ABSTRACT
Let H be a real Hilbert space and K a nonempty, closed convex subset of H.Let T : K → K be Lipschitz pseudo-contractive map with a nonempty fixed points set. We introduce a modified Ishikawa iterative algorithm for Lipschitz pseudo-contractive maps and prove that our new iterative algorithm converges strongly to a fixed point of T in real Hilbert space.
Chapter 1
Introduction
1.1 General Introduction
The contribution of this thesis falls under a branch of mathematics called Functional Analysis. Functional Analysis as an independent mathematical discipline started at the turn of the 19th century and was finally established in 1920’s and 1930’s, on one hand under the influence of the study of specific classes of linear operators-integral operators and integral equations connected with them-and on the other hand under the influence of the purely intrinsic development of modern mathematics with its desire to generalize and thus to clarify the true nature of some regular behaviour. Quantum Mechanics also had a great influence on the development of Functional Analysis, since its basic concepts, for example energy, turned out to be linear operators (which physicists at first rather loosely interpreted as infinite dimensional matrices) on infinite dimensional spaces. In the early stages of the development of Functional Analysis the problems studied were those that could be stated and solved in terms of linear operators on elements of the space alone. But as the concept of a space was being developed and deepened, the concept of a function was being developed and generalized. In the end, it became necessary to consider mapping (not necessary linear) from one space into another. One of the central problems in non- linear Functional Analysis is the study of such mappings. In the modern view, Functional Analysis is seen as the study of complete normed vector spaces over the real or complex
numbers. Such studies are narrowed to the study of Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product.
This project sets to solve the problem of constructing an iterative scheme for approximat- ing fixed points of Lipschitz Pseudo-contractive Maps in Hilbert spaces. We introduced a modified Ishikawa iterative algorithm and prove that if
F (T ) = {x ∈ H : T x = x} = ∅, then our proposed iterative algorithm converges strongly to a fixed point of T . No compactness assumption is imposed on T and no further require- ment is imposed on F (T ).
We proceed with the definitions of some basic terms, and the introduction of various non linear operators studied in this project.
Definition 1.1 : Let K be a non empty subset of a real normed space E and let T : K → K be a map. A point x ∈ K is said to be a fixed point of T if T x = x. We shall denote the set of fixed points of T by F (T ).
Definition 1.2 (Convex Set) : The set C of a real vector space X is called convex if, for any pair of points x, y ∈ C, the closed segment with extremities x, y ∈ C that is, the set {λx + (1 − λ)y : λ ∈ [0, 1]} is contained in C. A subset C of a real normed space is called bounded if there exists M > 0 such that kxk ≤ M ∀x ∈ C.
Definition 1.3 : Let K be a non-empty closed convex subset of a Hilbert space H. The (metric or nearest point) projection onto K is the mapping Pk : H → K which assigns to each x ∈ H the unique point Pkx in K with the property
kx − Pkxk = min{kx − yk : y ∈ K}.
Lemma 1.1: Given x ∈ H and z ∈ K. Then z = Pkx if and only if
hx − z, y − zi ≤ 0 for all y ∈ K. As a consequence we have that
k |
(i) kPkx − Pky 2
expansive;
≤ hx − y, Pkx − Pkyi for all x, y ∈ H; that is, the projection is non
k |
(ii) kx − Pkx 2
≤ kx − yk2
− ky − Pkxk2
∀x ∈ H and y ∈ K
2
(iii) If K is a closed subspace, then Pk coincides with the orthogonal projection from H onto K; that is, for x ∈ H, x − Pkx is orthogonal to K (i.e. hx − Pkx, yi = 0 for y ∈ K). If K is a closed convex subset with a particularly simple structure, then the projection Pk has a closed form expression as described below:
(a.) If K = {x ∈ H : kx − uk ≤ r} is a closed ball centred at u ∈ H with radius r > 0,
then
Pkx =
( u + r (x−u) , ifx / K
kx−uk
∈ |
x, ifx ∈ K.
(b.) If K = [a, b] is a closed rectangle in <n, where a = (a1, a2, …, an)T and b = (b1, b2, …, bn)T where T is the transpose, then, for 1 ≤ i ≤ n, Pkx has the ith coordinate
given by
(Pkx)i =
 ai, if xi < ai,
xi, if xi ∈ [ai, bi],
 bi, if xi > bi.
(c.) If K = {y ∈ H : ha, yi = α} is a hyperplane, with a = 0 and α ∈ <, then
kak2 a. |
Pkx = x − ha,xi−α
(d.) If K = {y ∈ H : ha, yi ≤ α} is a closed half space, with a = o and α ∈ <, then
( x − ha,xi−α a, if ha, xi > α
 |
Pkx =
kak2
x, if ha, xi ≤ α.
(e) If K is the range of an m × n matrix A with full column rank, then
Pkx = A(A∗A)−1A∗x
where A∗ is the adjoint of A.
1.2 Demiclosedness Principles
A fundamental result in the theory of nonexpansive mappings is Browder’s demiclosedness principle.
Definition 1.2.1 : A mapping T : K → H is said to be demiclosed (at y) if the conditions that {xn} converges weakly to x and that {T xn} converges strongly to y imply that x ∈ K
3
and T x = y. Moreover, we say that H satisfies the demiclosedness principle if for any closed convex subset K of H and any nonexpansive mapping T : K → H, the mapping I − T is demiclosed.
The demiclosedness principle plays an important role in the theory of non expansive map- pings (and other classes of non linear mappings as well). In 1965, Browder [9] gave the following demiclosed principle for non expansive mappings in Hilbert spaces.
Theorem 1.1(Browder [9]) Let K be a non empty closed convex subset of a real Hilbert space H. Let T be a non expansive mapping on K into itself, and let {xn} be a sequence in K. If xn * w and limn→∞ kxn − T xnk = 0, then T w = w.
This material content is developed to serve as a GUIDE for students to conduct academic research
WEAK AND STRONG CONVERGENCE OF AN ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDO-CONTRACTIVE MAPS IN HILBERT SPACES>
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