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## 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 xed points set. We introduce a

modied Ishikawa iterative algorithm for Lipschitz pseudo-contractive maps and prove

that our new iterative algorithm converges strongly to a xed point of T in real Hilbert

space.

** **

## TABLE OF CONTENTS

## Certication ii

Dedication iii

Acknowledgement iv

Abstract viii

1 Introduction 1

1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Demiclosedness Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Nonlinear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Iterative Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.1 The Picard iteration Method . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Krasnoselskii Iteration Method [6] . . . . . . . . . . . . . . . . . . . 13

1.4.3 The Mann Iteration Process [21] . . . . . . . . . . . . . . . . . . . . 14

1.4.4 The Ishikawa Iteration Process [19] . . . . . . . . . . . . . . . . . . 18

1.4.5 Mann Iteration Process with Errors in the Sense of Liu . . . . . . . 19

1.4.6 Ishikawa Iteration Process with Errors in the Sense of Liu . . . . . 19

vi

1.4.7 The Agarwal-O’Regan-Sahu Iteration Process . . . . . . . . . . . . 20

1.5 ORGANIZATION OF THESIS . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Preliminaries 22

2.1 Denitions and Technical Results About Convergent Sequences of Real

Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.1 Denition (Strong Convergence) . . . . . . . . . . . . . . . . . . . . 22

2.1.2 Denition (Weak Convergence) . . . . . . . . . . . . . . . . . . . . 22

2.2 Projections onto Convex Set . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3 Some Denitions and Results Used in the Main Work . . . . . . . . . . . . 36

2.4 Corollary (Demiclosedness Principle) . . . . . . . . . . . . . . . . . . . . . 38

3 Weak and Strong Convergence of an Iterative Algorithm for Lipschitz

Pseudo-Contractive Maps in Hilbert Spaces 44

3.1 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

References 52

## CHAPTER ONE

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 nally established in 1920’s and 1930’s, on one hand

under the in uence of the study of specic classes of linear operators-integral operators

and integral equations connected with them-and on the other hand under the in uence 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 in uence on the development of Functional Analysis, since its basic concepts, for

example energy, turned out to be linear operators (which physicists at rst rather loosely

interpreted as innite dimensional matrices) on innite 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

1

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 xed points of Lipschitz Pseudo-contractive Maps in Hilbert spaces. We introduced a

modied Ishikawa iterative algorithm and prove that if

F(T) = fx 2 H : Tx = xg 6= ;, then our proposed iterative algorithm converges strongly

to a xed point of T. No compactness assumption is imposed on T and no further require-

ment is imposed on F(T).

We proceed with the denitions of some basic terms, and the introduction of various non

linear operators studied in this project.

Denition 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 2 K is said to be a xed point of T if Tx = x. We shall denote the

set of xed points of T by F(T).

Denition 1.2 (Convex Set) : The set C of a real vector space X is called convex if,

for any pair of points x; y 2 C, the closed segment with extremities x; y 2 C that is, the

set fx + (1 )y : 2 [0; 1]g is contained in C. A subset C of a real normed space is

called bounded if there exists M > 0 such that kxk M 8x 2 C.

Denition 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 2 H the unique point Pkx in K with the property

kx Pkxk = minfkx yk : y 2 Kg.

Lemma 1.1: Given x 2 H and z 2 K. Then z = Pkx if and only if

hx z; y zi 0 for all y 2 K.

As a consequence we have that

(i) kPkx Pkyk2 hx y; Pkx Pkyi for all x; y 2 H; that is, the projection is non

expansive;

(ii) kx Pkxk2 kx yk2 ky Pkxk2 8x 2 H and y 2 K

2

(iii) If K is a closed subspace, then Pk coincides with the orthogonal projection from H

onto K; that is, for x 2 H; x Pkx is orthogonal to K (i.e. hx Pkx; yi = 0 for y 2 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 = fx 2 H : kx uk rg is a closed ball centred at u 2 H with radius r > 0,

then

Pkx =

(

u + r (xu)

kxuk ; ifx =2 K

x; ifx 2 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 =

8<

:

ai; if xi < ai;

xi; if xi 2 [ai; bi];

bi; if xi > bi:

(c.) If K = fy 2 H : ha; yi = g is a hyperplane, with a 6= 0 and 2 <, then

Pkx = x ha;xi

kak2 a.

(d.) If K = fy 2 H : ha; yi g is a closed half space, with a 6= o and 2 <, then

Pkx =

(

x ha;xi

kak2 a; if ha; xi >

x; if ha; xi :

(e) If K is the range of an m n matrix A with full column rank, then

Pkx = A(AA)1Ax

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.

Denition 1.2.1 : A mapping T : K ! H is said to be demiclosed (at y) if the conditions

that fxng converges weakly to x and that fTxng converges strongly to y imply that x 2 K

3

and Tx = y. Moreover, we say that H satises 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 fxng be a sequence

in K. If xn * w and limn!1 kxn Txnk = 0, then Tw = w.

1.3 Nonlinear Mappings

The following denitions contains the nonlinear mappings we are working with and that

will appear throughout the entire chapters.

Denition 1.3.1 Let K be a nonempty subset of a real Hilbert space H. The mapping

T : K ! H is called

(i) Lipschitz or Lipschitz continuous if there exists a constant L 0 such that

kTx Tyk Lkx yk 8 x; y 2 K (1:1)

If L = 1, then T is called nonexpansive; and if L < 1, then T is called a contraction.

It is easy to see from (1.1) that every contraction mapping is nonexpansive and every

nonexpansive mapping is Lipschitz.

Denition 1.3.2: Let K be a non empty subset of a real Hilbert space H.

(i) A mapping T : K ! H is called pseudo-contractive if

hTx Ty; x yi kx yk2 8 x; y 2 K (1:2)

(ii) A mapping T : K ! H is called a strict pseudo-contraction if for all x; y 2 K there

exists a constant 2 [0; 1) such that

kTx Tyk2 6 kx yk2 + k(I T)x (I T)yk2; 8 x; y 2 K (1:3)

4

Inequality (1.2) can be equivalently written as:

kTx Tyk2 kx yk2 + k(I T)x (I T)yk28x; y 2 K (1:4)

i.e. set A = I T

) k(I A)x (I A)yk2 kx yk2 + kAx Ayk2

) hx y (Ax Ay); x y (Ax Ay)i kx yk2 + kAx Ayk2

) kx yk2 + kAx Ayk2 2hAx Ay; x yi kx yk2 + kAx Ayk2

) 2hAx Ay; x yi 0

) hAx Ay; x yi 0

) h(I T)x (I T)y; x yi 0

) kx yk2 hTx Ty; x yi 0

) hTx Ty; x yi kx yk2

Nonexpansive ) strict pseudo-contraction ) pseudo-contraction. However, the following

examples show that the converse is not true.

Example 1.1: Take X = <2; B = fx 2 <2 : kxk 6 1g; B1 = fx 2 B : kxk 6 1

2g;

B2 = fx 2 B : 1

2 6 kxk 6 1g . If x = (a; b) 2 X we dened x? to be (b;a) 2 X.

Dene T : B ! B by

Tx =

x + x?; x 2 B1

x

kxk x + x?; x 2 B2

Then, T is Lipschitz and pseudo-contractive but not strictly pseudo-contractive. Example

(1.1) is due to Chidume and Mutangadura [12].

Proof

Trivially hx; x?i = 0; kx?k = kxk; hx?; yi + hx; y?i = 0; hx?; y?i = hx; yi; kx? y?k =

kx yk 8 x; y 2 H:

We now show that T is imfact Lipschitz. For x; y 2 B1 we have

kTx Tyk2 = kx + x? y y?k2 = hx y + x? y?; x y + x? y?i

= hx y; x y + x? y?i + hx? y?; x y + x? y?i

5

= hx y; x yi + hx y; x? y?i + hx? y?; x yi + hx? y?; x? y?i

= kx yk2 + kx? y?k2 + hx; x?i hx; y?i hy; x?i + hy; y?i = 2kx yk2 + 0

) kTx Tyk =

p

2kx yk

Next, for x; y 2 B2 we have

k(

x

kxk

y

kyk

)k2 = h

x

kxk

y

kyk

;

x

kxk

y

kyk

i

= h

x

kxk

;

x

kxk

y

kyk

i h

y

kyk

;

x

kxk

y

kyk

i

= h

x

kxk

;

x

kxk

i h

x

kxk

;

y

kyk

i h

y

kyk

;

x

kxk

i + h

y

kyk

;

y

kyk

i

=

kxk2

kxk2

2h

x

kxk

;

y

kyk

i +

kyk2

kyk2 = 2 2h

x

kxk

;

y

kyk

i = 2

2hx; yi

kxkkyk

=

2

kxkkyk

fkxkkyk hx; yig =

1

kxkkyk

f2kxkkyk 2hx; yig

=

1

kxkkyk

f2kxkkyk + kx yk2 (kxk2 + kyk2)g

=

1

kxkkyk

fkx yk2 (kxk kyk)2g

) k

x

kxk

y

kyk

k2 6 8kx yk2

) k

x

kxk

y

kyk

k 6

p

8kx yk

Hence, for x; y 2 B2 we have

kTx Tyk = k

x

kxk

x + x?

y

kyk

+ y y?k

6 k

x

kxk

y

kyk

k + kx yk + kx? y?k

=

p

8kx yk + kx yk + kx yk =

p

8kx yk + 2kx yk

6 3kx yk + 2kx yk = 5kx yk

So that T is Lipschitz on B2. Now let x and y be in the interiors of B1 and B2 respectively.

Then there exists 2 (0; 1) for which Z = x + (1 )y. Hence,

kTx Tyk 6 kTx Tzk + kTz Tyk 6

p

2kx zk + 5kz yk

6 5kx yk + 5kz yk = 5kx yk

6

Thus kTx Tyk 6 5kx yk 8 x; y 2 B as required.

We now show that T is a pseudo-contraction. First, we note that we may put j(x) = x,

since H is Hilbert. For x; y 2 B put (x; y) = kx yk2 hTx Ty; x yiand, if x and

y are both non zero, put (x; y) = hx;yi

kxkkyk . Hence to show that T is a pseudo-contraction,

we need to proof that (x; y) > 0 8 x; y 2 B. We only need examine the following three

cases

(i) for x; y 2 B1, we have

hTx Ty; x yi = hx + x? y y?; x yi

= hx y; x yi + hx? y?; x yi

= kx yk2 + hx?; xi hx?; yi hy?; xi + hy?; yi

= kx yk2 + 0 fhx?; yi + hy?; xig + 0 = kx yk2

) hTx Ty; x yi = kx yk2

so that (x; y) = 0;

(ii)For x; y 2 B2 we have

hTx Ty; x yi = h

x

kxk

x + x?

y

kyk

+ y y?; x yi

= h

x

kxk

; xi h

x

kxk

; yi hx; xi + hx; yi + hx?; xi hx?; yi h

y

kyk

; xi

+h

y

kyk

; yi + hx; yi hy; yi hy?; xi + hy?; yi

= kxk

1

kxk

hx; yi kxk2 + hx; yi + 0 hx?; yi

1

kyk

hx; yi + kyk + hx; yi kyk2 hy?; xi + 0

= kxk + kyk kxk2 kyk2

1

kxk

hx; yi + 2hx; yi

fhx?; yi + hy?; xig

1

kyk

hx; yi

= kxk + kyk kxk2 kyk2

1

kxk

hx; yi + 2hx; yi 0

1

kyk

hx; yi

= kxk kxk2 + kyk kyk2 + hx; yi(2

1

kxk

1

kyk

)

= kxk kxk2 + kyk kyk2 +

hx; yi

kxkkyk

f2kxkkyk kyk kxkg

7

= kxk kxk2 + kyk kyk2 + (x; y)f2kxkkyk kyk kxkg

Hence

(x; y) = 2kxk2 + 2kyk2 kxk kyk (x; y)f4kxkkyk kxk kykg

Since

f4kxkkyk kxk kykg > 0 8 x; y 2 B2

We have, for xed kxk and kyk; (x; y) has a minimum when (x; y) = 1.

) (x; y) 6 2kxk2 + 2kyk2 4kxkkyk = 2(kxk kyk)2

Again, we have (x; y) > 0 8 x; y 2 B2 as required.

(iii) For x 2 B1; y 2 B2 we have

hTx Ty; x yi = hx + x?

y

kyk

+ y + y?; x yi

= hx; yi hx; yi + hx?; xi hx?; yi

1

kyk

hy; xi

+

1

kyk

hy; yi + hy; xi hy; yi hy?; xi + hy?; yi

= kxk2 + 0 hx?; yi

1

kyk

hx; yi + kyk

kyk2 hy?; xi + 0

= kxk2 kyk2 + kyk [hx?; yi + hy?; xi]

1

kyk

hx; yi = kxk2 kyk2 + kyk

1

kyk

hx; yi

= kxk2 + kyk kyk2

1

kyk

hx; yikxkkyk

kxkkyk

= kxk2 + kyk kyk2 (x; y)kxk

Hence

(x; y) = 2kxk2 kyk + [kxk 2kxkkyk](x; y)

Since (kxk 2kxkkyk) 6 0 for x 2 B1 and y 2 B2; (x; y) has its minimum, for xed

kxk and kyk when (x; y) = 1. We conclude that

(x; y) > 2kxk2 kyk + kxk 2kxkkyk = (kyk kxk)(2kyk 1) > 0 8x 2 B1; y 2 B2

8

Therefore, T is a pseudo-contraction.

Next, we show that T is not a strictly pseudo-contraction

Let x; y 2 B1 be such that kx yk 6= 0 f for example x = ( 1

4 ; 0); y = (0; 0)g and let

2 [0; 1) be arbitrary. Then kx yk 6= 0 and

kTx Tyk2 = kx + x? (y y?)k2 = kx y + x? y?k2

= kx yk2 + kx? y?k2 + 2hx y; x? y?i = 2kx yk2 (1:5)

Furthermore,

kx Tx (y Ty)k2 = kx? y?k2 = kx yk2 (1:6)

(1.5) and (1.6) imply:

kTx Tyk2 = kx yk2 + kx Tx (y Ty)k2

> kx yk2 + kx Tx (y Ty)k2 8 2 [0; 1):

Therefore, T is not strictly pseudocontractive.

Example 1.2 Take X = <1 and dene T : X ! X by Tx = 3x: Then, T is a strict

pseudocontraction but not nonexpansive mapping.

Proof

jx Tx (y Ty)j2 = 16jx yj2:

Hence,

jTx Tyj2 = 9jx yj2 = jx yj2 + 8jx yj2

= jx yj2 +

1

2

jx Tx (y Ty)j2;

so that T is strictly pseudocontractive with = 1

2 :

Next, we show that T is not non-expansive.

jTx Tyj = j 3x + 3yj = j 3(x y)j = 3jx yj > jx yj; 8 x 6= y

Hence T is not non-expansive.

Remark 1.3 The following example shows that the class of pseudocontractive maps prop-

erly contains the class of nonexpansive maps.

9

It also shows that the class of pseudocontractive maps is more general than the class

of strictly pseudocontractive maps.

Example 1.4: Let < be the reals with the usual norm and K = [0; 1]. Let T : [0; 1] ! [0; 1]

be dened by

Tx = 1 x

2

3 (1:7):

We assert that T is not Lipschitz. To see this, let L > 0 be arbitrary, r = minf1; 1

L3 g, and

consider x 2 (0; r]; y = 0: Then jx yj = jxj = x and

jTx Tyj = j1 x

2

3 1j = x

2

3 = x

1

3 (x)

Since x < 1

L3 , then x

1

3 < 1

L. Thus 1

x

1

3

> L (i.e. x

1

3 > L). Thus

jTx Tyj = x

1

3 (x) > Lx = Ljxj = Ljx yj:

Since T is not Lipschitz, it is not nonexpansive. Let x; y;2 [0; 1], and let x y. Then

x

2

3 y

2

3 and x

2

3 y

2

3 . Thus 1 x

2

3 1 y

2

3 , and this yields Tx Ty.

It follows that if

x y 0; then Tx Ty 0:

Hence

hTx Ty; x yi 0 tjx yj2 8 t > 0

Thus T is pseudocontractive. T is not strictly pseudocontractive because it is not

Lipschitz.

Next, we show that the mapping dened in (1.7) has a xed point. To see this, let x be

the xed point of the given map. Then we have

1 x

2

3 = x

) (1 x)3 = x2

) 1 3x + 3×2 x3 = x2

) x3 2×2 + 3x 1 = 0 (1:8)

10

We want to solve a cubic equation of the form x3 + ax2 + bx + c = 0

Using Cardano’s method of solving a cubic equation, we have x = m a

3 where a is a

coecient of x2

x = m +

2

3

(1:9)

Then (1.8) becomes (m + 2

3 )3 2(m + 2

3 )2 + 3(m + 2

3 ) 1 = 0

) m3 + 3m2

2

3

+ 3m

4

9

+

8

27

2(m2 +

4m

3

+

4

9

) + 3m + 2 1 = 0

) m3 +

5m

3

+

11

27

= 0 (1:10)

Equation (1.10) can be re-written as

m3 + pm + q = 0 (1:11)

where

p =

5

3

; q =

11

27

(1:12)

letting m = u + v, we re-write the above equation (1.12) as

u3 + v3 + (u + v)(3uv + p) + q = 0 (1:13)

Next, we set 3uv + p = 0, and equation (1.13) becomes

u3 + v3 = q

Hence we are left with these equations

u3 + v3 = q and u3v3 = p3

27

Since the above equations are the product and sum of u3 and v3, then there is a quadratic

equation with roots u3 and v3. This quadratic equation is t2 + qt p3

27 = 0 with solutions

u3 =

q+

q

q2+4p3

27

2 and v3 =

q

q

q2+4p3

27

2

11

But from equation (1.12) q = 11

27 ; p = 5

3

) u3 =

11

27 +

q

( 11

27 )2 + 4( 5

3 )3

27

2

=

11

27 +

q

121

729 + 500

729

2

) u3 =

11 + 3

p

69

54

) u = (

11 + 3

p

69

54

)

1

3

and

v3 =

11 3

p

69

54

) v = (

11 3

p

69

54

)

1

3

Therefore,

m = (

11 + 3

p

69

54

)

1

3 + (

11 3

p

69

54

)

1

3

But from equation (1.9) x = m + 2

3

) x = (

11 + 3

p

69

54

)

1

3 + (

11 3

p

69

54

)

1

3 +

2

3

a xed point.

1.4 Iterative Algorithms

In this section, we will present several methods for solving xed point problems. We will

focus on iterative methods (we also call them iterative procedures or algorithms) which

are given in the form of the following recurrences:

1.4.1 The Picard iteration Method

Let X be any set and T : X ! X a self map. For any x0 2 X, the sequence fxngn>0 X

given by

xn = Txn1 = Tnx0; n = 1; 2; :::

12

is called the sequence of successive approximations with the initial value x0. It is also

known as the Picard iteration.

The theorem below, called the Banach xed point or the Banach theorem on contractions,

is widely applied in various areas of mathematics. The theorem holds for any complete

metric space, and hence, in particular, for every closed subset of a Hilbert space.

Theorem 1.2 (Banach, 1922) Let X be a complete metric space and T : X ! X be

a contraction. Then T has exactly one xed point x 2 X. Furthermore, for any x 2 X,

the orbit fTnxg1n

=0 converges to x at a rate of geometric progression.

The Banach xed point theorem is a widely applied tool for an iterative approximation

of xed points. Unfortunately, its application is restricted to contractions. We will need,

however, appropriate tools for an iterative approximation of xed points of non-expansive

operators T with F(T) 6= ?.

Below, we present several classical xed points theorems.

Theorem 1.3 (Brouwer, [8]) Let X < be non empty compact and convex and

T : X ! X be continuous. Then T has a xed point.

The Brouwer xed point theorem was generalized by Juliusz Schauder.

Theorem 1.4 (Schauder, [8]) Let X be a non empty compact and convex subset of a

Banach space and T : X ! X be continuous. Then T has a xed point.

For non-expansive operators in Hilbert space H the compactness of X H in the Schauder

Theorem (1.4) can be replaced by the boundedness and closeness of X.

1.4.2 Krasnoselskii Iteration Method [6]

For x0 2 K and 2 [0; 1] the sequence fxng1 n=0, dened by

xn+1 = (1 )xn + Txn; n = 0; 1; 2; :::

is called Krasnoselkii iterative method and is denoted by Kn(x0; ; T).

13

1.4.3 The Mann Iteration Process [21]

Mann iteration process is essentially an averaged algorithm which generates a sequence

recursively by

xn+1 = (1 n)xn + nTxn; n > 0 (1:14)

where the initial guess x0 2 K and fng is a sequence in (0; 1).

The Mann iteration method has been successfully employed in approximating xed points

(when they exist) of nonexpansive mappings. This success has not carried over to the

more general class of pseudo-contractions. If K is a compact convex subset of a Hilbert

space and T : K ! K is Lipschitz , then, by Schauder xed point theorem, T has a xed

point in K. All eorts to approximate such a xed point by means of the Mann sequence

when T is also assumed to be pseudo-contractive proved to be abortive.

Hicks and Kubicek [18], gave an example of a discontinuous pseudo-contraction with unique

xed point for which the Mann iteration does not always converge. Borwein and Borwein

[7] (proposition 8), gave an example of a Lipschitz map (which is not pseudo-contractive)

with a unique xed point for which the Mann sequence fails to converge. The problem

for Lipschitz pseudo-contraction still remained open. This was eventually resolved in the

negative by Chidume and Mutangadura [12] in the following example.

Example 1.4.3.1

Let H be a real Hilbert space <2 under the the usual Euclidean inner product.

If x = (a; b) 2 H we dene by x? 2 H to be (b;a). Trivially, we have

hx; x?i = 0; kx?k = kxk;

hx?; y?i = hx; yi; kx? y?k = kx yk

and hx?; yi + hx; y?i = 0 for all x; y 2 H.

We take our closed and bounded convex set K to be the closed unit ball in H and putK1 =

fx 2 H : kxk 1

2g, K2 = fx 2 H : 1

2 kxk 1g. We dene the map T : K ! K as

14

follows:

Tx =

x + x?; ifx 2 K1;

x

kxk x + x?; ifx 2 K2

Then T is a Lipschitz pseudo-contractive map of a compact convex set into itself with a

unique xed point for which no Mann sequence converges.

We notice that, for x 2 K1 \ K2, The two possible expressions for Tx coincide and that

T is continuous on both of K1 and K2. Hence T is continuous on all of K. We now show

that T is in fact, Lipschitz. One easily shows that

kTx Tyk =

p

2kx yk for x; y 2 K1. For x; y 2 K2, we have

k(

x

kxk

y

kyk

)k2 =

2

kxkkyk

(kxkkyk hx; yi)

=

1

kxkkyk

fkx yk2 (kxk kyk)2g

1

kxkkyk

2kx yk2 8kx yk2

Hence, for x; y 2 K2, we have

kTx Tyk k

x

kxk

y

kyk

k

+kx yk + kx? y?k 5kx yk;

So that T is Lipschitz on K2. Now let x and y be in the interiors of K1 and K2 respectively.

Then there exist 2 (0; 1) and z 2 K1 \ K2 for which z = x + (1 )y. Hence

kTx Tyk kTx Tzk + kTz Tyk

p

2kx zk + 5kz yk

5kx zk + 5kz yk = 5kx yk

Thus kTx Tyk 5kx yk8x; y 2 K, as required. The origin is clearly a xed point of

T. For x 2 K1; kTxk2 = 2kxk2, and for x 2 K2; kTxk2 = 1 + 2kxk2 2kxk. From these

expressions and from the fact that Tx = x? 6= x if kxk = 1, it is easy to show that the

origin is the only xed point of T. We now show that no Mann iteration sequence for T

15

is convergent for any non zero starting point.

First, we show that no Mann sequence converges to the xed point. Let x 2 K be such

that x 6= 0. Then, in case x 2 K1, any Mann iterate of x is actually further away from

the xed point of T than x is. This is because

k(1)x+Txk2 = (1+2)kxk2 > kxk2 for 2 (0; 1). If x 2 K2 then, for any 2 (0; 1)

k(1 )x + Txk2 = k(

kxk

+ 1 2)x + x?k2

= [(

kxk

+ 1 2)2 + 2]kxk2 > 0:

more generally, it is easy to see that for the recursion formula (1.2.8), if x0 2 K1 then

kxn+1k > kxnk for all integers n > 0, and if x0 2 K2, then kxn+1k >

p

2

2 kxnk for all

integers n > 0. We therefore conclude that, in addition, any Mann iterate of any non zero

vector in K is itself non zero. Thus any Mann sequence fxng, starting from a non zero

vector, must be innite. For such a sequence to converge to the origin, xn would have to

lie in the neighbourhood K1 of the origin for all n > N0, for some real N0. This is not

possible because, as already established for K1, kxnk < kxn+1k for all n > N0.

We now show that no Mann sequence converges to x 6= 0. We do this in the form of a

general lemma.

Lemma 1.2 Let M be a non empty closed and convex subset of a real Banach space

E and let S : M ! M be any continuous function. If a Mann sequence for S is norm

convergent, then the corresponding limit is a xed point for S.

proof. Let fxng be a Mann sequence in M for S, as dened in the recursion formula

(1.2.8). Assume, for proof by contradiction, that the sequence converges, in norm, to

x in M, where Sx 6= x. For each n 2 N put “n = xn Sxn x + Sx. Since S is

continuous, the sequence “n converges to 0. Pick p 2 N such that, if m > p and n > p,

then k”nk < 1

3kx Sxk kxn xmk < 1

3kx Sxk. Pick any positive integer q such that

Pp+q

n=p n > 1. We have that

kxp xp+q+1k = k

Xp+q

n=p

(xn xn+1)k = k

Xp+q

n=p

n(xn Sx + “n)k

16

> k

Xp+q

n=p

n(x Sx)k k

Xp+q

n=p

n”nk

>

Xp+q

n=p

n(kx Sxk)

1

3

kx Sxk) > 2

3

kx Sxk

The contradiction proves the result.

We now show that T is a peudo-contraction. First, we note that we may put j(x) = x,

since H is Hilbert. For x; y 2 K, put (x; y) = kx yk2 hTx Ty; x yi and, if x and

y are both non zero, put (x; y) = hx;yi

kxkkyk . Hence to show that T is a pseudo-contraction,

we need to proof that (x; y) > 0 for all x; y 2 K. We only need examine the following

three cases:

(i) x; y 2 K1: An easy computation shows that hTx Ty; x yi = kx yk2 so that

(x; y) = 0; thus we are home and dry for this case.

(ii)x; y 2 K2: Again, a straight forward calculation shows that

hTx Ty; x yi = kxk kxk2 + kyk kyk2

+hx; yi(2

1

kxk

1

kyk

)

= kxk kxk2 + kyk kyk2

+(x; y)(2kxkkyk kxk kyk):

Hence (x; y) = 2kxk2 + 2kyk2 kxk kyk (x; y)(4kxkkyk kxk kyk)

It is not hard to establish that (4kxkkykkxkkyk) > 08x; y 2 K2. Hence, for xed kxk

and kyk; (x; y) has a minimum when (x; y) = 1. This minimum is therefore 2kxk2 +

2kyk2 4kxkkyk = 2(kxk kyk)2.

Again, we have that (x; y) > 0 8 x; y 2 K2 as required.

(iii) x 2 K1; y 2 K2: we have

hTx Ty; x yi = kxk2 + kyk kyk2 (x; y)kxk. Hence (x; y) = 2kyk2 kyk +

(kxk 2kxkkyk)(x; y). Since kxk 2kxkkyk 0 for x 2 K1 and y 2 K2; (x; y) has its

17

minimum, for xed kxk and kyk when (x; y) = 1. We conclude that

(x; y) > 2kyk2 kyk + kxk 2kxkkyk

= (kyk kxk)(2kyk 1) > 08x 2 K1; y 2 K2:

This complete the proof.

1.4.4 The Ishikawa Iteration Process [19]

In 1974, Ishikawa introduced an iteration scheme which in some sense, is more general

than that of Mann and which converges under this setting, to a xed point of T. He

proved the following result.

Theorem 1.4.4.1 (Ishikawa [19]): If K is a compact convex subset of a Hilbert space

H, T : K ! K is a Lipschitzian pseudo-contractive map and x0 is any point of K, then the

sequence fxngn>0 converges strongly to a xed point of T, where xn is dened iteratively

for each integer n > 0 by

xn+1 = (1 n)xn + nTyn; yn = (1 n)xn + nTxn (1:15)

where fng; fng are sequences of positive numbers satisfying the conditions

(i) 0 n n < 1; (ii) limn!1 n = 0; (iii)

P

n>0 nn = 1.

The iteration method of Theorem 1.4.4.1 which is now referred to as the Ishikawa iterative

method has been studied extensively by various authors.The important thing here is that

Ishikawa yielded strong convergence (Normal Mann only yields weak convergence). But it

is still an open question whether or not this method can be employed to approximate xed

points of Lipschitz pseudo-contractive mapping without the compactness assumption on

K or T (see, e.g., [13,22,21]).

In order to obtain a strong convergence theorem for pseudo-contractive mappings without

the compactness assumption, Zhou [43] established the hybrid Ishikawa algorithm for

Lipschitz pseudo-contractive mappings as follows:

18

8>>>><

>>>>:

yn = (1 n)xn + nTxn;

Zn = (1 n)xn + nTyn;

Kn = fz 2 K : kzn zk2 kxn zk2 nn(1 2n L22n

)kxn Txnk2g Qn = fz 2 K : hxn z; x0 xni > 0g;

xn+1 = Pkn

T

Qnx0; n > 1:

He proved that the sequence fxng dened by (1.3.0) converges strongly to Pkx0, where Pk

is the metric projection from H into K.

In [31] Rhoades compared the performance of these two iteration processes and showed

that despite they are similar, they may exhibit dierent behaviours for dierent classes of

nonlinear mappings. In its original form the Ishikawa iteration process does not include

Mann process as a special case because of the condition 0 n n 1. In an eort to

have an Ishikawa type iteration process which does include the Mann process as a special

case, several authors have modied the inequality condition to read 0 n,n 1.

In 1995, Liu [23] introduced the Ishikawa and Mann iteration processes with “errors” as

follows:

1.4.5 Mann Iteration Process with Errors in the Sense of Liu

Let T : E ! E be a given map. Then the Mann iteration process with errors in the sense

of Liu is given by

xn+1 = (1 n)xn + nTxn + un; n 0

where fng1 n=1 is a suitable real sequence in [0; 1] and fung1 n=1 is a summable sequence in

E (i:e:;

1P

n=1

jjunjj < 1).

1.4.6 Ishikawa Iteration Process with Errors in the Sense of Liu

LetE be a real Banach Space, T : D(T) E ! E and x0 2 D(T). The Ishikawa iteration

process with errors is given by

yn = (1 n)xn + nTxn + un; n 0

19

xn+1 = (1 n)xn + nTyn + vn; n 0

where fng1 n=0, fng1 n=0 are suitable sequences in [0; 1] and fung1 n=0, fvng1 n=0 are summable

sequences in E.

Several authors have studied these iteration processes with errors and observed that it is

unsatisfactory because the conditions

X1

n=0

jjunjj < 1; and

X1

n=0

jjvnjj < 1

imply that error terms tend to zero. This is incompatible with the randomness of the

occurence of errors. Furthermore, if the operator T is dened on a nonempty convex

subset K of a normed linear space E, then the Mann and Ishikawa iteration process with

errors are not in general well dened.

1.4.7 The Agarwal-O’Regan-Sahu Iteration Process

The sequence fxng is dened by

8<

:

x1 2 K;

yn = nxn + (1 n)Txn;

xn+1 = nTxn + (1 n)Tyn; n > 1

where fng and fng are sequence in [0,1] is known as Agar-O’Regan-Sahu [19] iteration

scheme.

Theorem 1.4.7.1 [37] Let K be a non empty closed convex subset of a real Hilbert space

H, S : K ! K be non-expansive and T : K ! K be Lipschitz strongly pseudo-contractive

mappings respectively such that

F(S) \ F(T) = fx 2 K : Sx = x = Txg 6= ? and satisfying

kx Syk kSx Syk,

kx Tyk kTx Tyk for all x; y 2 K:

Let fngn>1; fngn>1 2 [0; 1] be such that

(i) n n

(ii)

P

n>1 n = 1 and

20

(iii)

P

n>1 n < 1:

For arbitrary x1 2 K let fxngn>1 be iteratively dened by

yn = (1 n)xn + nTxn;

xn+1 = (1 n)Syn + nTxn; n > 1:

Then fxngn > 1 converges strongly to the common xed point p of S and T.

corollary 1.4.7.2 [37] Let K be a nonempty closed convex subset of a real Hilbert

space H and T; S : K ! K be two Lipschitz pseudo-contractive mappings such that

p 2 F(T) \ F(S) = fx 2 K : Tx = x = Sxg. Let fng and fng be sequences in [0,1]

satisfying limn!1(1n) = 0: For arbitrary x1 2 K; let fxng1 n=1 be the sequence dened

iteratively by

yn = nxn + (1 n)Txn; n > 1

xn+1 = nTxn + (1 n)Syn;

Then the following are equivalent:

(a) fxng1 n=1 converges strongly to the common xed point q of T and S.

(b) fTxng1n

=1 and fSyng1 n=1 are bounded.

1.5 ORGANIZATION OF THESIS

We have introduced in Chapter one, various iterations methods for Lipschitz pseudo-

contractive maps and some existing results on them. We have also considered various

non-linear operators studied in this project.

In Chapter Two (the preliminary section), we presented most of the technical results about

convergent sequences of real numbers encountered in Operator Theory. We also considered

projection maps and some other results vital to this work.

Chapter Three featured the main results of this project. Using a modied Ishikawa it-

eration method we proved strong convergence theorem for Lipschitz pseudo-contractive

maps in Hilbert spaces. Our result extended the work of Yao and Li [20] from the class

of – strictly pseudo-contractive operators to the more general class of Lipschitz pseudo-

contractive operators.

21