OF RELAXED LIPSCHITZ OPERATORS

Journal

of

^Applied Mathematics and Stochastic Analysis, ^1{}:2

(1997),

^187-189.

AN ITERATIVE ALGORITHM ON FIXED POINTS

OF RELAXED LIPSCHITZ OPERATORS

RAM U. VERMA

International

Publications, 1205

^CoedDrive

Orlando,

^Florida 32825

USA

and

Istituto per la Ricerca di Base 1-85075Monteroduni

(IS),

Molise, Italy

(Received

^February,

^1996;

^Revised

November, 1996)

Fixed pointsofLipschitzian relaxed Lipschitz operators based on a

general-

ized iterative algorithm are approximated.

Key

words: Iterative Algorithm, Fixed

Point,

Relaxed Lipschitz

Operator.

AMS

subjectclassifications: 47H10.

1. Introduction

Recently, Wittman

[6,

^Theorem

2],

using an iterative procedure

x_n

(1 an)X

0q-

anTx

n 1 for n

_> 1, (1)

approximated fixed points of nonexpansive mappings T:K--,K from a nonempty closed convex subset

K

ofa real Hilbert space

H

into itself, where x₀is an element of g and

{an}

^isan increasing sequence in

[0, 1)

^{such that}

nliman

^{1 and}_n=l

E ^{(1 an)}

^oo.

⁽²⁾

This result refines anumber of results including

[1].

Here

our aim is to approximate the fixed points ofLipschitzian relaxed Lipschitz operators in a Hilbert space setting.

As such,

the iterative algorithm

(1)

^{is not}

suitable for our purpose, so we apply a modified iterative algorithm which reduces to

(1).

Let H

be a Hilbert space and

(u,v)

^and

II

^u

^{[I denote,}

respectively, the inner productand norm on

H

for u,^v in

H.

An

operator

T:H---.H

is said to be relaxed Lipschitz if, ^for all u,v in

H,

there exists a constant r

>

0 such that

Printed in theU.S.A. ()1997byNorth Atlantic SciencePublishing Company 187

188

RAM U. VERMA

The operator

T

is called Lipschitz continuous

(or Lipschitzian)

if there exists a constant s

>

^{0 such that}

IITu-Tvll <_s[lu-vII

^for

allu,

vinH.

(4) Next,

^we considerthe main result on the approximation of the fixed points ofLip- schitzian relaxed Lipschitz operators using a modified iterative algorithm which con- tains a number of iterative schemes including those considered by the author

[4, 5]

^as

special ^cases.

2. The Main Result

Theorem 1: Let

H

be a real Hilbert space and

K

be a nonempty closed convex subset

of ^{H. Let}

^{T:K-,K be a} relaxed Lipschitz and Lipschitz continuous operator on

K.

Let

r

>_

0 and s

>_

1 be constants

for

relaxed Lipschitzity and Lipschitz continuity

of T,

respectively.

Let F- {x

ⁱⁿ

^K:Tx- x}

^{be nonempty, and let}

{an}

^{be a sequence}

in

[0, 1]

^{such that}

E ^an

^oc

^for

^alln

_

^O.

⁽⁵⁾

n’-O

Then

for

^{any x}o ⁱⁿ

K

the sequence

{Xn} defined

^by

Xn --

¹

^{(1 an)x}

ⁿ

^{+ an[(1 t)x}

ⁿ

^{+ tTxn] for}

ⁿ

^{>_ 0, (6)}

0

<

k

((1 t)

^2-

2t(1 t)r + t2s2)

^1/2

^<

¹

for

^{all t such that 0}

^<

^t

< 2(1 +

(1 +

^2r

+

^s

2)

^andr

<_

s, converges to an element

ofF.

For {an}- ^1,

^{Theorem 1 reducesto:}

Corollary 1:

Let T:K--K

be relaxed Lipschitz and Lipschitz continuous.

Let F--{x

ⁱⁿ

K:Tx-x}

^{be a nonempty set.}

Then, for

^xo ⁱⁿ

K,

the sequence

{xn}

generated by an iterative algorithm

Xn --

¹

^{(1 t)x}

ⁿ

^{+ tTx}

ⁿ

⁽⁷⁾

for

⁰

<

^t

< 2(1 + r)/(1 +

^2r

+

^s

2)

converges to a unique

fixed

^point

of

^T.

Proof ofTheorem 1"

For

an element z in

F,

we have

II _{Xn +}

^{1 z}

^{I[ II(}

¹

^an)Xn -- ^{an[(1 t)Xn + tTxn]-}

^z

^II

_< (1 an)II (xn z)II + an ^]l(

¹

^{t)(xn z)} + t(Tx _{n- Tz)} II

Using the relaxed Lipschitzity and Lipschitz continuity of

T,

wefindthat

II ^{t(Tx,- Tz) + (1} t)(xn- z)II

²

(1 t)

²

II

^z

II

²

+ 2t(1 t)(Tx

n z,x_n

z} +

^t2

II Txn-

^z

II

²

_< (1 t)

²

]]

^x

_n-

^z

II

^2-

^{2t(1 t)r} II xn-

^z

^{I] + t2s}

²

II xn-

^z

II

²

An

Iterative Algorithm on Fixed Points

of

Relaxed Lipschitz

Operators

189

((1 t)

²

2t(1 t)r + ^t2s 2) I] _Xn

^z

]] 2.

It

follows that

II _{Xn +}

1 Z

I] (1

^an

+ a,((1 t)

²

2t(1 t)r + t2s2) 1/2) II xn

^z

II

(1 -(1- k)an)II xn-

^z

II

n

-< H(1-(1-k)aj)llx0-zll’

j=O

where 0

<

k

((1- t)

^2-

2t(1 t)r + t2s2)

^1/2

<

1 for all t such that 0

<

^t

< 2(1

(1 +

^2r

+

^s

2)

_(x)^{and r}

_

^s. _n

Since^j=0 aj diverges and k

< 1, nli__,m . ^{(1-(1- k)aj)-}

⁰

^and,

^{as a}

^result,

converges

strongly

to z. This completes theproof.

References [1]

[4]

Halpern,

B.,

Fixed points of nonexpanding maps, Bull.

A

mer. Math.

Soc.

73

(9),

^9V-91.

Reich,

S.,

Approximating fixed points of nonexpansive maps, PanAmerican Math

J.

4:2

(1994),

^23-28.

Verma, R.U.,

^Iterative algorithms for approximating fixed points of

strongly

monotone

operators,

Boletin

A

cad. Cien. Fis.

Mat. Natur. (to appear).

Verma, R.U., A

fixed point theorem involving Lipschitzian generalized pseudo-

contractions, Proc.

Royal Irish

A

cad.

(to appear).

Verma, R.U., An

iterative

procedure

for approximating fixed points of relaxed monotone

operators, Numer.

FunctionalAnal. Optimiz. 17

(1996),

^1045-1051.

Wittmann,

R.,

Approximation of fixed points ofnonexpansive mappings Arch.

Math. 58

(1992),

486-491.