatte [107,119]. Section 3.6 is devoted to the study of some ba-

sic qualitative properties of solutions of general nonlinear mixed Volterra-Fredholm-type

integral equations arising from the study of certain physical models and are taken from

[103,99]. Section 3.7 is dedicated to the presentation of a few results related to certain

selected topics, which may stimulate the interest of the readers in pursuing further devel-

opments.


Chapter 4

Parabolic-type integrodifferential equations

4.1 Introduction

The study of many physical processes arising in science and engineering leads to the

models of parabolic integrodifferential equations with different initial boundary condi-

tions. These equations occur in several applications, such as in heat flow in materi-

als with memory, control and optimization theories, reaction diffusion processes, epi-

demic models and various other fields of science and technology. Considerable re-

search on the special kinds of parabolic integrodifferential equations has been carried

out by using new mathematical ideas, approaches, and theories, see [1,5,8,9,31–33,59–

62,74,80,98,110,121,122,124,126,127,132,140,141] and the references given therein. The

main goal of this chapter is to discuss the wellposedness related to certain nonlinear

parabolic integrodifferential equations studied in [109,110,75,77,140]. There are other top-

ics related to such equations, which we did not include here due to space limitation e.g., see

the references quoted in [1,5,25,26,39,45,60,61,80,121,122,124,126,137,141]. A detailed

survey, including comprehensive list of references of the early developments to the topic

can be found in the monograph [5].

4.2 Basic integral inequalities

This section is concerned with some fundamental integral inequalities with explicit esti-

mates which play a vital role in certain applications. In what follows, we use some of the

notations and definitions in section 3.2 without further mention. Moreover, we denote by

I = [a,b] ⊂ R (a < b), G = R+ × I and Ω0 = {(t,x,s) : 0 � s � t < ∞, x ∈ B}.

In the following theorems we present the useful variants of the integral inequality given in

[98] and also the integral inequalities given in [109].

143


Theorem 4.2.1. Let u, f ∈C(G,R+), h ∈C(G× I,R+) and c � 0 is a constant.

(p1) If

u(t,x) � c+∫ t

0

[

f (s,x)u(s,x)+∫ b

ah(s,x,y)u(s,y)dy

]

ds, (4.2.1)

for (t,x) ∈ G, then

u(t,x) � cF(t,x)exp(∫ t

0

∫ b

ah(s,x,y)F(s,y)dyds

)

, (4.2.2)

for (t,x) ∈ G, where

F(t,x) = exp(∫ t

0f (σ ,x)dσ

)

. (4.2.3)

(p2) Let g∈C(R+,R+) be nondecreasing submultiplactive function and g(u) > 0 on (0,∞).

If

u(t,x) � c+∫ t

0

[

f (s,x)u(s,x)+∫ b

ah(s,x,y)g(u(s,y))dy

]

ds, (4.2.4)

for (t,x) ∈ G, then for 0 � t � t1; t, t1 ∈ R+, x ∈ I,

u(t,x) � F(t,x)W−1[

W (c)+∫ t

0

∫ b

ah(s,x,y)g(F(s,y))dyds

]

, (4.2.5)

where F(t,x) is given by (4.2.3) and W, W−1 are as in Theorem 3.2.5 part (c10) and t1 ∈R+

is chosen so that

W (c)+∫ t

0

∫ b

ah(s,x,y)g(F(s,y))dyds ∈ Dom

(W−1) ,

for all t ∈ R+ lying in the interval 0 � t � t1 and x ∈ I.

Theorem 4.2.2. Let u ∈C(D0,R+); r,D1r ∈C(Ω0,R+), k,D1k ∈C(Ω,R+) and c � 0 is

a constant.

(p3) If

u(t,x) � c+∫ t

0r(t,x,s)u(s,x)ds+

∫ t

0

∫

Bk(s,x,s,y)u(s,y)dyds, (4.2.6)


u(t,x) � cP(t,x)exp(∫ t

0A(σ ,x)dσ

)

, (4.2.7)


P(t,x) = exp(Q(t,x)), (4.2.8)

in which

Q(t,x) =∫ t

0

[

r(η ,x,η)+∫ η

0D1r(η ,x,ξ )dξ

]

dη, (4.2.9)

Parabolic-type integrodifferential equations 145

and

A(t,x) =∫

Bk(t,x, t,y)P(t,y)dy+

∫ t

0

∫

BD1k(t,x,s,y)P(s,y)dyds, (4.2.10)

for (t,x) ∈ D0.

(p4) Let g(u) be as in Theorem 4.2.1 part (p2). If

u(t,x) � c+∫ t

0r(t,x,s)u(s,x)ds+

∫ t

0

∫

Bk(t,x,s,y)g(u(s,y))dyds, (4.2.11)


u(t,x) � P(t,x)W−1[

W (c)+∫ t

0A(σ ,x)dσ

]

, (4.2.12)

where P(t,x) is given by (4.2.8); W, W−1 are as in part (p2) and t2 ∈ R+ is chosen so that

W (c)+∫ t


(W−1) ,

for all t ∈ R+ lying in the interval 0 � t � t2 and x ∈ B, where A(t,x) is given by the right

hand side of (4.2.10) by replacing k(t,x,s,y)P(s,y) by k(t,x,s,y)g(P(s,y)).

Proofs of Theorems 4.2.1 and 4.2.2. We shall give the details of the proofs of (p2), (p3)

only, the proofs of (p1), (p4) can be completed by following the proofs of these inequalities

and the results given in Chapter 3, sections 3.2 and 3.3.

(p2) Define a function m(t,x) by

m(t,x) = c+∫ t

0

∫ b

ah(s,x,y)g(u(s,y))dyds, (4.2.13)


u(t,x) � m(t,x)+∫ t

0f (s,x)u(s,x)ds. (4.2.14)

It is easy to observe that m(t,x) is nonnegative for (t,x) ∈ G and nondecreasing in t ∈ R+

for every x ∈ I. Treating (4.2.14) as one-dimensional integral inequality in t ∈R+ for every

x ∈ I and a suitable application of the inequality given in [82, Theorem 1.3.1, p. 12] yields

u(t,x) � m(t,x)F(t,x). (4.2.15)


m(t,x) � c+∫ t

0

∫ b

ah(s,x,y)g(F(s,y)m(s,y)) dyds

� c+∫ t

0

∫ b

ah(s,x,y)g(F(s,y))g(m(s,y)) dyds. (4.2.16)

Setting

e(s) =∫ b

ah(s,x,y)g(F(s,y))g(m(s,y)) dy, (4.2.17)


for every x ∈ I, the inequality (4.2.16) can be restated as

m(t,x) � c+∫ t

0e(s)ds. (4.2.18)

Let

z(t) = c+∫ t

0e(s)ds, (4.2.19)

then z(0) = c and

m(t,x) � z(t), (4.2.20)

for t ∈ R+ and for every x ∈ I. From (4.2.19), (4.2.17) and (4.2.20), we observe that

z′(t) = e(t)

=∫ b

ah(t,x,y)g(F(t,y))g(m(t,y))dy

� g(z(t))∫ b

ah(t,x,y)g(F(t,y))dy. (4.2.21)

Now by following the proof of Theorem 2.3.1 given in [82, p. 107], we get

z(t) � W−1[

W (c)+∫ t

0

∫ b

ah(s,x,y)g(F(s,y))dyds

]

. (4.2.22)

The desired inequality in (4.2.5) follows from (4.2.22), (4.2.20) and (4.2.15).

(p3) Define a function n(t,x) by

n(t,x) = c+∫ t

0

∫



u(t,x) � n(t,x)+∫ t

0r(t,x,s)u(s,x)ds. (4.2.24)

From the hypotheses, it is easy to observe that n(t,x) is nonnegative for (t,x) ∈ D0 and

nondecreasing in t ∈ R+ for every x ∈ B. Treating (4.2.24) as one-dimensional integral

inequality for every x ∈ B and a suitable application of the inequality given in [87, Theo-

rem 1.2.1, Remark 1.2.1, p. 11] yields

u(t,x) � n(t,x)P(t,x). (4.2.25)

From (4.2.23) and (4.2.25), we obtain

n(t,x) � c+∫ t

0

∫

Bk(t,x,s,y)P(s,y)n(s,y)dyds. (4.2.26)

Now a suitable application of Theorem 3.2.5 part (c9) to (4.2.26) yields

n(t,x) � cexp(∫ t

0A(σ ,x)dσ

)

. (4.2.27)



4.3 Integrodifferential equation of Barbashin-type

E.A. Barabashin [8] first initiated the study of the integrodifferential equation of the form

∂∂ t

u(t,x) = c(t,x)u(t,x)+∫ b

ak(t,x,y)u(t,y)dy+ f (t,x), (B)

which arise in mathematical modeling of many applied problems (see [5, section 19]). This

equation attracted the attention of many researchers and is now known in the literature as

integrodifferential equation of Barbashin-type or simply Barbashin equation, see [5, p. 1].

For a detailed account on the study of such equations by using various techniques, see [5]

and the references cited therein. In this section, we consider the nonlinear integrodifferen-

tial equation of Barbashin-type (see [110])

∂∂ t

u(t,x) = f (t,x,u(t,x))+∫ b

ag(t,x,y,u(t,y))dy+h(t,x), (4.3.1)

which satisfies the initial condition

u(0,x) = u0(x), (4.3.2)

for (t,x) ∈ Δ, where f ∈ C(Δ ×R,R), g ∈ C(Δ× I ×R,R), h ∈ C (Δ,R), u0 ∈ C(I,R)

are given functions and u is the unknown function to be found, in which I = [a,b] ⊂ R

(a < b) and Δ = R+×I. Here, we focus our attention to study some fundamental qualitative

properties of solutions of problem (4.3.1)–(4.3.2). Let E be the space of functions z ∈C(Δ,R) which fulfil the condition

|z(t,x)| = O(exp(λ (t + |x|))), (4.3.3)

where λ > 0 is a constant. In the space E we define the norm

|z|E = sup(t,x)∈Δ

|z(t,x)|(exp(−λ (t + |x|))). (4.3.4)

It is easy to see that E is a Banach space and |z|E � N, where N � 0 is a constant.

The existence and uniqueness of solutions of problem (4.3.1)–(4.3.2) is given in the fol-

lowing theorem.


(i) the functions f , g in equation (4.3.1) satisfy the conditions

| f (t,x,u)− f (t,x,u)| � c(t,x)|u−u|, (4.3.5)

|g(t,x,y,u)−g(t,x,y,u)| � k(t,x,y)|u−u|, (4.3.6)

where c ∈C(Δ,R+), k ∈C(Δ× I,R+),



(l1) there exists a nonnegative constant α such that α < 1 and∫ t

0

[

c(s,x)exp(λ (s+ |x|))+∫ b

ak(s,x,y)exp(λ (s+ |y|))dy

]

ds

� α exp(λ (t + |x|)), (4.3.7)

(l2) there exists a nonnegative constant β such that

|φ(t,x)|+∫ t

0

[

| f (s,x,0)|+∫ b

a|g(s,x,y,0)|dy

]

ds � β exp(λ (t + |x|)), (4.3.8)

for (t,x) ∈ Δ, where

φ(t,x) = u0(x)+∫ t

0h(s,x)ds, (4.3.9)

in which h, u0 are as in (4.3.1), (4.3.2).

Under the assumptions (i) and (ii), the problem (4.3.1)–(4.3.2) has a unique solution u(t,x)

on Δ in E.

Proof. Let u ∈ E and define the operator T by

(Tu)(t,x) = φ(t,x)+∫ t

0

[

f (s,x,u(s,x))+∫ b

ag(s,x,y,u(s,y))dy

]

ds,

for (t,x) ∈ Δ, where φ(t,x) is given by (4.3.9). The proof that T maps E into itself and is

a contraction map can be completed by following the proof of Theorem 1.3.1 in Chapter 1

with suitable modifications. We leave the details to the reader.

The following theorem deals with the estimate on the solution of problem (4.3.1)–(4.3.2).

Theorem 4.3.2. Suppose that the functions f , g in equation (4.3.1) satisfy the conditions

(4.3.5), (4.3.6) and

d = sup(t,x)∈Δ

[

|φ(t,x)|+∫ t

0

[

| f (s,x,0)|+∫ b

a|g(s,x,y,0)|dy

]

ds

]

< ∞, (4.3.10)

where φ(t,x) is given by (4.3.9). If u(t,x) is any solution of problem (4.3.1)–(4.3.2) on Δ,

then

|u(t,x)| � dC(t,x)exp(∫ t

0

∫ b

ak(s,x,y)C(s,y)dyds

)

, (4.3.11)

for (t,x) ∈ Δ, where

C(t,x) = exp(∫ t

0c(s,x)ds

)

. (4.3.12)


Proof. Using the fact that u(t,x) is a solution of problem (4.3.1)–(4.3.2) and hypotheses,

we observe that

|u(t,x)| � |φ(t,x)|+∫ t

0

[

| f (s,x,0)|+∫ b

a|g(s,x,y,0)|dy

]

ds

+∫ t

0

[

| f (s,x,u(s,x))− f (s,x,0)|+∫ b

a|g(s,x,y,u(s,y))−g(s,x,y,0)|dy

]

ds

� d +∫ t

0

[

c(s,x)|u(s,x)|+∫ b

ak(s,x,y)|u(s,y)|dy

]

ds. (4.3.13)

Now an application of the inequality in Theorem 4.2.1 part (p1) to (4.3.13) yields (4.3.11).

Remark 4.3.1. We note that the estimate obtained in (4.3.11) provides the bound on the

solution u(t,x) of problem (4.3.1)–(4.3.2) on Δ. In Theorem 4.3.2 in addition, if we assume

that∫ ∞

0c(s,x)ds < ∞,

∫ ∞

0

∫ b

ak(s,x,y)C(s,y)dyds < ∞,

for every x ∈ I, then the solution u(t,x) of problem (4.3.1)–(4.3.2) is bounded on Δ.

A slight variant of Theorem 4.3.2 is embodied in the following theorem.

Theorem 4.3.3. Suppose that the functions f , g, h, u0 in (4.3.1)–(4.3.2) satisfy the con-

ditions (4.3.5), (4.3.6) and

d = sup(t,x)∈Δ

[∫ t

0

[

| f (s,x,φ(s,x))|+∫ b

a|g(s,x,y,φ(s,y))|dy

]

ds

]

< ∞, (4.3.14)

where φ(t,x) is defined by (4.3.9). If u(t,x) is any solution of problem (4.3.1)–(4.3.2) on

Δ, then

|u(t,x)−φ(t,x)| � dC(t,x)exp(∫ t

0

∫ b

ak(s,x,y)C(s,y)dyds

)

, (4.3.15)

for (t,x) ∈ Δ, where C(t,x) is given by (4.3.12).

Proof. Let e(t,x) = |u(t,x)−φ (t,x)|, (t,x) ∈ Δ. Using the fact that u(t,x) is a solution of

problem (4.3.1)–(4.3.2) and hypotheses, we observe that

e(t,x) �∫ t

0

[

| f (s,x,u(s,x))− f (s,x,φ(s,x))+ f (s,x,φ(s,x))|

+∫ b

a|g(s,x,y,u(s,y))−g(s,x,y,φ(s,y))+g(s,x,y,φ(s,y))|dy

]

ds


�∫ t

0

[

| f (s,x,φ(s,x))|+∫ b

a|g(s,x,y,φ(s,y))|dy

]

ds

+∫ t

0

[

| f (s,x,u(s,x))− f (s,x,φ(s,x))|

+∫ b

a|g(s,x,y,u(s,y))−g(s,x,y,φ(s,y))|dy

]

� d +∫ t

0

[

c(s,x)e(s,x)+∫ b

ak(s,x,y)u(s,y)dy

]

ds. (4.3.16)


Let u(t,x) ∈C (Δ,R) ; ∂∂ t u(t,x) exists on Δ and satisfies the inequality

∣∣∣∣

∂∂ t

u(t,x)− f (t,x,u(t,x))−∫ b

ag(t,x,y,u(t,y))dy−h(t,x)

∣∣∣∣ � ε,

for a given constant ε � 0, where it is supposed that (4.3.2) holds. Then we call u(t,x) the

ε-approximate solution with respect to the equation (4.3.1).

The following theorem estimates the difference between the two approximate solutions of

problem (4.3.1)–(4.3.2).

Theorem 4.3.4. Suppose that the functions f , g in (4.3.1) satisfy the conditions (4.3.5),

(4.3.6). Let ui(t,x) (i = 1, 2), (t,x) ∈ Δ be respectively εi-approximate solutions of (4.3.1)

with

ui(0,x) = ui(x), (4.3.17)

where ui ∈C (I,R) and

φi(t,x) = ui(x)+∫ t

0h(s,x)ds. (4.3.18)

Suppose that

|φ1(t,x)−φ2(t,x)| � δ , (4.3.19)

where δ � 0 is a constant and

M = supt∈R+

[(ε1 + ε2)t +δ ] < ∞. (4.3.20)

Then

|u1(t,x)−u2(t,x)| � MC(t,x)exp(∫ t

0

∫ b

ak(s,x,y)C(s,y)dyds

)

, (4.3.21)



Proof. Since ui(t,x) (i = 1, 2), (t,x) ∈ Δ are respectively εi-approximate solutions of

(4.3.1) with (4.3.17), we have∣∣∣∣

∂∂ t

ui(t,x)− f (t,x,ui(t,x))−∫ b

ag(t,x,y,ui(t,y))dy−h(t,x)

∣∣∣∣ � εi. (4.3.22)

By taking t = s in (4.3.22) and integrating both sides with respect to s from 0 to t for t ∈R+,

we get

εit �∫ t

0

∣∣∣∣

∂∂ s

ui(s,x)− f (s,x,ui(s,x))−∫ b

ag(s,x,y,ui(s,y))dy−h(t,x)

∣∣∣∣ds

�∣∣∣∣

∫ t

0

{∂∂ s

ui(s,x)− f (s,x,ui(s,x))−∫ b

ag(s,x,y,ui(s,y)) dy−h(t,x)

}

ds∣∣∣∣

=∣∣∣∣ui(t,x)−φi(t,x)−

∫ t

0

[

f (s,x,ui(s,x))+∫ b

a|g(s,x,y,ui(s,y))|dy

]

ds∣∣∣∣ . (4.3.23)

From (4.3.23) and using the elementary inequalities in (1.3.25), we observe that

(ε1 + ε2)t

�∣∣∣∣u1(t,x)−φ1(t,x)−

∫ t

0

[

f (s,x,u1(s,x))+∫ b

ag(s,x,y,u1(s,y)) dy

]

ds∣∣∣∣

+∣∣∣∣u2(t,x)−φ2(t,x)−

∫ t

0

[

f (s,x,u2(s,x))+∫ b


]

ds∣∣∣∣

�∣∣∣∣

{

u1(t,x)−φ1(t,x)−∫ t

0

[

f (s,x,u1(s,x))+∫ b


]

ds}

−{

u2(t,x)−φ2(t,x)−∫ t

0

[

f (s,x,u2(s,x))+∫ b


]

ds}∣

∣∣∣

� |u1(t,x)−u2(t,x)|− |φ1(t,x)−φ2(t,x)|

−∣∣∣∣

∫ t

0

[

f (s,x,u1(s,x))+∫ b


]

ds

−∫ t

0

[

f (s,x,u2(s,x))+∫ b


]

ds∣∣∣∣ . (4.3.24)

Let u(t,x) = |u1(t,x)− u2(t,x)|, (t,x) ∈ Δ. From (4.3.24) and using the hypotheses, we

observe that

u(t,x) � (ε1 + ε2)t +δ +∫ t

0

[

c(s,x)u(s,x)+∫ b

ak(s,x,y)u(s,y)dy

]

ds

� M +∫ t

0

[

c(s,x)u(s,x)+∫ b

ak(s,x,y)u(s,y)dy

]

ds. (4.3.25)



Remark 4.3.2. In case u1(t,x) is a solution of (4.3.1) with u1(0,x) = u1(x), then we have

ε1 = 0 and from (4.3.21), we see that u2(t,x) → u1(t,x) as ε2 → 0 and δ → 0. Furthermore,

if we put

(i) ε1 = ε2 = 0, u1(x) = u2(x) in (4.3.21), then the uniqueness of solutions of (4.3.1) is

established and

(ii) ε1 = ε2 = 0 in (4.3.21), then we get the bound which shows the dependency of solutions

of (4.3.1) on given initial values.

Consider (4.3.1)–(4.3.2) together with the following integrodifferential equation

∂∂ t

v(t,x) = f (t,x,v(t,x))+∫ b

ag(t,x,y,v(t,y))dy+h(t,x), (4.3.26)

with the initial condition

v(0,x) = v0(x), (4.3.27)

for (t,x) ∈ Δ, where f ∈C(Δ×R,R), g ∈C(Δ× I ×R,R), h ∈C(Δ,R), v0 ∈C(I,R).

In the following theorem we provide conditions concerning the closeness of solutions of

problems (4.3.1)–(4.3.2) and (4.3.26)–(4.3.27).


(4.3.6) and there exist constants ε i � 0, δ i � 0 (i = 1, 2) such that

| f (t,x,u)− f (t,x,u)| � ε1, (4.3.28)

|g(t,x,y,u)−g(t,x,y,u)| � ε2, (4.3.29)

|h(t,x)−h(t,x)| � δ 1, (4.3.30)

|u0(x)− v0(x)| � δ 2, (4.3.31)

where f , g, h, u0 and f , g, h, v0 are the functions in (4.3.1)–(4.3.2) and (4.3.26)–(4.3.27)

and

M = supt∈R+

[[δ 1 + ε1 + ε2(b−a)

]t +δ 2

]< ∞. (4.3.32)

Let u(t,x) and v(t,x) be respectively the solutions of (4.3.1)–(4.3.2) and (4.3.26)–(4.3.27)

for (t,x) ∈ Δ. Then

|u(t,x)− v(t,x)| � MC(t,x)exp(∫ t

0

∫ b

ak(s,x,y)C(s,y)dyds

)

, (4.3.33)



Proof. Let z(t,x) = |u(t,x)− v(t,x)|, (t,x) ∈ Δ. Using the facts that u(t,x), v(t,x) are

respectively the solutions of (4.3.1)–(4.3.2), (4.3.26)–(4.3.27) and hypotheses, we observe

that

z(t,x) �∣∣∣∣u0(x)+

∫ t

0h(s,x)ds− v0(x)−

∫ t

0h(s,x)ds

∣∣∣∣

+∫ t

0

[

| f (s,x,u(s,x))− f (s,x,v(s,x))|+ | f (s,x,v(s,x))− f (s,x,v(s,x))|

+∫ b

a{|g(s,x,y,u(s,y))−g(s,x,y,v(s,y))|+ |g(s,x,y,v(s,y))−g(s,x,y,v(s,y))|}dy

]

ds

� |u0(x)− v0(x)|+∫ t

0|h(s,x)−h(s,x)|ds

+∫ t

0

[

c(s,x)z(s,x)+ ε1 +∫ b

a{k(s,x,y)z(s,y)+ ε2}dy

]

ds

�[δ 2 +δ 1t + ε1t + ε2(b−a)t

]+

∫ t

0

[

c(s,x)z(s,x)+∫ b

ak(s,x,y)z(s,y)dy

]

ds

� M +∫ t

0

[

c(s,x)z(s,x)+∫ b

ak(s,x,y)z(s,y)dy

]

ds. (4.3.34)


Remark 4.3.3. We note that the result given in Theorem 4.3.5 relates the solutions of

(4.3.1)–(4.3.2) and (4.3.26)–(4.3.27) in the sense that if f , g, h, u0 are respectively close

to f , g, h, v0; then the solutions of (4.3.1)–(4.3.2) and (4.3.26)–(4.3.27) are also close to-

gether.

Next, we consider the following integrodifferential equations of Barbashin-type

∂∂ t

u(t,x) = f (t,x,u(t,x),μ)+∫ b

ag(t,x,y,u(t,y),μ) dy+h(t,x), (4.3.35)

∂∂ t

u(t,x) = f (t,x,u(t,x),μ0)+∫ b

ag(t,x,y,u(t,y),μ0) dy+h(t,x), (4.3.36)

with the given initial condition (4.3.2), where f ∈C(Δ×R×R,R), g∈C(Δ×I×R×R,R),

h ∈C(Δ,R) and μ , μ0 are parameters.

The following theorem deals with the dependency of solutions of (4.3.35)–(4.3.2) and

(4.3.36)–(4.3.2) on parameters.


Theorem 4.3.6. Suppose that the functions f , g in (4.3.35), (4.3.36) satisfy the conditions

| f (t,x,u,μ)− f (t,x,u,μ)| � c(t,x)|u−u|, (4.3.37)

| f (t,x,u,μ)− f (t,x,u,μ0)| � n1(t,x)|μ −μ0|, (4.3.38)

|g(t,x,y,u,μ)−g(t,x,y,u,μ)| � k(t,x,y)|u−u|, (4.3.39)

|g(t,x,y,u,μ)−g(t,x,y,u,μ0)| � n2(t,x,y)|μ −μ0|, (4.3.40)

where c, n1 ∈C(Δ,R+), k, n2 ∈C(Δ× I,R+) and

N = sup(t,x)∈Δ

∫ t

0

[

n1(s,x)+∫ b

an2(s,x,y)dy

]

ds < ∞.

Let u1(t,x) and u2(t,x) be respectively, the solutions of (4.3.35)–(4.3.2) and (4.3.36)–

(4.3.2) on Δ. Then

|u1(t,x)−u2(t,x)| � |μ −μ0|NC(t,x)exp(∫ t

0

∫ b

ak(s,x,y)C(s,y)dyds

)

, (4.3.41)

for (t,x) ∈ Δ, where C(t,x) is given by the right hand side of (4.3.12) by replacing c(t,x)

by c(t,x).

Proof. Let w(t,x) = |u1(t,x)−u2(t,x)|, (t,x)∈Δ. Using the facts that u1(t,x) and u2(t,x)

are respectively the solutions of (4.3.35)–(4.3.2) and (4.3.36)–(4.3.2) and hypotheses, we

observe that

w(t,x) �∫ t

0[| f (s,x,u1(s,x),μ)− f (s,x,u2(s,x),μ)|

+| f (s,x,u2(s,x),μ)− f (s,x,u2(s,x),μ0)|

+∫ b

a{|g(s,x,y,u1(s,y),μ)−g(s,x,y,u2(s,y),μ)|

+|g(s,x,y,u2(s,y),μ)−g(s,x,y,u2(s,y),μ0)|}dy]ds

�∫ t

0[c(s,x)|u1(s,x)−u2(s,x)| +n1(s,x)|μ −μ0|

+∫ b

a

{k(s,x,y)|u1(s,y)−u2(s,y)|+n2(s,x,y)|μ −μ0|

}dy

]ds

� |μ −μ0|N +∫ t

0

[

c(s,x)w(s,x)+∫ b

ak(s,x,y)w(s,y)dy

]

ds. (4.3.42)

Now an application of the inequality in Theorem 4.2.1 part (p1) to (4.3.42) yields (4.3.41),

which shows the dependency of solutions of (4.3.35)–(4.3.2) and (4.3.36)–(4.3.2) on pa-

rameters.


Remark 4.3.4. We note that by using the inequality in Theorem 4.2.2 part (p3), one can

obtain results similar to those given above to the following Barbashin-type integrodifferen-

tial equation

∂∂ t

u(t,x) = f (t,x,u(t,x))+∫

Bk(t,x,y,u(t,y))dy+h(t,x), (4.3.43)

with the given initial condition, under some suitable conditions on the functions involved

in (4.3.43) and the given initial condition, where B is a compact subset of Rn. As far

as equations of the forms (4.3.1)–(4.3.2) and (4.3.43) is concerned, we believe that the

numerous models (see [5]) whose detailed treatment is desired.

4.4 General integral equation of Barbashin-type

Integrodifferential equations of Barbashin-type (B) and partial integral equations are con-

nected to each other in several ways. For example, suppose we are intersted in findining a

solution u of equation (B) satisfying the initial condition (4.3.2). Putting ∂∂ t u(t,x) := w(t,x)

we arrive at the equation

w(t,x) = g(t,x)+∫ t

0c(t,x)w(τ,x)dτ +

∫ t

0

∫ b

ak(t,x,y)w(τ,y)dydτ, (4.4.1)

where

g(t,x) = f (t,x)+c(t,x)u0(x)+∫ b

ak(t,x,y)u0(y)dy. (4.4.2)

Throughout this section, we shall use the notations and definitions given in section 3.2

without further mention. The monograph [5] contains the study of many variants and gen-

eralizations of equations (B) and (4.4.1) by using different techniques. In this section we

consider the following general partial integral equation of the form (see [109])

u(t,x) = h(t,x)+∫ t

0f (t,x,s,u(s,x))ds+

∫ t

0

∫

Bg(t,x,s,y,u(s,y))dyds, (4.4.3)

for (t,x)∈D0, where h∈C(D0,R), f ∈C(Ω0×R,R), g ∈C(Ω×R,R) are given functions

and u is the unknown function to be found, in which Ω0 = {(t,x,s) : 0 � s � t < ∞, x ∈ B}.

The problems of existence and uniqueness of solutions of equation (4.4.3) can be dealt with

the method employed in Chapter 3, section 3.4. Here, we present some basic qualitative

properties of solutions of (4.4.3) which provide simple and elegant results and thus have a

wider scope of applicability.

First we shall give the following theorem concerning the estimate on the solution of equa-

tion (4.4.3).


Theorem 4.4.1. Suppose that the functions f , g,h in (4.4.3) satisfy the conditions

| f (t,x,s,u)− f (t,x,s,u)| � r(t,x,s)|u−u|, (4.4.4)

|g(t,x,s,y,u)−g(t,x,s,y,u)| � k(t,x,s,y)|u−u|, (4.4.5)

and

d = sup(t,x)∈D0

[

|h(t,x)|+∫ t

0| f (t,x,s,0)|ds +

∫ t

0

∫

B|g(t,x,s,y,0)|dyds

]

< ∞, (4.4.6)

where r ∈ C(Ω0,R+), k ∈ C(Ω,R+); D1r, D1k exist and D1r ∈ C(Ω0,R+), D1k ∈C(Ω,R+). If u(t,x) is any solution of equation (4.4.3) on D0, then

|u(t,x)| � dP(t,x)exp(∫ t

0A(σ ,x)dσ

)

, (4.4.7)

for (t,x) ∈ D0, where P and A are given by (4.2.8) and (4.2.10).

Proof. Using the fact that u(t,x) is a solution of (4.4.3) and hypotheses, we observe that

|u(t,x)| � |h(t,x)|+∫ t

0| f (t,x,s,u(s,x))− f (t,x,s,0)+ f (t,x,s,0)|ds

+∫ t

0

∫

B|g(t,x,s,y,u(s,y))−g(t,x,s,y,0)+g(t,x,s,y,0)|dyds

� |h(t,x)|+∫ t

0| f (t,x,s,0)|ds+

∫ t

0

∫

B|g(t,x,s,y,0)|dyds

+∫ t

0| f (t,x,s,u(s,x))− f (t,x,s,0)|ds+

∫ t

0

∫

B|g(t,x,s,y,u(s,y))−g(t,x,s,y,0)|dyds

� d +∫ t

0r(t,x,s)|u(s,x)|ds+

∫ t

0

∫

Bk(t,x,s,y)|u(s,y)|dyds. (4.4.8)

Now an application of Theorem 4.2.2 part (p3) to (4.4.8) yields (4.4.7).

The following theorem deals with a slight variant of Theorem 4.4.1.

Theorem 4.4.2. Suppose that the functions f , g and h in (4.4.3) satisfy the conditions

(4.4.4), (4.4.5) and

d = sup(t,x)∈D0

[∫ t

0| f (t,x,s,h(s,x))|ds +

∫ t

0

∫

B|g(t,x,s,y,h(s,y))|dyds

]

< ∞. (4.4.9)

If u(t,x) is any solution of equation (4.4.3) on D0, then

|u(t,x)−h(t,x)| � dP(t,x)exp(∫ t

0A(σ ,x)dσ

)

, (4.4.10)



Proof. Let z(t,x) = |u(t,x)−h(t,x)|, (t,x) ∈ D0. Using the fact that u(t,x) is a solution

of (4.4.3) and hypotheses, we observe that

z(t,x) �∫ t

0| f (t,x,s,u(s,x))− f (t,x,s,h(s,x))+ f (t,x,s,h(s,x))|ds

+∫ t

0

∫

B|g(t,x,s,y,u(s,y))−g(t,x,s,y,h(s,y))+g(t,x,s,y,h(s,y))|dyds

�∫ t

0| f (t,x,s,h(s,x))|ds+

∫ t

0

∫

B|g(t,x,s,y,h(s,y))|dyds

+∫ t

0| f (t,x,s,u(s,x))− f (t,x,s,h(s,x))|ds

+∫ t

0

∫

B|g(t,x,s,y,u(s,y))−g(t,x,s,y,h(s,y))|dyds

� d +∫ t

0r(t,x,s)z(s,x)ds+

∫ t

0

∫



In the next theorem we formulate conditions on the functions involved in (4.4.3) which

shows that the solutions of (4.4.3) tends to zero as t → ∞.


(i) the functions f , g in (4.4.3) satisfy the conditions (4.4.4), (4.4.5) and f (t,x,s,0) = 0,

g(t,x,s,y,0) = 0,

(ii) the function h in (4.4.3) satisfies the condition

|h(t,x)| � M exp(−αt), (4.4.12)

for (t,x) ∈ D0, where α > 0, M � 0 are constants,

(iii) the functions r,D1r,k,D1k be as in Theorem 4.4.1 and

sup(t,x)∈D0

Q(t,x) < ∞,∫ ∞

0A(σ ,x)dσ < ∞, (4.4.13)

where Q and A are given respectively by the right hand sides of (4.2.9) and (4.2.10)

by replacing r and k by r exp(α(t − s)) and k exp(α(t − s)). If u(t,x) is any solution

of (4.4.3) on D0, then it tends exponentially toward zero as t → ∞.


Proof. From the hypotheses, we observe that

|u(t,x)| � |h(t,x)|+∫ t

0| f (t,x,s,u(s,x))− f (t,x,s,0)|ds

+∫ t

0

∫

B|g(t,x,s,y,u(s,y))−g(t,x,s,y,0)|dyds

� M exp(−αt)+∫ t

0r(t,x,s)|u(s,x)|ds+

∫ t

0

∫

Bk(t,x,s,y)|u(s,y)|dyds. (4.4.14)


|u(t,x)|exp(αt) � M +∫ t

0r(t,x,s)exp(α(t − s))|u(s,x)|exp(αs)ds

+∫ t

0

∫

Bk(t,x,s,y)exp(α(t − s))|u(s,y)|exp(αs)dyds. (4.4.15)

Now an application of Theorem 4.2.2 part (p3) to (4.4.15) yields

|u(t,x)|exp(αt) � MP(t,x)exp(∫ t

0A(σ ,x)dσ

)

, (4.4.16)

where P is given by the right hand side of (4.2.8) by replacing Q by Q. Multiplying both

sides of (4.4.16) by exp(−αt) and in view of (4.4.13), the solution u(t,x) tends to zero as

t → ∞.

The following theorem deals with the estimate on the difference between the two approxi-

mate solutions of equation (4.4.3).


(4.4.5) respectively. Let ui(t,x) (i = 1, 2) be respectively εi-approximate solutions of equa-

tion (4.4.3) on D0, i.e.,∣∣∣∣ui(t,x)−h(t,x)−

∫ t

0f (t,x,s,ui(s,x))ds −

∫ t

0

∫

Bg(t,x,s,y,ui(s,y))dyds

∣∣∣∣ � εi, (4.4.17)

on D0 for constants εi � 0. Then

|u1(t,x)−u2(t,x)| � (ε1 + ε2)P(t,x)exp(∫ t

0A(σ ,x)dσ

)

, (4.4.18)


The proof follows by the arguments as in the proof Theorem 3.4.4 and making use of the

inequality in Theorem 4.2.2 part (p3). We leave the details to the reader.


Remark 4.4.1. In case u1(t,x) is a solution of equation (4.4.3), then we have ε1 = 0 and

from (4.4.18) we see that u2(t,x) → u1(t,x) on D0 as ε2 → 0. Moreover, from (4.4.18) it

follows that if ε1 = ε2 = 0, then the uniqueness of solutions of (4.4.3) is established.

Consider the equation (4.4.3) together with the following integral equation

v(t,x) = h(t,x)+∫ t

0f (t,x,s,v(s,x))ds+

∫ t

0

∫

Bg(t,x,s,y,v(s,y))dyds, (4.4.19)

for (t,x)∈D0, where h∈C(D0,R), f ∈C(Ω0×R,R), g∈C(Ω×R,R), are given functions

and v is the unknown function.

In the following theorem we provide conditions concerning the closeness of solutions of

equations (4.4.3) and (4.4.19).


(4.4.5) respectively and there exist constants ε i � 0, (i = 1, 2), δ � 0 such that

| f (t,x,s,u)− f (t,x,s,u)| � ε1, (4.4.20)

|g(t,x,s,y,u)−g(t,x,s,y,u)| � ε2, (4.4.21)

|h(t,x)−h(t,x)| � δ , (4.4.22)

where f , g, h and f , g, h are functions involved in (4.4.3) and (4.4.19) and

M = supt∈R+

[

δ + t

{

ε1 + ε2

n

∏i=1

(bi −ai)

}]

< ∞. (4.4.23)

Let u(t,x) and v(t,x) be respectively the solutions of (4.4.3) and (4.4.19) on D0. Then

|u(t,x)− v(t,x)| � MP(t,x)exp(∫ t

0A(σ ,x)dσ

)

, (4.4.24)


Proof. Let z(t,x) = |u(t,x)− v(t,x)|, (t,x) ∈ D0. Using the hypotheses, we observe that

z(t,x) � |h(t,x)−h(t,x)|

+∫ t

0| f (t,x,s,u(s,x))− f (t,x,s,v(s,x)).

+ f (t,x,s,v(s,x))− f (t,x,s,v(s,x))|ds

+∫ t

0

∫

Bg(t,x,s,y,u(s,y))−g(t,x,s,y,v(s,y))

+g(t,x,s,y,v(s,y))−g(t,x,s,y,v(s,y))|dyds


� δ +∫ t

0| f (t,x,s,u(s,x))− f (t,x,s,v(s,x))|ds

+∫ t

0| f (t,x,s,v(s,x))− f (t,x,s,v(s,x))|ds

+∫ t

0

∫

B|g(t,x,s,y,u(s,y))−g(t,x,s,y,v(s,y))|dyds

+∫ t

0

∫

B|g(t,x,s,y,v(s,y))−g(t,x,s,y,v(s,y))|dyds

� δ +∫ t

0r(t,x,s)z(s,x)ds+

∫ t

0ε1ds+

∫ t

0

∫

Bk(t,x,s,y)z(s,y)dyds+

∫ t

0

∫

Bε2dyds

� M +∫ t

0r(t,x,s)z(s,x)ds+

∫ t

0

∫



Remark 4.4.2. The result given in Theorem 4.4.5 relates the solutions of (4.4.3) and

(4.4.19) in the sense that if f , g, h are respectively close to f , g, h; then the solutions of

(4.4.3) and (4.4.19) are also close together.

In the following theorem we formulate conditions for continuous dependence of solutions

of equations

u(t,x) = hi(t,x)+∫ t

0f (t,x,s,u(s,x))ds+

∫ t

0

∫

Bg(t,x,s,y,u(s,y))dyds, (4.4.26)

for (i = 1, 2), (t,x) ∈ D0, where hi ∈ C(D0,R) and the functions f , g are as defined in

(4.4.3).


(4.4.5) and there exists a constant δ � 0 such that

|h1(t,x)−h2(t,x)| � δ , (4.4.27)

for (t,x) ∈ D0. Let ui(t,x) (i = 1, 2) be respectively εi – approximate solutions of (4.4.26).

Then

|u1(t,x)−u2(t,x)| � (δ + ε1 + ε2)P(t,x)exp(∫ t

0A(σ ,x)dσ

)

, (4.4.28)



Proof. Let z(t,x) = |u1(t,x)−u2(t,x)|, (t,x) ∈ D0. Following the proof of Theorem 3.4.4

and using the hypotheses, we obtain

z(t,x) � |h1(t,x)−h2(t,x)|+ ε1 + ε2 +∫ t

0| f (t,x,s,u1(s,x))− f (t,x,s,u2(s,x))|ds

+∫ t

0

∫

B|g(t,x,s,y,u1(s,y))−g(t,x,s,y,u2(s,y))|dyds

� δ + ε1 + ε2 +∫ t

0r(t,x,s)z(s,x)ds+

∫ t

0

∫


Now applying Theorem 4.2.2 part (p3) to (4.4.29) yields (4.4.28), which shows that the so-

lutions of (4.4.26) depends continuously on the functions on the right hand side of (4.4.26).

Remark 4.4.3. As noted in [5, p. 12 and p. 476], another significant source of parabolic

type integrodifferential equations is provided by the study of equation of the form

u(t,x) = f (t,x)+∫ t

0P(t,x,s,u(s,x))ds+

∫

BQ(t,x,y,u(s,y))dy

+∫ t

0

∫


and variants thereof under some suitable conditions. It is hoped and expected that the

analysis presented above in Theorems 4.4.1–4.4.6 will also be equally useful in the study

of equations like (4.4.30). Indeed, it involves the task of desiging a new inequality similar

to the one given in Theorem 4.2.2 part (p3), which will allow applications in the discussion

of equation (4.4.30). In fact, it is not an easy matter but a challenging problem for future

investigations.

4.5 Integrodifferential equation of the type arising in reactor dynamics

Parabolic integrodifferential equations of various types occurring in reactor dynamics have

been extensively investigated in the literature by using different techniques. In this sec-

tion we offer some basic results from [75, 77] concerning the existence, uniqueness and

asymptotic behavior of solutions of the following general nonlinear diffusion system

∂u∂ t

−Lu = f (x,t,u)+(Hu)(x,t) (t > 0, x ∈ Ω), (4.5.1)

with the given boundary conditions

B[u] = d(x)∂ u∂τ

+u = 0 (t > 0, x ∈ ∂ Ω), (4.5.2)

u(x,0) = u0(x), x ∈ Ω, (4.5.3)


where Ω is a bounded domain in Rn, ∂Ω the sufficiently smooth boundary of Ω, τ is the

outward normal unit vector on the boundary ∂ Ω and

L =n

∑i, j=1

ai j(x)∂ 2

∂xi∂x j+

n

∑i=1

ai(x)∂

∂xi−a0(x), (4.5.4)

is the uniformly elliptic operator.

Let D = Ω× (0,T ], D = Ω× [0,T ], S = ∂Ω× (0,T ], where T > 0 is a finite but can be

arbitrarily large and Ω is the closure of Ω. Denote by C2(D) the class of functions which

are continuous in D, continuously differentiable in t and twice continuously differentiable

in x and for (x,t) ∈ D and ∂u∂τ exists on ∂ Ω. Throughout assume that the coefficients of L

and the first order derivatives of ai j are Holder continuous in Ω (of exponent α , 0 < α < 1),

the matrix(ai j

)is symmetric positive definite in Ω, a0(x) � 0 in Ω, f is Holder continuous

in (x,t,u) in every bounded subset of Ω×R+ ×Rn, the operator Hu is uniformly Holder

continuous with respect to u ∈ Rn and for (x, t) ∈ Ω×R+, the boundary ∂ Ω is of class

C2+α , d ∈ H1+α(∂ Ω) and u0 ∈ H2+α(Ω

)and satisfies boundary condition (4.5.2) at t = 0,

where C2+α ,H2+α(Ω

),H1+α(∂ Ω) are the function spaces defined in Chapter 1 of [57] (see

also [37,69,122]). The above smoothness assumptions are required only for ensuring the

existence of a solution for the corresponding linear problem. A function u ∈C2(D) which

satisfies (4.5.1)–(4.5.3) is called a solution of problem (4.5.1)–(4.5.3). System (4.5.1)–

(4.5.3) occur in nuclear reactor dynamics with the operator Hu may in general be of the

form

(Hu)(x,t) = u(x,t)h(

x,t,∫ t

0k(x,t,s)u(x,s)ds

)

,

or

(Hu)(x,t) = h(

x, t,∫ t

0k(x,t,s)u(x,s)ds

)

,

or variants thereof (see [59,60,75 ]), where all the functions are Holder continuous and

defined on the respective domains of their definitions.

Below we employ the notion of upper and lower solutions and monotone method to estab-

lish the existence of maximal and minimal solutions of (4.5.1)–(4.5.3) which also yields

the upper and lower estimates on the exact solution of the problem (4.5.1)–(4.5.3). In what

follows, we assume that all inequalities between vectors are componentwise.

The function w ∈C2(D) is called an upper solution of (4.5.1)–(4.5.3) if

∂w∂ t

−Lw � f (x,t,w)+(Hw)(x,t), (x,t) ∈ D, (4.5.5)


B [w] � 0, (x, t) ∈ S, (4.5.6)

w(x,0) � u0(x), x ∈ Ω. (4.5.7)

Similarly, the function v ∈C2(D) is called a lower solution of (4.5.1)–(4.5.3) if it satisfies

all the reversed inequalities in (4.4.5)–(4.4.7).

The functions u, u ∈ C2(D) are called maximal and minimal solutions of (4.5.1)–(4.5.3),

respectively, if every other solution u ∈ C2(D) of (4.5.1)–(4.5.3) satisfies the relation

u(x,t) � u(x,t) � u(x,t) for (x,t) ∈ D.

We list for convenience the following assumptions:

(H1) the functions v, w ∈C2(D) with v � w on D are lower and upper solutions of (4.5.1)–

(4.5.3),

(H2) for each i, fi(x,t,u1)− fi(x, t,u2) �−M(u1i−u2i) whenever v(x,t) � u2 � u1 � w(x,t)

for (x, t) ∈ D, where M � 0 is a constant,

(H3) for each i, the operator Hiu is monotone nondecreasing in u, whenever v(x,t) � u �w(x,t) for (x,t) ∈ D.

For any η ∈C2(D) such that v � η � w on D, consider the linear system∂ ui

∂ t−Lui = fi(x,t,η)+(Hiη)(x,t)−M(ui −ηi), (x, t) ∈ D, (4.5.8)

B[ui] = 0, (x,t) ∈ S, (4.5.9)

ui(x,0) = u0i(x), x ∈ Ω. (4.5.10)

For a known η the assumptions on f and H ensures (see [57,69,122]) the existence of a

unique solution of (4.5.8)–(4.5.10). For each η ∈C2(D) such that v � η � w on D, define

the mapping A by

Aη = u, (4.5.11)

where u is the unique solution of (4.5.8)–(4.5.10). This mapping will be used to define

the sequences that converge to the minimal and maximal solutions of (4.5.1)–(4.5.3). To

achieve this, we first prove the following lemma.

Lemma 4.5.1. Assume that the hypotheses (H1)–(H3) hold. Then

(i) the unique solution u of (4.5.8)–(4.5.10) satisfies v(x, t) � u(x,t) � w(x,t), (x,t) ∈ D;

(ii) v � Av, w � Aw, for (x,t) ∈ D;

(iii) A is a monotone operator on the set of functions

U(v,w) ={

u ∈C2(D) : v(x,t) � u � w(x,t), (x,t) ∈ D}

.


Proof. (i) We shall first show that v(x,t) � u(x,t) for (x,t) ∈ D. Setting pi = vi −ui, we

get

∂ pi

∂ t� Lvi + fi(x,t,v)+(Hiv)(x,t)−Lui − fi(x,t,η)− (Hiη)(x,t)+M(ui −ηi)

� Lpi +M(ηi − vi)+(Hiη)(x, t)− (Hiη)(x,t)+M(ui −ηi)

= Lpi −Mpi.

This implies

∂ pi

∂ t−Lpi +Mpi � 0,

and also we observe that B[pi] � 0, pi(x,0) � 0. Hence by the maximum principle (see

[125]) pi � 0 for i = 1, 2, . . . ,n. This proves that v(x,t) � u(x,t) for (x, t) ∈ D. Similarly,

we can show that u(x, t) � w(x,t) for (x,t) ∈ D. This proves (i).

(ii) Let Av = u, where u is the unique solution of (4.5.8)–(4.5.10) with η = v. Now, we

shall prove that v � u for (x, t) ∈ D. Setting pi = vi −ui, we get

∂ pi

∂ t� Lvi + fi(x,t,v)+(Hiv)(x,t)−Lui − fi(x,t,v)− (Hiv)(x,t)+M(ui − vi)

= Lpi −Mpi.

This implies

∂ pi


and also we observe that B[pi] � 0, pi(x,0) � 0. Hence by the maximum principle pi � 0,

i = 1, 2, . . . ,n. This implies v(x,t) � u(x,t) for (x,t) ∈ D. This proves v � Av. In a similar

way we can prove w � Aw.

(iii) Let η1, η2 ∈ C2(D) be such that η1, η2 ∈ U(v,w) and η1 � η2. Let Aη1 = u1 and

Aη2 = u2, where u1 and u2 are the unique solutions of (4.5.8)–(4.5.10) corresponding to η1

and η2, respectively. Then setting pi = u1i −u2i, we get

∂ pi

∂ t= Lu1i + fi(x,t,η1)+(Hiη1)(x, t)−M(u1i −η1i)

−Lu2i − fi(x,t,η2)− (Hiη2)(x,t)+M(u2i −η2i)

� Lpi −{ fi(x,t,η2)− fi(x,t,η1)}+(Hiη2)(x,t)− (Hiη2)(x,t)

−M(u1i −η1i)+M(u2i −η2i)

� Lpi +M(η2i −η1i)−M(u1i −η1i)+M(u2i −η2i)


= Lpi −Mpi.

This implies∂ pi


and also we have B[pi] = 0, pi(x,0) = 0. Hence, by the maximum principle pi � 0, i =

1, 2, . . . ,n, which implies u1(x,t) � u2(x,t) for (x,t) ∈ D. It therefore follows that the

mapping A is monotone on U(v,w). The proof is complete.

In view of Lemma 4.5.1, we can define the sequences

vn = Avn−1,wn = Awn−1,

with v0 = v and w0 = w. It is easy to observe that the sequences {vn} and {wn} are mono-

tone nondecreasing and nonincreasing respectively, and v � vn � wn � w on D. Further-

more, using the standard arguments, it follows that these sequences converge uniformly

and monotonically to solutions u and u of (4.5.1)–(4.5.3).

Let u be any solution of (4.5.1)–(4.5.3) such that u ∈ U(v,w). Then, by the induction

argument, it is easily seen that u � wn and u � vn for every n = 0, 1, 2, . . .. Hence, we

have u � u � u. This shows that u is a maximal solution and u is a minimal solution of

(4.5.1)–(4.5.3) on D. Thus we have proved the following theorem.

Theorem 4.5.1. Assume that the hypotheses (H1)–(H3) hold. Then the sequence {wn}converges uniformly from above to a maximal solution u of (4.5.1)–(4.5.3), while the se-

quence {vn} converges uniformly from below to a minimal solution u of (4.5.1)–(4.5.3).

Furthermore, if u is any solution of (4.5.1)–(4.5.3) such that u ∈U(v,w), then

v � v1 � · · · � vn � · · · � u � u � u � · · · � wn · · · � w1 � w,

on D.

We note that the maximal and minimal solutions established in Theorem 4.5.1 are not

necessarily the same. However, if f and Hu satisfy the conditions

(H4) f (x,t,ξ1)− f (x,t,ξ2) � c1 (ξ1 −ξ2),

(Hξ1)(x,t)− (Hξ1)(x,t) � c2 (ξ1 −ξ2),

whenever v � ξ2 � ξ1 � w on D, where c1 and c2 are nonnegative constants, then we have

the uniqueness result.

Theorem 4.5.2. Assume that the hypotheses (H1)–(H4) hold. Then the maximal and

minimal solutions of (4.5.1)–(4.5.3) coincide. Furthermore, if (H2) and (H4) hold for v = 0

and every finite w, then (4.5.1)–(4.5.3) has a unique nonnegative solution.


Proof. Let λ be a constant satisfying λ > c+ c1 + c2 and let z = e−λ t(u−u). Then

∂ z∂ t

−Lz +(λ − c1 − c2)z

= e−λ t [ f (x,t,u)− f (x,t,u)+(Hu)(x,t)− (Hu)(x,t)]− (c1 + c2)z

� e−λ t [c1(u−u)+ c2(u−u)]− (c1 + c2)z = 0,

and B[z] = 0, z(x,0) = 0. Hence by the maximum principle z = 0, that is u = u. Let u be

any nonnegative solution of (4.5.1)–(4.5.3). Then u is also an upper solution and thus by

Theorem 4.5.1 with w = u we have u � u. By replacing u by u in the definition of z the

same argument leads immediately to u = u. Hence, problem (4.5.1)–(4.5.3) cannot have

more than one nonnegative solution. The proof is complete.

Consider the eigenvalue problem

Lφ +λφ = 0, x ∈ Ω; B [φ ] = 0, x ∈ ∂ Ω, (4.5.12)

where L and B are defined in (4.5.4) and (4.5.2). The next result depends on the construction

of the upper solution which depends upon the use of the eigenfunction φ corresponding to

the least eigenvalue λ0 of (4.5.12). It is well known that λ0 is real and positive and φis (or can be chosen) positive in Ω (see [125]). When d(x) > 0 the maximum principle

implies that φ(x) > 0 on Ω. We normalize φ so that max{φ(x) : x ∈ Ω} = 1 and write

φm ≡ min{φ(x) : x ∈ Ω}. Notice that φm > 0 when d > 0 on ∂Ω.

Theorem 4.5.3. Assume that the hypotheses of Theorem 4.5.1 hold. Let u(x,t) be

nonnegative solution of (4.5.1)–(4.5.3) and d(x) > 0. Let β > 0, b < λ0, α = (λ0−b)2 ,

δ = (λ0−b)2

4β be such that for t > 0, x ∈ Ω,

f (x,t,ξ ) � bξ , ξ � 0, (4.5.13)

(Hξ )(x,t) � βδξ 2∫ t

0e−αsds, ξ � 0. (4.5.14)

Then for any constant k > 0,

0 � u(x, t) � ke−αtφ(x) (t � 0, x ∈ Ω), (4.5.15)

whenever 0 � u0(x) � kφ(x).


Proof. To prove (4.5.15) it suffices to show that w(x,t) = δ e−αtφ (x) is an upper solution.

Clearly, the inequalities (4.5.6), (4.5.7) are satisfied. We observe that

∂w∂ t

−Lw = δ e−αt [−αφ(x)−Lφ(x)] ,

and in view of the assumptions (4.5.13) and (4.5.14) we need only to show that

−αφ(x)−Lφ(x) � bφ(x)+βδφ 2(x)∫ t

0e−αsds.

Since φ � 1, −Lφ = λ0φ , the inequality holds if

(λ0 −b−α) �(

βδα

)(1− e−αt) , t > 0.

An optimal choice of α, δ is given by α = (λ0−b)2 , δ = (λ0−b)2

4β . Hence w is an upper

solution of (4.5.1)–(4.5.3). Conclusion (4.5.15) is now a consequence of Theorem 4.5.1.

We note that the result in Theorem 4.5.3 gives the rate of the exponential decay of the

solutions of (4.5.1)–(4.5.3) and u(x, t) → 0 as t → ∞.

In [124,77] the authors have studied the integrodifferential equation of the form

∂u∂ t

−Lu = Φ(x, t,u,Fu), (x,t) ∈ D, (4.5.16)

B[u] = a∂ u∂τ

+bu = h(x,t), (x,t) ∈ S, (4.5.17)

u(x.0) = u0(x), x ∈ Ω, (4.5.18)

where

L =n

∑i, j=1

ai j(x,t)∂ 2

∂xi∂ x j+

n

∑i=1

ai(x, t)∂

∂xi,

is the uniformly elliptic operator whose coefficients are as defined above with suitable

modifications (see also [37,57,122,125]), a, b are nonnegative constants, h ∈ H1+α(s), Φis Holder continuous in (x,t,ω ,ξ ) in every bounded subset of Ω×R+ ×R

n ×Rn and the

operator F may in particular be of the form

Fu(x,t) =∫ t

0k1(x,t,s,u(x,s))ds,

or

Fu(x,t) =∫

Ωk2(x, t,y,u(y, t))dy,

or variants thereof, in which k1, k2 are defined on the respective domains of their definitions.

In [77] the problem of existence of maximal and minimal solution is investigated by using


the concept of upper and lower solutions and monotone method. Below we explore in brief

the results given in [77].

The function σ+ ∈C2(D) is called an upper solution of problem (4.5.16)–(4.5.18) if

∂σ+

∂ t−Lσ+ � Φ(x, t,σ+,Fσ+), (x,t) ∈ D, (4.5.19)

B[σ+] � h(x,t), (x, t) ∈ S, (4.5.20)

σ+(x,0) � u0(x), x ∈ Ω. (4.5.21)

Similarly, the function σ− ∈ C2(D) is called a lower solution of (4.5.16)–(4.5.18) if the

inequalities in (4.5.19)–(4.5.21) are reversed.

We list the following assumptions for convenience:

(G1) the functions σ−, σ+ ∈ C2(D) with σ−(x,t) � σ+(x, t) on D are lower and upper

solutions of (4.5.16)–(4.5.18),

(G2) for each i, Φi(x,t,u,Fu)−Φi(x, t,u,Fu) �−N(ui−ui), for every (x, t)∈D, whenever

σ−(x, t) � u � u � σ+(x,t), where N � 0 is a constant,

(G3) the operator Fu is monotone increasing on U(σ−,σ+) and for each i, Φi(x, t,u,v) is

monotone nondecreasing in v.

For any η ∈C2(D) such that σ− � η � σ+ on D, consider the following linear system

∂ ui

∂ t−Lui = Φi(x, t,η,Fη)−N(ui −ηi), (x,t) ∈ D, (4.5.22)

B [ui] = h(x,t), (x,t) ∈ S, (4.5.23)

ui(x,0) = u0i(x), x ∈ Ω. (4.5.24)

For a known function η , the assumptions on Φ and F ensure (see [37,57,122]) the existence

of a unique solution of (4.5.22)–(4.5.24). For each η ∈C2(D) such that σ− � η � σ+ on

D, define the mapping P by

Pη = u, (4.5.25)

where u is the unique solution of (4.5.22)–(4.5.24).

Using the techniques of proof of Lemma 4.5.1 and Theorem 4.5.1, one can easily prove the

following results.


Lemma 4.5.2. Assume that the hypotheses (G1)–(G3) hold. Then

(i) the unique solution u of (4.5.22)–(4.5.24) satisfies σ−(x,t) � u(x,t) � σ+(x,t), (x,t) ∈D;

(ii) σ− � Pσ−, σ+ � Pσ+ on D;

(iii) P is monotone operator on the set of functions U(σ−,σ+).

From Lemma 4.5.2, we can define the sequences

σ−n = Pσ−

n−1, σ+n = Pσ+

n−1,

with σ−0 = σ− and σ+

0 = σ+. By following the similar observations below Lemma 4.5.1

on the sequences {σ−n }, {σ+

n }, we have the following theorem on the existence of maximal

and minimal solutions of (4.5.16)–(4.5.18).

Theorem 4.5.4. Assume that the hypotheses (G1)–(G3) hold. Then the sequence {σ+n },

converges uniformly from above to the maximal solution u+ of (4.5.16)–(4.5.18), while

the sequence {σ−n } converges uniformly from below to a minimal solution u− of (4.5.16)–

(4.5.18). Furthermore, if u is any solution of (4.5.16)–(4.5.18) such that u ∈ U(σ−,σ+),

then

σ− � σ−1 � · · · � σ−

n � · · · � u− � u � u+ � · · · � σ+n � · · · � σ+

1 � σ+,

on D.

4.6 Initial-boundary value problem for integrodifferential equations

In this section, we deal with solvability in the classical sense of a nonlinear integrodiffer-

ential initial-boundary value problem:

ut = a(x, t,u,ux)uxx +b(x, t,u,ux)+∫ t

0c(x,τ,u,ux)dτ, (4.6.1)

in QT ,

u(0,t) = f1(t), 0 � t � T, (4.6.2)

u(1,t) = f2(t), 0 � t � T, (4.6.3)

u(x,0) = u0(x), 0 � x � 1, (4.6.4)

where QT = [0,1]× [0,T ] with T > 0 arbitrary. One of the characteristics of this kind

of problem is that the maximum principle is no longer valid in general. In dealing with

this problem in [140], integral estimates in conjunction with Schauder estimate theory to

derive an a priori bound for the solution of (4.6.1)–(4.6.4) in the norm of the Banach space


C2+α ,1+ α2 (QT ) is used. The notations of the norms in Banach spaces C(QT ), C2,1(QT ),

etc. are those of Ladyzenskaya et.al. [57]. The method of continuity, which is similar to

that applicable for a regular parabolic boundary value problem is then applied in [140] to

establish a global solvability of (4.6.1)–(4.6.4) in the classical sense. Below, the main goal

is to present the results obtained in [140] concerning the solvability of (4.6.1)–(4.6.4).

The following hypotheses are assumed throughout the discussion.

(H1) the functions a(x,t,u, p),b(x, t,u, p) and c(x,t,u, p) are differentiable with respect to

all of their arguments. Furthermore,

(i) a(x,t,u, p) � A1 > 0,

(ii) |b(x,t,u, p)| � A2[1+ |u|+ |p|],(iii) |c(x,t,u, p)| � A3[1 + |u|+ |p|], for (x,t,u, p) ∈ QT ×R

2, where A1, A2 and A3 are

three absolute constants.

(H2) the functions f1(t), f2(t)∈C2[0,T ], u0(x)∈C2+α [0,1] and the consistency conditions

f1(0) = u0(0), f2(0) = u0(1),

f ′1(0) = a(0,0,u0(0),u′0(0))u′′0(0)+b(0,0,u0(0),u′0(0)),

and

f ′2(0) = a(1,0,u0(1),u′0(1))u′′0(1)+b(1,0,u0(1),u′0(1)),

are satisfied.

We need the following well known inequalities to establish the results.

1. Young’s inequality: If a � 0, b � 0, then, for any η > 0,

ab � ηar

r+η− s

rbs

s,

where r > 1, s > 1 and 1r + 1

s = 1.

2. Interpolation inequalities: If u(x) ∈ H1(0,1), then

‖u‖L∞(0,1) � C‖u‖23H1(0,1)‖u‖

13L1(0,1).

We first establish a global a priori bound for the solution u(x,t) in C2+α ,1+ α2 (QT ) based on

an integral calculation, imbedding inequalities and Schauder estimates under the hypothe-

ses (H1)–(H2).

Let T > 0 be arbitrary and assume that u(x,t) is an arbitrary solution of the problem (4.6.1)–

(4.6.4). We first deduce the following result.


Lemma 4.6.1. Under the assumptions (H1)–(H2), u(x,t) satisfies the following inequal-

ity:∫ ∫

QT

u2xxdxdt + sup

0�t�T

∫ 1

0u2

x(x, t)dx � C1, (4.6.5)

where C1 depends only on the Ai (i = 1, 2, 3), the known data and the upper bound of T .

Proof. In what follows, various constants which appear during the process of the proof

will be denoted by C; their dependency is the same as final constants unless stated other-

wise.

Let v(x, t) = (1− x) f1(t)+ x f2(t) and w(x,t) = u(x,t)− v(x,t),(x, t) ∈ QT . Then w(x,t) is

a solution of the following problem:

wt = awxx +b− vt +∫ t

0c(x,τ,w+ v,wx + vx) dτ, (4.6.6)

w(0,t) = w(1,t) = 0, 0 � t � T, (4.6.7)

w(x,0) = u0(x)− [(1− x) f1(0)+ x f2(0)] = w0(x) (say), 0 � x � 1. (4.6.8)

Multiplying equation (4.6.6) by wxx and integrating it over QT , we obtain, employing the

Cauchy–Schwarz inequality with small parameter ε > 0 and the assumption (H1) that

A1

∫ ∫

QT

w2xxdxdt −

∫ ∫

QT

wtwxxdxdt

� ε∫ ∫

QT

w2xxdxdt +C(ε)

∫ ∫

QT

{

1+w2 +w2x

+[∫ t

0A3(1+ |w|+ |wx|)dτ

]2}

dxdt. (4.6.9)

Observe that

−∫ ∫

QT

wtwxxdxdt =12

∫ 1

0wx(x,T )2dx− 1

2

∫ 1

0w′

0(x)2dx, (4.6.10)

∫ ∫

QT

w2dxdt � C∫ ∫

QT

w2xdxdt, (4.6.11)

and that∫ ∫

QT

[∫ t

0(1+ |w|+ |wx|)dτ

]2

dxdt �∫ T

0

∫ 1

0

[

2t∫ t

0(1+w2 +w2

x)dτ]

dxdt

� 2T∫ T

0

∫ 1

0

∫ t

0[1+w2 +w2

x ]dτ dxdt


≡ 2T∫ T

0

∫ 1

0(T − τ)[1+w2 +w2

x ]dxdt

� 2T 2∫ T

0

∫ 1

0[1+w2 +w2

x ]dxdt. (4.6.12)

Combining (4.6.10)–(4.6.12) by choosing ε = 14A1

, we have from (4.6.9) that

A1

2

∫ ∫

QT

w2xxdxdt +

∫ 1

0wx(x,T )2dx � (1+T 2)C

∫ ∫

QT

w2xdxdt +(1+T 2)C.

Since T is arbitrary, Gronwall’s inequality (see [82]) implies that∫ 1

0wx(x,t)2dx � C(T ).

Therefore,∫ ∫

QT

wx(x,t)2dxdt � C, (4.6.13)

and∫ ∫

QT

w2xxdxdt + sup

0�t�T

∫ 1

0w2

x(x,t)dx � C. (4.6.14)

This concludes the estimate (4.6.5) since u(x,t) = w(x,t)+ v(x,t) on QT .

Corollary 4.6.1. There exists a positive constant C2 such that

‖u(x,t)‖C(QT ) � C2, (4.6.15)

where C2 depends on the same quantities as C1.

Proof. This can be obtained directly from the estimate (4.6.5).

Next we establish the following lemma which deals with the estimate on the norm of ux.

Lemma 4.6.2. There exists a constant C3 such that

‖ux‖C(QT ) � C3, (4.6.16)

where the dependency of C3 is the same as C1.

Proof. Let p > 2 be an arbitrary even integer. Since∫ T

0

ddt

[∫ 1

0wp

x dx]

dt =∫ T

0

∫ 1

0pwp−1

x wxtdxdt

= −∫ T

0

∫ 1

0p(p−1)wp−2

x wxxwtdxdt +∫ T

0pwp−1

x wt

∣∣∣∣

x=1

x=0dt


= −∫ T

0

∫ 1

0p(p−1)wp−2

x wxx

[

awxx +b− vt +∫ t

0cdτ

]

dxdt, (4.6.17)

it follows that∫ 1

0wp

x (x,T )dx+A1

∫ T

0

∫ 1

0p(p−1)wp−2

x w2xxdxdt

�∫ 1

0w′

0(x)pdx+

∫ T

0

∣∣∣∣p(p−1)wp−2

x wxx

(

b− vt +∫ t

0cdτ

)∣∣∣∣dxdt

�∫ 1

0w′

0(x)pdx+ ε

∫ T

0

∫ 1

0p(p−1)wp−2

x w2xxdxdt

+C(ε)∫ T

0

∫ 1

0p(p−1)wp−2

x

[

b− vt +∫ t

0cdτ

]2

dxdt. (4.6.18)

Choosing ε = A22 and using (H1), we find

∫ 1

0wp

x (x,T )dx+A1

2

∫ T

0

∫ 1

0p(p−1)wp−2

x w2xxdxdt

�∫ 1

0w′

0(x)pdx +C

∫ T

0

∫ 1

0p(p−1)wp−2

x

[

1+w2 +w2x

+(∫ t

0(1+ |w|+ |wx|)dτ

)2 ]

dxdt. (4.6.19)

Let

I =∫ T

0

∫ 1

0wp−2

x

[∫ t

0(1+ |w|+ |wx|)dτ

]2

dxdt.

Then

I �∫ T

0

∫ 1

0wp−2

x

[

2T(

T +C22 +

∫ t

0w2

xdτ)]

dxdt

� CT (1+T )∫ T

0

∫ 1

0wp−2

x dxdt +2T∫ T

0

∫ 1

0wp−2

x

(∫ t

0w2

xdτ)

dxdt

≡CT (1+T )I1 +2T I2.

Using Young’s inequality with r = pp−2 , s = p

2 and η = 1, we have

I2 =∫ T

0

∫ 1

0wp−2

x

(∫ t

0w2

xdτ)

dxdt

�∫ T

0

∫ 1

0

[p−2

pwp

x +2p

(∫ t

0w2

xdτ) p

2]

dxdt

�∫ T

0

∫ 1

0wp

x dxdt +∫ T

0

∫ 1

0

[

tp−2

2

(∫ t

0wp

x dτ)]

dxdt

�∫ T

0

∫ 1

0wp

x dxdt +Tp−2

2

∫ T

0

∫ 1

0

∫ t

0wp

x dτ dxdt

�∫ T

0

∫ 1

0wp

x dxdt +Tp−2

2

∫ T

0

∫ 1

0(T − τ)wp

x dxdτ

�(

1+Tp2

)∫ T

0

∫ 1

0wp

x dxdt. (4.6.20)


For the moment, we restrict T by 0 < T � T0 = 1 (say). Under this condition, it follows

from (4.6.19)–(4.6.20) and T ∈ [0,T0] arbitrary that

sup0�t�T

∫ 1

0wp

x (x, t)dx+A1

2

∫ ∫

QT

p(p−1)wp−2x wxxdxdt

�∫ 1

0w′

0(x)pdx+C

∫ T

0

∫ 1

0p(p−1)wp−2

x[1+w2

x]

dxdt, (4.6.21)

where C depends only on C2 and known data. Assume that ‖w(x,t)‖L∞(QT ) �max

{1, 1

T ‖w′0(x)‖0

}(here 0 < T � T0 = 1 is a fixed number). Otherwise, we already

have the estimate (4.6.16) on the interval [0,T0]. Then

sup0�t�T

∫ 1

0wp

x dx+A1

2

∫ T

0

∫ 1

0p(p−1)wp−2

x w2xxdxdt � C

∫ T

0

∫ 1

0p(p−1)wp

x dxdt

� C∫ T

0p(p−1)‖wx(·,t)‖p

L∞(0,1)dt. (4.6.22)

If the interpolation inequality is employed, we have∥∥∥w

p2x

∥∥∥

L∞(0,1)� C

∥∥∥w

p2x

∥∥∥

23

H1(0,1)

∥∥∥w

p2x

∥∥∥

13

L1(0,1),

i.e.,

‖wx‖pL∞(0,1) � C

∥∥∥w

p2x

∥∥∥

43

H1(0,1)

∥∥∥w

p2x

∥∥∥

23

L1(0,1)

� Cη∥∥∥w

p2x

∥∥∥

2

H1(0,1)+Cη−2‖wx‖p

Lp2 (0,1)

,

where the last inequality is obtained from Young’s inequality for r = 32 and s = 3. Note that

∥∥∥w

p2x

∥∥∥

2

H1(0,1)=

∫ 1

0

[( p2

)w

p2 −1x wxx

]2dx+

∫ 1

0wp

x dx.

As a consequence, it follows that

sup0�t�T

∫ 1

0wp

x dx +A1

2

∫ T

0

∫ 1

0p(p−1)wp−2

x w2xxdxdt

� Cp(p−1)[

p2

4η

∫ T

0

∫ 1

0wp−2

x w2xxdxdt +η

∫ T

0

∫ 1

0wp

x dxdt]

+Cp(p−1)η−2∫ T

0‖wx‖p

Lp2 (0,1)

dt.

If now η is chosen as η = min{ 1

2CT0, A1

p2C

}, then

sup0�t�T

∫ 1

0wp

x dx+ p(p−1)∫ T

0

∫ 1

0wp−2

x w2xxdxdt � Cp(p−1)η−2T sup

0�t�T‖wx‖p

Lp2 (0,1)


� Cp4 sup0�t�T

‖wx‖p

Lp2 (0,1)

,

where C is constant which depends only on known data.

To complete the proof we want to consider large value of p. Let

p = pk = 2k and αk = sup0�t�T

{∫ 1

0wpk

x dx} 1

pk.

If we take the pk-th root of both sides of above inequality, we obtain

αk �(Cp4

k) 1

pk αk−1.

Now+∞

∏k=1

C1pk = C∑+∞

k=11pk = C∑+∞

k=1 21k � C,

and+∞

∏k=1

p4pkk = 2∑+∞

k=14k2k � C,

since+∞

∑k=1

4k2k = 4

+∞

∑k=1

k2k ,

is convergent. Thus it follows that, for dk =(Cp4

k

) 1pk ,

αk � dkαk−1 �[

k

∏�=1

d�

]

α1 � Cα1.

As

limk→+∞

αk = ‖wx‖L∞(QT ),

and α1 � C1 by Lemma 4.6.1, it follows that

‖wx‖QT0 � Cα1 � C. (4.6.23)

Note that for the interval [T0,2T0], we can repeat the above procedure and obtain previously

the same inequality (4.6.23). After finitely many steps, one has the estimate (4.6.16).

Lemma 4.6.3. There exist constants C4 and α (0 < α < 1), which depend on the same

quantities as Ci (i = 1, 2, 3), such that

‖u‖C1+α , 1+α

2 (QT )� C4, (4.6.24)

and hence

‖ux‖Cα , α2 (QT )

� C4. (4.6.25)


Proof. Let

μ = max(x,t)∈QT|u|�C2|ux|�C3

[

|a(x,t,u,ux)|+ |b(x,t,u,ux)|+∫ t

0|c(x,τ,u,ux)|dτ

]

.

Lemma 4.6.2 implies that μ is uniformly bounded and that the bound depends only on the

known data. The desired result then follows from Theorem 5.1 in Ladyzenskaya et.al. [57,

p. 561] as a regular parabolic equation case.

Lemma 4.6.4. There exists a constant C5 such that

‖u‖C2+α ,1+ α

2 (QT )� C5, (4.6.26)

where C5 depends only on the same quantities as Ci, i = 1, 2, 3, 4.

Proof. By Lemma 4.6.3, we know that a(x,t,u(x,t),ux(x,t)) and b(x,t,u(x,t),ux(x,t))

are uniformly Holder continuous in QT with exponents α and α2 with respect to x and t,

respectively. Considering equation (4.6.1) as a linear equation

ut = auxx +b+∫ t

0cdτ,

with initial-boundary conditions (4.6.2)–(4.6.4), we employ the Schauder estimate to obtain

‖u‖C2+α ,1+ α

2 (QT )� C

[

1+∥∥∥∥

∫ t

0cdτ

∥∥∥∥

Cα , α2 (QT )

]

. (4.6.27)

Note that, for any function g(x,t) ∈Cα, α2 (QT ), we have the property

∥∥∥∥

∫ t

0g(x,τ)dτ

∥∥∥∥

Cα , α2 (QT )

�[‖g(x,0)‖C[0,1] +

(T +T 1− α

2

)‖g‖

Cα , α2 (QT )

]. (4.6.28)

As a consequence∥∥∥∥

∫ t

0c(x,τ,u,ux)dτ

∥∥∥∥

Cα , α2 (QT )

� C[1+

(T +T 1− α

2

)‖c(x,t,u,ux)‖Cα, α

2 (QT )

]

� C[

1+(

T +T 1− α2

)‖u‖

C1+α, 1+α2 (QT )

]

,

is uniformly bounded by Lemma 4.6.3, and the bound depends only on the known data.

Hence the estimate (4.6.26) follows from (4.6.27) and the above inequality.

From Lemmas 4.6.1–4.6.4, we establish the following existence theorem.

Theorem 4.6.1. Under hypotheses (H1)–(H2), there exists a solution u(x,t) ∈C2+α ,1+ α

2 (QT ) to the problem (4.6.1)–(4.6.4).


Proof. Define the operator Lλ by

Lλ u = ut − [auxx +b+λ∫ t

0cdτ].

Let

∑(λ ) = {λ ∈ [0,1] : the problem (4.6.1)λ –(4.6.4) is solvable},

where (4.6.1)λ is the equation Lλ u = 0. By the standard continuation method (∑(λ ) is not

empty, ∑(λ ) is open and also closed), it follows that ∑(λ ) ≡ [0,1].

The following theorem deals with the regularity of the solution for the problem (4.6.1)–

(4.6.4).

Theorem 4.6.2. Assume that a(x, t,u, p), b(x,t,u, p) and c(x, t,u, p) are infinitely differ-

entiable in all of their arguments and that the boundary values f1(t) and f2(t) belong to

C∞(0,T ]. Then the solution u(x,t) is infinitely differentiable with respect to x and t on the

region QT ∩{(x, t) : t > 0}.

Proof. Since u ∈ C2+α ,1+ α2 (QT ), we can differentiate equation (4.6.1) with respect to t

and then v = ut satisfies

vt = avxx +[apuxx +bp]vx +[auuxx +bu]v+[atuxx +bt + c(x, t,u,ux)] in QT , (4.6.29)

v(0,t) = f ′1(t), 0 � t � T, (4.6.30)

v(1,t) = f ′2(t), 0 � t � T. (4.6.31)

Since the coefficients of equation (4.6.29) are Holder continuous with respect to x and t,

the Schauder estimate for a parabolic equation implies that the solution v ∈C2+α ,1+ α2 (QT ).

Hence u ∈C4+α,2+ α2 (QT ). We can repeat this procedure and obtain v ∈C+∞,+∞(QT ∩{t :

t > 0}). It follows that u(x,t) ∈C+∞,+∞(QT ∩{t : t > 0}).

Next, we give the following theorem on continuous dependence of the solution of (4.6.1)–

(4.6.4) upon the known data.

Theorem 4.6.3. Assume that ( f1(t), f2(t),u0(x)) and ( f ∗1 (t), f ∗2 (t),u∗0(x)) are two known

sets of initial-boundary values which satisfy (H2). Let u(x,t) and u∗(x,t) be two solutions

of the problem (4.6.1)–(4.6.4) corresponding, respectively, to the above data. Then

‖u(x, t)−u∗(x,t)‖C2+α ,1+ α

2 (QT )

� C[‖ f1(t)− f ∗1 (t)‖

C1+ α2 [0,T ]

+‖ f2(t)− f ∗2 (t)‖C1+ α

2 [0,T ]

+‖u0(x)−u∗0(x)‖C2+α [0,1]

], (4.6.32)

where C depends only on known data.


Proof. Let w(x,t) = u(x,t)−u∗(x,t), (x,t) ∈ QT . Then w(x, t) satisfies

wt = awxx +b∗(x,t)wx + c∗(x,t)w +d∗(x,t) in QT , (4.6.33)

w(0,t) = f1(t)− f ∗1 (t), 0 � t � T, (4.6.34)

w(1,t) = f2(t)− f ∗2 (t), 0 � t � T, (4.6.35)

w(x,0) = u0(x)−u∗0(x), 0 � x � 1, (4.6.36)

where

b∗(x, t) =∫ 1

0bp(x,tu∗,zux +(1− z)u∗x)dz

+[∫ 1

0ap(x, t,u∗,zux +(1− z)u∗x)dz

]

u∗xx,

c∗(x,t) =∫ 1

0bu(x,t,zu+(1− z)u∗,ux)dz

+[∫ 1

0au(x,t,zu+(1− z)u∗,ux)dz

]

u∗xx,

d∗(x,t) =∫ t

0[d1(x,τ)wx +d2(x,τ)w]dτ,

d1(x,t) =∫ 1

0cp(x,t,u∗,zux +(1− z)u∗x)dz,

d2(x,t) =∫ 1

0cz(x,t,zu+(1− z)u∗,ux)dz.

The estimate (4.6.24) implies that all the Holder moduli of the coefficients in (4.6.33) are

dominated by known data. From the Schauder estimate for the linear parabolic equation

(4.6.33), we have

‖u‖C2+α,1+ α

2 (QT )� C

2

∑i=1

‖ fi(t)− f ∗i (t)‖C1+ α

2 [0,T ]

+‖u0(x)−u∗0(x)‖C2+α [0,1] +‖d∗(x,t)‖Cα , α

2 (QT ).

The inequalities (4.6.26) and (4.6.28) yield

‖d∗(x,t)‖Cα , α

2 (QT )� C

[‖w(x,0)‖C[0,1] +

(T +T 1− α

2

)‖u‖

C2+α ,1+ α2 (QT )

].

Therefore, when T is small enough so that(

T +T 1− α2

)C � 1

2 we have the desired result.

By taking a finite number of steps, we establish (4.6.32) for arbitrary T .


Corollary 4.6.2. The solution of the problem (4.6.1)–(4.6.4) is unique.

4.7 Miscellanea

4.7.1 Corduneanu [28]

Consider the integrodifferential equation of the form

∂∂ t

f (t,x) =12

∫ x

0φ (x− y,y) f (t,x− y) f (t,y)dy− f (t,x)

∫ ∞

0φ(x,y) f (t,y)dy, (4.7.1)

which is encountered in the mathematical description of coagulation processes, with the

initial condition

f (0,x) = f0(x), x � 0. (4.7.2)

For the interpretation of equation (4.7.1) and detailed meanings of the functions therein,

see [28, p. 274]. Assume that

(H1) the function φ(x,y) is continuous and symmetric in the quadrant x � 0, y � 0, and

verifies 0 � φ(x,y) � A, x � 0, y � 0, where A is a positive number,

(H2) f0(x) is continuous, bounded and integrable on x � 0,

(H3) f0(x) � 0 for x � 0.

Under the hypotheses (H1)–(H3), there exists a solution f (t,x) of equation (4.7.1), satisfy-

ing (4.7.2), which is defined for x � 0, t � 0, is continuous, bounded, nonnegative, analytic

in t for each fixed x � 0 and integrable in x for each t � 0. Moreover, the solution is unique.


Consider the parabolic-type Fredholm integral equation

∂∂ t

u(x,t) = F(

x,t,u(x,t),∫ b

aK(x,t,y,u(y,t))dy

)

, (4.7.3)

with the given initial condition

u(x,0) = φ(x), (4.7.4)

for (x,t)∈ Δ, where φ ∈C(I,R), K ∈C(Δ× I×R,R), F ∈C(Δ×R2,R) in which I = [a,b]

(a < b), Δ = I ×R+. Assume that

(G1) the functions F, K in equation (4.7.3) satisfy the conditions

|F(x,t,u,v)−F(x,t,u,v)| � [p(x,t)|u−u|+ |v− v|],

|K(x,t,y,u)−K(x,t,y,u)| � r(x, t,y)|u−u|,


where p ∈C(Δ,R+), r ∈C(Δ× I,R+);

(G2) for λ as in section 4.3:

(i) there exists a nonnegative constant α such that α < 1 and∫ t

0

[

p(x,s)exp(λ (|x|+ s))+∫ b

ar(x,s,y)exp(λ (|y|+ s))dy

]

ds � α exp(λ (|x|+ t)),

(ii) there exists a nonnegative constant β such that

|φ(x)|+∫ t

0

∣∣∣∣F

(

x,s,0,

∫ b

aK(x,s,y,0)dy

)∣∣∣∣ds � β exp(λ (|x|+ t)),

for (x, t) ∈ Δ.

Under assumptions (G1)–(G2), the problem (4.7.3)–(4.7.4) has a unique solution on Δ in

E, where E is the space of functions as defined in section 4.3.


Under the notations as in section 4.5, consider the following nonlinear integrodifferential

system of the form

∂ u∂ t

−Lu = F(

x,t,u,∫ t

0

∫

ΩK(x,t,s,u(x,s))dxds

)

(t > 0, x ∈ Ω), (4.7.5)


B[u] = α1(x)∂u∂τ

+α2(x)u = 0 (t > 0, x ∈ ∂ Ω), (4.7.6)

u(x,0) = u0(x), x ∈ Ω, (4.7.7)

where L denote the uniformly elliptic operator defined by

L =n

∑i, j=1

ai j(x)∂ 2

∂xi∂ x j+

n

∑i=1

ai(x)∂

∂ xi, (4.7.8)

on the bounded domain Ω in Rn, τ is the outward normal unit vector on the boundary

∂ Ω; αi(x) � 0 (i = 1, 2) with α1(x)+ α2(x) �= 0 on ∂ Ω; the coefficients of L and the first

partial derivatives of ai j are Holder continuous (of exponent α ∈ (0,1)) in Ω; the matrix

(ai j) is symmetric positive definite in Ω; K and F are Holder continuous in every bounded

subset of Ω×R2+×R and Ω×R+×R

2 respectively; the boundary ∂Ω is of class of C2+α ,

αi ∈ H1+α(∂Ω), and u0 ∈ H1+α(Ω

)satisfies (4.7.6) at t = 0.

Assume that

(H1) there exist pair of functions v, w ∈C2(D) with v � w on D such that

∂w∂ t

−Lw � F(

x,t,w,∫ t

0

∫

ΩK(x,t,s,w(x,s))dxds

)

, (x,t) ∈ D,


B[w] � 0, (x,t) ∈ S,

w(x,0) � u0(x), x ∈ Ω;

and∂ v∂ t

−Lv � F(

x,t,v,∫ t

0

∫

ΩK(x, t,s,v(x,s))dxds

)

, (x,t) ∈ D,

B[v] � 0, (x,t) ∈ S,

v(x,0) � u0(x), x ∈ Ω;

(H2) the function K(x,t,s,u) is monotone nondecreasing in u for fixed x, t, s and the func-

tion F(x, t,u,ξ ) is monotone nondecreasing in ξ for fixed x, t, u;

(H3) for a given constant M � 0,

F(x,t,u,ξ )−F(x,t,u,ξ ) � −M(u−u),

whenever v(x,t) � u � u � w(x,t) for (x,t) ∈ D.

For any η ∈C2(D) such that v � η � w on D, consider the following linear system

∂u∂ t

−Lu = F(

x, t,η,∫ t

0

∫

ΩK(x,t,s,η)dxds

)

−M(u−η), (x,t) ∈ D, (4.7.9)

B[u] = 0, (x,t) ∈ S, (4.7.10)

u(x,0) = u0(x), x ∈ Ω. (4.7.11)

For a known η the above assumptions ensures the existence of a unique solution of (4.7.9)–

(4.7.11) (see [37,57 ]). For each η ∈C2(D) such that v � η � w on D, define the mapping

A by

Aη = u,

where u is the unique solution of (4.7.9)–(4.7.11).

(g1) Assume that the hypotheses (H1)–(H3) hold. Then

(i) the unique solution u of (4.7.9)–(4.7.11) satisfies v(x,t) � u(x,t) � w(x,t), for (x,t)∈D;

(ii) v � Av, w � Aw for (x, t) ∈ D;

(iii) A is monotone operator on the set of functions

U(v,w) = {u ∈C2(D) : v(x,t) � u � w(x,t),(x,t) ∈ D}.

In view of (g1), we can define the sequences

vn = Avn−1, wn = Awn−1,


with v0 = v and w0 = w.

(g2) Assume that the hypotheses (H1)–(H3) hold. Then the sequence {wn} converges uni-

formly from above to a maximal solution u of (4.7.5)–(4.7.7), while the sequence {vn}converges uniformly from below to a minimal solution u of (4.7.5)–(4.7.7). Furthermore,

if u is any solution of (4.7.5)–(4.7.7) such that u ∈U(v,w), then

v � v1 � · · · � vn � · · · � u � u � u � · · · � wn � · · · � w1 � w,

on D


Under the notations as in section 4.5, consider the following nonlinear coupled parabolic

integrodifferential equations of the form

∂u∂ t

−Lu = F(x,t,u,v,M[u,v](x,t)), (4.7.12)

∂v∂ t

= H(x,t,u,v,N[u,v](x,t)), (4.7.13)

for (x, t) ∈ D, with the given boundary and initial conditions

B[u] =∂ u∂τ

+b(x)u = c0(x,t), (x,t) ∈ S, (4.7.14)

u(x,0) = u0(x), v(x,0) = v0(x), x ∈ Ω, (4.7.15)

where

M[u,v](x,t) =∫ t

0

∫

Ωf (x,t,y,s,u(y,s),v(y,s))dyds,

N[u,v](x,t) =∫ t

0

∫

Ωh(x,t,y,s,u(y,s),v(y,s))dyds,

and L is the uniformly elliptic operator as defined in (4.7.8); f , h and F, H are real

valued Holder continuous in (x,t,y,s,u,v) and (x,t,u,v,r) in every bounded subset of

Ω×R+ ×Ω×R+ ×R2 and Ω×R+ ×R

3 respectively; b(x) � 0 on ∂Ω, b ∈ H1+α(∂Ω),

c0 ∈ H1+α(s), u0,v0 ∈ H2+α(Ω

)and u0 satisfies the boundary condition u(x,0) = u0(x) at

t = 0. Assume that

(H4) there exist pair of functions σ = (u,v), σ = (u,v); u, u ∈C2(D) and v,v ∈C(D) with

σ � σ on D such that

∂ u∂ t

−Lu � F(x, t,u,v,M[u,v](x,t)), (x,t) ∈ D,

∂v∂ t

� H(x, t,u,v,N[u,v](x,t)), (x,t) ∈ D,


B[u] � c0(x,t), (x,t) ∈ S,

u(x,0) � u0(x), v(x,0) � v0(x), x ∈ Ω,

and∂ u∂ t

−Lu � F(x, t,u,v,M[u,v](x,t)), (x,t) ∈ D,

∂v∂ t

� H(x, t,u,v,N[u,v](x,t)), (x,t) ∈ D,

B[u] � c0(x,t), (x,t) ∈ S,

u(x,0) � u0(x), v(x,0) � v0(x), x ∈ Ω;

(H5) the functions f (x,t,y,s,u,v), h(x, t,y,s,u,v) both are monotone nondecreasing in u

and v;

(H6) the function F(x,t,u,v,r) is monotone nondecreasing in v and r, and the function

H(x,t,u,v,r) is monotone nondecreasing in u and r;

(H7) the functions F and H satisfy

F(x,t,u1,v,r)−F(x,t,u2,v,r) � −Q(u1 −u2),

whenever u(x, t) � u2 � u1 � u(x,t) for (x,t) ∈ D, and

H(x,t,u,v1,r)−H(x,t,u,v2,r) � −Q(v1 − v2),

whenever v(x,t) � v2 � v1 � v(x,t) for (x, t) ∈ D, where Q � 0 is a constant.

For a given function z = (ξ ,η); ξ ∈ C2(D), η ∈C(D) such that σ � z � σ on (x,t) ∈ D,

where σ = (u,v), σ = (u,v), consider the following linear system∂ u∂ t

−Lu = F(x,t,ξ ,η ,M[ξ ,η ](x,t))−Q(u−ξ ), (4.7.16)

∂ v∂ t

= H(x,t,ξ ,η,N[ξ ,η ](x,t))−Q(v−η), (4.7.17)

for (x, t) ∈ D, with the given boundary and initial conditions

B[u] = c0(x,t), (x,t) ∈ S, (4.7.18)

u(x,0) = u0(x), v(x,0) = v0(x), x ∈ Ω. (4.7.19)

For a known function z = (ξ ,η) the above assumptions ensures (see [37, 57]) the existence

of a unique solution of (4.7.16)–(4.7.19). For each z = (ξ ,η), ξ ∈C2(D), η ∈C(D) such

that σ � z � σ on D, define the mapping A by

Az = σ ,

where σ = (u,v) is the solution of (4.7.16)–(4.7.19).

(g3) Assume that the hypotheses (H4)–(H7) hold. Then


(i) the unique solution σ = (u,v) of (4.7.16)–(4.7.19) satisfies σ(x, t) � σ(x,t) � σ(x,t),

(x,t) ∈ D;

(ii) σ � Aσ , σ � Aσ on D;

(iii) A is monotone operator on the set of functions

Δ ={

σ = (u,v) : σ � σ � σ ; u ∈C2(D), v ∈C(D)}

,

on D.

In view of (g3), we can define the sequences

σ n = Aσ n−1, σn = Aσ n−1,

where σ n = (un,vn), σ n = (un,vn) with σ 0 = σ and σ 0 = σ .

(g4) Assume that the hypotheses (H4)–(H7) hold. Then the sequence {(un,vn)} con-

verges uniformly from above to a maximal solution (α,β ) of (4.7.12)–(4.7.15), while

the sequence {(un,vn)} converges uniformly from below to a minimal solution (α,β ) of

(4.7.12)–(4.7.15). Furthermore, if σ = (u,v) is any solution of (4.7.12)–(4.7.15) such that

σ ∈ Δ, then

u � u1 � · · · � un � · · · � α � u � α � · · · � un � · · · � u1 � u,

v � v1 � · · · � vn � · · · � β � v � β � · · · � vn � · · · � v1 � v,

on D.

4.7.5 Cannon and Lin [23]

Under the standard notations for Holder classes defined in Chapter 1 of [57], consider the

linear integrodifferential equation of parabolic type

ut(x,t) = A(x,t)u(x, t)+∫ t

0B(x,t,τ)u(x,τ)dτ + f (x,t), (4.7.20)

in QT = Ω × (0,T ], where T > 0 and Ω is a bounded open subset of Rn with regular

boundary ∂ Ω and QT is the closure of QT ,

Au =n

∑i, j=1

ai j(x,t)uxix j(x,t)+n

∑i=1

ai(x,t)uxi(x,t)+a(x,t)u(x,t), (4.7.21)

B(x,t,τ)u(x,τ) =n

∑i, j=1

bi j(x,t,τ)uxix j(x,τ)

+n

∑i=1

bi(x,t,τ)uxi(x,τ)+b(x,t,τ)u(x,τ). (4.7.22)


The equation (4.7.20) is considered together with the following three types of boundary

conditions:

(L) u(x,0) = u0(x), x ∈ Ω,

u(x,t) = φ(x,t), (x,t) ∈ ST = ∂Ω× [0,T ];

(M) u(x,0) = u0(x), x ∈ Ω,n

∑i=1

ci(x,t)uxi(x,t)cos(xi,ν)+ c(x,t)u(x,t) = Φ(x,t),(x,t) ∈ ST ;

(N) u(x,0) = u0(x), x ∈ Ω,n

∑i=1

ci(x,t)uxi(x,t)cos(xi,ν)+ c(x, t)u(x,t)

+∫ t

0

(n

∑i=1

di(x,t,τ)uxi(x,τ)+d(x,t,τ)u(x,τ)

)

dτ = Φ(x,t), (x, t) ∈ ST ;

where∣∣∣∣∣

n

∑i=1

ci(x, t)cos(xi,ν)

∣∣∣∣∣� δ ,

for some positive constant δ and ν(x) = (ν1(x), . . . ,νn(x)) is the outer normal direction to

∂ Ω.

The equation (4.7.20) will be treated as a parabolic equation with integral term as a pertur-

bation and employing the basic classical theory of parabolic partial differential equation

ut(x, t) = A(x, t)u(x,t)+ f (x,t). (4.7.23)

Assume that

(H1) u0 ∈ H2+α(Ω

); ai j, ai, a, f ∈ Hα, α

2 (QT ) and

α0|ξ |2 � ai jξiξ j � α1|ξ |2,

for (x, t) ∈ QT , α0, α1 are positive constants and ξ ∈ Rn;

(H2) Φ ∈ H2+α ,1+ α2(ST

);

(H3) bi j(x,t, ·), bi(x,t, ·), b(x,t, ·) ∈ C([0,T ]) uniformly for (x,t) ∈ QT and bi j(·, ·,τ),

bi(·, ·,τ), b(·, ·,τ) ∈ Hα, α2 (QT ) uniformly for τ ∈ [0,T ];

(H4) ci, c, Φ ∈ H1+α ,(1+α)

2(ST

);

(H5) di(x, t, ·), d(x,t, ·) ∈ C([0,T ]) uniformly on ST and di(·, ·,τ), d(·, ·,τ) ∈H1+α ,

(1+α)2 (ST ) uniformly for τ ∈ [0,T ].

Problem (4.7.20)–(L) or (4.7.23)–(L) is said to satisfy condition (C1) if

(C1) u0(x) = Φ(x,0), Φt(x,0) = Au0 + f (x,0),


and problem (4.7.20)–(M) or (4.7.20)–(N) or (4.7.23)–(M) is said to satisfy condition (C2)

if

(C2) ∑ni=1 ci(x,0)u0xi(x)cos(xi,ν)+ c(x,0)u0(x) = Φ(x,0).

(g5) For 0 < α < 1, assume that ∂ Ω∈H2+α . Under assumptions (H1)–(H3) and (C1), there

exists a unique smooth solution u in H2+α ,1+ α2 (QT ) for problem (4.7.20)–(L).

(g6) For 0 < α < 1, assume that ∂Ω∈H2+α . Under assumptions (H1), (H3)–(H4) and (C2),

there exists a unique smooth solution u in H2+α ,1+ α2 (QT ) for problem (4.7.20)–(M).

(g7) For 0 < α < 1, assume that ∂Ω∈H2+α . Under assumptions (H1), (H3)–(H5) and (C2),

there exists a unique smooth solution u in H2+α ,1+ α2 (QT ) for problem (4.7.20)–(N).

4.7.6 Roux and Thomee [126]

Consider the semilinear parabolic integrodifferential equation

∂∂ t

u(x, t)+Au(x,t) =∫ t

0f (t,s,x,u(x,s))ds, (4.7.24)

for x ∈ Ω, 0 � t � T , where A is a self-adjoint elliptic second order partial differential

operator with smooth, time-independent coefficients of the form

Au =d

∑i, j=1

∂∂ xi

(

ai j∂ u∂x j

)

+a0u, a0 � 0,

in a domain Ω ⊂ Rd with smooth boundary ∂ Ω, and f is a given smooth function of its

arguments that is bounded together with a sufficient number of its derivatives. The equation

(4.7.24) is considered together with the Dirichlet type boundary condition

u = 0 on ∂Ω for 0 � t � T, (4.7.25)

and with the initial condition

u(x,0) = v(x) in Ω. (4.7.26)

For the numerical solution of this problem by the Galerkin finite element method we shall

assume that we are given a family of subspaces Sh of H10 (Ω) with the approximation prop-

erty that, for some r � 2,

infχ∈Sh

{‖v− χ‖+h‖v− χ‖1} � C hs‖v‖s,

for v ∈ Hs(Ω)∩H10 , 1 � s � r. Here ‖ · ‖ denotes the norm in L2(Ω) and ‖ · ‖s that in

Hs(Ω) (see [57]). With (·, ·) the standard inner product in L2(Ω) and A(·, ·) the positive


definite bilinear form on H10 (Ω)×H1

0 (Ω) defined by the operator A, we may then pose the

semidiscrete problem of finding uh : [0,T ]→ Sh such that, with f (t,s,u) = f (t,s,x,u(x,s)),(

∂ uh

∂ t,χ

)

+A(uh,χ) =∫ t

0( f (t,s,uh),χ)ds, (4.7.27)

for all χ ∈ Sh, 0 � t � T ,

uh(0) = vh, (4.7.28)

where vh is a suitable approximation of v in Sh.

(g8) Let uh and u be the solutions of (4.7.27)–(4.7.28) and (4.7.24)–(4.7.26), respectively.

Then, if u is appropriately smooth, the estimate

‖uh(t)−u(t)‖ � C‖vh − v‖+Chr‖v‖r +∫ t

0‖ut‖rds,

holds for 0 � t � T .

4.7.7 Yin [141]

Let T > 0 and QT = Ω× (0,T ], where Ω is an open bounded region in Rn with a smooth

boundary S = ∂Ω and ST = S× [0,T ]. Consider the following integrodifferential equation

ut −∂

∂xiai(x, t,u,ux)−a(x,t,u,ux)

=∫ t

0

[∂

∂xibi(x,t,τ,u,ux)+b(x,t,τ,u,ux)

]

dτ, (x, t) ∈ QT , (4.7.29)


u(x,t) = 0, (x, t) ∈ ST , (4.7.30)

u(x,0) = u0(x), x ∈ Ω. (4.7.31)

A function u(x,t) in V2(QT ) = L∞(0,T ;L2(Ω))∩L2(0,T ;H10 (Ω)) is said to be a weak so-

lution of (4.7.29)–(4.7.31), if u(x,t) satisfies∫ T

0

∫

Ω

[

−uφt +(

ai +∫ t

0bidτ

)

φxi −(

a+∫ t

0bdτ

)

φ]

dxdt

=∫

Ωu0(x)φ(x,0)dx−

∫

Ωu(x,T )φ(x,T )dx,

for any φ (x,t) ∈ H1(0,T ;H10 (Ω)).

Assume that

(H1) the functions ai, a and bi, b are differentiable with respect to all of their arguments in

QT ×R×Rn and QT ×R+ ×R×R

n, respectively;


(H2) the following ellipticity assumption and the growth conditions hold:

n

∑i=1

[ai(x,t,u, p1)−ai(x, t,u, p2)](p1 − p2) � a0(p1 − p2)2,

n

∑i=1

[|ai(x,t,u, p)|+ |bi(x,t,τ,u, p)|]+ |a|+ |b| � A0 [1+ |u|+ |p|] ;

n

∑i=1

|bip(x,t,τ,u, p)| � B0,

where a0, A0 and B0 are positive constants;

(H3) u0(x) ∈ H10 (Ω).

Under the hypotheses (H1)–(H3), the problem (4.7.29)–(4.7.31) admits a unique weak so-

lution.

4.7.8 Yin [141]

Let T > 0 and QT = Ω× (0,T ], where Ω is an open bounded region on Rn with a smooth

boundary S = ∂ Ω and ST = S× [0,T ]. Consider the following initial-boundary value prob-

lem:

ut −Δu− f (u) =∫ t

0[b(t,τ)Δu]dτ, (x,t) ∈ QT , (4.7.32)

u(x, t) = 0, (x,t) ∈ ST , u(x,0) = u0(x), x ∈ Ω, (4.7.33)

with the assumptions

(G1) the function b(t,τ) is differentiable on R2;

(G2) f (u) is differentiable and there exist positive constants A0 and M such that

f ′(u) � A0 for |u| � M;

(G3) u0(x) ∈ C2+α(Ω

)(Ω is the closure of Ω) and the following compatibility conditions

hold

u0(x) = 0, Δu0(x)+ f (u0) = 0 on S.

Under the hypotheses (G1)–(G3), the problem (4.7.32)–(4.7.33) admits a unique classical

solution on [0,T ] for any T > 0.


4.8 Notes

Many phenomena in physical, chemical, biological sciences and engineering can be

modeled via parabolic integrodifferential equations, for which elegant theories and pow-

erful techniques have been developed, see [1,5,27,31–33,59–63,121,122,124,126,131–

134,140,141] and the references cited therein. The results in section 4.2 provides some

basic integral inequalities with explicit estimates which can be used as tools in handling

the study of qualitative properties of solutions of certain parabolic partial integrodifferential

and integral equations and are adapted from Pachpatte [98,109]. Sections 4.3 and 4.4 are

concerned with the study of some fundamental qualitative properties of solutions of an inte-

grodifferential equation of Barbashin-type and the general integral equation of Barbashin-

type, and are taken from Pachpatte [110,109]. Here, the treatment of results is essentially

different from those used in [5] to study such equations. Section 4.5 deals with the exis-

tence, uniqueness and asymptotic behavior of solutions of an integrodifferential equation

of the type arising in reactor dynamics. The notion of upper and lower solutions based on

the monotone iterative method is used to study the problem and the results are taken from

Pachpatte [75,77]. The section 4.6 is devoted to the solvability in the classical sense of a

class of nonlinear one-dimensional integrodifferential equation of parabolic type and the

results are due to Yin [140]. Section 4.7 contains results related to certain aspects of some

selected equations, which we hope will motivate the reader’s interest in further study of

related topics.


Chapter 5

Multivariable sum-difference inequalities andequations

5.1 Introduction

Multivariable difference equations occur in numerous settings and forms in various appli-

cations. By computing the value of a unknown function recursively for a set of equidistant

points from a given set of values, leads to the equation which may be viewed as sum-

difference equation on the fixed region of summation, see [2,3,14,16,46,47,51,67,68,85].

The problems of existence of solutions for these equations can be dealt with the use of the

well known fixed point theorems, see [51,54]. Besides the existence problems, there are

many basic questions which are significant with respect to the theory itself or to applica-

tions to it. In practice it is often difficult to obtain explicitly the solutions, and thus need a

new insight to handle the qualitative properties of their solutions. The method of finite dif-

ference inequalities with explicit estimates provides the most powerful and widely used an-

alytic tool in the study of various discrete dynamic equations. It enable us to obtain valuable

information about solutions without the need to know in advance the solution explicitly. In

this chapter, we focus our attention to present some fundamental sum-difference inequal-

ities with explicit estimates recently established in [108,104,98,95,102,111,116,114,106],

which can be used as tools for handling the qualitative properties of solutions of various

types of multivariable sum-difference equations . Furthermore, some important basic qual-

itative aspects related to the solutions of certain sum-difference equations investigated in

[100,103,116,101,99,107,109] are also given.

5.2 Sum-difference inequalities in two variables

In this section we offer some basic sum-difference inequalities in two variables which can

be used as tools in certain applications when the earlier inequalities in the literature do not

191


apply directly.

The inequalities given in the following theorems are adapted from [108,104,98].

Theorem 5.2.1. Let u, f , e ∈ D(N

20,R+

)and c � 0 is a real constant.

(q1) If

u(n,m) � c+n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)u(σ ,τ), (5.2.1)

for (n,m) ∈ N20, then

u(n,m) � cn−1

∏s=0

[

1+s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)

]

, (5.2.2)

for (n,m) ∈ N20.

(q2) Let e(n,m) be nondecreasing in each variable n,m ∈ N0. If

u(n,m) � e(n,m)+n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)u(σ ,τ), (5.2.3)


u(n,m) � e(n,m)n−1

∏s=0

[

1+s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)

]

, (5.2.4)

for (n,m) ∈ N20.

Theorem 5.2.2. Let u, p, q, f ∈ D(N20,R+).

(q3) Let L ∈ D(N20 ×R+,R+) be such that

0 � L(n,m,u)−L(n,m,v) � M(n,m,v)(u− v), (5.2.5)

for u � v � 0, where M ∈ D(N20 ×R+,R+). If

u(n,m) � p(n,m)+q(n,m)n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

L(σ ,τ,u(σ ,τ)), (5.2.6)


u(n,m) � p(n,m)+q(n,m)

(n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

L(σ ,τ, p(σ ,τ))

)

×n−1

∏s=0

[

1+s−1

∑σ=0

m−1

∑τ=0

M(σ ,τ, p(σ ,τ))q(σ ,τ)

]

, (5.2.7)

for (n,m) ∈ N20.

Multivariable sum-difference inequalities and equations 193

(q4) If


∑s=0

s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)u(σ ,τ), (5.2.8)



(n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)p(σ ,τ)

)

×n−1

∏s=0

[

1+s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)q(σ ,τ)

]

, (5.2.9)

for (n,m) ∈ N20.

Theorem 5.2.3. Let u, f ∈ D(E,R+) and c � 0 is a real constant, where E = N0 ×Nα,β .

(q5) If

u(n,m) � c+n−1

∑s=0

β

∑t=α

f (s,t)u(s,t), (5.2.10)

for (n,m) ∈ E, then

u(n,m) � cn−1

∏s=0

[

1+β

∑t=α

f (s,t)

]

, (5.2.11)

for (n,m) ∈ E.

(q6) Let g ∈C(R+,R+) be nondecreasing function, g(u) > 0 on (0,∞). If

u(n,m) � c+n−1

∑s=0

β

∑t=α

f (s,t)g(u(s,t)), (5.2.12)

for (n,m) ∈ E, then for 0 � n � n1; n, n1 ∈ N0,m ∈ Nα ,β ,

u(n,m) � W−1

[

W (c)+n−1

∑s=0

β

∑t=α

f (s,t)

]

, (5.2.13)

where

W (r) =∫ r

r0

dzg(z)

, r > 0, (5.2.14)

r0 > 0 is arbitrary and W−1 is the inverse of W and n1 ∈ N0 is chosen so that

W (c)+n−1

∑s=0

β

∑t=α

f (s,t) ∈ Dom(W−1),

for all n ∈ N0 lying in 0 � n � n1 and m ∈ Nα ,β .


Theorem 5.2.4. Let u, f ,c be as in Theorem 5.2.3.

(q7) If

u2(n,m) � c+n−1

∑s=0

β

∑t=α

f (s,t)u(s,t), (5.2.15)


u(n,m) �√

c+12

n−1

∑s=0

β

∑t=α

f (s,t), (5.2.16)

for (n,m) ∈ E

(q8) Let g be as in part (q6). If

u2(n,m) � c+n−1

∑s=0

β

∑t=α

f (s,t)u(s,t)g(u(s, t)), (5.2.17)

for (n,m) ∈ E, then for 0 � n � n2; n, n2 ∈ N0, m ∈ Nα ,β ,

u(n,m) � W−1

[

W (√

c)+12

n−1

∑s=0

β

∑t=α

f (s,t)

]

, (5.2.18)

where W, W−1 are as in part (q6) and n2 ∈ N0 is chosen so that

W(√

c)+

12

n−1

∑s=0

β

∑t=α

f (s,t) ∈ Dom(W−1),

for all n ∈ N0 lying in 0 � n � n2 and m ∈ Nα ,β .

Theorem 5.2.5. (q9) Let u, p, q, f ∈ D(E,R+). If


∑s=0

s−1

∑σ=0

β

∑τ=α

f (σ ,τ)u(σ ,τ), (5.2.19)



(n−1

∑s=0

s−1

∑σ=0

β

∑τ=α

f (σ ,τ)p(σ ,τ)

)

×n−1

∏s=0

[

1+s−1

∑σ=0

β

∑τ=α

f (σ ,τ)q(σ ,τ)

]

, (5.2.20)

for (n,m) ∈ E.

(q10) Let u, f , c be as in Theorem 5.2.3 and h ∈ D(E ×Nα ,β ,R+). If

u(n,m) � c+n−1

∑s=0

[

f (s,m)u(s,m)+β

∑y=α

h(s,m,y)u(s,y)

]

, (5.2.21)


u(n,m) � cF(n,m)n−1

∏s=0

[

1+β

∑y=α

h(s,m,y)F(s,m)

]

, (5.2.22)

for (n,m) ∈ E, where

F(n,m) =n−1

∏σ=0

[1+ f (σ ,m)].


Proofs of Theorems 5.2.1-5.2.5. We give the proofs of (q1), (q3), (q6), (q7), (q10) only;

the proofs of other inequalities can be completed by following the proofs of the above noted

inequalities and closely looking at the similar results given in [85,87]. To prove (q1), (q5)–

(q7), (q10), it is sufficient to assume that c > 0, since the standard limiting arguments can

be used to treat the remaining case, see [85, p. 300].

(q1) Let c > 0 and define a function z(n,m) by the right hand side of (5.2.1). Then z(n,0) =

z(0,m) = c, u(n,m) � z(n,m),

Δ2z(n,m) =n−1

∑s=0

s−1

∑σ=0

f (σ ,m)u(σ ,m),

Δ1z(n,m) =n−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)u(σ ,τ),

Δ21z(n,m) =

m−1

∑τ=0

f (n,τ)u(n,τ),

and

Δ2Δ21z(n,m) = f (n,m)u(n,m) � f (n,m)z(n,m). (5.2.23)

From (5.2.23) and in view of the facts that z(n,m) � z(n,m + 1) and Δ21 z(n,m) � 0 we

observe that (see [85, p. 324])

Δ21z(n,m+1)z(n,m+1)

− Δ21z(n,m)z(n,m)

� f (n,m). (5.2.24)

Keeping n fixed in (5.2.24), set m = τ and sum over τ from 0 to m−1 and use the fact that

Δ21z(n,0) = 0, to obtain the estimate

Δ21z(n,m)z(n,m)

�m−1

∑τ=0

f (n,τ). (5.2.25)

From (5.2.25) and in view of the facts that z(n,m) � z(n + 1,m) and Δ1z(n,m) � 0, we

observe that

Δ1z(n+1,m)z(n+1,m)

− Δ1z(n,m)z(n,m)

�m−1

∑τ=0

f (n,τ). (5.2.26)

Keeping m fixed in (5.2.26), set n = σ and sum over σ from 0 to n−1 and use the fact that

Δ1z(0,m) = 0, to obtain the estimate

Δ1z(n,m)z(n,m)

�n−1

∑σ=0

m−1

∑τ=0

f (σ ,τ). (5.2.27)



z(n+1,m) �[

1+n−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)

]

z(n,m). (5.2.28)

Now keeping m fixed in (5.2.28), set n = s and substitute s = 0, 1, 2, . . . ,n−1 successively

to obtain the estimate

z(n,m) � cn−1

∏s=0

[

1+s−1

∑σ=0

m−1

∑τ=0

f (σ ,τ)

]

. (5.2.29)

Using (5.2.29) in u(n,m) � z(n,m), we get (5.2.2).

(q3) Define a function z(n,m) by

z(n,m) =n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

L(σ ,τ,u(σ ,τ)), (5.2.30)

then z(0,m) = z(n,0) = 0 and (5.2.6) can be restated as

u(n,m) � p(n,m)+q(n,m)z(n,m). (5.2.31)


z(n,m) �n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

{L(σ ,τ, p(σ ,τ)+q(σ ,τ)z(σ ,τ))

−L(σ ,τ, p(σ ,τ))+L(σ ,τ, p(σ ,τ))}

�n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

L(σ ,τ, p(σ ,τ))

+n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

M(σ ,τ, p(σ ,τ))q(σ ,τ)z(σ ,τ). (5.2.32)

Clearly the first term on the right hand side in (5.2.32) is nonnegative and nondecreasing in

n, m ∈ N0. Now a suitable application of the inequality in part (q2) to (5.2.32) yields

z(n,m) �(

n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

L(σ ,τ, p(σ ,τ))

)

×n−1

∏s=0

[

1+s−1

∑σ=0

m−1

∑τ=0

M(σ ,τ, p(σ ,τ))q(σ ,τ)

]

. (5.2.33)


(q6) Setting

e(s) =β

∑t=α

f (s,t)g(u(s,t)), (5.2.34)



u(n,m) � c+n−1

∑s=0

e(s). (5.2.35)

Let c > 0 and define

z(n) = c+n−1

∑s=0

e(s), (5.2.36)

then z(0) = c and

u(n,m) � z(n). (5.2.37)

From (5.2.36), (5.2.34), (5.2.37) and using the fact that z(n) is nondecreasing in n ∈N0, we

observe that

Δz(n) = e(n) =β

∑t=α

f (n,t)g(u(n,t)) � g(z(n))β

∑t=α

f (n,t).

Now, by following the proof of Theorem 2.3.1 given in [85], we get

z(n) � W−1

[

W (c)+n−1

∑s=0

β

∑t=α

f (s, t)

]

, (5.2.38)

for 0 � n � n1. Using (5.2.38) in (5.2.37), we get (5.2.13).

(q7) Setting

E(s) =β

∑t=α

f (s,t)u(s,t), (5.2.39)


u2(n,m) � c+n−1

∑s=0

E(s). (5.2.40)

Let c > 0 and define by z(n) the right hand side of (5.2.40), then z(0) = c, u(n,m) �√

z(n)

and we observe that

Δz(n) = E(n) =β

∑t=α

f (n,t)u(n,t) �√

z(n)β

∑t=α

f (n, t).

Now by following the proof of Theorem 3.3.1 given in [85], we obtain

√z(n) �

√c+

12

n−1

∑s=0

β

∑t=α

f (s,t), (5.2.41)

for n ∈ N0. Using (5.2.41) in u(n,m) �√

z(n), we get the required inequality in (5.2.16).

(q10) The proof can be completed by following the arguments as in the proof of Theo-

rem 4.2.1 in Chapter 4 and closely looking at the proof of (q6) given above with suitable

modifications, see also [85,87].


5.3 Sum-difference inequalities in three variables

In this section, we shall deal with some fundamental sum-difference inequalities in three

variables which will also be equally important in the situations for which other available

inequalities fail to apply directly.

In the following theorems we present the inequalities established by Pachpatte in

[95,102,111].

Theorem 5.3.1. Let u, p, q, f ∈ D(H,R+) and c � 0 is a real constant, where H = N20 ×

Nα ,β .

(r1) If

u(m,n,k) � c+m−1

∑s=0

n−1

∑t=0

β

∑r=α

f (s,t,r)u(s,t,r), (5.3.1)

for (m,n,k) ∈ H, then

u(m,n,k) � cn−1

∏s=0

[

1+n−1

∑t=0

β

∑r=α

f (s,t,r)

]

, (5.3.2)

for (m,n,k) ∈ H.

(r2) If

u(m,n,k) � p(m,n,k)+q(m,n,k)m−1

∑s=0

n−1

∑t=0

β

∑r=α

f (s,t,r)u(s, t,r), (5.3.3)


u(m,n,k) � p(m,n,k)+q(m,n,k)

(m−1

∑s=0

n−1

∑t=0

β

∑r=α

f (s,t,r)p(s,t,r)

)

×m−1

∏s=0

[

1+n−1

∑t=0

β

∑r=α

f (s,t,r)q(s,t,r)

]

, (5.3.4)

for (m,n,k) ∈ H.

Theorem 5.3.2. Let u, f ∈ D(H,R+) and c � 0,d � 1 are real constants.

(r3) If

u2(m,n,k) � c+m−1

∑s=0

n−1

∑t=0

β

∑r=α

f (s,t,r)u(s,t,r), (5.3.5)


u(m,n,k) �√

c+12

m−1

∑s=0

n−1

∑t=0

β

∑r=α

f (s,t,r), (5.3.6)


for (m,n,k) ∈ H.

(r4) If u(m,n,k) � 1 and

u(m,n,k) � d +m−1

∑s=0

n−1

∑t=0

β

∑r=α

f (s,t,r)u(s,t,r) logu(s,t,r), (5.3.7)


u(m,n,k) � d∏m−1s=0

[1+∑n−1

t=0 ∑βr=α f (s,t,r)

]

, (5.3.8)

for (m,n,k) ∈ H.

Theorem 5.3.3. Let u, p, q, f ∈ D(H,R+).

(r5) If


∑s=0

∞

∑t=n+1

β

∑r=α

f (s,t,r)u(s,t,r), (5.3.9)



(m−1

∑s=0

∞

∑t=n+1

β

∑r=α

f (s,t,r)p(s, t,r)

)

×m−1

∏s=0

[

1+∞

∑t=n+1

β

∑r=α

f (s,t,r)q(s,t,r)

]

, (5.3.10)

for (m,n,k) ∈ H.

(r6) If

u(m,n,k) � p(m,n,k)+q(m,n,k)∞

∑s=m+1

∞

∑t=n+1

β

∑r=α

f (s,t,r)u(s,t,r), (5.3.11)



(∞

∑s=m+1

∞

∑t=n+1

β

∑r=α

f (s,t,r)p(s, t,r)

)

×∞

∏s=m+1

[

1+∞

∑t=n+1

β

∑r=α

f (s,t,r)q(s,t,r)

]

, (5.3.12)

for (m,n,k) ∈ H.


Theorem 5.3.4. Let u, p, q, c, f , g ∈ D(H,R+).

(r7) Suppose that


∑s=0

∞

∑t=n+1

β

∑r=α

f (s,t,r)u(s,t,r)

+c(m,n,k)∞

∑s=0

∞

∑t=0

β

∑r=α

g(s,t,r)u(s, t,r), (5.3.13)

for (m,n,k) ∈ H. If

β1 =∞

∑s=0

∞

∑t=0

β

∑r=α

g(s, t,r)B1(s,t,r) < 1, (5.3.14)

then

u(m,n,k) � A1(m,n,k)+D1B1(m,n,k), (5.3.15)

for (m,n,k) ∈ H, where

A1(m,n,k) = p(m,n,k)+q(m,n,k)

(m−1

∑s=0

∞

∑t=n+1

β

∑r=α

f (s,t,r)p(s,t,r)

)

×m−1

∏s=0

[

1+∞

∑t=n+1

β

∑r=α

f (s,t,r)q(s,t,r)

]

, (5.3.16)

B1(m,n,k) = c(m,n,k)+q(m,n,k)

(m−1

∑s=0

∞

∑t=n+1

β

∑r=α

f (s,t,r)c(s,t,r)

)

×m−1

∏s=0

[

1+∞

∑t=n+1

β

∑r=α

f (s,t,r)q(s,t,r)

]

, (5.3.17)

and

D1 =1

1−β1

∞

∑s=0

∞

∑t=0

β

∑r=α

g(s,t,r)A1(s,t,r). (5.3.18)

(r8) Suppose that

u(m,n,k) � p(m,n,k)+q(m,n,k)∞

∑s=m+1

∞

∑t=n+1

β

∑r=α

f (s,t,r)u(s,t,r)

+c(m,n,k)∞

∑s=0

∞

∑t=0

β

∑r=α

g(s,t,r)u(s, t,r), (5.3.19)

for (m,n,k) ∈ H. If

β2 =∞

∑s=0

∞

∑t=0

β

∑r=α

g(s, t,r)B2(s,t,r) < 1, (5.3.20)


then

u(m,n,k) � A2(m,n,k)+D2B2(m,n,k), (5.3.21)

for (m,n,k) ∈ H, where

A2(m,n,k) = p(m,n,k)+q(m,n,k)

(∞

∑s=m+1

∞

∑t=n+1

β

∑r=α

f (s,t,r)p(s,t,r)

)

×∞

∏s=m+1

[

1+∞

∑t=n+1

β

∑r=α

f (s,t,r)q(s,t,r)

]

, (5.3.22)

B2(m,n,k) = c(m,n,k)+q(m,n,k)

(∞

∑s=m+1

∞

∑t=n+1

β

∑r=α

f (s,t,r)c(s,t,r)

)

×∞

∏s=m+1

[

1+∞

∑t=n+1

β

∑r=α

f (s,t,r)q(s,t,r)

]

, (5.3.23)

and

D2 =1

1−β2

∞

∑s=0

∞

∑t=0

β

∑r=α

g(s,t,r)A2(s,t,r). (5.3.24)

Theorem 5.3.5. Let u, p ∈ D(H,R+), q ∈ D(H ×Nα ,β ,R+), and c � 0 is a real constant.

If

u(m,n,z) � c+m−1

∑s=0

n−1

∑t=0

[

p(s,t,z)u(s,t,z)+β

∑r=α

q(s,t,z,r)u(s,t,r)

]

, (5.3.25)

for (m,n,z) ∈ H, then

u(m,n,z) � cL(m,n,z)m−1

∏s=0

[

1+n−1

∑t=0

β

∑r=α

q(s, t,z,r)L(s,t,r)

]

, (5.3.26)

for (m,n,z) ∈ H, where

L(m,n,z) =m−1

∏σ=0

[

1+n−1

∑τ=0

p(σ ,τ,z)

]

.

Proofs of Theorems 5.3.1–5.3.5. We give the details of the proofs for (r1), (r4) and (r5)

only; the proofs of other inequalities can be completed by following the proofs of these

inequalities and the ideas employed in the proofs of the results in Chapter 2, section 2.3

with suitable modifications, see also [85,87].

(r1) Introducing the notation

e(s,t) =β

∑r=α

f (s,t,r)u(s,t,r), (5.3.27)


in (5.3.1), we get

u(m,n,k) � c+m−1

∑s=0

n−1

∑t=0

e(s, t), (5.3.28)

for (m,n,k) ∈ H. Let c > 0 and define

v(m,n) = c+m−1

∑s=0

n−1

∑t=0

e(s,t), (5.3.29)

then v(m,0) = v(0,n) = c and

u(m,n,k) � v(m,n). (5.3.30)

From (5.3.29), (5.3.27) and (5.3.30), we observe that (see [85, p. 299])

Δ2Δ1v(m,n) = e(m,n)

=β

∑r=α

f (m,n,r)u(m,n,r)

� v(m,n)β

∑r=α

f (m,n,r). (5.3.31)

The rest of the proof can be completed by following similar arguments as in the proof of

Theorem 4.2.1 given in [85] with suitable modifications.


e(s,t) =β

∑r=α

f (s, t,r)u(s,t,r) logu(s,t,r), (5.3.32)

in (5.3.7), we get

u(m,n,k) � d +m−1

∑s=0

n−1

∑t=0

e(s,t). (5.3.33)

Define

w(m,n) = d +m−1

∑s=0

n−1

∑t=0

e(s,t), (5.3.34)

then w(m,0) = w(0,n) = d and

u(m,n,k) � w(m,n). (5.3.35)

From (5.3.34), (5.3.32) and (5.3.35), we observe that (see [85, p. 437])

Δ2Δ1w(m,n) = e(m,n)

=β

∑r=α

f (m,n,r)u(m,n,r) logu(m,n,r)


� w(m,n)β

∑r=α

f (m,n,r) logw(m,n). (5.3.36)

The remaining proof can be completed by following the proof of Theorem 5.5.1 given in

[85].


e0(s,t) =β

∑r=α

f (s,t,r)u(s, t,r), (5.3.37)



∑s=0

∞

∑t=n+1

e0(s,t). (5.3.38)

Define

ψ(m,n) =m−1

∑s=0

∞

∑t=n+1

e0(s,t), (5.3.39)

then ψ(0,n) = 0 and

u(m,n,k) � p(m,n,k)+q(m,n,k)ψ(m,n). (5.3.40)


Δ1ψ(m,n) =∞

∑t=n+1

e0(m,t)

=∞

∑t=n+1

{β

∑r=α

f (m,t,r)u(m, t,r)

}

�∞

∑t=n+1

{β

∑r=α

f (m,t,r)[p(m,t,r)+q(m, t,r)ψ(m,t)]

}

=∞

∑t=n+1

β

∑r=α

f (m,t,r)p(m,t,r)+∞

∑t=n+1

{β

∑r=α

f (m,t,r)q(m,t,r)ψ(m,t)

}

. (5.3.41)

By taking m = s in (5.3.41) and then taking sum over s from 0 to m−1, m ∈ N0, we get

ψ(m,n) �m−1

∑s=0

∞

∑t=n+1

β

∑r=α

f (s,t,r)p(s,t,r)+m−1

∑s=0

∞

∑t=n+1

{β

∑r=α

f (s, t,r)q(s,t,r)ψ(s,t)

}

= E(m,n)+m−1

∑s=0

∞

∑t=n+1

ψ(s,t)

{β

∑r=α

f (s, t,r)q(s,t,r)

}

, (5.3.42)

where

E(m,n) =m−1

∑s=0

∞

∑t=n+1

β

∑r=α

f (s,t,r)p(s, t,r). (5.3.43)

Clearly, E(m,n) is nonnegative, nondecreasing in m and nonincreasing in n for m, n ∈ N0.

Now a suitable application of the inequality given in Theorem 5.4.1 part (a1) in [87, p. 266]

to (5.3.42) yields

ψ(m,n) � E(m,n)m−1

∏s=0

[

1+∞

∑t=n+1

β

∑r=α

f (s,t,r)q(s, t,r)

]

. (5.3.44)

Using (5.3.44), (5.3.43) in (5.3.40), we get (5.3.10).


5.4 Multivariable sum-difference inequalities

Let Ni[αi,βi] = {αi, αi + 1, . . . ,βi} (αi < βi), αi, βi ∈ N0 for i = 1, . . . ,m and G =

∏mi=1 Ni[αi,βi] ⊂ R

m. Let E = N0 ×G, Ω = {(n,x,s · y) : 0 � s � n < ∞, x, y ∈ G} for

s, n ∈ N0 and for the function r defined on Ω, we define Δ1r(n,x,s,y) = r(n + 1,x,s,y)−r(n,x,s,y). For any function w defined on G we denote the m-fold sum over G with respect

to the variable y = (y1, . . . ,ym) ∈ G by ∑G w(y) = ∑β1y1=α1 · · ·∑

βmym=αm w(y1, . . . ,ym). Clearly,

∑G w(y) = ∑G w(x) for x, y ∈ G. The main objective of this section is to provide some basic

finite difference inequalities which can be used as tools in the development of the theory of

certain new classes of sum-difference equations.

The following theorems deals with the inequalities recently established by Pachpatte

[116,114,106].

Theorem 5.4.1. Let u ∈ D(E,R+), h, Δ1h ∈ D(Ω,R+) and c � 0 is a real constant.

(s1) If

u(n,x) � c+n−1

∑s=0

∑G

h(n,x,s,y)u(s,y), (5.4.1)

for (n,x) ∈ E, then

u(n,x) � cn−1

∏σ=0

[1+A(σ ,x)

], (5.4.2)

for (n,x) ∈ E, where

A(n,x) = ∑G

h(n+1,x,n,y)+n−1

∑s=0

∑G

Δ1h(n,x,s,y), (5.4.3)

for (n,x) ∈ E.

(s2) Let g ∈C(R+,R+) be nondecreasing function, g(u) > 0 on (0,∞). If

u(n,x) � c+n−1

∑s=0

∑G

h(n,x,s,y)g(u(s,y)), (5.4.4)

for (n,x) ∈ E, then for 0 � n � n1; n, n1 ∈ N0, x ∈ G,

u(n,x) � W−1

[

W (c)+n−1

∑σ=0

A(σ ,x)

]

, (5.4.5)

where

W (r) =∫ r

r0

dsg(s)

, r > 0, (5.4.6)

r0 > 0 is arbitrary and W−1 is the inverse of W , A(n,x) is given by (5.4.3) and n1 ∈ N0 be

chosen so that

W (c)+n−1

∑σ=0

A(σ ,x) ∈ Dom(W−1) ,

for all n ∈ N0 lying in 0 � n � n1 and x ∈ G.


Theorem 5.4.2. Let u, h, Δ1h and c be as in Theorem 5.4.1.

(s3) If

u2(n,x) � c+n−1

∑s=0

∑G

h(n,x,s,y)u(s,y), (5.4.7)


u(n,x) �√

c+12

n−1

∑σ=0

A(σ ,x), (5.4.8)

for (n,x) ∈ E, where A(n,x) is given by (5.4.3).

(s4) Let g be as in Theorem 5.4.1 part (s2). If

u2(n,x) � c+n−1

∑s=0

∑G

h(n,x,s,y)u(s,y)g(u(s,y)), (5.4.9)


u(n,x) � W−1

[

W(√

c)+

12

n−1

∑σ=0

A(σ ,x)

]

, (5.4.10)

where W, W−1, A(n,x) are as in part (s2) and n2 ∈ N0 be chosen so that

W(√

c)+

12

n−1

∑σ=0

A(σ ,x) ∈ Dom(W−1) ,


Theorem 5.4.3. Let u ∈ D(E,R1), h, Δ1h ∈ D(Ω,R+) and c � 1 be a real constant.

(s5) If

u(n,x) � c+n−1

∑s=0

∑G

h(n,x,s,y)u(s,y) logu(s,y), (5.4.11)


u(n,x) � c∏n−1σ=0[1+A(σ ,x)], (5.4.12)

for (n,x) ∈ E, where A(n,x) is given by (5.4.3).

(s6) Let g be as in Theorem 5.4.1 part (s2). If

u(n,x) � c+n−1

∑s=0

∑G

h(n,x,s,y)u(s,y)g(logu(s,y)), (5.4.13)


u(n,x) � exp

(

W−1

[

W (logc)+n−1

∑σ=0

A(σ ,x)

])

, (5.4.14)

where W, W−1, A(n,x) are as in part (s2) and n3 ∈ N0 be chosen so that

W (logc)+n−1

∑σ=0

A(σ ,x) ∈ Dom(W−1) ,



Theorem 5.4.4. (s7) Let u, p, q, f , g ∈ D(E,R+) and

u(n,x) � p(n,x)+q(n,x)n−1

∑s=0

∑G

f (s,y)

[

u(s,y)+q(s,y)s−1

∑τ=0

∑G

g(τ,z)u(τ,z)

]

, (5.4.15)

for (n,x) ∈ E. Then


∑s=0

∑G

p(s,y)[ f (s,y)+g(s,y)]

×n−1

∏σ=s+1

[

1+∑G

q(σ ,z)[ f (σ ,z)+g(σ ,z)]

]

, (5.4.16)

for (n,x) ∈ E.

(s8) Let u, p, q, f , g ∈ D(E,R+) where E = N20 ×G and suppose that

u(n,m,x) � p(n,m,x)+q(n,m,x)n−1

∑s=0

m−1

∑t=0

∑G

f (s,t,y)

×[

u(s,t,y)+q(s,t,y)s−1

∑σ=0

t−1

∑τ=0

∑G

g(σ ,τ,z)u(σ ,τ,z)

]

, (5.4.17)

for (n,m,x) ∈ E. Then

u(n,m,x) � p(n,m,x)+q(n,m,x)

×(

n−1

∑s=0

m−1

∑t=0

∑G

p(s, t,y)[ f (s,t,y)+g(s,t,y)]

)

×n−1

∏s=0

[

1+m−1

∑t=0

∑G

q(s,t,y)[ f (s, t,y)+g(s,t,y)]

]

, (5.4.18)

for (n,m,x) ∈ E.

Theorem 5.4.5. Let u, a, b, c, p, q ∈ D(E,R+).

(s9) Suppose that

u(n,x) � a(n,x)+b(n,x)n−1

∑s=0

∑G

p(s,y)u(s,y)+ c(n,x)∞

∑s=0

∑G

q(s,y)u(s,y), (5.4.19)

for (n,x) ∈ E. If

d =∞

∑s=0

∑G

q(s,y)B(s,y) < 1, (5.4.20)

then

u(n,x) � A(n,x)+DB(n,x), (5.4.21)



A(n,x) = a(n,x)+b(n,x)n−1

∑s=0

∑G

p(s,y)a(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]

, (5.4.22)

B(n,x) = c(n,x)+b(n,x)n−1

∑s=0

∑G

p(s,y)c(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]

, (5.4.23)

for (n,x) ∈ E and

D =1

1−d

∞

∑s=0

∑G

q(s,y)A(s,y). (5.4.24)

(s10) If


∑s=0

∑G

p(s,y)u(s,y), (5.4.25)



∑s=0

∑G

p(s,y)a(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]

, (5.4.26)

for (n,x) ∈ E.

Proofs of Theorems 5.4.1–5.4.5. Below, we give the proofs of the inequalities in (s1),

(s2), (s7), (s9) only; the proofs of other inequalities can be completed by following the

proofs of these inequalities and closely looking at the proofs of the inequalities in Chapter 3,

see also [85,87]. To prove (s1)–(s4), it is sufficient to assume that c > 0, since the standard

limiting argument can be used to treat the remaining case, see [85, p. 300].

(s1) Setting

e(n,s) = ∑G

h(n,x,s,y)u(s,y), (5.4.27)

for every x ∈ G, the inequality (5.4.1) can be restated as

u(n,x) � c+n−1

∑s=0

e(n,s). (5.4.28)

Let c > 0 and define

z(n) = c+n−1

∑s=0

e(n,s), (5.4.29)

then z(0) = c and

u(n,x) � z(n). (5.4.30)


From (5.4.29), (5.4.27), (5.4.30) and the fact that z(n) is nondecreasing in n ∈ N0, we

observe that

Δz(n) = e(n+1,n)+n−1

∑s=0

Δ1e(n,s)

= ∑G

h(n+1,x,n,y)u(n,y)+n−1

∑s=0

Δ1

{

∑G

h(n,x,s,y)u(s,y)

}

� ∑G

h(n+1,x,n,y)z(n)+n−1

∑s=0

∑G

Δ1h(n,x,s,y)z(s)

�[

∑G

h(n+1,x,n,y)+n−1

∑s=0

∑G

Δ1h(n,x,s,y)

]

z(n)

= A(n,x)z(n). (5.4.31)

Now a suitable application of Theorem 1.2.1 given in [85, p. 11] to (5.4.31) yields

z(n) � cn−1

∏σ=0

[1+A(σ ,x)

]. (5.4.32)


(s2) Setting

e(n,s) = ∑G

h(n,x,s,y)g(u(s,y)), (5.4.33)

for every x ∈ G, the inequality (5.4.4) can be restated as

u(n,x) � c+n−1

∑s=0

e(n,s). (5.4.34)

Let c > 0 and define by z(n) the right hand side of (5.4.34). Following the proof of (s1)

given above, we get

Δz(n) � A(n,x)g(z(n)). (5.4.35)

Now by following the proof of Theorem 2.3.1 given in [85, p. 104] from (5.4.35), we get

z(n) � W−1

[

W (c)+n−1

∑σ=0

A(σ ,x)

]

, (5.4.36)

on 0 � n � n1. Using (5.4.36) in u(n,x) � z(n), we get the desired inequality in (5.4.5).

(s7) Introducing the notation

e(s) = ∑G

f (s,y)

[

u(s,y)+q(s,y)s−1

∑τ=0

∑G

g(τ,z)u(τ,z)

]

, (5.4.37)


in (5.4.15), we get


∑s=0

e(s), (5.4.38)

for (n,x) ∈ E. Define

r(n) =n−1

∑s=0

e(s), (5.4.39)

for n ∈ N0, then r(0) = 0 and from (5.4.38), we get

u(n,x) � p(n,x)+q(n,x)r(n) (5.4.40)

for (n,x) ∈ E. From (5.4.39), (5.4.37) and (5.4.40), we observe that

Δr(n) = e(n)

= ∑G

f (n,y)

[

u(n,y)+q(n,y)n−1

∑τ=0

∑G

g(τ,z)u(τ,z)

]

� ∑G

f (n,y)

[

p(n,y)+q(n,y)r(n)

+q(n,y)n−1

∑τ=0

∑G

g(τ,z)[p(τ,z)+q(τ,z)r(τ)]

]

= ∑G

f (n,y)

[

p(n,y)+q(n,y)+

{

r(n)+n−1

∑τ=0

∑G

g(τ,z)[p(τ,z)+q(τ,z)r(τ)]

}]

. (5.4.41)

Define

v(n) = r(n)+n−1

∑τ=0

∑G

g(τ,z)[p(τ,z)+q(τ,z)r(τ)], (5.4.42)

for n ∈ N0, then v(0) = r(0) = 0, r(n) � v(n) and from (5.4.41), we get

Δr(n) � ∑G

f (n,y)[p(n,y)+q(n,y)v(n)]. (5.4.43)

From (5.4.42), (5.4.43) and the fact that r(n) � v(n), n ∈ N0, we observe that

Δv(n) = Δr(n)+∑G

g(n,z)[p(n,z)+q(n,z)r(n)]

� v(n)∑G

q(n,y)[ f (n,y)+g(n,y)]+∑G

p(n,y)[ f (n,y)+g(n,y)]. (5.4.44)


Now a suitable application of Theorem 1.2.1 given in [85, p. 11] with v(0) = 0 to (5.4.44)

yields

v(n) �n−1

∑s=0

∑G

p(s,y)[ f (s,y)+g(s,y)]

×n−1

∏σ=s+1

[

1+∑G

q(σ ,z)[ f (σ ,z)+g(σ ,z)]

]

. (5.4.45)

Using the fact that r(n) � v(n) and (5.4.40), (5.4.45) we get the required inequality in

(5.4.16).

(s9) Let

z(n) =n−1

∑s=0

∑G

p(s,y)u(s,y), (5.4.46)

λ =∞

∑s=0

∑G

q(s,y)u(s,y). (5.4.47)

Then (5.4.19) can be restated as

u(n,x) � a(n,x)+b(n,x)z(n)+ c(n,x)λ . (5.4.48)

Introducing the notation

e0(s) = ∑G

p(s,y)u(s,y), (5.4.49)

in (5.4.46), we get

z(n) =n−1

∑s=0

e0(s). (5.4.50)


Δz(n) = e0(n)

= ∑G

p(n,y)u(n,y)

� ∑G

p(n,y)[a(n,y)+b(n,y)z(n)+ c(n,y)λ ]

= z(n)∑G

p(n,y)b(n,y)+∑G

p(n,y)[a(n,y)+ c(n,y)λ ]. (5.4.51)

Now, applying the inequality in Theorem 1.2.1 given in [85, p. 11] with z(0) = 0 to (5.4.51)

yields

z(n) �n−1

∑s=0

∑G

p(s,y)[a(s,y)+ c(s,y)λ ]n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]


=n−1

∑s=0

∑G

p(s,y)a(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]

+λn−1

∑s=0

∑G

p(s,y)c(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]

. (5.4.52)

From (5.4.48) and (5.4.52), we have

u(n,x) � a(n,x)+b(n,x)

{n−1

∑s=0

∑G

p(s,y)a(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]

+λn−1

∑s=0

∑G

p(s,y)c(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]}

+ c(n,x)λ

= A(n,x)+λB(n,x). (5.4.53)


λ �∞

∑s=0

∑G

q(s,y)[A(s,y)+λB(s,y)],

which implies

λ � D. (5.4.54)

Using (5.4.54) in (5.4.53), we get (5.4.21).

5.5 Sum-difference equations in two variables

In this section, first we shall deal with the following initial value problem (see [100])

Δ2Δ1u(m,n) = f (m,n,u(m,n),Gu(m,n)), (5.5.1)

with

u(m,0) = α(m), u(0,n) = β (n), α(0) = β (0), (5.5.2)

for m, n ∈ N0, where

Gu(m,n) :=m−1

∑σ=0

n−1

∑τ=0

g(m,n,σ ,τ,u(σ ,τ)), (5.5.3)

f , g are given functions and u is the unknown function. The equations of the form

(5.5.1) arise naturally in the approximation of solutions of partial integrodifferential equa-

tions by finite difference methods and also appear in their own right. We assume that

f ∈ D(N20 ×R

2,R), g ∈ D(N40 ×R,R), α, β ∈ D(N0,R). Here, it is to be noted that the


problem (5.5.1)–(5.5.2) under some suitable conditions admits a unique solution. Below,

we offer the conditions for the error evaluation of approximate solutions of equation (5.5.1)

and convergence properties of solutions of approximate problems and also dependency of

solutions of equations of the form (5.5.1) on parameters, by employing a certain finite

difference inequality with explicit estimate given in [87].

Let u ∈ D(N20,R) and Δ2Δ1u(m,n) for m, n ∈ N0 exist and satisfy the inequality

|Δ2Δ1u(m,n)− f (m,n,u(m,n),Gu(m,n))| � ε,

for a given constant ε � 0, where it is assumed that (5.5.2) holds. Then we call u(m,n) an

ε-approximate solution of (5.5.1).

The following finite difference inequality given in [87] (see also [85, Theorem 5.3.2]) is

crucial in the study of problem (5.5.1)–(5.5.2).

Lemma 5.5.1. Let u, a, p ∈ D(N20,R+); q, Δ1q, Δ2q, Δ2Δ1q ∈ D(N4

0,R+). If a(m,n) is

nondecreasing in each variable m, n ∈ N0 and

u(m,n) � a(m,n)+m−1

∑s=0

n−1

∑t=0

p(s,t)

[

u(s,t)+s−1

∑σ=0

t−1

∑τ=0

q(s,t ·σ ,τ)u(σ ,τ)

]

, (5.5.4)

for m, n ∈ N0, then

u(m,n) � a(m,n)

[

1+m−1

∑s=0

n−1

∑t=0

p(s,t)s−1

∏ξ=0

[

1+t−1

∑η=0

[p(ξ ,η)+T q(ξ ,η)]

]

, (5.5.5)


T q(m,n) := q(m+1,n+1,m,n)+m−1

∑σ=0

Δ1q(m,n+1,σ ,n)

+n−1

∑τ=0

Δ2q(m+1,n,m,τ)+m−1

∑σ=0

n−1

∑τ=0

Δ2Δ1q(m,n,σ ,τ). (5.5.6)

The following theorem estimates the difference between the two approximate solutions of

(5.5.1).

Theorem 5.5.1. Suppose that f , g in (5.5.1) satisfy the conditions

| f (m,n,u,v)− f (m,n,u,v)| � p(m,n) [|u−u|+ |v− v|] , (5.5.7)

|g(m,n,σ ,τ,u)−g(m,n,σ ,τ,u)| � q(m,n,σ ,τ)|u−u|, (5.5.8)


where p ∈ D(N20,R+), q ∈ D(N4

0,R+) with Δ1q, Δ2q, Δ2Δ1q ∈ D(N40,R+). For i = 1, 2, let

ui(m,n) (m, n ∈ N0) be respectively εi-approximate solutions of (5.5.1) with

ui(m,0) = αi(m), ui(0,n) = βi(n), αi(0) = βi(0), (5.5.9)

for m, n ∈ N0, where αi, βi ∈ D(N0,R) satisfy

|α1(m)−α2(m)+β1(n)−β2(n)| � δ , (5.5.10)

in which δ � 0 is a constant. Then

|u1(m,n)−u2(m,n)| � ((ε1 + ε2)mn+δ )

×[

1+m−1

∑s=0

n−1

∑t=0

p(s,t)s−1

∏ξ=0

[

1+t−1

∑η=0

[p(ξ ,η)+T q(ξ ,η)]

]]

, (5.5.11)

for m, n ∈ N0, where T q(m,n) is given by (5.5.6).

Proof. Since ui(m,n) (i = 1, 2) for m, n ∈ N0, are respectively εi-approximate solutions

of equation (5.5.1) with (5.5.9), we have

|Δ2Δ1ui(m,n)− f (m,n,ui(m,n),Gui(m,n))| � εi.

Now keeping m fixed in the above inequality, setting n = t and taking sum on both sides

over t from 0 to n−1, then keeping n fixed in the resulting inequality and setting m = s and

taking sum over s from 0 to m−1 and using (5.5.9), we observe that

εimn �m−1

∑s=0

n−1

∑t=0

|Δ2Δ1ui(s,t)− f (s,t,ui(s,t),Gui(s,t))|

�∣∣∣∣∣

m−1

∑s=0

n−1

∑t=0

{Δ2Δ1ui(s,t)− f (s,t,ui(s,t),Gui(s,t))}∣∣∣∣∣

=

∣∣∣∣∣

{

ui(m,n)− [αi(m)+βi(n)]−m−1

∑s=0

n−1

∑t=0

f (s,t,ui(s,t),Gui(s,t))

}∣∣∣∣∣. (5.5.12)

From this inequality and using the elementary inequalities in (1.3.25), we observe that

(ε1 + ε2)mn �∣∣∣∣∣

{

u1(m,n)− [α1(m)+β1(n)]−m−1

∑s=0

n−1

∑t=0

f (s, t,u1(s,t),Gu1(s,t))

}∣∣∣∣∣

+

∣∣∣∣∣

{

u2(m,n)− [α2(m)+β2(n)]−m−1

∑s=0

n−1

∑t=0

f (s,t,u2(s,t),Gu2(s, t))

}∣∣∣∣∣

�∣∣∣∣∣

{

u1(m,n)− [α1(m)+β1(n)]−m−1

∑s=0

n−1

∑t=0

f (s, t,u1(s,t),Gu1(s,t))

}


−{

u2(m,n)− [α2(m)+β2(n)]−m−1

∑s=0

n−1

∑t=0

f (s,t,u2(s,t),Gu2(s,t))

}∣∣∣∣∣

� |u1(m,n)−u2(m,n)|− |α1(m)+β1(n)−{α2(m)+β2(n)}|

−∣∣∣∣∣

m−1

∑s=0

n−1

∑t=0

f (s,t,u1(s,t),Gu1(s,t))−m−1

∑s=0

n−1

∑t=0

f (s,t,u2(s, t),Gu2(s, t))

∣∣∣∣∣. (5.5.13)

Let u(m,n) = |u1(m,n)−u2(m,n)| for m, n ∈ N0. From (5.5.13) and using the hypotheses,

we observe that

u(m,n) � (ε1 + ε2)mn+δ +m−1

∑s=0

n−1

∑t=0

| f (s,t,u1(s,t),Gu1(s,t))− f (s,t,u2(s,t),Gu2(s,t))|

� (ε1 + ε2)mn+δ +m−1

∑s=0

n−1

∑t=0

p(s,t)

[

u(s, t)+s−1

∑σ=0

t−1

∑τ=0

q(s,t,σ ,τ)u(σ ,τ)

]

. (5.5.14)


Remark 5.5.1. In case u1(m,n) is a solution of (5.5.1)–(5.5.2), then we have ε1 = 0 and

from (5.5.11), we see that u2(m,n) → u1(m,n) as ε2 → 0 and δ → 0. Furthermore, if we

put

(i) ε1 = ε2 = 0, α1(m) = α2(m), β1(n) = β2(n) in (5.5.11), then the uniqueness of solutions

of (5.5.1)–(5.5.2) is established, and

(ii) ε1 = ε2 = 0 in (5.5.11),

then we get the bound which shows the dependency of solutions of (5.5.1) on given initial

values.

Consider the problem (5.5.1)–(5.5.2) together with

Δ2Δ1v(m,n) = f (m,n,v(m,n),Gv(m,n)), (5.5.15)

v(m,0) = α(m),v(0,n) = β (n),α(0) = β (0), (5.5.16)

for m, n ∈ N0, where G is given by (5.5.3) and f ∈ D(N20 ×R

2,R), α, β ∈ D(N0,R).

The next theorem concerns the closeness of solutions of (5.5.1)–(5.5.2) and of (5.5.15)–

(5.5.16).


Theorem 5.5.2. Suppose that f , g in (5.5.1) satisfy (5.5.7), (5.5.8) and there exist con-

stants ε � 0, δ � 0 such that

| f (m,n,u,w)− f (m,n,u,w)| � ε, (5.5.17)

|α(m)−α(m)+β (n)−β (n)| � δ , (5.5.18)

where f , α , β and f , α, β are as in (5.5.1)–(5.5.2) and (5.5.15)–(5.5.16). Let u(m,n) and

v(m,n) be respectively the solutions of (5.5.1)–(5.5.2) and of (5.5.15)–(5.5.16) for m, n ∈N0. Then

|u(m,n)− v(m,n)|

� (εmn+δ )

[

1+m−1

∑s=0

n−1

∑t=0

p(s,t)s−1

∏ξ=0

[

1+t−1

∑η=0

[p(ξ ,η)+Tq(ξ ,η)]

]]

, (5.5.19)

for m, n ∈ N0, where T q(m,n) is given by (5.5.6).

Proof. Let e(m,n) = |u(m,n)− v(m,n)| for m, n ∈ N0. Using the facts that u(m,n),

v(m,n) are the solutions of (5.5.1)–(5.5.2), (5.5.15)–(5.5.16) and the hypotheses, we ob-

serve that

e(m,n) � |α(m)−α(m)+β (n)−β (n)|

+m−1

∑s=0

n−1

∑t=0

| f (s,t,u(s,t),Gu(s,t))− f (s,t,v(s,t),Gv(s,t))|

+m−1

∑s=0

n−1

∑t=0

| f (s,t,v(s,t),Gv(s,t))− f (s,t,v(s,t),Gv(s,t))|

� (εmn+δ )+m−1

∑s=0

n−1

∑t=0

p(s,t)

[

e(s,t)+s−1

∑σ=0

t−1

∑τ=0

q(s,t,σ ,τ)e(σ ,τ)

]

. (5.5.20)


Remark 5.5.2. The result given in Theorem 5.5.2 relates the solutions of (5.5.1)–(5.5.2)

and of (5.5.15)–(5.5.16) in the sence that if f is close to f , α is close to α , β is close to β ,

then the solutions of (5.5.1)–(5.5.2) and of (5.5.15)–(5.5.16) are also close to each other.

Now we consider (5.5.1)–(5.5.2) and sequence of initial value problems

Δ2Δ1w(m,n) = fk(m,n,w(m,n),Gw(m,n)), (5.5.21)

w(m,0) = αk(m), w(0,n) = βk(n), αk(0) = βk(0), (5.5.22)

for m, n ∈ N0, k = 1, 2, . . ., where G is given by (5.5.3) and fk ∈ D(N20 ×R

2,R), αk, βk ∈D(N0,R).

As an immediate consequence of Theorem 5.5.2 we have the following corollary.


Corollary 5.5.1. Suppose that f , g in (5.5.1) satisfy (5.5.7), (5.5.8) and there exist con-

stants εk � 0, δk � 0 (k = 1, 2, . . .) such that

| f (m,n,u,v)− fk(m,n,u,v)| � εk, (5.5.23)

|α(m)−αk(m)+β (n)−βk(n)| � δk, (5.5.24)

with εk → 0 and δk → 0 as k → ∞, where f , α, β and fk, αk, βk are respectively as in

(5.5.1)–(5.5.2) and in (5.5.21)–(5.5.22). If wk(m,n) (k = 1, 2, . . .) and u(m,n) are respec-

tively the solutions of (5.5.21)–(5.5.22) and (5.5.1)–(5.5.2) for m, n ∈ N0, then wk(m,n) →u(m,n) as k → ∞.

Proof. For k = 1, 2, . . ., the conditions of Theorem 5.5.2 hold. An application of Theo-

rem 5.5.2 yields

|wk(m,n)−u(m,n)|

� (εkmn+δk)

[

1+m−1

∑s=0

n−1

∑t=0

p(s, t)s−1

∏ξ=0

[

1+t−1

∑η=0

[p(ξ ,η)+Tq(ξ ,η)]

]]

, (5.5.25)

for m, n ∈ N0, where T q(m,n) is given by (5.5.6) and k = 1, 2, . . .. The required result

follows from (5.5.25).

Remark 5.5.3. We note that the result obtained in Corollary 5.5.1 provides sufficient

conditions that ensures, solutions of (5.5.21)–(5.5.22) will converge to the solutions to

(5.5.1)–(5.5.2).

We now consider the sum-difference equations

Δ2Δ1u(m,n) = f (m,n,u(m,n),Gu(m,n),μ), (5.5.26)

Δ2Δ1u(m,n) = f (m,n,u(m,n),Gu(m,n),μ0), (5.5.27)

with the initial conditions (5.5.2), where Gu(m,n) is given by (5.5.3), f ∈ D(N20 ×R

2 ×R,R) and μ, μ0 are parameters.

The next theorem shows the dependency of solutions of (5.5.26)–(5.5.2) and (5.5.27)–

(5.5.2) on the parameters μ , μ0.

Theorem 5.5.3. Suppose that g and f in (5.5.26), (5.5.27) satisfy respectively (5.5.8) and

| f (m,n,u,v,μ)− f (m,n,u,v,μ)| � p(m,n) [|u−u|+ |v− v|] , (5.5.28)

| f (m,n,u,v,μ)− f (m,n,u,v,μ0)| � r(m,n)|μ −μ0|, (5.5.29)


where p, r ∈ D(N20,R+). Let u1(m,n) and u2(m,n) be the solutions of (5.5.26)–(5.5.2) and

of (5.5.27)–(5.5.2) respectively. Then

|u1(m,n)−u2(m,n)|

� a(m,n)

[

1+m−1

∑s=0

n−1

∑t=0

p(s,t)s−1

∏ξ=0

[

1+t−1

∑η=0

[p(ξ ,η)+Tq(ξ ,η)]

]]

, (5.5.30)


a(m,n) = |μ −μ0|m−1

∑s=0

n−1

∑t=0

r(s, t), (5.5.31)

for m, n ∈ N0.

Proof. Let e(m,n) = |u1(m,n)− u2(m,n)| for m, n ∈ N0. Using the facts that u1(m,n)

and u2(m,n) are respectively the solutions of (5.5.26)–(5.5.2) and of (5.5.27)–(5.5.2) and

the hypotheses, we observe that

e(m,n) �m−1

∑s=0

n−1

∑t=0

| f (s,t,u1(s,t),Gu1(s,t),μ)− f (s,t,u2(s,t),Gu2(s,t),μ)|

+m−1

∑s=0

n−1

∑t=0

| f (s, t,u2(s,t),Gu2(s,t),μ)− f (s,t,u2(s, t),Gu2(s, t),μ0)|

� a(m,n)+m−1

∑s=0

n−1

∑t=0

p(s,t)

[

e(s,t)+s−1

∑σ=0

t−1

∑τ=0

q(s,t,σ ,τ)e(σ ,τ)

]

. (5.5.32)


dency of solutions of (5.5.26)–(5.5.2) and (5.5.27)–(5.5.2) on the parameters μ , μ0.

Remark 5.5.4. We note that the results given above can be extended very easily to study

the sum-difference equation

Δ2Δ1u(m,n)+Δ2(b(m,n)u(m,n)) = f (m,n,u(m,n),Gu(m,n),Hu(m,n)), (5.5.33)

with the given initial conditions in (5.5.2), where G is defined by (5.5.3) and H is given by

Hu(m,n) :=∞

∑σ=0

∞

∑τ=0

h(m,n,σ ,τ,u(σ ,τ)), (5.5.34)

under some suitable conditions on b, f , g, h involved in (5.5.33), (5.5.34), (5.5.2) by mak-

ing use of the finite difference inequality given in [87, Theorem 5.2.3].

Next, we shall study some fundamental qualitative properties of solutions of the following

Fredholm type sum-difference equation (see [113])

u(m,n) = f (m,n)+a

∑s=0

b

∑t=0

g(m,n,s,t,u(s,t),Δ1u(s,t),Δ2u(s,t)), (5.5.35)


where f , g are given functions and u is the unknown function. Let H = N0,a ×N0,b and for

any function w : H → R, we denote by |w(m,n)|0 = |w(m,n)|+ |Δ1w(m,n)|+ |Δ2w(m,n)|and assume that w(m,n) = 0 for (m,n) /∈ H. It is assumed that f , Δi f ∈ D(H,R) and

g, Δig ∈ D(H2 ×R3,R) for i = 1, 2. The origin of equation (5.5.35) can be traced back to

the study of its one variable integral analogue in [11] (see also [90,91]). By a solution of

equation (5.5.35) we mean a function u : H → R for which Δiu(m,n) (i = 1, 2) exist and

satisfies the equation (5.5.35). It is easy to observe that the solution u(m,n) of equation

(5.5.1) satisfy for i = 1, 2 the following sum-difference equations

Δiu(m,n) = Δi f (m,n)+a

∑s=0

b

∑t=0

Δig(m,n,s,t,u(s,t),Δ1u(s,t),Δ2u(s,t)), (5.5.36)

for (m,n) ∈ H. The problem of existence of solutions for equations of the forms (5.5.35)

can be dealt with the method employed in in Chapter 1, section 1.6, see also [51,54]. Here

we present some basic qualitative aspects of solutions of equation (5.5.35) under some

suitable conditions on the functions involved therein.

We recall the following special version of the finite difference inequality given in [87,

Theorem 5.5.1, p. 286] that will be needed to establish the results.

Lemma 5.5.2. Let z, p, q, r ∈ D(H,R+) and

z(m,n) � p(m,n)+q(m,n)a

∑s=0

b

∑t=0

r(s,t)z(s,t), (5.5.37)

for (m,n) ∈ H. If

d =a

∑s=0

b

∑t=0

r(s,t)q(s, t) < 1, (5.5.38)

then

z(m,n) � p(m,n)+q(m,n)

{1

1−d

a

∑s=0

b

∑t=0

r(s,t)p(s,t)

}

, (5.5.39)

for (m,n) ∈ H.

The results concerning estimates, uniqueness, and dependency of solutions of (5.5.35) are

given in the following theorems.

Theorem 5.5.4. Suppose that the function g in (5.5.35) and Δig for i = 1, 2 in (5.5.36)

satisfy the conditions

|g(m,n,s,t,u,v,w)−g(m,n,s,t,u,v,w)|

� c(m,n)h(s,t) [|u−u|+ |v− v|+ |w−w|] , (5.5.40)


and

|Δig(m,n,s,t,u,v,w)−Δig(m,n,s,t,u,v,w)|

� c(m,n)hi(s,t) [|u−u|+ |v− v|+ |w−w|] , (5.5.41)

where c, h, hi ∈ D(H,R+) and

d1 =a

∑s=0

b

∑t=0

[h(s,t)+h1(s,t)+h2(s,t)]c(s,t) < 1. (5.5.42)

If u(m,n) is any solution of equation (5.5.35) on H, then

|u(m,n)− f (m,n)|0 � Q(m,n)+ c(m,n)

×{

11−d1

a

∑s=0

b

∑t=0

[h(s,t)+h1(s, t)+h2(s,t)]Q(s,t)

}

, (5.5.43)

for (m,n) ∈ H, where

Q(m,n) =a

∑s=0

b

∑t=0

|g(m,n,σ ,τ, f (σ ,τ),Δ1 f (σ ,τ),Δ2 f (σ ,τ))|0, (5.5.44)

for (m,n) ∈ H.


satisfy the conditions (5.5.40) and (5.5.41) and the condition (5.5.42) holds. Then equation

(5.5.35) has at most one solution on H.

We next consider the equation (5.5.35) together with the following Fredholm type sum-

difference equation

v(m,n) = F(m,n)+a

∑s=0

b

∑t=0

G(m,n,s,t,v(s, t),Δ1v(s,t),Δ2v(s,t)), (5.5.45)

for (m,n) ∈ H, where F, ΔiF ∈ D(H,R); G, ΔiG ∈ D(H2 ×R3,R) for i = 1, 2.


satisfy the conditions (5.5.40) and (5.5.41) and the condition (5.5.42) holds. Then for

every given solution v ∈ D(H,R) of (5.5.45) and any solution u ∈ D(H,R) of (5.5.35), the

estimate

|u(m,n)− v(m,n)|0 � [| f (m,n)−F(m,n)|0 +M(m,n)]

+c(m,n)

{1

1−d1

a

∑s=0

b

∑t=0

[h(s,t)+h1(s,t)+h2(s,t)]

× [| f (s,t)−F(s,t)|0 +M(s,t)]

}

, (5.5.46)


holds for (m,n) ∈ H, where

M(m,n) =a

∑σ=0

b

∑τ=0

|g(m,n,σ ,τ,v(σ ,τ),Δ1v(σ ,τ),Δ2v(σ ,τ))

−G(m,n,σ ,τ,v(σ ,τ),Δ1v(σ ,τ),Δ2v(σ ,τ))|0, (5.5.47)

for (m,n) ∈ H.

We next consider the following Fredholm type sum-difference equations

z(m,n) = f (m,n)+a

∑s=0

b

∑t=0

g(m,n,s, t,z(s,t),Δ1z(s,t),Δ2z(s,t),μ), (5.5.48)

z(m,n) = f (m,n)+a

∑s=0

b

∑t=0

g(m,n,s, t,z(s,t),Δ1z(s,t),Δ2z(s,t),μ0), (5.5.49)

for (m,n) ∈ H, where f , Δi f ∈ D(H,R); g, Δig ∈ D(H2×R3×R,R) for i = 1, 2 and μ , μ0

are parameters.

Theorem 5.5.7. Suppose that the function g in (5.5.48), (5.5.49) and Δig for i = 1, 2

satisfy the conditions

|g(m,n,s,t,u,v,w,μ)−g(m,n,s,t,u,v,w,μ)|

� c(m,n)h(s,t) [|u−u|+ |v− v|+ |w−w|] , (5.5.50)

|g(m,n,s,t,u,v,w,μ)−g(m,n,s,t,u,v,w,μ0)| � γ(m,n,s,t)|μ −μ0|, (5.5.51)

and

|Δig(m,n,s, t,u,v,w,μ)−Δig(m,n,s,t,u,v,w,μ)|

� c(m,n)hi(s,t) [|u−u|+ |v− v|+ |w−w|] , (5.5.52)

|Δig(m,n,s, t,u,v,w,μ)−Δig(m,n,s,t,u,v,w,μ0)| � γi(m,n,s,t)|μ −μ0|, (5.5.53)

where c, h, hi ∈ D(H,R+); γ ,γi ∈ D(H2,R+). Let

P(m,n) = |μ −μ0|a

∑σ=0

b

∑τ=0

[γ(m,n,σ ,τ)+ γ1(m,n,σ ,τ)+ γ2(m,n,σ ,τ)] , (5.5.54)

and suppose that

d2 =a

∑s=0

b

∑t=0

[h(s,t)+h1(s,t)+h2(s,t)

]c(s,t) < 1. (5.5.55)

Let z1(m,n) and z2(m,n) be the solutions of (5.5.48) and (5.5.49) respectively on H. Then

|z1(m,n)− z2(m,n)|0 � P(m,n)+c(m,n)

×{

11−d2

a

∑s=0

b

∑t=0

[h(s,t)+h1(s,t)+h2(s, t)]P(s,t)

}

, (5.5.56)

for (m,n) ∈ H.


Proofs of Theorems 5.5.4–5.5.7. Here we present the proof of Theorem 5.5.7 only; the

proofs of Theorems 5.5.4–5.5.6 can be completed by following the proof of Theorem 5.5.7

and closely looking at the proofs of the results given in Chapter 1, section 1.6.

Let w(m,n) = z1(m,n)− z2(m,n). Using the facts that z1(m,n) and z2(m,n) are the solu-

tions of (5.5.48) and (5.5.49) on H and the hypotheses, we have

|w(m,n)|0 �a

∑s=0

b

∑t=0

|g(m,n,s,t,z1(s,t),Δ1z1(s,t),Δ2z1(s,t),μ)

−g(m,n,s,t,z2(s,t),Δ1z2(s,t),Δ2z2(s, t),μ)|

+a

∑s=0

b

∑t=0

|g(m,n,s,t,z2(s, t),Δ1z2(s,t),Δ2z2(s,t),μ)

−g(m,n,s,t,z2(s,t),Δ1z2(s,t),Δ2z2(s,t),μ0)|

+a

∑s=0

b

∑t=0

|Δ1g(m,n,s, t,z1(s,t),Δ1z1(s,t),Δ2z1(s,t),μ)

−Δ1g(m,n,s,t,z2(s,t),Δ1z2(s,t),Δ2z2(s,t),μ)|

+a

∑s=0

b

∑t=0


−Δ1g(m,n,s,t,z2(s, t),Δ1z2(s,t),Δ2z2(s,t),μ0)|

+a

∑s=0

b

∑t=0


−Δ2g(m,n,s,t,z2(s,t),Δ1z2(s,t),Δ2z2(s,t),μ)|

+a

∑s=0

b

∑t=0


−Δ2g(m,n,s,t,z2(s, t),Δ1z2(s,t),Δ2z2(s,t),μ0)|

�a

∑s=0

b

∑t=0

c(m,n)h(s,t)|w(s,t)|0 +a

∑s=0

b

∑t=0

γ(m,n,s,t)|μ −μ0|

+a

∑s=0

b

∑t=0

c(m,n)h1(s, t)|w(s,t)|0 +a

∑s=0

b

∑t=0

γ1(m,n,s,t)|μ −μ0|

+a

∑s=0

b

∑t=0

c(m,n)h2(s, t)|w(s,t)|0 +a

∑s=0

b

∑t=0

γ2(m,n,s,t)|μ −μ0|

= P(m,n)+ c(m,n)a

∑s=0

b

∑t=0

[h(s,t)+h1(s,t)+h2(s,t)

]|w(s, t)|0. (5.5.57)




5.6 Volterra-Fredholm-type sum-difference equations

The numerical methods are often used very effectively in studying the behavior of solutions

of equations of the forms (3.3.1) which often leads to the study of Volterra-Fredholm-type

sum-difference equation of the form

u(n,x) = h(n,x)+n−1

∑s=0

∑G

F(n,x,s,y,u(s,y)), (5.6.1)

where h, F are known functions and u is the unknown function. In this section, we

adopt the notations as given earlier in section 5.4 without further mention and assume

that h ∈ D(E,R), F ∈ D(E2 ×R,R

). Here we note that by modifying the idea employed

in Theorem 3.4.1, Chapter 3, one can formulate existence result for the solution of equa-

tion (5.6.1), see also [51,54]. The main goal here is to study some fundamental qualitative

properties of solutions of equation (5.6.1) under some suitable conditions on the functions

involved therein.

We call the function u ∈ D(E,R) an ε-approximate solution to equation (5.6.1), if there

exists a constant ε � 0 such that∣∣∣∣∣u(n,x)−

{

h(n,x)+n−1

∑s=0

∑G

F(n,x,s,y,u(s,y))

}∣∣∣∣∣� ε,

for (n,x) ∈ E.

First we shall give the following theorem which deals with the relation between an ε-

approximate solution and a solution of equation (5.6.1).


(i) the function F in (5.6.1) satisfies the condition

|F(n,x,s,y,u)−F(n,x,s,y,v)| � b(n,x)p(s,y)|u− v|, (5.6.2)

where b, p ∈ D(E,R+);

(ii) the functions uε (n,x), u(n,x) ∈ D(E,R) are respectively, an ε-approximate solution

and any solution of equation (5.6.1).

Then

|uε(n,x)−u(n,x)| � ε

[

1+b(n,x)n−1

∑s=0

∑G

p(s,y)n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]]

, (5.6.3)

for (n,x) ∈ E.


Proof. Let e(n,x) = |uε(n,x)− u(n,x)| for (n,x) ∈ E. From the hypotheses, we observe

that

e(n,x) =

∣∣∣∣∣uε (n,x)−h(n,x)−

n−1

∑s=0

∑G

F(n,x,s,y,uε (s,y))

+n−1

∑s=0

∑G{F(n,x,s,y,uε(s,y))−F(n,x,s,y,u(s,y))}

∣∣∣∣∣

�∣∣∣∣∣uε (n,x)−

{

h(n,x)+n−1

∑s=0

∑G

F(n,x,s,y,uε (s,y))

}∣∣∣∣∣

+n−1

∑s=0

∑G|F(n,x,s,y,uε (s,y))−F(n,x,s,y,u(s,y))|

� ε +b(n,x)n−1

∑s=0

∑G

p(s,y)e(s,y). (5.6.4)

Now an application of the inequality in Theorem 5.4.5 part (s10) to (5.6.4) yields (5.6.3).

The next theorem deals with the estimate on the difference between the two approximate



(i) the function F in (5.6.1) satisfies the condition (5.6.2);

(ii) the functions ui(n,x)∈D(E,R) for i = 1, 2 are the εi-approximate solutions of (5.6.1).

Then

|u1(n,x)−u2(n,x)| � (ε1 + ε2)

[

1+b(n,x)n−1

∑s=0

∑G

p(s,y)

×n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]]

, (5.6.5)

for (n,x) ∈ E.

The proof follows by the similar arguments as in the proof of Theorem 3.4.4, Chapter 3

and using the inequality in Theorem 5.4.5 part (s10). We omit the details.

Remark 5.6.1. If we take ε1 = ε2 = 0 in Theorem 5.6.2, then the uniqueness of solutions

of equation (5.6.1) follows.

The following theorem deals with the estimate on the solution of equation (5.6.1) by as-

suming that the function F satisfies the Lipschitz type condition (5.6.2).



(5.6.2). If u(n,x) is any solution of equation (5.6.1) on E , then

|u(n,x)−h(n,x)| � a(n,x)+b(n,x)n−1

∑s=0

∑G

p(s,y)a(s,y)

×n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]

(5.6.6)


a(n,x) =n−1

∑s=0

∑G|F(n,x,s,y,h(s,y))| ,

for (n,x) ∈ E.

Proof. Using the fact that u(n,x) is any solution of equation (5.6.1) and the condition

(5.6.2), we have

|u(n,x)−h(n,x)| �n−1

∑s=0

∑G|F(n,x,s,y,u(s,y))−F(n,x,s,y,h(s,y))|

+n−1

∑s=0

∑G|F(n,x,s,y,h(s,y))|

� a(n,x)+b(n,x)n−1

∑s=0

∑G

p(s,y)|u(s,y)−h(s,y)|, (5.6.7)

for (n,x) ∈ E . Now an application of the inequality in Theorem 5.4.5 part (s10) to (5.6.7)

yields (5.6.6).

Consider the equation (5.6.1) and the following Volterra-Fredholm-type sum-difference

equation

v(n,x) = h(n,x)+n−1

∑s=0

∑G

F(n,x,s,y,v(s,y)), (5.6.8)

for (n,x) ∈ E where h ∈ D(E,R), F ∈ D(E2 ×R,R).

The following theorem deals with the continuous dependence of solution of equation

(5.6.1) on the functions involved therein.

Theorem 5.6.4. Suppose that the function F in (5.6.1) satisfies the condition (5.6.2).

Furthermore, suppose that

|h(n,x)−h(n,x)|+n−1

∑s=0

∑G|F(n,x,s,y,v(s,y))−F(n,x,s,y,v(s,y))| � ε, (5.6.9)

where h, F and h, F are as in equations (5.6.1) and (5.6.8) respectively, v(n,x) ∈ D(E,R)

is a given solution of equation (5.6.8) and ε > 0 is an arbitrary small constant. Then

the solution u(n,x) ∈ D(E,R) of equation (5.6.1) depends continuously on the functions

involved in equation (5.6.1).


Proof. Let z(n,x) = |u(n,x)−v(n,x)| for (n,x)∈E. Using the facts that u(n,x) and v(n,x)

are the solutions of equations (5.6.1) and (5.6.8) respectively and the hypotheses, we have

z(n,x) � |h(n,x)−h(n,x)|

+n−1

∑s=0

∑G|F(n,x,s,y,u(s,y))−F(n,x,s,y,v(s,y))|

+n−1

∑s=0

∑G|F(n,x,s,y,v(s,y))−F(n,x,s,y,v(s,y))|

� ε +b(n,x)n−1

∑s=0

∑G

p(s,y)z(s,y). (5.6.10)

Now an application of the inequality in Theorem 5.4.5 part (s10) to (5.6.10) yields

|u(n,x)− v(n,x)| � ε

[

1+b(n,x)n−1

∑s=0

∑G

p(s,y)

×n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)]]

, (5.6.11)

for (n,x) ∈ E. From (5.6.11), it follows that the solution of equation (5.6.1) depends con-

tinuously on the functions involved therein.

We next consider the following Volterra-Fredholm-type sum-difference equations

z(n,x) = g(n,x)+n−1

∑s=0

∑G

F(n,x,s,y,u(s,y),μ), (5.6.12)

z(n,x) = g(n,x)+n−1

∑s=0

∑G

F(n,x,s,y,u(s,y),μ0), (5.6.13)

for (n,x) ∈ E, where g ∈ D(E,R), F ∈ D(E2 ×R2,R) and μ , μ0 are parameters.

The following theorem shows the dependency of solutions of equations (5.6.12), (5.6.13)

on parameters.

Theorem 5.6.5. Suppose that the function F in equations (5.6.12), (5.6.13) satisfy the

conditions

|F(n,x,s,y,u,μ)−F(n,x,s,y,v,μ)| � b(n,x)p(s,y)|u− v|, (5.6.14)

|F(n,x,s,y,u,μ)−F(n,x,s,y,u,μ0)| � c(s,y)|μ −μ0|, (5.6.15)

where b, p, c ∈ D(E,R+). Let z1(n,x) and z2(n,x) be the solutions of equations (5.6.12)

and (5.6.13) respectively on E. Assume thatn−1

∑s=0

∑G

c(s,y) � M, (5.6.16)


where M � 0 is a constant. Then

|z1(n,x)− z2(n,x)| � M|μ −μ0|[

1+b(n,x)n−1

∑s=0

∑G

p(s,y)

×n−1

∏σ=s+1

[

1+∑G

p(σ ,y)b(σ ,y)

]]

, (5.6.17)

for (n,x) ∈ E.

Proof. Let z(n,x) = |z1(n,x)− z2(n,x)| for (n,x) ∈ E. Using the facts that z1(n,x) and

z2(n,x) are the solutions of equations (5.6.12) and (5.6.13) and hypotheses, we have

z(n,x) �n−1

∑s=0

∑G|F(n,x,s,y,z1(s,y),μ)−F(n,x,s,y,z2(s,y),μ)|

+n−1

∑s=0

∑G|F(n,x,s,y,z2(s,y),μ)−F(n,x,s,y,z2(s,y),μ0)|

� b(n,m)n−1

∑s=0

∑G

p(s,y)|z1(s,y)− z2(s,y)|+n−1

∑s=0

∑G

c(s,y)|μ −μ0|

� M|μ −μ0|+b(n,m)n−1

∑s=0

∑G

p(s,y)z(s,y). (5.6.18)

Now an application of the inequality in Theorem 5.4.5 part (s10) to (5.6.18) yields (5.6.17),

which shows the dependency of solutions of equations (5.6.12) and (5.6.13) on parameters.

Next we study some basic qualitative ascepts of solutions of general Volterra-Fredholm-

type sum-difference equation

w(n,x) = f (n,x)+n−1

∑s=0

∑G

F(n,x,s,y,w(s,y))+∞

∑s=0

∑G

H(n,x,s,y,w(s,y)), (5.6.19)

for (n,x) ∈ E, where f , F, H are known functions and u is the unknown function. We

assume that f ∈ D(E,R) and F, H ∈ D(E2 ×R,R).

The following theorems hold.

Theorem 5.6.6. Suppose that the functions F, H in equation (5.6.19) satisfy the condi-

tions

|F(n,x,s,y,u)−F(n,x,s,y,v)| � b(n,x)p(s,y)|u− v|, (5.6.20)

|H(n,x,s,y,u)−H(n,x,s,y,v)| � c(n,x)q(s,y)|u− v|, (5.6.21)

where b, p, c, q ∈ D(E,R+). Let d, D, A(n,x), B(n,x) be defined as in Theorem 5.4.5 part

(s9). Then the equation (5.6.19) has at most one solution on E.


Theorem 5.6.7. Suppose that the functions F, H in equation (5.6.19) satisfy the condi-

tions (5.6.20), (5.6.21). Let d, B(n,x) be defined as in Theorem 5.4.5 part (s9) and

r(n,x) =n−1

∑s=0

∑G|F(n,x,s,y, f (s,y))|+

∞

∑s=0

∑G|H(n,x,s,y, f (s,y))|, (5.6.22)

D2 =1

1−d

∞

∑s=0

∑G

q(s,y)A2(s,y), (5.6.23)

where A2(n,x) is defined by the right hand side of (5.4.22) by replacing a(n,x) by r(n,x).

If w(n,x) is any solution of equation (5.6.19), then

|w(n,x)− f (n,x)| � A2(n,x)+D2B(n,x), (5.6.24)

for (n,x) ∈ E.

Theorem 5.6.8. Suppose that the functions F,H in equations (5.6.1), (5.6.19) satisfy the

conditions (5.6.20), (5.6.21) and H(n,x,s,y,0) = 0. Let u ∈ D(E,R) be a solution of equa-

tion (5.6.1) such that |u(n,x)| � Q for (n,x) ∈ E, where Q � 0 is a constant. Let d, B(n,x)

be defined as in Theorem 5.4.5 part (s9) and

a(n,x) = | f (n,x)−h(n,x)|+Qc(n,x)∞

∑s=0

∑G

q(s,y), (5.6.25)

D3 =1

1−d

∞

∑s=0

∑G

q(s,y)A3(s,y), (5.6.26)

where A3(n,x) is defined by the right hand side of (5.4.22) by replacing a(n,x) by a(n,x).

If w ∈ D(E,R) is any solution of equation (5.6.19), then

|w(n,x)−u(n,x)| � A3(n,x)+D3B(n,x), (5.6.27)

for (n,x) ∈ E.

Proofs of Theorems 5.6.6–5.6.8. Below, we give the proof of Theorem 5.6.8 only; the

proofs of Theorems 5.6.6 and 5.6.7 can be completed by following the ideas used to prove

the Theorems 1.4.2 and 1.4.4 in Chapter 1.

Using the facts that w(n,x) and u(n,x) are the solutions of equations (5.6.19) and (5.6.1)

and the hypotheses, we observe that

|w(n,x)−u(n,x)| � | f (n,x)−h(n,x)|

+n−1

∑s=0

∑G|F(n,x,s,y,w(s,y))−F(n,x,s,y,u(s,y))|

+∞

∑s=0

∑G|H(n,x,s,y,w(s,y))−H(n,x,s,y,u(s,y))|


+∞

∑s=0

∑G|H(n,x,s,y,u(s,y))−H(n,x,s,y,0)|

� | f (n,x)−h(n,x)|+b(n,x)n−1

∑s=0

∑G

p(s,y)|w(s,y)−u(s,y)|

+c(n,x)∞

∑s=0

∑G

q(s,y)|w(s,y)−u(s,y)|+ c(n,x)∞

∑s=0

∑G

q(s,y)|u(s,y)|

� a(n,x)+b(n,x)n−1

∑s=0

∑G

p(s,y)|w(s,y)−u(s,y)|

+c(n,x)∞

∑s=0

∑G

q(s,y)|w(s,y)−u(s,y)|. (5.6.28)

Now an application of the inequality in Theorem 5.4.5 part (s9) to (5.6.28) yields (5.6.27).

Remark 5.6.2. We note that, one can use the inequality in Theorem 5.4.4 part (s7) to for-

mulate results similar to those given in Theorems 5.6.1–5.6.5 for the solutions of Volterra-

Fredholm-type sum-difference equation of the form

u(n,x) = h(n,x)+n−1

∑s=0

∑G

F(n,x,s,y,u(s,y),Tu(s,y)), (5.6.29)


Tu(n,x) :=n−1

∑τ=0

∑G

L(n,x,τ,z,u(τ,z)), (5.6.30)

under some suitable conditions on the functions involved in (5.6.29).

5.7 Miscellanea


Let u, r ∈ D(N20,R+) and c � 0 is a real constant.

(p1) If

u2(n,m) � c+n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

r(σ ,τ)u(σ ,τ),


u(n,m) �√

c+12

n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

r(σ ,τ),

for (n,m) ∈ N20.


(p2) Suppose that u � 1 and c � 1. If

u(n,m) � c+n−1

∑s=0

s−1

∑σ=0

m−1

∑τ=0

r(σ ,τ)u(σ ,τ) logu(σ ,τ),


u(n,m) � cB(n,m),

for (n,m) ∈ N20, where

B(n,m) =n−1

∏s=0

[

1+s−1

∑σ=0

m−1

∑τ=0

r(σ ,τ)

]

.


Let u, p, q, f , g ∈D(H,R+), where H = N20×Nα,β ; k � 0 is a real constant and L ∈D(H×

R+,R+) be such that

0 � L(x,y,z,u)−L(x,y,z,v) � M(x,y,z,v)(u− v),

for u � v � 0, where M ∈ D(H ×R+,R+).

(p3) If

u(x,y,z) � p(x,y,z)+q(x,y,z)x−1

∑s=0

y−1

∑t=0

β

∑r=α

L(s,t,r,u(s,t,r)),

for (x,y,z) ∈ H, then

u(x,y,z) � p(x,y,z)+q(x,y,z)

(x−1

∑s=0

y−1

∑t=0

β

∑r=α

L(s,t,r, p(s,t,r))

)

×x−1

∏s=0

[

1+y−1

∑t=0

β

∑r=α

M(s,t,r, p(s,t,r))q(s,t,r)

]

,

for (x,y,z) ∈ H.

(p4) If

u2(x,y,z) � k2 +2x−1

∑s=0

y−1

∑t=0

β

∑r=α

[g(s,t,r)u(s, t,r)+ f (s,t,r)u(s,t,r)L(s,t,r,u(s,t,r))] ,

for (x,y,z) ∈ H, then

u(x,y,z) � n0(x,y)+

(x−1

∑s=0

y−1

∑t=0

β

∑r=α

f (s,t,r)L(s,t,r,n0(s,t))

)

×x−1

∏s=0

[

1+y−1

∑t=0

β

∑r=α

f (s, t,r)M(s,t,r,n0(s,t))

]

,

for (x,y,z) ∈ H, where

n0(x,y) = k +x−1

∑σ=0

y−1

∑τ=0

β

∑ω=α

g(σ ,τ ,ω),

for x, y ∈ N0.



Under the notations as in section 5.4, let u, p, q, f ∈ D(E,R+) and c � 0 is a real constant.

(p5) Let L ∈ D(E ×R+,R+) be such that

0 � L(n,x,u)−L(n,x,v) � M(n,x,v)(u− v),

for u � v � 0, where M ∈ D(E ×R+,R+). If


∑s=0

s−1

∑τ=0

∑G

L(τ,y,u(τ,y)),


u(n,x) � p(n,x)+q(n,x)

(n−1

∑s=0

s−1

∑τ=0

∑G

L(τ,y, p(τ,y))

)

×n−1

∏s=0

[

1+s−1

∑τ=0

∑G

M(τ,y, p(τ,y))q(τ,y)

]

,

for (n,x) ∈ E.

(p6) Let g be as in Theorem 5.4.1 part (s2). If

u(n,x) � c+n−1

∑s=0

s−1

∑τ=0

∑G

f (τ,y)g(u(τ,y)),

for (n,x) ∈ E, then for 0 � n � n1; n, n1 ∈ N0, x ∈ G

u(n,x) � W−1

[

W (c)+n−1

∑s=0

s−1

∑τ=0

∑G

f (τ,y)

]

,

where W, W−1 are as in Theorem 5.4.1 part (s2) and n1 ∈ N0 be chosen so that

W (c)+n−1

∑s=0

s−1

∑τ=0

∑G

f (τ,y) ∈ Dom(W−1) ,



Under the notations as in section 5.4, let u, p, q, f , g ∈ D(E,R+) and k � 0 is a real con-

stant.

(p7) Let L ∈ D(E ×R+,R+) be such that

0 � L(n,x,u)−L(n,x,v) � M(n,x,v)(u− v),

for u � v � 0, where M ∈ D(E ×R+,R+). If


∑s=0

∑G

L(s,y,u(s,y)),




∑s=0

∑G

L(s,y, p(s,y))n−1

∏σ=s+1

[

1+∑G

M(σ ,y, p(σ ,y))q(σ ,y)

]

,

for (n,x) ∈ E.

(p8) If

u2(n,x) � k2 +2n−1

∑s=0

∑G

[f (s,y)u2(s,y)+g(s,y)u(s,y)

],


u(n,x) � kn−1

∏s=0

[

1+∑G

f (s,y)

]

+n−1

∑s=0

∑G

g(s,y)n−1

∏σ=s+1

[

1+∑G

f (σ ,y)

]

,

for (n,x) ∈ E.


Consider the initial value problem

Δ2Δ1u(m,n) = f (m,n,u(m,n),Δ1u(m,n)), (5.7.1)

with

u(m,0) = σ(m),u(0,n) = τ(n),u(0,0) = 0, (5.7.2)

for m, n ∈ N0, where f , σ , τ are given functions and u is the unknown function and f ∈D

(N

20 ×R

2,R), σ , τ ∈ D(N0,R).

(p9) Assume that

| f (m,n,u,v)− f (m,n,u,v)| � p(m,n) [|u−u|+ |v− v|] ,

where p ∈ D(N20,R+). Then the problem (5.7.1)–(5.7.2) has at most one solution on N

20.

(p10) Assume that

| f (m,n,u,v)| � r(m,n) [|u|+ |v|] ,

|σ(m)|+ |τ(n)|+ |Δσ(m)| � k,

where r ∈ D(N20,R+) and k � 0 is a constant. If u(m,n) is any solution of problem (5.7.1)–

(5.7.2), then

|u(m,n)|+ |Δ1u(m,n)| � kq(m,n)m−1

∏s=0

[

1+n−1

∑t=0

r(s,t)q(s, t)

]

,


q(m,n) =m−1

∏t1=0

[1+ r(m,t1)],

for m, n ∈ N0.



Consider the initial value problem

Δ2Δ1u(m,n) = F(m,n,u(m,n),Δ2Δ1u(m,n)), (5.7.3)

with

u(m,0) = α(m), u(0,n) = β (n), u(0,0) = 0, (5.7.4)

for m, n ∈ N0, where f , α, β are given functions, u is the unknown function and F ∈D(N2

0 ×R2,R), α , β ∈ D(N0,R).

(p11) Let ui(m,n)∈D(N20,R) (i = 1, 2) be respectively εi-approximate solutions of equation

(5.7.3) i.e.

|Δ2Δ1ui(m,n)−F(m,n,ui(m,n),Δ2Δ1ui(m,n))| � εi

for given constants εi � 0, with

ui(m,0) = αi(m), ui(0,n) = βi(n), ui(0,0) = 0,

where αi, βi ∈ D(N0,R) and such that

|α1(m)−α2(m)+β1(n)−β2(n)| � δ ,

where δ � 0 is a constant. Suppose that the function F in (5.7.3) satisfies the condition

|F(m,n,u,v)−F(m,n,u,v)| � p(m,n) [|u−u|+ |v− v|] , (5.7.5)

where p ∈ D(N20,R+) satisfies p(m,n) < 1 for m, n ∈ N0. Then

|u1(m,n)−u2(m,n)|+ |Δ2Δ1u1(m,n)−Δ2Δ1u2(m,n)|

� L(m,n)+E(m,n)

(m−1

∑s=0

n−1

∑t=0

p(s,t)L(s,t)

)n−1

∏s=0

[

1+n−1

∑t=0

p(s,t)E(s,t)

]

,


L(m,n) =(ε1 + ε2)(mn+1)+δ

1− p(m,n), E(m,n) =

11− p(m,n)

. (5.7.6)

Consider the initial value problem (5.7.3)–(5.7.4) together with the following initial value

problem

Δ2Δ1v(m,n) = G(m,n,v(m,n),Δ2Δ1v(m,n)), (5.7.7)

with

v(m,0) = α(m), v(0,n) = β (n), v(0,0) = 0, (5.7.8)


for m, n ∈ N0, where G ∈ D(N20 ×R

2,R), α, β ∈ D(N0,R).

(p12) Suppose that the function F in (5.7.3) satisfies the condition (5.7.5) and there exist

constants ε � 0, δ � 0 such that

|F(m,n,u,v)−G(m,n,u,v)| � ε,

|α(m)−α(m)+β (n)−β (n)| � δ ,

where F, α, β and G, α, β are as in (5.7.3)–(5.7.4) and (5.7.7)–(5.7.8). Let u(m,n) and

v(m,n) be respectively the solutions of (5.7.3)–(5.7.4) and (5.7.7)–(5.7.8) for m, n ∈ N0.

Then

|u(m,n)− v(m,n)|+ |Δ2Δ1u(m,n)−Δ2Δ1v(m,n)|

� L(m,n)+E(m,n)

(m−1

∑s=0

n−1

∑t=0

p(s,t)L(s,t)

)n−1

∏s=0

[

1+n−1

∑t=0

p(s,t)E(s,t)

]

,


L(m,n) =ε(mn+1)+δ

1− p(m,n),

and E(m,n) is as in (5.7.6).


Consider the sum-difference equation

v(m,x) = f (m,x)+m−1

∑s=0

s−1

∑τ=0

β

∑y=α

K(m,x,s,τ,y,v(τ,y)), (5.7.9)

for (m,x) ∈ H = Na,b ×Nα ,β , where f , K are given functions, v is the unknown function

and f ∈ D(H,R), K ∈ D(H ×Na,b ×H ×R,R).

(p13) Assume that the function K in (5.7.9) satisfies the condition

|K(m,n,s,τ,y,v)−K(m,n,s,τ,y,w)| � q(m,x)g(τ,y)|v−w|,

where q, g ∈ D(H,R+). Then the equation (5.7.9) has at most one solution on H.

(p14) Assume that the function K in (5.7.9) satisfies the condition

|K(m,n,s,τ,y,v)| � q(m,x)g(τ,y)|v|,

where q, g ∈ D(H,R+). If v(m,x) is any solution of equation (5.7.9) on H, then

|v(m,n)|� | f (m,x)|+q(n,x)

[m−1

∑s=0

s−1

∑τ=0

β

∑y=α

g(τ,y)| f (τ,y)|]

m−1

∏s=0

[

1+s−1

∑τ=0

β

∑y=α

g(τ,y)q(τ,y)

]

,

for (m,x) ∈ H.



Under the notations as in section 5.4, consider the sum-difference equation of the form

u(n,x) = h(n,x)+n−1

∑s=0

∑G

H(n,x,s,y,u(s,y)), (5.7.10)

for (n,x) ∈ E, where h, H are given functions, u is the unknown function and h ∈ D(E,R),

H ∈ D(E2 ×R,R).

(p15) Suppose that the function H in equation (5.7.10) satisfies the condition

|H(n,x,s,y,u)| � b(n,x)L(s,y, |u|),

where b ∈ D(E,R+) and L is as in part (p7). Then for every solution u ∈ D(E,R) of

equation (5.7.10), the estimate

|u(n,x)|� |h(n,x)|+b(n,x)n−1

∑s=0

∑G

L(s,y, |h(s,y)|)n−1

∏σ=s+1

[

1+∑G

M(σ ,y, |h(σ ,y)|)b(σ ,y)

]

,

holds for (n,x) ∈ E, where M is as in (p7).

Consider the equation (5.7.10) together with the following sum-difference equation

w(n,x) = h(n,x)+n−1

∑s=0

∑G

K(n,x,s,y,w(s,y)), (5.7.11)

for (n,x) ∈ E, where h ∈ D(E,R), K ∈ D(E2 ×R,R).

(p16) Suppose that the function H in (5.7.10) satisfies the condition

|H(n,x,s,y,u)−H(n,x,s,y,w)| � b(n,x)L(s,y, |u−w|),

where b ∈ D(E,R+) and L is as in part (p7). Then for every given solution w ∈ D(E,R) of

equation (5.7.11) and u ∈ D(E,R) any solution of equation (5.7.10), the estimate

|u(n,x)−w(n,x)| � [h0(n,x)+ r(n,x)]

+b(n,x)n−1

∑s=0

∑G

L(s,y, [h0(s,y)+ r(s,y)])n−1

∏σ=s+1

[1+M(σ ,y, [h0(σ ,y)+ r(σ ,y)])] ,

holds for (n,x) ∈ E, where

h0(n,x) = |h(n,x)−h(n,x)|,

r(n,x) =n−1

∑s=0

∑G|H(n,x,s,y,w(s,y))−K[h0(s,y)+ r(s,y)]|,

and M is as in part (p7).


5.8 Notes

Although some books and papers contains some basic results on partial finite difference

equations (see [3,17,46,47,51,67,85,87]), it seems, much of the qualitative and quantita-

tive theory of these equations remain to be developed in various directions. The results

in sections 3.2–3.4 deals with a large number of fundamental finite difference inequali-

ties with explicit estimates involving functions of two, three and many variables recently

investigated, as a response to the needs of diverse applications and are adapted from Pach-

patte [98,104,108,95,102,111,116,114,106]. Section 5.5 contains results related to error

evaluation of approximate solutions of a certain sum-difference equation in two variables

and also some basic results on Fredholm-type sum-difference equation in two variables re-

cently obtained in [100,113]. Section 5.6 is devoted to present the basic theory of mixed

sum-difference equations typically arise while studying some initial boundary value prob-

lems for partial differential equations of parabolic type by using discretization methods

and are taken from Pachpatte [116,101,99,107,109]. Section 5.7 addresses some selected

results which, we hope, will provide a clue to effective methods for future important devel-

opments.


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Subject Index

Aanalytic tools, 59approximate solutions, 17, 18, 27, 28, 52, 75,

76, 83, 92, 119, 122–124, 132–134, 150,151, 158, 160, 212, 213, 222, 223, 232, 235

approximation of solutions, 211

BBanach fixed point theorem, 6, 9, 15, 34, 39,

44, 86, 117space of bounded functions, 45spaces, 14, 16, 32, 37, 48, 50, 73, 79, 84,

116, 135, 136, 147, 169, 170Barbashin-type, 4

equation, 4, 147, 155, 189integrodifferential equation, 147, 153, 155,

189basic integral inequalities, 9, 57, 59, 95, 97,

143, 189behavior of solutions, 5, 59, 161, 189, 222Bielecki-type norm, 6, 15boundary and initial conditions, 4, 35, 36, 51,

52, 55, 140, 143, 176, 180, 182, 183, 187

Ccharacteristic data, 1, 2classical mechanics, 1closeness of solutions, 29, 124, 152, 159, 214compactness of the operator, 5continuous dependence of solutions, 24, 35, 44,

76, 83, 122, 131, 160, 177, 224continuously differentiable, 37, 45, 162contraction mapping, 15, 16, 21, 45, 74, 86,

148convergence properties, 27, 212coupled parabolic integrodifferential equations,

182

DDarboux problem, 55dependency of solutions, 19, 20, 26, 36, 76, 88,

89, 125, 126, 152–154, 212, 214, 216–218,221, 225, 226

difference equation, 5, 191, 217–220, 222,224–226, 228, 233–235

differential equations, 1, 15, 87, 97difference between two approximate

solutions, 17, 27, 119, 123, 132,150, 158, 212, 223

diffusion, 97, 141, 143, 161Dirichlet boundary data, 3

type boundary condition, 186dynamic equations, 1, 95, 97, 191dynamical systems, 59

Eelementary inequalities, 18, 28, 119, 124, 151,

213elliptic second order partial differential

operator, 186empty sums and products, 7epidemic models, 143epidemiology, 97error evaluation, 27, 212, 235estimate on the solution, 16, 22, 23, 34, 40, 41,

74, 80, 117, 127, 129, 148, 155, 223Euclidean space, 6existence of a solution, 1, 122, 162, 191, 218

and uniqueness of solutions, 14, 15, 21, 22,27, 83, 87, 116, 147, 155

of a unique solution, 14, 32, 44, 73, 79, 84,87, 129, 163, 168, 181, 183

243


explicit estimates, 5, 6, 9, 57, 59, 65, 95, 97,126, 141, 143, 189, 191, 212, 235

Ffinite difference equations, 5, 235

difference methods, 211Fredholm type, 3, 97

integral equation, 37, 42integrodifferential equation, 3, 37, 83, 87type integral equation, 179type sum-difference equations, 217, 219,

220

GGalerkin finite element method, 186Green’s function, 3

HHolder continuous, 162, 167, 176, 177, 180,

182classes, 184

Hammerstein type integral equation, 46hyperbolic equation, 1

integrodifferential equation, 31partial integrodifferential equation, 55type, 3

Iinequalities, 5, 18, 28, 54, 119, 124, 213

with explicit estimates, 6, 9, 59, 97, 141,191, 235

initial value problem, 211, 215, 231, 232boundary conditions, 1, 4, 35, 36, 51, 52, 55,

140, 143, 176, 187boundary value problem, 169, 188, 235

integral equation, 97, 128, 135, 136, 141, 189equation of Barbashin-type, 189inequalities, 97, 101, 102, 141

integrodifferential equations, 1–4, 6, 9, 20, 31,51, 52, 83, 87, 141, 143, 147, 152, 153, 155,161, 167, 169, 179, 182, 184, 186, 187, 189,211equation of Barbashin-type, 4, 147, 153,

155, 189system, 4, 180

interpolation inequalities, 174

LLipschitz condition, 55

type conditions, 23, 41, 223

lower solution, 162, 163, 168, 189

Mmathematical models, 1maximal solution, 165, 169, 182, 184

and minimal solution, 162, 163, 165, 167,169

method of continuity, 170of integral inequalities, 97of successive approximations, 5

minimal solution, 162, 163, 165, 167, 169, 182,184

mixed Volterra-Fredholm integral equationsVolterra-Fredholm type integral inequalities,

108Volterra-Fredholm-type integral equations,

3, 6, 97, 116, 117, 141Volterra-Fredholm-type integral inequalities,

6monotone method, 162, 168multidimensional, 1, 95multivariable integral and integrodifferential

equations, 1sum-difference equations, 191sum-difference inequalities, 6, 204sum-difference inequalities and equations, 6

NNeumann series, 136neutral type hyperbolic integrodifferential

equation, 31nonexpansive and monotone mappings, 5numerical methods, 222

Ooutward normal unit vector, 162, 180

Pparabolic differential equations, 97

boundary value problem, 170integrodifferential equations, 143, 161, 182,

184, 186, 189type Fredholm integral equation, 179type integrodifferential equations, 4, 6

partial differential equations, 141, 185, 235derivative, 6, 98, 180integral equation, 155integral operators, 1integrodifferential equations, 141, 189, 211

physical and biological phenomena, 1, 116, 189

Subject Index 245

pseudo-parabolic equation, 2

Qqualitative theory, 1, 5, 122, 235

behavior, 5, 59properties, 1, 5, 6, 9, 13, 21, 37, 57, 59, 83,

87, 95, 97, 122, 129, 132, 141, 147,155, 189, 191, 217, 222

Rreaction diffusion processes, 143reactor dynamics, 4, 161, 162, 189

SSchauder estimate, 169, 170, 176–178science and engineering, 143, 189self-adjoint, 186sum-difference equations, 6, 191, 204,

216–220, 233–235inequalities, 6, 191, 204inequalities in three variables, 198inequalities in two variables, 191

Ttime-independent coefficients, 186

two variables, 6, 9, 27, 44, 46, 57, 191, 235and three independent variables, 6, 95

Uuniqueness of solutions, 14, 15, 19, 21, 22, 27,

29, 39, 40, 74, 79, 83, 87, 116, 120, 124,126, 130, 147, 152, 155, 159, 214, 223

upper solution, 162, 166–168and lower solutions, 162, 163, 168, 189

VVolterra type, 3, 57, 97

-Fredholm-type integral equation, 3, 6, 21,24, 26, 91, 97, 116, 117, 120–122,128, 135, 139–141

-Fredholm-type integral inequalities, 6, 108-Fredholm-type integrodifferential

equations, 27-Fredholm-type sum-difference equations,

222, 224–226, 228type integral equation, 23

YYoung’s inequality, 170, 173, 174

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