Weak and Strong Convergence Theorems for Nonspreading Type Mapping in Hilbert Spaces
Chapter One
PREAMBLE OF THE STUDY
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 fields 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 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:
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}∞
n=1 to a point
x ∈ H by xn * x, and the strong convergence of {xn}∞
n=1 to a point x ∈ H by xn → x.
Definition 3.1. Let C be a nonempty subset of a real Hilbert space H, and let T : C −→ C be a
mapping. A point x0 is a fixed point of C if T x0 = x0.
Definition 3.2. A mapping T : D(T) ⊆ H −→ H is said to be l-Lipschitzian if there exists l > 0
such that
kT x − T yk ≤ lkx − yk, ∀x, y ∈ D(T) (3.1)
Definition 3.3. If l < 1 in inequality (3.1) then T is said to be strictly contractive.
Definition 3.4. If l = 1 in inequality (3.1) then T is nonexpansive.
Definition 3.5. T is strongly nonexpansive if T is nonexpansive and xn − yn − (T xn − T yn) → 0
as n → ∞, whenever {xn}∞
n=1 and {yn}∞
n=1 ⊂ C such that {xn − yn}∞
n=1 is a bounded sequence
and kxn − ynk − kT xn − T ynk → 0 as n → ∞.
Definition 3.6. T is said to be quasi-nonexpansive if F(T) 6= ∅ and
kT x − pk ≤ kx − pk, ∀ x ∈ D(T), ∀ p ∈ F(T).
CHAPTER FOUR
MAIN RESULT
k-strictly Pseudo nonspreading Mapping
Let H be a real Hilbert space, following the terminology of Browder-Petryshyn, we say that a mapping T : D(T) ⊆ H −→ H is k-strictly pseudo nonspreading if there exists k ∈ [0, 1) such that
kT x − T yk
2 ≤ kx − yk
2 + kkx − T x − (y − T y)k
2 + 2hx − T x, y − T yi ∀ x, y ∈ D(T).
Clearly, every non spreading mapping is 0-strictly pseudo nonspreading. The following example shows that the class of k-strictly pseudo nonspreading is a generalization of the class of non spreading mappings.
CHAPTER FIVE
SUMMARY
In the first chapter, we discussed fixed point theory and its applications to several areas of research such as in Optimization, Economics, Evolution Equations e.t.c; we discussed the intimate relationship between fixed points of nonexpansive mappings and equilibrium points of certain dynamical systems. We also introduced the history of the class of nonspreading mappings and the class of strictly pseudononspreading mappings in Hilbert spaces.
In the second chapter, we gave a review of other works done in the area of research carried out in
this thesis.
In the third chapter, we gave the definitions of some basic terms used in this thesis; we gave some basic facts concerning the class of k-strictly pseudononspreading mappings in Hilbert spaces with examples and at the end of the third chapter we presented some preliminary results used in this thesis.
In the fourth chapter, we presented with examples, the class of k-strictly pseudononspreading mappings in Hilbert spaces and its relationship with the class of nonspreading mappings in Hilbert spaces. We further presented the main results in the work of Osilike and Isiogugu, Nonlinear Analysis, 74 (2011), 1814-1822 (i.e weak and strong convergence theorems for nonspreading type mapping in a Hilbert space).
In the last chapter, we gave a summary of the thesis in general.
BIBLIOGRAPHY
- [Kohsaka et al., 2008] F. Kohsaka and W. Takahashi (2008); existence and approximation of fixed points of firmly nonexpansive-type mapping in Banach spaces Arch. Math. 91, 166 177.
- [Bruck,1980] R.E Bruck (1980); asymptotic behaviour of nonexpansive mappings,contemporary mathematics, 18, fixed points and nonexpansive mappings, (R. C. sine, editor), AMS, providence, RI.
- [Bruck, 1973] R.E Bruck; properties of fixed point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math.Soc., vol. 179.
- [Kirk, 1965] W. A. Kirk; a fixed point theorem for mappings which do not increase, Amer. Math. Soc. 72 , 1004 – 1006.
- [Karlovitz, 1976] L. A Karlovitz; existence of fixed points of nonexpansive mappings in a space without normal structure, pacci. J. math , vol. 66, no. 1 , 153 – 159.
- [Soardi, 1979] P. M .Soardi; existence of fixed points of nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. soc, vol. 73, no. 1, pp. 25 – 29.
- [Chidume, 2009] C. E. Chidume,Geometric properties of Banach spaces and nonlinear iterations, vol. 1965 of lecture notes in mathematics, springer, London, UK.
