**TABLE OF CONTENTS**

Acknowledgment i

Certication ii

Approval iii

Dedication v

Abstract vi

1 General Introduction and Literature Review 1

CHAPTER ONE 1

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

1.1.1 Variational Inequality Problems . . . . . . . . . . . . . . . . . 1

1.1.2 Fixed Point Problems . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.3 Variational Inequality and Fixed Point Problems . . . . . . . . 3

1.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Motivation of Research and Objectives . . . . . . . . . . . . . . . . . 3

1.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Preliminaries 6

2.1 Denition of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Important Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Results of Kraikaew and Saejung 13

CHAPTER THREE 13

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 The Subgradient Extragradient Algorithm . . . . . . . . . . . . . . . 13

3.3 The Modied Subgradient Extragradient Algorithm . . . . . . . . . . 17

4 Our Contributions 21

viii

CHAPTER FOUR 21

4.1 Approximating a solution of a variational inequality problem . . . . . 21

4.2 Approximating a common element of solutions of a variational inequality

problem and a xed point of a relatively nonexpansive map . 25

4.3 Approximating a common element of variational inequality and convex

feasibility problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.5 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Appendix 33

5.1 Analytical representations of duality maps in Lp; lp; and Wpm

; spaces,

1 < p < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Bibliography 38

ix

**CHAPTER ONE**

General Introduction and Literature Review

In this chapter, we give a general introduction on variational inequality problem,

xed point problem and nally we give a brief review of existing results on variational

inequality and xed point problem.

1.1 Background of study

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

analysis and applications, in particular, nonlinear operator theory. We are interested

in nding or approximating solution(s) of a variational inequality problem for a

monotone k-Lipschitz map and a convex feasibility problem for a countable family

of relatively nonexpansive maps, in Banach spaces.

1.1.1 Variational Inequality Problems

Variational inequality problems were formulated in the late 1960s by Lions and Stampacchia,

and since then, they have been studied extensively. Numerous researchers

have proposed and analyzed various iterative schemes for approximating solutions of

variational inequality problems. The literature on this is extensive (see, for example,

[Chidume, 2009], [Nilsrakoo and Saejung, 2011], [Buong, 2010],[Hieu et al., 2006],

[Iiduka and Takahashi, 2008],[Censor et al., 2012], [Censor et al., 2011], [Dong et al., 2016],

[Gibali et al., 2015], [Chidume et al., 2017], [Censor et al., 2010], and the references

contained in them).

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 if there exists a constant k 0, such that

kAx Ayk kkx yk; 8x; y 2 C: (1.1.1)

Remark 1.1.1 If k 2 (0; 1), A is called a contraction. If k = 1, A is called

nonexpansive.

1

• monotone if the following inequality holds:

x y; Ax Ay

0; 8x; y 2 C: (1.1.2)

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

x y; Ax Ay

kAx Ayk2; 8x; y 2 C: (1.1.3)

• 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.

The problem of nding a point u 2 C, such that

hv u; Aui 0; 8 v 2 C; (1.1.4)

is called a variational inequality problem. We denote the set of solutions to the

variational inequality problem (1:1:4) by V I(C;A).

Remark 1.1.2 It is easy to see that if u is a solution of the variational inequality

problem (1:1:4) then,

hx u; Axi 0; 8x 2 C:

1.1.2 Fixed Point Problems

The theory of xed point proves to be a useful tool in modern mathematics. This

comes from the fact that most important nonlinear problems in applications can be

transformed to a xed point problem.

Theorems concerning the existence and properties of xed points are known as xed

point theorems. Several theorems have been proved on the existence and uniqueness

of xed point(s) of self-maps. These theorems include the Banach contraction

mapping principle, Brouwer xed point theorem, Schauder xed point theorems

and a host of other authors (see for example [Asati et al., 2013], [Khamsi, 2002],

[Lee, 2013], [Smith, 2015])

Example 1.1.3 Let E be a real normed space and A : E ! E; be an accretive

operator; most real life problems can be modelled into an equation of the form

du

dt

+ Au = 0: (1.1.5)

At equilibrium, du

dt = 0. Thus, (1:1:5) reduces to

Au = 0: (1.1.6)

[Browder, 1967], introduced an operator T : E ! E, by T = I A and called the

map T, pseudo-contractive . It is easy to see that zeros of A corresponds to xed

points of T (i.e., Au = 0 if and only if Tu = u).

Also, several existence theorems have been proved for the equation (1:1:6), where A is

of the monotone-type (or accretive-type) (see for example, [Brezis, 1974], [Browder, 1967],

[Deimling, 1974], [Pascali and Sburian, 1978], and the references contained in them).

2

1.1.3 Variational Inequality and Fixed Point Problems

In numerous models for solving real-life problems, such as in signal processing, networking,

resource allocation, image recovery, and so on, the constraints can be expressed

as variational inequality problems and (or) as xed point problems. Consequently,

the problem of nding common elements of the set of solutions of variational

inequality problems and the set of xed points of operators has become a ourishing

area of contemporary research for numerous mathematicians working in nonlinear

operator theory (see, for example, [Mainge, 2010a, Mainge, 2008, Ceng et al., 2010]

and the references contained in them).

1.2 Statement of the Problem

Let A : E ! E be a monotone and k-Lipschitz map and S : E ! E be a nonexpansive

map. In studying variational inequality problems and xed point problems on

real Banach spaces more general than Hilbert spaces, several algorithms have been

constructed for approximating solutions of variational inequality problems and xed

point problems (see, e.g., the following monographs: [Alber, 1996], [Berinde, 2007],

[Browder, 1967], [Chidume, 2009], [Goebel and Reich, 1994] and the references contained

in them). Consequently, since most real life problems exist in spaces more

general than Hilbert spaces, this induced mathematicians to ask if such results can

be obtained for a monotone, k-Lipschitz map and a nonexpansive map in Banach

spaces.

However, the pursuit of analogous results for variational inequality problems and

xed point problems in more general Banach space with nonexpansive maps seem

not to be feasible. The main diculty (or challenge) is that most properties of the

Lyapunov functional and generalized projection are proved using relatively nonexpansive

maps.

1.3 Motivation of Research and Objectives

Motivated by the results of [Kraikaew and Saejung, 2014], and [Nakajo, 2015], it is

our purpose in this thesis to introduce a Krasnoselskii-type algorithm in a uniformly

smooth and 2-uniformly convex real Banach space and prove strong convergence of

the sequence generated by our algorithm to a point q 2 F(S) \ V I(C;A). The

objectives are:

• To use the normalized duality map and Lyapunov functional for estimations;

• To extend the class of maps from one nonexpansive to a countable family of

relatively nonexpansive maps; and

• To propose an algorithms with less computational cost when compared with

existing algorithms in the Banach space.

3

1.4 Literature Review

Numerous researchers in nonlinear operator theory have studied various iterative

methods for approximating solutions of variational inequality problems, approximating

xed points of nonexpansive maps and their generalizations (see, e.g., the following

monographs: [Alber, 1996], [Berinde, 2007], [Browder, 1967], [Chidume, 2009],

[Goebel and Reich, 1994] and the references contained in them). In most of the early

results on iterative methods for approximating these solutions, the map A was often

assumed to be inverse-strongly monotone (see, e.g., [Buong, 2010], [Censor et al., 2012],

[Chidume et al., 2016], and the references contained in them). To relax the inversestrong

monotonicity condition on A, [Korpelevic, 1967] introduced, in a nite dimensional

Euclidean space Rn, the following extragradient method

(

x1 = x 2 C;

xn+1 = PC(xn A[PC(xn Axn)]); 8 n 2 N;

(1.4.1)

where A was assumed to be monotone and Lipschitz. The extragradient method

has since then been studied and improved on by many authors in various ways.

However, we observe that in the extragradient method, two projections onto a closed

and convex subset C of H need to be computed in each step of the iteration process.

As mentioned by [Censor et al., 2011], this may aect the eciency of the method

if the set C is not simple enough. Therefore, to improve on the extragradient

method, [Censor et al., 2011] modied the the extragradient method and proposed

the following iterative algorithm:

8>>><

>>>:

x0 2 H;

yn = PC(xn Axn);

Tn = fw 2 H : hxn Axn yn;w yni 0g;

xn+1 = PTn(xn Ayn):

(1.4.2)

The method (1:4:2) replaces the second projection onto the closed and convex subset

C in (1:4:1) with a projection on to the half-space Tn. Algorithm (1:4:2) is the socalled

subgradient extragradient method. We note that, the set Tn is a half-space, and

hence algorithm (1:4:2) is easier to execute than algorithm (1:4:1). Under some mild

assumptions, [Censor et al., 2011] proved that algorithm (1:4:2) converges weakly to

a solution of variational inequality (1:1:4) in a real Hilbert space.

In order to obtain the strong convergence, [Kraikaew and Saejung, 2014] combined

the subgradient extragradient method (1:4:2) with the method introduced by [Halpern, 1967]

and proposed the following iterative algorithm:

8>>><

>>>: x0 2

H

;

yn = PC(xn Axn);

Tn = fw 2 H : hxn Axn yn;w yni 0g;

xn+1 = nx0 + (1 n)PTn(xn Ayn);

(1.4.3)

where fng is a sequence in [0; 1] satisfying limn!1 n = 0 and

P1

n=1 n = 1.

They proved that the sequence generated by algorithm (1:4:3) converges strongly to

4

a solution of the variational inequality problem (1:1:4) in a real Hilbert space. We

remark, however, that convergence theorems have also been proved in real Banach

spaces more general than Hilbert space. For instance, [Iiduka and Takahashi, 2008],

using the following scheme,

(

x1 2 C;

xn+1 = CJ1(Jxn nAxn);

(1.4.4)

obtained weak convergence of the sequence fxng generated by equation (1:4:4) to

a solution of the variational inequality problem (1.1.4) in a 2-uniformly convex,

uniformly smooth real Banach space whose duality map J is weakly sequentially

continuous, under the conditions that,

(A1) A is -inverse-strongly-monotone;

(A2) V I(C;A) 6= ;; and

(A3) kAyk kAu Ayk; 8 y 2 C and u 2 V I(C;A).

An example of such a real Banach space is lp; 1 < p 2. The space Lp; 1 < p 2

is excluded since the duality map on it is not weakly sequentially continuous.

Motivated by the result of [Iiduka and Takahashi, 2008], in 2015, [Nakajo, 2015] proposed

and studied the following CQ method in a 2-uniformly convex and uniformly

smooth real Banach space.

8>>>>>>>><

>>>>>>>>:

x1 = x 2 E;

yn = CJ1[Jxn nA(xn)];

zn = Tyn;

Cn = fz 2 C : (z; zn) (z; xn) (yn; xn) 2nhyn z; Axn Aynig;

Qn = fz 2 C : hxn z; Jx Jxni 0g;

xn+1 = Cn\Qnx; n 0:

(1.4.5)

He signicantly improved the result of [Iiduka and Takahashi, 2008] in the following

sense:

• The operator A is assumed to be monotone and Lipschitz.

• The sequence fxng generated by his scheme converges strongly to an element

of V I(C;A).

• The requirement that J be weakly sequentially continuous is dispensed with;

consequently, the result of Nakajo is applicable in Lp spaces, 1 < p 2.

• The condition (A3) is also dispensed with.

• The sequences fxng and fzng generated by his algorithm, not only converge

to a point in V I(C;A) but also to a xed point of a relatively nonepxansive

self-map of C.

However, we note that the algorithm (1.4.5) of Nakajo, at each step of the iteration

process, requires the computation of two convex subsets, Cn and Qn, their intersection

Cn \ Qn and the projection of the initial vector onto this intersection. This is

certainly not convenient in several possible applications.

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