Open Access

On -Stability of Picard Iteration in Cone Metric Spaces

Fixed Point Theory and Applications20092009:751090

DOI: 10.1155/2009/751090

Received: 28 March 2009

Accepted: 19 October 2009

Published: 28 October 2009

Abstract

The aim of this work is to investigate the -stability of Picard's iteration procedures in cone metric spaces and give an application.

1. Introduction and Preliminary

Let be a real Banach space. A nonempty convex closed subset is called a cone in if it satisfies the following:

(i) is closed, nonempty, and ,

(ii) , and imply that

(iii) and imply that

The space can be partially ordered by the cone ; by defining, if and only if . Also, we write if int , where int denotes the interior of .

A cone is called normal if there exists a constant such that implies .

In the following we always suppose that is a real Banach space, is a cone in , and is the partial ordering with respect to .

Definition 1.1 (see [1]).

Let be a nonempty set. Assume that the mapping satisfies the following:

(i) for all and if and only if ,

(ii) for all ,

(iii) for all .

Then is called a cone metric on , and is called a cone metric space.

Definition 1.2.

Let be a map for which there exist real numbers satisfying . Then is called a Zamfirescu operator if, for each pair , satisfies at least one of the following conditions:

(1) ,

(2) ,

(3) .

Every Zamfirescu operator satisfies the inequality:
(1.1)

for all , where max , with . For normed spaces see [2].

Lemma 1.3 (see [3]).

Let and be nonnegative real sequences satisfying the following inequality:
(1.2)

where for all and as Then

Remark 1.4.

Let and be nonnegative real sequences satisfying the following inequality:
(1.3)

where for all and for some positive integer number . If as Then

Lemma 1.5.

Let be a normal cone with constant , and let and be sequences in satisfying the following inequality:
(1.4)

where and as Then

Proof.

Let be a positive integer such that By recursion we have
(1.5)
so
(1.6)

and then by Remark 1.4?? Therefore

2. -Stability in Cone Metric Spaces

Let be a cone metric space, and a self-map of . Let be a point of , and assume that is an iteration procedure, involving , which yields a sequence of points from .

Definition 2.1 (see [4]).

The iteration procedure is said to be -stable with respect to if converges to a fixed point of and whenever is a sequence in with we have

In practice, such a sequence could arise in the following way. Let be a point in . Set . Let . Now . Because of rounding or discretization in the function , a new value approximately equal to might be obtained instead of the true value of . Then to approximate , the value is computed to yield , an approximation of . This computation is continued to obtain an approximate sequence of .

One of the most popular iteration procedures for approximating a fixed point of is Picard's iteration defined by . If the conditions of Definition 2.1 hold for then we will say that Picard's iteration is -stable.

Recently Qing and Rhoades [5] established a result for the -stability of Picard's iteration in metric spaces. Here we are going to generalize their result to cone metric spaces and present an application.

Theorem 2.2.

Let be cone metric space, a normal cone, and with If there exist numbers and such that
(2.1)

for each and in addition, whenever is a sequence with as , then Picard's iteration is -stable.

Proof.

Suppose and We shall show that Since
(2.2)

if we put and in Lemma 1.5, then we have

Note that the fixed point of is unique. Because if is another fixed point of , then
(2.3)

which implies

Corollary 2.3.

Let be a cone metric space, a normal cone, and with If there exists a number such that for each then Picard's iteration is -stable.

Corollary 2.4.

Let be a cone metric space, a normal cone, and is a Zamfirescu operator with and whenever is a sequence with as , then Picard's iteration is -stable.

Definition 2.5 (see [6]).

Let be a cone metric space. A map is called a quasicontraction if for some constant and for every there exists such that

Lemma 2.6.

If is a quasicontraction with , then is a Zamfirescu operator and so satisfies (2.1).

Proof.

Let for every we have for some In the case that we have
(2.4)
So
(2.5)

Put so The other cases are similarly proved. Therefore is a Zamfirescu operator.

Theorem 2.7.

Let be a nonempty complete cone metric space, be a normal cone, and a quasicontraction and self map of with some Then Picard's iteration is -stable.

Proof.

By [6, Theorem??2.1], has a unique fixed point Also satisfies (2.1). So by Theorem 2.2 it is enough to show that We have
(2.6)
Put and Therefore as and
(2.7)
where
(2.8)

Hence we have or where or and or Therefore by (2.7), by Now by Lemma 1.5 we have

3. An Application

Theorem 3.1.

Let with for and let be a self map of defined by where

(a) is a continuous function,

(b)the partial derivative of with respect to exists and for some

(c)for every real number one has for every

Let be a normal cone and the complete cone metric space defined by where Then,

(i)Picard's iteration is -stable if ,

(ii)Picard's iteration fails to be -stable if and

Proof.
  1. (i)

    We have being a continuous quasicontraction map with so by Theorem 2.7, Picard's iteration is -stable.

     
  2. (ii)
    Put so and where Also since
    (3.1)
     

as But and is not a fixed point for Therefore Picard's iteration is not -stable.

Example 3.2.

Let and Therefore and satisfy the hypothesis of Theorem 3.1 where has property (i) and has property (ii). So the self maps of defined by and have unique fixed points but Picard's iteration is -stable for but not -stable for

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU)
(2)
Department of Mathematics, Amirkabir University of Technology
(3)
Department of Mathematics, Newcastle University
(4)
Department of Mathematics, Indiana University

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Copyright

© The Author(s). 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.