Applied Mathematics, 2011, 2, 1369-1371

doi:10.4236/am.2011.211192 Published Online November 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

The Equivalence between the Mann and Ishikawa

Iterations for Generalized Contraction Mappings in a Cone

L. Jones Tarcius Doss, T. Esakkiappan

Department of Mat hematics, Anna University, Chennai, India

E-mail: tesakki@gmail.com

Received July 22, 201 1; revised Septe mber 13, 2011; accepted September 20, 2011

Abstract

In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping

in cone subset of a real Banach space is discussed.

Keywords: Mann Iteration, Ishikawa Iterations, Generalized Contraction, Cone

1. Introduction

Generally, the iteration techniques of W.R.Mann [1] and

Shiro Ishikwa [2] are used to find the approximation of

fixed point of a contraction mapping. These iterations are

quite useful even for the cases of where Picard iteration

fails. In this paper, we see the equivalence between these

Mann and Ishikawa iterations for a generalized contra-

ction mapping in a cone. First, we recall the definition of

a cone (refer Huang Long-guang and Zhang Xian [3])

and some of its properties.

Definition 1.1: Let be a real Banach space and a

subset of is said to be a cone if satisfies the

following:

E

P E

1) , P is closed and ;

P

ax by {0}P

,2) for every

P

yP and ; ,0ab

3) .

()=PP{0}

The partial ordering with respect to the cone P is

defines by

y if and only if

xP. We shall

write <

y to indicate that

y

but

y. Further

y will stand for

xintP, where in de-

notes interior of . We now define the generalized

contraction mapping. Let be a real Banach space,

a nonempty convex cone subset of E. Let a self

map of with the property that

tP

PE

P T

P

(, )TxTyMx y

(1.1)

where

and (, )

xy satisfy the following:

1) :[0,)[0,)isareal-valued,nondecreasing,

rightcontinuous function;

(1.2)

2) ()<foreach>0;

ttt (1.3)

3) isnondecreasingon(0,);

(1.4)

4) ():=(())isnonincreasingon(0,);

gtt tt (1.5)

5) (,):=

max, ,,,

Mxy

yxTxyTyxTyyTx

(1.6)

T satisfying above conditions is said to be a Generalized

contraction. Below, we see the definition of the two

iteration schemes due to Mann [1] and Ishikawa [2]. Further,

these two iterations are applied to a class of generalized

contraction mapping which is mentioned just above.

Let 00

uP

. The Mann iteration is defined by

1=(1)

nnnn

uu

n

Tu

. (1.7)

The Ishikawa iteration is defined by

1=(1),

=(1),

nnnn

nnnn

n

n

xTy

yxT

x

(1.8)

where

(0,1), 0,1

nn

. Clearly, the sequences

n

x,

n

u and

n

are in because 00

P=

uP

and

(0,1)

n

and

[0

n

,1) and from the defi-

nition of cone.

Let

n

w

li

be a sequence in P which is a subset of a

real Banach space. We say that converges to

n

ww

and write if

m =

n

nww

lim= 0

n

nww where .

is the norm associated with .

E

The main aim of this paper is to show that the con-

vergence of Mann iteration is equivalent to the con-

vergence of Ishikawa iteration in the cone .

P

Below, we sate two results without proof which are

very much useful for our analysis. for proof, one may

refer [4] and [5] respectively.

Lemma 1 [4]

Let

n

a be a nonnegative sequence which satisfies