Weak and Strong Convergence Theorems for Nonspreading Type Mapping in Hilbert Spaces Chapter One PREAMBLE OF THE STUDY

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 definitions 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 five, conclusions will be given.

CHAPTER TWO
LITERATURE REVIEW
Introduction
In this chapter, we review other works done in the area of research carried out in this thesis; so that the results obtained in this work will be well appreciated.
Review
The origin of nonspreading type mapping is traced as far as 2008, when Fumiaki Kohsaka and Wataru Takahashi considered the class of nonspreading mappings to study the resolvents of a maximal monotone operator in Banach spaces [Kohsaka et al., 2008]. This class of mappings contains the important class of firmly nonexpansive mappings, (i.e., a mapping T : D(T) ⊂ H −→ H such that kT x − T yk
2 ≤ hx − y, T x − T yi, ∀ x, y ∈ D(T)).
Firmly nonexpansive mappings have intimate connection with maximal monotone operators on Hilbert spaces where an operator T : D(T) ⊂ H −→ 2
H with effective domain
D(T) = {x ∈ H : T x 6= ∅} is maximal monotone if
hu − v, x − yi ≥ 0, ∀ x, y ∈ D(T), u ∈ T x, v ∈ T y and its graph
G(T) = {(X, u) : x ∈ D(T), u ∈ T x} is not properly contained in the graph of any other monotone
operator.
It is proved in [Kohsaka et al., 2008] that if T is maximal monotone then the resolvent Jλ =
(I + λT)
−1 is singled valued and firmly nonexpansive, where λ > 0 and I is the identity in H.
Furthermore, F(Jλ) = T
−10 = {x ∈ D(T) : 0 ∈ T x}. Thus, the problem of finding zeros of maximal monotone operator in Hilbert space is reduced to fixed point problem for firmly nonexpansive mappings.
The class of firmly nonexpansive mappings is a proper subclass of nonexpansive mappings. For the class of nonexpansive mappings, apart from being an obvious generalization of the contraction mappings, they are important, as has been observed by Bruck [Bruck,1980], for the following two
reasons:
• Nonexpansive maps are intimately connected with the monotonicity methods developed since the early 1960s and constitute one of the first classes of nonlinear mappings for which fixed point theorems were obtained by using the geometric properties of the underlying Banach spaces instead of compactness properties.
• Nonexpansive mappings appear in applications as the transition operators for initial value problems of differential inclusions of the form 0 ∈ du
dt + A(t)u, where the operator {A(t)} are, in general, set-valued and are accretive or dissipative and minimally continuous.
Nonexpansive mappings have been studied extensively by numerous authors (see e.g. [Bruck, 1973],
[Kirk, 1965],[Karlovitz, 1976], [Soardi, 1979]). Unlike as in the case of contraction mappings, trivial example shows that for a nonexpansive map T, mapping from a complete metric space into itself with F(T) = {x ∈ D(T) : T x = x} 6= ∅, the Picard iterative sequence xn+1 = T xn, x0 ∈ D(T), n ≥ 0, may fail to converge even when T has a unique fixed point. It suffices, for example, to take T to be the anti-clockwise rotation of the closed unit ball in
R 2 around the origin of coordinates through a fixed acute angle. Clearly, T is nonexpansive, zero is the fixed point of T and the Picard iterative sequence xn+1 = T xn, x0 ∈ D(T), n ≥ 0, does
not converge to zero [Chidume, 2009].
Krasnoselskii [Krasnoselskii et al., 1957], however, showed that in this example, if for any fixed element of the ball, the sequence {xn}∞ n=1 is generated by
xn+1 = (xn + T xn), n ≥ 0, (2.1)
then {xn} converges strongly to the unique fixed point of T. Schaefer [Schaefer, 1957], gave a generalization of this scheme which has successfully been employed in approximating fixed points of nonexpansive maps mapping from a nonempty closed convex subset of a real normed space into itself. The recurrence relation of Schaefer is given by:

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CHAPTER THREE
PRELIMINARIES
 Definition of some terms
Let H be a real Hilbert space, D(T) be domain of T, R(T) be range of T, and F(T) be fixed point
set of T. Let {xn}∞
n=1 be a sequence in H, we denote the weak convergence of {xn}&i

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