A Modified Subgradient Extragradeint Method for Variational Inequality Problems and Fixed Point Problems in Real Banach Spaces

A Modified Subgradient Extragradeint Method for Variational Inequality Problems and Fixed Point Problems in Real Banach Spaces

ABSTRACT

Let E be a 2-uniformly convex and uniformly smooth real Banach space with dual space E. Let A : C ! E be a monotone and Lipschitz continuous mapping and U : C ! C be relatively non-expansive.

An algorithm for approximating the common elements of the set of fixed points of a relatively non-expansive map U and the set of solutions of a variational inequality problem for the monotone and Lipschitz continuous map A in E is constructed and proved to converge strongly.

TABLE OF CONTENTS

Certification i
Approval ii
Abstract iii
Acknowledgement iv
Dedication v
1 Introduction 1
1.1 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Variational inequality problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Fixed Point Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Literature Review 4
2.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Non-expansive Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Theory and Methods 9
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Metric Projection Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Calculating the projection onto a closed convex set in Hilbert spaces . . . . . 19
4 Main Result 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Convergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5 Application 29
5.1 Strong Convergence Theorem for a Countable Family of Relatively Non-expansive
Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6 Conclusion 31
Bibliography 32

CHAPTER ONE

1.1 Background of study

The notion of monotone operators was introduced by Zarantonello [Zarantonello, 1960], Minty [Minty, 1962] and Ka˘ curovskii [Ka˘ curovskii, 1960]. Monotonicity conditions in the context of variational methods for nonlinear operator equations were also used by Vainberg and Ka˘ curovskii [Vainberg et al., 1959].

A map A : D(A) H !H is monotone if hAx 􀀀 Ay;x 􀀀 yi 0 8x;y 2 H: Consider the problem of finding the equilibrium states of the system described by
du
dt
+ Au = 0; (1.1)

where A is a monotone-type mapping on a real Hilbert space. This equation describes the evolution of many physical phenomena which generate energy over time. It is known that many physically significant problems in different areas of research can be transformed into an equation of the form

Au = 0: (1.2)

At equilibrium state, equation (1.1) reduces to equation (1.2) whose solutions, in this case, correspond to the equilibrium state of the system described by equation (1.1). Such equilibrium points are very desirable in many applications, for example, economics, ecology, physics and so on.

1.2 Variationally inequality problem

Let C be a nonempty, closed and convex subset of a real normed space E with dual space E. Let A : C E ! E be a nonlinear operator. The classical variational inequality problem is the following: find x 2 C such that
hAx;y 􀀀 xi 0 8 y 2 C: (1.3)

The set of solutions of inequality (1.3) is denoted by V I(C; A). The variational inequality problem is connected with convex minimization, fixed point problem, zero of nonlinear operator and so on.

Variational inequality has been shown to be an important mathematical model in the study of many real problems, in particular equilibrium problems. It provides us with a tool for formulating and qualitatively analyzing the equilibrium problems in terms of existence and uniqueness of solutions, stability, and sensitivity analysis, and provides us with algorithms for computational purposes.