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## 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 < 1 such that for u 2 X

hAu; ui Rkuk2

X:

Then, there exists exactly one solution w in X of the nonlinear equation

w + ANw = 0:

Existence and uniqueness is also proved using variational methods.

vii

** **

## TABLE OF CONTENTS

Dedication iii

Preface iv

Acknowledgement vi

Abstract vii

1 General Introduction

1

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

13

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 G^ateaux derivative and gradient . . . . . . . . . . . . . . . . . . 20

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

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

3.4 Extension of Vainberg’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

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, integrodifferential

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.

2

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 :

5

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;

7

(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 < p < 1 are examples of uniformly

convex spaces.

Definition 1.2.12 (Strictly convex spaces): A normed linear space X is

said to be strictly convex if for all x; y 2 X; x 6= y; kxk = kyk = 1, we

have kx + (1 )yk < 1 for all 2 (0; 1).

Theorem 1.2.2 Milman-Pettis Theorem: Every uniformly convex Banach

space X is reflexive.

For the proof of theorem (1.2.2), see, for instance, Chidume [1].

Definition 1.2.13 (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).

Definition 1.2.16 (Measure Space): If M is a algebra of subsets of

, and is a measure on M, then the tripple (

; M; ) is referred to as a

measure space.

8

Definition 1.2.17 (Measurable Functions): Let (

; M) be a measurable

space. A function f :

! R is measurable or Mmeasurable if the set

fx 2

: f(x) > g 2M for all 2 R.

Definition 1.2.18 (finite ) : A measure space (

; M; ) is said to be

finite if there exists a countable family (

n)n1 inMsuch that

= [1 n=1

n

and (

n) < 1; 8 n:

Definition 1.2.19 (Green’s Function): This is a function associated with

a given boundary value problem, which appears as an integrand for an integral

representation of the solution of the problem.

Let L be a differential operator and assume that

L(y) =

Xn

p=0

aP (t)y(p)(t) = an(t)yn(t) + ::: + a(t)y(1)(t) + a0(t)y(t):

Suppose that an(t) is not zero on [0; 1] and that each term of the sequence

ap(t); p = 0; :::; n, has at least n continuous derivatives. Also suppose that

B is the given boundary conditions associated with L and denote by M, the

manifold associated with (L;B). (Manifold simply refers to the differential

equation together with the associated boundary conditions.) We present the

algorithm for constructing the Green’s function, G(t; x) for nth order equations.

For x 2 [0; 1], we denote by x, the values of t 2 [0; x) and by x+, the

values of t 2 (x; 1] .

(a) L(G(:; x)) (t) = 0 for 0 < t < x and for x < t < 1;

(b) G(:; x) is in M;

(c) for 0 p n 2, @pG(t;x)

@tp =t=x+ = @pG(t;x)

@tp =t=x ;

(d) @n1G(t;x)

@tn1 =t=x+ @n1G(t;x)

@tn1 =t=x = 1

an(x) .

Definition 1.2.20 (Caratheodory Condition): Let m and n be positive

integers,

be a nonempty subset of Rm and let f be a function from

Rn

into R. A function f :

Rn ! R is said to satisfy the Caratheodory

conditions if

(i) f(x; 🙂 : Rn ! R is a continuous function for almost all x 2

;

(ii) f(:; u) :

! R is a measurable function for all u 2 Rn.

Definition 1.2.21 (Nemystkii Operators): Let f be a function from

Rn into R. We denote by F(X; Y ), the set of all maps from X to Y . The

Nemystkii operator associated to f is the operator Nf : F(

;Rn) ! F(

;R)

defined by

u 7! Nf (u)

where (Nfu)(x) = f (x; u(x)) 8 u 2 F (

; Rn) ; 8 x 2

: For simplicity, we

shall write Nuf (x) instead of (Nfu)(x).

9

Example 1.2.10 Given a map f : R R ! R defined by

f(x; s) = jsj 8 (x; s) 2 R R;

the Nemystkii operator associated to f is given by the expression Nfu(x) =

ju(x)j for any map u : R ! R and for any x 2 R.

Example 1.2.11 Given a map g : R R ! R defined by

g(x; s) = xes 8 (x; s) 2 R R;

the Nemystkii operator associated to g is given by the expression Nfu(x) =

xeu(x) for any map u : R ! R and for any x 2 R.

Observe that by the continuity of f and g, Nf and Ng map the set of

real-valued continuous function on

; C(

) into itself. Moreover, they map

the set of real-valued measurable function into itself.

1.3 Hammerstein Equations

A nonlinear integral equation of Hammerstein type on

is one of the form

u(x) +

Z

k(x; y)f(y; u(y))dy = h(x) (1.3.1)

where dy stands for