ABSTRACT
The work of Osilike and Isiogugu, Nonlinear Analysis, 74 (2011), 1814-1822 on weak and strong
convergence theorems for a new class of k-strictly pseudononspreading mappings in real Hilbert
spaces is reviewed. We studied in detail this new class of mappings which is more general than
the class of nonspreading mappings studied by Kurokawa and Takahashi, Nonlinear Analysis 73
(2010) 1562-1568. Many incisive examples establishing the relationship of the class of k-strictly
pseudononspreading mappings and several other important classes of operators are presented. In-
teresting properties of k-strictly pseudononspreading mappings and weak and strong convergence
theorems for approximation of its xed points which appeared in the cited work of Osilike and
Isiogugu were studied and presented.
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TABLE OF CONTENTS
Acknowledgment i
Certication ii
Approval iii
Abstract v
Dedication vi
1 Introduction 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background of study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Literature Review 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Preliminaries 11
3.1 Denition of some terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.1 Basic facts in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.2 Demiclosedness Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Main Result 20
4.1 k-strictly Pseudononspreading Mapping . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Properties of k-strictly Pseudononspreading Mappings . . . . . . . . . . . . . . . . 22
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4.2.1 Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Weak and strong convergence theorem for nonspreading-type mappings in a Hilbert
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Summary 35
CHAPTER ONE
INTRODUCTION
1.1 General Introduction
The content of this thesis falls within the general area of functional analysis, in particular, nonlinear
operator theory; an area which has attracted the attention of prominent mathematicians due to its
diverse applications in numerous elds of science. In this thesis, we concentrate on an important
topic in this area { Weak and strong convergence theorems of nonspreading type mappings in a
Hilbert space.
In this chapter, the background of our research work will be given; this will reveal how relevant our
work is. Then in chapter two, we shall review the research work carried out in the area of research
described in this thesis. Some basic denitions and fundamental tools we used in our work will be
given in chapter three as preliminaries, while our main results will be presented in chapter four.
In chapter ve, conclusions will be given.
1.2 Background of study
Fixed point theory has been an important area of mathematics due to its applications in several
areas of research such as in Optimization, Economics, and Evolution Equations, to mention but a
few. For example, consider the problem of nding the equilibrium points of the system described
by the following equation
du
dt
+ Au(t) = 0 (1.1)
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where A : D(A) H ! H is a nonlinear map and H a real Hilbert space. At equilibrium,
du
dt = 0. Thus, the original problem is reduced to the problem of nding solutions of the equation:
Au = 0 (1.2)
i.e, nding the zeros of A. Several problems arising in Reservoir Engineering, Economics, Physics
to mention but a few, can be modelled in the form of equation (1.2). Since generally A is nonlinear,
there is no closed form solution of this equation. To solve equation (1.2), where A is a multi-valued
monotone map, Felix Browder dened an operator T : H ! H by T := I A, where I is the
identity map on H. He called such T a pseudocontraction. It is easy to see that xed points of
T correspond to zeros of A which in turn correspond to equilibrium points of dynamical system
described by equation(1.1). As a result of this, the study of xed point theory of pseudocontractive
maps and their types has attracted the interest of numerous scientists and researchers.
Kohsaka and Takahashi introduced an important class of mappings called the class of nonspread-
ing mappings. They obtained a xed point theorem for a single valued nonspreading mapping
in Banach space. Furthermore, they obtained a common xed point theorem for a commutative
family of nonspreading mappings in Banach space.
Let C be a nonempty closed and convex subset of a real smooth, strictly convex and re exive
real Banach space, a map T : C ! C is nonspreading if 8 x; y 2 C
(Tx; Ty) + (Ty; Tx) (Tx; y) + (Ty; x): (1.3)
where (x; y) = kxk2 2hx; j(y)i + kyk2; 8 x; y 2 E and J : E ! 2E
dened by
Jx = fj(x) 2 E : hx; j(x)i = kxkkj(x)kg; 8x 2 E, where h:; :i is the duality pairing between
x 2 E and j(x) 2 E, so that hy; j(x)i = (j(x))(y).
We observe that if E is a real Hilbert space, then j is the identity and
(x; y) = kxk2 2hx; yi + kyk2 = kx yk2.
Thus, if C is a nonempty closed and convex subset of a real Hilbert space, then
T : C ! C is nonspreading if
2kTx Tyk2 kTx yk2 + kTy xk2; 8 x; y 2 C: (1.4)
It is shown in Lemma 3.22 that inequality 1.4 is equivalent to
kTx Tyk2 kx yk2 + 2hx Tx; y Tyi 8x; y 2 C.
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S. Lemoto and W. Takahashi obtained some fundamental properties for nonspreading mapping
in a Hilbert space. Furthermore, they studied the approximation of common xed points of non-
expansive mappings and nonspreading mappings in a Hilbert space.
Y. Kurokawa and W. Takahashi obtained a weak convergence theorem of Bailon’s type for non-
spreading mapping in Hilbert space, using an idea of mean convergence they proved a strong
convergence theorem for nonspreading mapping in a Hilbert space.
Osilike and Isiogugu [Osilike and Isiogugu, 2011] introduced a new class of nonspreading type
mappings which is more general than the class studied by Kurokawa and Takahashi. Following the
terminology of Browder and Petryshn they called a mapping
T : C ! C k strictly pseudononspreading if there exists k 2 [0; 1) such that
kTx Tyk2 kx yk2 + kkx Tx (y Ty)k2 + 2hx Tx; y Tyi 8x; y 2 C (1.5)
Our main focus in this thesis, is to review the work done by Osilike and Isiogugu in their paper
titled “Weak and strong convergence theorem for nonspreading type mapping in Hilbert space”
which appeared in Nonlinear Analysis, Vol 74 (2011), 1814-1822.
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