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ABSTRACT

Let E be a real normed space with dual space E and let A : E ! 2E be any map. Let J : E ! 2E be

the normalized duality map on E. A new class of mappings, J-pseudocontractive maps, is introduced

and the notion of J-fixed points is used to prove that T := (J ? A) is J-pseudocontractive if and

only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach

space with dual E, T : E ! 2E is a bounded J-pseudocontractive map with a nonempty J-fixed

point set, and J ? T : E ! 2E is maximal monotone, a sequence is constructed which converges

strongly to a J-fixed point of T. As an immediate consequence of this result, an analogue of a recent

important result of Chidume for bounded m-accretive maps is obtained in the case that A : E ! 2E is

bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and

Rockafellar. Furthermore, this analogue is applied to approximate solutions of Hammerstein integral

equations and is also applied to convex optimization problems.

viii

## TABLE OF CONTENTS

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Certification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

1 INTRODUCTION 1

1.1 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Zeros of Monotone operators on Hilbert spaces . . . . . . . . . . . . . . . . . . 1

1.1.2 Extension of Hilbert space Monotonicity to arbitrary normed spaces . . . . . . . 4

1.1.3 Application of Fixed Point Techniques . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Aim and Objectives of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 LITERATURE REVIEW 8

2.0.1 Accretive-type mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.0.2 Monotone-type mappings in arbitrary normed spaces . . . . . . . . . . . . . . . 9

3 PRELIMINARY CONCEPTS AND RESULTS 12

3.1 Geometry of Some Banach spaces. Duality Mappings . . . . . . . . . . . . . . . . . . . 12

3.1.1 Strictly Convex and Uniformly Convex Spaces . . . . . . . . . . . . . . . . . . 13

3.1.2 Smooth and Uniformly smooth spaces . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.3 Classical Banach spaces: Lp; 1 p 1 . . . . . . . . . . . . . . . . . . . . . 16

3.1.4 Moduli. p-uniformly convex and q-uniformly smooth spaces . . . . . . . . . . . 17

3.1.5 Duality Mapping of Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1.6 Important Banach space Identities and Characterizations . . . . . . . . . . . . . 21

3.2 Nonlinear Operators. Maximal Monotone Mappings . . . . . . . . . . . . . . . . . . . 24

3.2.1 Topological Properties of Nonlinear Operators . . . . . . . . . . . . . . . . . . 24

3.2.2 Accretive Operators and Pseudocontractive Mappings . . . . . . . . . . . . . . 25

3.2.3 Monotone and Maximal monotone Operators . . . . . . . . . . . . . . . . . . . 26

3.2.4 Some Characterizations and Properties of Maximal Operators . . . . . . . . . . 28

3.2.5 Semigroup of Operators. Resolvents . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.6 Approximation of the Nonlinear Equation Au = 0 . . . . . . . . . . . . . . . . 30

3.3 Convex Analysis: Subdifferential and Optimization . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Basic Definitions and Results in Convex Analysis . . . . . . . . . . . . . . . . . 31

3.3.2 Subdifferential and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Fixed Point Theory: Approximate Fixed Points . . . . . . . . . . . . . . . . . . . . . . 35

v

3.4.1 Approximation and Iterative Algorithm . . . . . . . . . . . . . . . . . . . . . . 35

3.4.2 Important Recurrent Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 MAIN RESULTS AND APPLICATIONS 40

4.1 Application to zeros of maximal monotone maps . . . . . . . . . . . . . . . . . . . . . 51

4.2 Complement to proximal point algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Application to solutions of Hammerstein integral equations . . . . . . . . . . . . . . . . 52

4.4 Application to convex optimization problem . . . . . . . . . . . . . . . . . . . . . . . . 57

## CHAPTER ONE

INTRODUCTION

1.1 Background of study

The contributions of this thesis work fall within the general area of nonlinear functional analysis and

applications, in particular, nonlinear operator theory. We are interested in the solution or approximation

of solutions of nonlinear equations or inclusions (i.e., equations or inclusions defined by nonlinear operators)

in Banach spaces.

Problems in the area involve methods of fixed point theory and application of iterative algorithms to

approximate zeros or fixed points of nonlinear mappings. Research in the area is enormous due to varied

classification of Banach spaces, operators and topological assumptions on them (e.g., continuity, boundedness,

compactness, closedness e.t.c). The literature of the last four decades abounds with papers which

establish fixed point theorems for selfmaps or nonselfmaps satisfying a variety of contractive type conditions

on several ambient spaces. See figures 1.1, 1.2 and 3.1.

(1 < p 2) Lp (2 p < 1)

[

H

[

Rn

p-Uniformly convex q-Uniformly smooth

Uniformly convex Uniformly smooth

Reflexive Smooth

Unif. Gat. Diff. norm

Strictly convex

(a) Lattice of Banach spaces

k-Contractive maps

Nonexpansive maps

Strictly

pseudocontractive

Lipschitz

pseudocontractive

(b) Metric Fixed-point Operator lattice

Figure 1.1: Lattice of Spaces and Metric fixed-point Operator

1

k-Contractive maps

Nonexpansive maps

Strictly

pseudocontractive

Asymptotically

nonexpansve

Asymp. Nonexp. in the

Intermediate sense Pseudocontractive

Asymp. Strictly

pseudocontractive

Asymp. Strictly

pseudocontr. in the

Interm. Sense

Asymp.

pseudocontractive

Asymp. pseudocontr. in

the Interm. sense

Firmly

Quasi-nonexpansive

Quasi-nonexpansive

Demi-contractive

(a) Contractive-type self map Operator lattice

Figure 1.2: Lattice of Operators

Let H be a real inner product space. A map A : H ! 2H is called monotone if for each x; y 2 H,

? ; x ? y

0 8 2 Ax; 2 Ay: (1.1.1)

Monotone mappings were first studied in Hilbert spaces by Zarantonello [120], Minty [84], Kaˇcurovskii

[64] and a host of other authors. Interest in such mappings stems mainly from their usefulness in applications.

1.1.1 Zeros of Monotone operators on Hilbert spaces

We consider the problem given by

Au 3 0 (1.1.2)

where A : H ! 2H is a monotone map on a Hilbert space. Problems of this kind find relevance in

several areas of applications. In particular, we have the following examples:

Convex optimisation problems

Let g : H ! R [ f1g be a proper convex function. The subdifferential of g, @g : H ! 2H, is defined

for each x 2 H by

@g(x) =

x 2 H : g(y) ? g(x)

y ? x; x

8 y 2 H

:

It is easy to check that @g is a monotone operator on H, and that 0 2 @g(u) if and only if u is a minimizer

of g. Setting @g A, it follows that solving the inclusion 0 2 Au, in this case, is solving for a minimizer

of g.

2

In particular, as an example of the above, where g(x) = jxj, the subdifferential of g at zero, @g(0) =

[?1; 1], which trivially contains zero. Hence, zero is the minimizer of g.

Equilibrium problem of dynamical systems: Evolution equation

The equation 0 2 Au when A is a monotone map from a real Hilbert space to itself also appears in

evolution systems. Consider the evolution equation for a single-valued operator,

du

dt

+ Au = 0

where A is a monotone map from a real Hilbert space to itself. At equilibrium state, du

dt = 0 so that

Au = 0, whose solutions correspond to the equilibrium state of the dynamical system.

In particular, consider the following diffusion equation

8<

:

@u

@t (t; x) = 4u(t; x) + g(u(t; x)); t 0; x 2

;

u(t; x) = 0; t 0; x 2 @

;

u(0; x) = u0(x); u0 2 L2(

);

(1.1.3)

where

is an open subset of Rn.

By simple transformation i.e., by setting v(t) = u(t; :); where v : [0;1) ?! L2(

) is defined by

v(t)(x) = u(t; x) and f(‘)(x) = g(‘(x)); such that f : L2(

) ?! L2(

); we see that equation (1.1.3)

is equivalent to

v0(t) = Av(t) + f(v(t)); t 0;

v(0) = u0;

(1.1.4)

where A is a nonlinear monotone-type mapping defined on L2(

).

Setting f to be identically zero, at an equilibrium state (i.e., when the system becomes independent of

time) we see that equation (1.1.4) reduces to

Au = 0: (1.1.5)

Thus, approximating zeros of equation (1.1.5) is equivalent to the approximation of solutions of the

diffusion equation (1.1.3) at equilibrium state.

Hammerstein integral equations

Definition 1.1.1. Let

Rn be bounded. Let k :

! R and f :

R ! R be measurable

real-valued functions. An integral equation (generally nonlinear) of Hammerstein-type has the form

u(x) +

Z

k(x; y)f(y; u(y))dy = w(x); (1.1.6)

where the unknown function u and inhomogeneous function w lie in a Banach space E of measurable

real-valued functions.

By simple transformation (1.1.6) can put in the abstract form

u + KFu = 0; (1.1.7)

Interest in Hammerstein integral equations stems mainly from the fact that several problems that arise in

differential equations, for instance, elliptic boundary value problems whose linear part possesses Green’s

function can, as a rule, be transformed into the form (1.1.6) (see e.g., Pascali and Sburian [88], p. 164).

3

1.1.2 Extension of Hilbert space Monotonicity to arbitrary normed spaces

We recall that for a Hilbert space H, H = H. So, in the definition of a monotone operator in a Hilbert

space, the map A : H ! H could have been A : H ! H. Thus, the notion of monotone mappings has

been extended to real normed spaces. We now briefly examine two well-studied extensions of Hilbert

space monotonicity to arbitrary normed spaces, say , E.

A

A : E ! E A : E ! E

Accretive Monotone

Figure 1.3: Extension of Hilbert space monotonicity

Accretive-type mappings

Let E be a real normed space with dual space E. A map

J : E ! 2E defined by

Jx :=

x 2 E :

x; x

= kxk:kxk; kxk = kxk

is called the normalized duality map on E. We denote J?1 by J

A map A : D(A) ! 2E is called accretive if for each x; y 2 D(A), there exists j(x ? y) 2 J(x ? y)

such that

? ; j(x ? y)

0 8 2 Ax; 2 Ay: (1.1.8)

Roughly speaking, accretive mappings acting in a space E are generalizations of non-decreasing realvalued

functions. More precisely, A is said to be accretive if for all x1; x2 2 D(A), y1 2 Ax1, y2 2 Ax2

and 0,

kx1 ? x2k kx1 ? x2 + (y1 ? y2)k:

A is called maximal accretive if, in addition, the graph of A is not properly contained in the graph of any

other accretive operator. It is m-accretive if and only if A is accretive and R(I + tA) = E for all t > 0.

In a normed space, “m-accretive” implies “maximal accretive” . The converse assertion need not be

true. The first counterexample was constructed in lp by B.D. Calvert (1970). Moreover, A. Cernes

(1974) showed that even if both E and E are uniformly convex, but E is not a Hilbert space, then there

are maximal accretive mappings which are not m-accretive. However, it was proved by G. Minty (1962)

that in Hilbert spaces, the notions of ”m-accretive” and ”maximal accretive” are equivalent (see e.g.,

[80]) In a Hilbert space, the normalized duality map is the identity map, and so, in this case, inequality

(1.1.8) and inequality (1.1.1) coincide. Hence, accretivity is one extension of Hilbert space monotonicity

to general normed spaces.

Monotone-type mappings in arbitrary normed spaces

Let E be a real normed space with dual E. A map A : E ! 2E is called monotone if for each x; y 2 E,

the following inequality holds:

? ; x ? y

0 8 2 Ax; 2 Ay: (1.1.9)

4

It is called maximal monotone if, in addition, the graph of A is not properly contained in the graph of

any other monotone operator. Also, A is m-monotone if and only if it is monotone and R(J +tA) = E

for all t > 0. When E is a strictly convex Banach space with a Fr´echet differentiable norm, a maximal

monotone operator from E into E is m-monotone (see e.g., Kido [71]).

It is obvious that monotonicity of a map defined from a normed space to its dual is another extension of

Hilbert space monotonicity to general normed spaces.

The extension of the monotonicity condition from a Banach space into its dual has been

the starting point for the development of nonlinear functional analysis…: The monotone

mappings appear in a rather wide variety of contexts, since they can be found in many

functional equations. Many of them appear also in calculus of variations, as subdifferential

of convex functions (Pascali and Sburian [88], p. 101).

1.1.3 Application of Fixed Point Techniques

The theory of fixed point proves to be one of the most useful tools of modern mathematics. This comes

from earlier development of the theory and the fact that most important nonlinear problems in applications

can be transformed to fixed point problems.

Definition 1.1.2. Let X be a non-empty set and f be a self-map on X. A fixed point of f is a point

x 2 X such that f(x) = x. If f is a multivalued then a point p in X is called a fixed point of f if p 2 fp.

Theorems concerning the existence and properties of fixed points are known as fixed point theorems.

Several fixed point theorems include the Banach contraction mapping principle, Brouwer fixed point

theorem, Schauder fixed point theorem and a host of others (see e.g., Asati et al. [5], Khamsi [67], Smith

[107], Lee [74]).

Let E be a real normed space and A : E ! E be an accretive operator. Assume that Au = 0 has

a solution. Browder [14] introduced an operator T : E ! E by T = I ? A and called the map T,

pseudo-contractive. It is clear that zeros of A correspond to fixed points of T (i.e., Au = 0 if and

only if Tu = u). The class of pseudocontractive maps properly contains the class of nonexpansive maps

which are a generalisation of contraction maps. A map T : E ! E is called nonexpansive if for each

x; y 2 E, the inequality kTx ? Tyk kx ? yk is true.

Several existence theorems have been proved for the equation Au = 0; where A is of the monotone-type

(or accretive-type) (see e.g., Brezis [11], Browder [14], Deimling [50], Pascali and Sburian [88], e.t.c.).

Likewise, several results have appeared in the literature for approximating zeros of accretive-type (or

fixed points of pseudo-contractive) mappings in certain Banach spaces (see e.g., Chidume et al.[20],

Takahashi [113], Bruck [17], and host of other authors).

Let E be a real normed space and T := I ? A : E ! E a pseudocontractive mapping. If K is a

nonempty convex subset of E and F(T) := fx 2 K : Tx = xg 6= ;, the following recursion formula

has been used to approximate fixed points of T, x0 2 K,

xn+1 = (1 ? n)xn + nTxn; n 0;

where fng is a real sequence satisfying appropriate conditions. The most general iterative scheme for

bounded pseudocontractive maps seems to be that obtained from the following

Theorem 1.1.3 (C. E. Chidume [23]). Let E be a uniformly smooth real Banach space with modulus of

smoothness E, and let A : E ! 2E be a multi-valued bounded m?accretive operator with D(A) = E

such that the inclusion 0 2 Au has a solution. For arbitrary x1 2 E, define a sequence fxng by,

xn+1 = xn ? nun ? nn(xn ? x1); un 2 Axn; n 1;

5

where fng and fng are sequences in (0; 1) satisfying the following conditions:

(i) limn!1 n = 0; fng is decreasing; (ii)

P

nn = 1;

P

E(nM1) < 1, for some

constantM1 > 0; (iii) limn!1

h

n?1

n

?1

i

nn

= 0. There exists a constant 0 > 0 such that E(n)

n

0n.

Then, the sequence fxng converges strongly to a zero of A.

1.2 Statement of Problem

In studying the inclusion (1.1.2) on real Banach spaces more general than Hilbert spaces when A is

of accretive-type mapping, several iterative algorithms have been constructed and results obtained for

approximating solutions of problems of the equation (see e.g., the following monographs: Berinde [9],

Browder [14], Chidume [22], Reich [90], and the references contained in them). Consequently, this has

generated interests and the question asked if similar results for the case of monotone-type mappings in

arbitrary Banach spaces can be obtained, where A maps a space into its dual.

Regrettably, the pursuit of analogous results has only been greeted with very little progress and seemingly

unpropitious prospects as the success for the accretive-type case doesn’t quite easily carry over to

the case of monotone-type mappings. The difficulty, for the most part, seems to be that all efforts made

to apply directly known geometric properties of Banach spaces proved abortive; also developing and

understanding concepts with applying knowledge of the structure and geometry of the dual space, existence

and uniqueness theorems for monotone-type mappings in arbitrary Banach spaces, weak topology

and relevant tools of functional analysis, and other notions of operator theory were rather too slow for

the ambitious researcher. Also, defining the iterative sequence to make sense posed a challenge.

Furthermore, the technique of converting the inclusion (1.1.2) into a fixed point problem of defining the

map T := I ? A is not applicable since, in this case when A is monotone, A maps E into E and such

T is never well-defined as the identity map does not make sense.

1.3 Aim and Objectives of Study

The aim of this work is to contribute to the efforts being made to approximate solutions of inclusion

(1.1.2) where A is of monotone-type. We consider the problem of solving zeros of nonlinear equations

of maximal monotone-type mappings with no continuity assumption. We proceed thus.

1. We introduce, as far as we know, a class of mappings called J-Pseudocontractive mappings and

study the concept of J-fixed points. We establish the relationship between monotone mappings

and J-pseudocontractive mappings and between J-fixed points and zeros of operators.

2. We construct an iterative algorithm which converges to a J-fixed point of a J-Pseudocontractive

mappings and hence, by extension, to a zero of a monotone mapping.

3. We apply our results to:

Zeros of maximal monotone mappings. ( This corresponds, as noted earlier, to the equilibrium

state of some dynamical system)

Proximal point algorithm

Solutions Hammerstein integral equations

Convex minimization problems

6

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