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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