**Existence and Uniqueness of Solutions of Integral Equations of Hammerstein Type**

**ABSTRACT**

Let X be a real Banach space, X its conjugate dual space. Let A be a monotone angle-bounded continuous linear mapping of X into X with constant of angle-boundedness c 0. Let N be a hemicontinuous (possibly nonlinear) mapping of X into X such that for a given constant k 0;

hv1 v2; Nv1 Nv2i kkv1 v2k2

X

for all v1 and v2 in X. Suppose finally that there exists a constant R with

k(1 + c2)R TABLE OF CONTENTS

Dedication iii

Preface iv

Acknowledgement vi

Abstract vii

1 General Introduction

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Definition and examples of some basic terms . . . . . . . . . . . 2

1.3 Hammerstein Equations . . . . . . . . . . . . . . . . . . . . . . 10

2 Existence and Uniqueness Results Using Factorization of Operators

2.1 Existence and uniqueness theorem . . . . . . . . . . . . . . . . . 13

2.2 Result of Minty [5] . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Proof of theorem (2.1.1) . . . . . . . . . . . . . . . . . . . . . . 17

3 Existence and Uniqueness Results Using Variational Methods 20

3.1 Gateaux derivative and gradient . . . . . . . . . . . . . . . . . . 20

3.2 Maxima and minima of functions . . . . . . . . . . . . . . . . . 22

3.3 Fundamental theorems of optimization . . . . . . . . . . . . . . 23

3.4 Extension of Vain-berg’s result to real

Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Bibliography 30

**CHAPTER ONE**

**General Introduction**

**1.1 Introduction**

The contribution of this thesis falls within the general area of nonlinear functional analysis. Within this area, our attention is focused on the topic: “Existence and Uniqueness of Solutions of Nonlinear Hammerstein Integral Equations” in Banach spaces. We study theorems that establish existence and uniqueness of solutions of these equations using factorization of operators and variational methods.

Several classical problems in the theory of differential equations lead to integral equations. In many cases, these equations can be dealt with in a more satisfactory manner using the integral form than directly with differential

equations.

Interest in Hammerstein equations stem mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green’s function can, as a rule be transformed into a nonlinear integral equation of Hammerstein type. Elliptic boundary value problems are a class of problems which do not involve time variable but only depend on the space variables. That is, they are class of problems which are typically associated with steady state behaviour. An example is a Laplace’s equation: r2u = 0 e.g @2u

@x2 +

@2u

@y2 = 0 in 2D :

Consequently, solvability of such differential equations is equivalent to the solvability of the corresponding Hammerstein equation.

1.2 Definition and examples of some basic terms

In this section, definitions of basic terms used are given. Throughout this chapter, X denotes a real Banach space and X denotes its corresponding dual. We shall denote by the pairing hx; xi or x(x) the value of the functional x 2 X at x 2 X: The norm in X is denoted by k:k, while the norm in X is denoted by k:k. If there is no danger of confusion, we omit the asterisk and denote both norms in X and X by the symbol k:jj. We shall use the symbol ! to indicate strong and * to indicate weak convergence.

We shall also use w! to indicate the weak-star convergence.

The first term we define is monotone map. The concept of monotonicity pertains to nonlinear functional analysis, and its use in the theory of functional equations (ordinary differential equations, integral equations, integro differential equations, delay equations) is probably the most powerful method in obtaining existence theorems.

Definition 1.2.1 (Monotone Operator): A map A : D(A) X ! 2X is said to be monotone if 8 x; y 2 D(A); x 2 Ax; y 2 Ay, we have hx y; x yi 0:

From the definition above, a single-valued map A : D(A) X ! X is monotone if

hAx Ay; x yi 0; 8 x; y 2 D(A):

Remark 1.2.1 For a linear map A, the above definition reduces to hAu; ui 0 8 u 2 D(A):

The following are some examples of monotone operators.

Example 1.2.1 Every nondecreasing function on R is monotone.

Proof.

Let f : R ! R be a nondecreasing function. Then for arbitrary x; y 2 R, both

(f(x) f(y)) and (x y) have the same sign. Thus we see that

hf(x) f(y); x yi = (f(x) f(y))(x y) 0 8 x; y 2 R. Hence, f is monotone.

Example 1.2.2 Let h : R2 ! R2 be defined as h(x; y) = (2x; 5y); 8 (x; y) 2 R2. Then h is montone.

Proof.

For arbitrary (x1; y1); (x2; y2) 2 R2; we have hh(x1; y1) h(x2; y2); (x1; y1) (x2; y2)i = 2(x1 x2)2 + 5(y1 y2)2 0:

Thus, h is monotone.

Example 1.2.3 Let H be a real Hilbert space, I is the identity map of H and

T : H ! H be a non-expansive map (i:e kTx Tyk kx yk 8 x; y 2 H).

Then the operator I T is monotone.

Proof.

Let x; y 2 H; then

h(I T)x (I T)y; x yi = h(x y) (Tx Ty); x yi

= kx yk2 hTx Ty; x yi

kx yk2 kTx Tyk:kx yk

kx yk2 kx yk2 = 0 (T is nonexpansive).

Thus we have that I T is monotone on H.

Example 1.2.4 Let A = (1 0

0 0) and x = (xy

). Consider the function

g : R2 ! R2 defined by g(x) = Ax: Then g is monotone.

Proof.

Since g is linear, by remark (1.2.1) it suffices to show that hg(x); xi 0. For

arbitrary x = (xy

) 2 R2; we have Ax = (1 0

0 0)(xy

) = (x0

).

Thus hg(x); xi = hAx; xi = x2 + 0 = x2 0. Hence g is monotone.

Example 1.2.5 Let X be a real Banach space. The duality map J : X ! 2X

defined by

Jx := fx 2 X : hx; xi = kxk:kxk; kxk = kxk; x 2 Xg

is monotone.

Proof.

Let x; y 2 X and x 2 Jx; y 2 Jy. Then

hx y; x yi = hx y; xi hx y; yi

= hx; xi hy; xi hx; yi + hy; yi

= kxk2 + kyk2 hy; xi hx; yi

kxk2 + kyk2 kyk:kxk kxk:kyk

= kxk2 + kyk2 2kxk:kyk

= (kxk kyk)2 0:

Thus, J is monotone.

Example 1.2.6 Let f : X ! R[f+1g be convex and proper. The subdifferential

of f; @f : X ! 2X defined as

@f(x) =

fx 2 X : hy x; xi f(y) f(x); y 2 Xg ; if f(x) 6= 1

;; if f(x) = 1;

is monotone.

3

Proof.

Let x; y 2 X; x 2 @f(x) and y 2 @f(y).

x 2 @f(x) ) hy x; xi f(y) f(x) 8 y 2 X: (1.2.1)

y 2 @f(y) ) hx y; yi f(x) f(y) 8 x 2 X

) hy x; yi f(x) f(y) 8 x 2 X: (1.2.2)

Adding inequalities (1.2.1) and (1.2.2), we have

hy x; xi hy x; yi 0:

This implies that hy x; x yi 0, i.e hx y; x yi 0.

Definition 1.2.2 (Hemicontinuity): A mapping A : D(A) X ! X is

said to be hemicontinuous if it is continuous from each line segment of X to

the weak topology of X. That is, 8 u 2 D(A); 8 v 2 X and (tn)n1 R+

such that tn ! 0+ and u + tnv 2 D(A) for n sufficiently large, we have

A(u + tnv) * A(u).

Proposition 1.2.1 Let X denote a Banach space and X its corresponding

dual. Let A : D(A) X ! X be a continuous mapping . Then A is

hemicontinuous.

Proof

Let u 2 D(A); v 2 X, (tn)n1 be a sequence of positive numbers such that

tn ! 0+ as n ! 1 and (u + tnv) 2 D for n large enough. We observe that

(u + tnv) ! u as n ! 1 because tn ! 0+ as n ! 1. By the continuity

of A, we have A(u + tnv) ! A(u) as n ! 1. Since strong convergence

implies weak convergence we have A(u + tnv) * A(u) as n ! 1: Hence A is

hemicontinuous.

Remark 1.2.2 The converse of proposition (1.2.1) is false.

Consider the function f : R2 ! R2 defined by

f(x; y) =

(

( x2+xy2

x2+y4 ; x); if (x; y) 6= (0; 0)

(1; 0); if (x; y) = (0; 0):

Clearly, f is not continuous at (0; 0). For,

f(x; 0) = ( x2

x2 ; x) = (1; x) for all x 6= 0: This implies lim

x!0

f(x; 0) = (1; 0).

f(0; y) = (0; 0); 8y 6= 0. This implies lim

y!0

f(0; y) = (0; 0). Thus, the

limit does not exist at (0; 0). Hence, f is not continuous at (0,0).

4

However, f is hemicontinuous. Indeed, let u = (0; 0); v = (v1; v2) and

ftngn1 be arbitrary such that tn ! 0+ as n ! 1. Then,

f(u + tnv) = f(tnv1; tnv2)) =

v2

1+tnv1v2

2

v2

1+t2

nv4

2

; tnv1

! (1; 0); as n ! 1: Therefore,

lim

n!1

f(u + tnv) = (1; 0) = f(0; 0). Thus, f(u + tnv) ! f(u) as tn ! 0+.

Hence, f is hemicontinuous on R2 since strong and weak convergence are the

same on R2.

Definition 1.2.3 (Coercivity): An operator A : X ! X is said to be

coercive if for any x 2 X; hx;Axi

kxk ! 1 as kxk ! 1:

Example 1.2.7 Let H be a real Hilbert space and f : H ! H be defined by

f(x) = 1

2u. Then, f is coercive.

Proof.

Let x 2 H be arbitrary. Then,

hf(x); xi

kxk

=

1

2 hx; xi

kxk

=

1

2kxk2

kxk

=

1

2

kxk ! +1 as kxk ! 1:

Hence f is coercive.

Definition 1.2.4 (Symmetry): Let A : X ! X be a bounded linear mapping.

A is said be symmetric if for all u and v in X, we have hAu; vi = hAv; ui :

Example 1.2.8 Let A : l2(R) ! l2(R) be a map defined by Au = 1

2u. Then

A is symmetric.

Proof.

For arbitrary u; v 2 l2;

hAu; vi =

1

2

u; v

=

1

2

h(u1; u2; :::); (v1; v2; :::)i =

1

2

X1

i=1

uivi

=

1

2

X1

i=1

viui =

1

2

h(v1; v2; :::); (u1; u2; :::)i

=

1

2

v; u

= hu; Avi :

Hence A is symmetric.

Definition 1.2.5 (Skew-symmetry): Let A : X ! X be a bounded linear mapping. A is said be skew-symmetric if for all u and v in X, we have hAu; vi = hAv; ui :

Definition 1.2.6 (Angle-boundedness): Let A : X ! X be a bounded monotone linear mapping . A is said be angle-bounded with constant c 0 if for all u, v in X, j hAu; vihAv; ui j 2c fhAu; uig

1

2 fhAv; vig

1

2 . (This is well defined since hAu; ui 0 and hAv; vi 0 by the linearity and monotonicity of A).

Example 1.2.9 A symmetric map. It follows that every symmetric mapping A of X into X is angle-bounded with constant of angle-boundedness c = 0:

Definition 1.2.7 (Adjoint Operators): Let X and Y be normed linear spaces and A 2 B(X; Y ): The adjoint of A, denoted by A, is the operator A : Y ! X defined by hAy; xi = hy; Axi for all y 2 Y and all

x 2 X.

We note that A is well-defined. Indeed, 8 y 2 Y ; x1; x2 2 X and 2 R,

we have

hAy; x1 + x2i = hy;A(x1 + x2)i = hy; Ax1i + hy; Ax2i

= hy; Ax1i + hy; Ax1i

which shows that Ay is linear.

For boundedness, given y 2 Y and x 2 X;

j hAy; xi j = j hy; Axi j

kyk:kAxk since y 2 Y .

kyk:kAk:kxk since A 2 B(X; Y ).

Therefore, for all y 2 Y ,

j hAy; xi j Kykxk 8 x 2 X; where Ky = kyk:kAk 0:

Hence, for all y 2 Y ;Ay 2 X:

Theorem 1.2.1 Let A : X ! Y be a bounded linear maps with adjoint A.

Then,

(a) A 2 B(Y ;X);

(b) kAk = kAk.

Proof.

(a) Let y; z 2 Y and 2 R. We show that

A (y + z) = Ay + Az;

6

i.e

8 x 2 X; hA (y + z) ; xi = hAy; xi + hAz; xi :

Let x 2 X: Then

hA (y + z) ; xi = hy + z; Axi = hy; Axi + hz; Axi

= hAy; xi + hAz; xi :

So, A is linear.

Furthermore, for any y 2 Y and x 2 X,

j hAy; xi j = j hy; Axi j kyk:kAk:kxk; since A 2 B(X; Y ) :

Thus, kAyk = sup

kxk=1

j hAy; xi j kAk:kyk: Therefore, kAyk

Kkyk; where K = kAk 0: Hence A 2 B(Y ;X).

(b)

kAk = sup

kxk=1

kAxk = sup

kxk=1

sup

kyk=1

hy; Axi

!

= sup

kxk=1

sup

kyk=1

hAy; xi

!

= sup

kyk=1

sup

kxk=1

hAy; xi

!

= sup

kyk=1

kAyk = kAk:

Definition 1.2.8 (Weak Topology): Let (X; !) denote a Banach space endowed with the weak topology. For an arbitrary sequence fxngn1 X and x 2 X, we say that fxng converges weakly to x if f(xn) ! f(x) for each

f 2 X. We denote this by xn * x:

Definition 1.2.9 (Weak Star Topology): Let (X; !) denote a Banach space endowed with the weak star topology. For an arbitrary sequence ffngn1 X and f 2 X we say that ffng converges to f in weak-star topology, denoted fn!

! f, if fn(x) ! f(x) for each x 2 X.

Proposition 1.2.2 Let fxng be a sequence and x a point in X. Then the following hold.

(a) xn ! x ) xn * x;

(b) xn * x ) fxng is bounded and kxk lim inf kxnk;

(c) xn * x (in X), fn ! f (in X) ) fn(xn) ! f(x) (in R).

Definition 1.2.10 (Reflexive Space): Let X be a Banach space and let

J : X ! X be the canonical injection from X into X, that is hJ(x); fi =

hf; xi ; 8 x 2 X; f 2 X. Then X is said to be reflexive if J is surjective, i.e

J(X) = X:

Definition 1.2.11 (Uniformly convex Banach spaces): A Banach space X is called uniformly convex if for any 2 (0; 2], there exists a = () > 0

such that if x; y 2 X, with kxk 1; kyk 1 and kx yk , then

k1

2 (x + y)k 1 .

Hilbert spaces, Lp and lp spaces, 1 algebra): A collection M of subsets of a nonempty

set

is called a algebra if

(a) ;

2M,

(b) A2 M ! Ac 2 M,

(c) [1 n=1An 2 M whenever An 2 M 8 n.

Definition 1.2.14 (Measurable Space): If M is a algebra of, then the pair (;M) is referred to as a measurable space.

Definition 1.2.15 (Measure): A measure on (; M) is a function

: M! [0; 1] such that

(a) (A) 0 for all A 2M;

(b) () = 0;

(c) if Ai 2M are pairwise disjoint, then ([1i

Ai) =

P1

i=1 (Ai).

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