ABSTRACT
In this thesis, a hybrid extragradient-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 nonexpansive 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 signicant 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
Certication 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 Denition 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 phe-
nomena 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 nonexpansive 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 conguration 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 diculty 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]).
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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 nding 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
unication (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 dierential 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 nonexpansive map S and this is denoted by F(S).
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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 signicance.
[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 nding 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 nding a solution of a variational inequality problem that
satises 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 nonexpansive 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
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of maps (see, Nadezhkina and Takahashi [2006]).
In order to weaken the -inverse monotonicity condition on A, Ceng and Yao [2007] and Nadezhkina
and Takahashi [2006], proved the following strong convergence theorems.
Theorem 1.4.1 (Ceng and Yao [2007]) Let C be a nonempty closed and convex subset of a real
Hilbert space H. Let f : C ! C be a contractive mapping with a contractive constant L 2 (0; 1). Let
A : C ! H be a monotone and k-Lipschitz continuous mapping and S : C ! C be a nonexpansive
mapping such that F(S) \ V I(C;A) 6= ;. Let fxng; fyng be the sequence generated by
8><
>:
x0 2 C;
yn = (1 n)xn + nPC(xn nAxn);
xn+1 = (1 n n)xn + nf(yn) + nSPC(xn Ayn);
(1.4.1)
where fng (0; 1) with
P1
n=0 n < 1 and fng; fng and f ng are sequences in [0; 1] satisfying
the following conditions:
1. n + n 1 8 n 0;
2. limn!1 n = 0 and
P1
n=0 n = 1;
3. 0 < lim infn!1 n lim supn!1 n < 1.
Then, the sequences fxng and fyng converge strongly to the same point q = PF(S)\V I(C;A)f(q) if
and only if fAxng is bounded and lim infn!1hy xn; Axni 0; 8 y 2 C:
Theorem 1.4.2 (Nadezhkina and Takahashi [2006]) Let C be a nonempty closed and convex
subset of a real Hilbert space H. Let A : C ! C be a montone k-Lipschitz continuous mapping
and S : C ! C be a nonexpansive mapping such that F(S) \ V I(C;A) 6= ;. Let fxng; fyng and
fzng be sequences generated by
8>>>>>>>>><
>>>>>>>>>:
x0 2 C;
yn = PC(xn nAxn);
zn = nxn + (1 n)SPC(xn nAyn);
Cn = fz 2 C : kzn zk kxn zkg;
Qn = fz 2 C : hxn z; x0 xni 0g;
xn+1 = PCn\Qnx0;
(1.4.2)
for all n 0 wherefng [a; b] for some a; b 2 (0; 1
k ) and fng [0; c] for some c 2 [0; 1). Then
the sequences fxng; fyng and fzng converge strongly to the same point q = PF(S)\V I(C;A)x0.
Motivated by these two results, Ceng et al. [2010] in 2010 introduced a hybrid extragradient-like
approximation method and proved the following strong convergence theorem.
Theorem 1.4.3 (Ceng et al. [2010]) Let C be a nonempty closed convex subset of a real Hilbert
space H. Let A : C ! H be a monotone and k-Lipschitz continuous mapping and S : C ! C
be a nonexpansive mapping such that F(S) \ V I(C;A) 6= ;. Dene inductively the sequence
(xn); (yn); (zn) by
8>>>>>>>>><
>>>>>>>>>:
x0 2 C;
yn = (1 n)xn + nPC(xn nAxn);
zn = (1 n n)xn + nyn + nSPC(xn nAyn);
Cn = fz 2 C : kzn zk2 kxn zk2 + (3 3 n + n)b2kAxnk2g;
Qn = fz 2 C : hxn z; x0 xni 0g;
xn+1 = PCn\Qnx0;
(1.4.3)
for all n 0 where fng is a sequence in [a; b] with a > 0 and b < 1
2k , and fng; fng; f ng are
three sequences in [0; 1] satisfying the conditions:
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1. n + n 1 8 n 0;
2. limn!1 n = 0;
3. lim infn!1 n > 0;
4. limn!1 n = 1 and n > 3
4 8 n 0.
Then, the sequences fxng; fyng; fzng are well dened and converge strongly to the same point
q = PF(S)\V I(C;A)x0.
In this thesis, motivated by the result of Ceng et al. [2010], we introduce a hybrid extragradientlike
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 u 2 F(S) \ V I(C;A).
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