WEAK AND STRONG CONVERGENCE OF AN ITERATIVE ALGORITHM FOR LIPSCHITZ PSEUDO-CONTRACTIVE MAPS IN HILBERT SPACES

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.

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