Project Materials

On J-fixed Points of J-pseudo Contractions with Applications

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-pseudo contractive maps, is introduced and the notion of J-fixed points is used to prove that T:= (J 􀀀 A) is J-pseudo contractive 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-pseudo contractive 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 analog 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 analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems.

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
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 the 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 the application of iterative algorithms to
approximate zeros or fixed points of nonlinear mappings. Research in the area is enormous due to the varied
classification of Banach spaces, operators, and topological assumptions on them (e.g., continuity, boundedness,
compactness, closeness e.t.c).

The literature of the last four decades abounds with papers that
establish fixed point theorems for self-maps or nonself-maps satisfying a variety of contractive type conditions
on several ambient spaces. See figures 1.1, 1.2, and 3.1.