Project Materials

A Hybrid Algorithm for Approximating a Common Element of Solutions of a Variational Inequality Problem and a Convex Feasibility Problem

ABSTRACT

In this thesis, a hybrid extra-gradient like iteration algorithm for approximating a common element of the set of solutions of a variational inequality problem for a monotone, k-Lipschitz map, and common xed points of a countable family of relatively non-expansive maps in a uniformly smooth and 2-uniformly convex real Banach space is introduced.

A strong convergence theorem for the sequence generated by this algorithm is proved. The theorem obtained is a significant improvement of the results of Ceng et al. (J. Glob. Optim. 46(2010), 635-646). Finally, some applications of the theorem are presented

TABLE OF CONTENTS

Certification i
Approval ii
Abstract iv
Acknowledgment vi
Dedication viii
1 General Introduction and Literature Review 2
1.1 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Variational Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Fixed Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Variational Inequality and Fixed Point Problem . . . . . . . . . . . . . . . . 4
1.1.4 Convex Feasibility Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Statement of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Objective of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Preliminaries 7
2.1 Definition of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Results of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Results of Ceng et al. 12
4 Main Results 17
4.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Bibliography 26

CHAPTER ONE

GENERAL INTRODUCTION AND LITERATURE REVIEW

1.1 Background of study

There is no branch of mathematics, however abstract which may not someday be applied to phenomena of the real world” | Lobachevsky.

Attesting to the authenticity of Lobachevsky’s claim, the vast applicability of mathematical models whose constraints can be expressed as xed point and (or) variational inequality problems in solving real life problems, such as in signal processing, networking, resource allocation, image recovery and so on, makes the eld of variational inequality and xed point theory a worthwhile area of research [See for example Maainge [2008] and Maainge [2010b] and the references contained in them.

In this thesis, we concentrate on approximating a common element of solutions of a variational inequality problem and common xed point of a countable family of relatively non-expansive maps in real Banach spaces. Hence, the results of this thesis will form major contributions to nonlinear operator theory, which falls within the general area of nonlinear functional analysis and applications.

1.1.1 Variational Inequality

It was the year 1958, in a classroom, at the Instituto Nazionale di Alta Mathematica Italy, that Antonio Signorini posed the problem what will be the equilibrium conjuration of a spherically shaped elastic body resting on a rigid frictionless plane?”

A natural question is: what is special about this problem? Its ambiguous boundary condition. In fact, Signorini himself called it problem with ambiguous boundary condition”.

The statement of the problem involves inequalities and according to Antman (1983), the essential difficulty is that the point of contact between the body and the plane is not known a-priori, conceivably too, the contact set could be especially complicated.

Nevertheless, Signorini warmly invited young analyst to study the problem (Signorini [1959]).

Gaetano Fichera, a student in that class, decided to investigate the problem using the virtual work principle and in January 1963, he produced a complete proof of the existence and uniqueness of a solution for the problem. In honour of his teacher, Fichera renamed the problem as Signorini problem” (Fichera [1963]).

Fichera’s solution of the Signorini problem became the bedrock that has metamorphosed into the eld known as variational inequality today. Indeed, just as Antman puts it, the solution of the Signorini problem coincides with the birth of the eld of variational inequalities (Antman [1983]).

Let E be a real Banach space with dual space E. Let C be a nonempty, closed and convex subset of E and A : C ! E. Let h; i : E E ! R be the duality pairing.

The problem of ending a point u 2 C such that

hv 􀀀 u; Aui 0; 8 v 2 C; (1.1.1)

is called a variational inequality problem.

Let C be a nonempty, closed and convex subset of a real Banach space E with dual space E and

A : C ! E be a map. Then, A is said to be:

• k-Lipschitz continuous if there exists a constant k 0 such that kAx 􀀀 Ayk kkx 􀀀 yk; 8x; y 2 C: (1.1.2)

• monotone if the following inequality holds:

x 􀀀 y; Ax 􀀀 Ay

0; 8x; y 2 C: (1.1.3)

• -inverse strongly monotone if there exists a 0, such that

x 􀀀 y; Ax 􀀀 Ay

kAx 􀀀 Ayk2; 8x; y 2 C: (1.1.4)

• maximal monotone if A is monotone and the graph of A is not properly contained in the graph of any other monotone map.

It is immediate that if A is -inverse strongly monotone, then A is monotone and Lipschitz continuous.

In this thesis, we shall assume that the subset C of E is nonempty, closed and convex and the map A is monotone and k-Lipschitz. We shall denote the set of solutions of the variational inequality problem by V I(C;A).

Remark 1.1.1 It is easy to see that if u is a solution of the variational inequality problem (1:1:1) then, hx 􀀀 y; Axi 0; 8x 2 C:

1.1.2 Fixed Point Theory

Fixed point theory is one of the most important and useful tools of modern mathematics. Its interconnectedness with other elds such as game theory, optimization theory, approximation theory and variational inequality points to the fact that xed point theory is a show piece of mathematical communication (Vandana and Chetan [2017]). Fixed point theory is based on a very simple mathematical setting.

A point is called a xed point, if it remains invariant under any form of transformation. For a self map f, i.e., f : E ! E, a xed point is a point x0 2 E such that f(x0) = x0. This point however, may or may not exist. This gives rise to the problem: what condition(s) guarantees existence of a xed point?”

This problem has been of interest since the 19th century and no doubt has attracted huge research from many mathematicians ranging from the contributions of Cauchy Fredhlin, Liouville, Lipschitz, Peano and Picard in establishing existence and uniqueness of solutions, particularly to differential equations using successive approximations (Vandana and Chetan [2017]).

Several theorems on existence and properties of xed points have been proved, amongst them include Banach xed point theorem and Brouwer xed point theorem referred to as the two fundamental theorem of xed points (Vandana and Chetan [2017]).

In recent years, books, monographs and scholarly articles on xed point theory abound (see e.g., Chidume [2009], Chidume et al. [2016], Berinde [2007], Zeidler [1985]). This thesis work focuses on the set of xed points of a relatively non-expansive map S and this is denoted by F(S).

1.1.3 Variational Inequality and Fixed Point Problem

Many models for solving real life problems have their associated constraints captured as xed point and variational inequality problems. Consequently, the problem of approximating a solution of the variational inequality problem that is also a xed point of some operator is of great significance.

[See e.g., Maainge [2010a], Ceng et al. [2010] and the references contained in them].

1.1.4 Convex Feasibility Problem

Let E be a real Banach space, and let fCngn1 be a countable family of closed convex subsets of E. The problem of ending a point x0 2

1T
n=1
Cn is called the convex feasibility problem.

1.2 Statement of Problem

This thesis is concerned with the problem of approximating a common element of the set of solutions of the variational inequality problem for a monotone and k-Lipschitz map A and the set of xed points of a relatively non-expansive map S, in a uniformly smooth and 2-uniformly convex real Banach space.

1.3 Objective of the Study

It is our aim in this thesis to:

• Study and analyse the work done in Hilbert space by [Ceng et al, 2010].

• Introduce an iterative algorithm for approximating an element of V I(C;A) F(S) in a uniformly smooth and 2-uniformly convex real Banach space.

• Establish the well-denedness of our algorithm.

• Prove a strong convergence theorem for the sequence generated by our algorithm.

• Give some applications of our theorem.

1.4 Literature Review

The evolution of variational inequality problems dates back to the late 1960’s by Lions and Stampacchia Lions and Stampacchia [1967] and over the years, extensive study, analysis and generalisation of these problems have been done by numerous researchers in the eld of nonlinear operator theory.

The literature abounds with iterative algorithms for approximating solutions of variational inequality problems and xed points of some operators (see, for example, Chidume [2009], Nilsrakoo and Saejung [2011], Buong [2010], Censor et al. [2012, 2011], Hieu et al. [2006], Dong et al. [2016], Gibali et al. [2015], Iiduka and Takahashi [2008], Chidume et al., Censor et al. [2010], Xu and Kim [2003], and the references contained in them).

Antipin [2000] studied methods for ending a solution of a variational inequality problem that satisfies some additional constraints in a nite dimensional space. Takahashi and Toyoda [2003], investigated the problem of nding a solution of a variational inequality problem which is also a xed point of some map in a Hilbert space.

By assuming A to be -inverse strongly monotone, S to be a non-expansive map of C into C, and V I(C;A) F(S) 6= ;, they proposed an iterative algorithm and established weak convergence result of the sequence generated by their algorithm to an element of V I(C;A)F(S), where F(S) is the set of xed points of S.

Later, Iiduka and Takahashi [2008], using a modied algorithm, while retaining the same assumptions on A and S proved strong convergence of the sequence generated by their algorithm to a point of V I(C;A) F(S).

However, the assumption that A is -inverse strongly monotone excludes some important classes of maps (see, Nadezhkina and Takahashi [2006]).