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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 381980, 7 pages
http://dx.doi.org/10.1155/2013/381980
Research Article

Data Dependence Results for Multistep and CR Iterative Schemes in the Class of Contractive-Like Operators

1Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey
2Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey

Received 17 February 2013; Accepted 10 July 2013

Academic Editor: Adem Kilicman

Copyright © 2013 Vatan Karakaya et al. 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.

Abstract

We intend to establish some results on the data dependence of fixed points of certain contractive-like operators for the multistep and CR iterative processes in a Banach space setting. One of our results generalizes the corresponding results of Soltuz and Grosan (2008) and Chugh and Kumar (2011).

1. Introduction

Throughout this paper, denotes the set of all nonnegative integers. Let be a Banach space, a nonempty closed, convex subset of , and a self-map on . Suppose that is the set of all fixed points of . Iterative schemes abound in the literature of fixed point theory for which the fixed points of operators have been approximated over the years by many authors.

It is well known that the Picard iteration procedure [1] is defined by Let , , and , , be the real sequences in satisfying certain conditions.

The Mann iterative scheme [2] is defined by If (constant) in (2), then the resulting iteration will be called Krasnoselkij iteration procedure [3].

A sequence , defined by is commonly known as the Ishikawa iterative method [4].

The Noor iterative procedure [5] is defined by In 2004, Rhoades and oltuz [6] introduced a multistep iterative process as follows: The iteration processes (2), (3), and (4) can be viewed as the special cases of the iteration procedure (5).

Recently, Chugh et al. introduced a CR iterative scheme in [7] as follows:

Now we mention some important contractive type operators.

Any mapping is called a Kannan mapping, see [8], if there exists such that, for all ,

Similar mapping is called a Chatterjea mapping, see [9], if there exists such that, for all ,

In [10] Zamfirescu collected these classes of operators and proved an important result which states that an operator satisfies condition Z (Zamfirescu condition) if and only if there exist the real numbers satisfying , such that, for each pair , at least one of the following conditions is true: (z1), (z2), (z3). Then has a unique fixed point and the Picard iteration defined by (1) converges to , for any .

It is well known, see [11], that the conditions (), (), and () are independent.

Let . Since satisfies condition Z, at least one of the conditions from (), (), and () is satisfied. Then satisfies the inequalities for all where , , and it was shown that this class of operators is wider than the class of Zamfirescu operators; see [12]. Any mapping satisfying condition (9) or (10) is called a quasi-contractive operator.

Osilike and Udomene [13] extended the previous definition by considering an operator satisfying the condition that there exist and such that, for all , Thereafter, Imoru and Olatinwo [14] further generalized and extended the previous definition as follows: an operator is called contractive-like operator if there exist a constant and a strictly increasing and continuous function with , such that, for each ,

Remark 1 (see [15]). A map satisfying (12) need not have a fixed point. However, using (12) it is obvious that if has a fixed point, then it is unique.

It is important to know whether an iterative scheme converges to fixed points of its associated map. In this context, there are numerous works dealing with the convergence of various iterative schemes in the literature, such as [6, 10, 12, 1627].

As shown by Soltuz and Grosan [26, Theorem 3.1], in a real Banach space , the Ishikawa iteration given by (3) converges to the fixed point of , where is a mapping satisfying condition (12).

It is known from [28, Corollary 2] that there is equivalence between convergence of iterative procedures (3), (5) and that of some other well-known iterative procedures for the class of operators satisfying (12).

Hussain et al. [29] introduced a Kirk-CR iterative scheme and proved the convergence of this iteration for the class of operators satisfying (12).

Remark 2. Putting in [29, Theorem 2.5], convergence of the CR iteration to a fixed point of contractive-like operators satisfying (12) can be obtained easily.

2. Preliminaries

Definition 3 (see [30]). Let be a Banach space and two operators. We say that is an approximate operator of if for all and for a fixed we have

Suppose that there exists a certain fixed point iteration that converges to some fixed point and has a fixed point which can be computed by certain method. If it cannot compute fixed point of   due to various results, then approximate operator can be used. One can find some of works done under this title in the following list [15, 2426, 31].

In this paper, we prove the data dependence results for the multistep and CR iterative procedures utilizing the contractive-like operators satisfying (12).

The following lemma will be useful to prove the main results of this work.

Lemma 4 (see [26]). Let be a nonnegative sequence for which one assumes there exists , such that for all one has satisfied the inequality where , for all , and , for all . Then the following holds:

3. Main Results

For simplicity we use the following notation through this section.

For any iterative process, and denote iterative sequences associated to and , respectively.

Theorem 5. Let be a map satisfying (12) with , and let be an approximate operator of as in Definition 3. Let , be two iterative sequences defined by the multistep iteration (5) and with real sequences , , and satisfying . If and , then one has

Proof. For a given and we consider the following multistep iteration for and :
Then from (17), we get Thus, we have the following estimates by using (18) and (12): Combining (19), (20), (21), and (22) we obtain Thus, inductively, we get Using now (17) and (12), we get Substituting (25) in (24) we have If this inequality is rearranged using , for each , then we get the following inequality as follows: Denote From [26, Theorem 3.1] and [28, Corollary 2] we have . Since satisfies condition (12), and , that is, , it follows from (12) that Considering , for all , , and using (12) and (5) we have It is easy to see from (30) that this result is also valid for .
Making use of the fact that is a continuous map we have Hence an application of Lemma 4 to (27) leads to

Now we prove result on data dependence for the CR iterative procedure.

Theorem 6. Let be a map satisfying (12) with , and let be an approximate operator of as in Definition 3. Let , be two iterative sequences defined by the CR iteration (6) and with real sequences , , satisfying (i), for all , and (ii). If and , then one has

Proof. For a given and we consider the following iteration for and : Then from (34) we have Thus, by considering (35), it follows from (6) and (12) that Combining (36), (37), and (38) we obtain It may be noted that for , , and the following inequalities are always true: Using now the inequality we get Therefore an application of (40) and (41) to (39) gives us Now, by the condition (i) , for all we have Utilizing (43) in (42) we obtain Denote
From Remark 2, we have . Since satisfies condition (12), and , that is, , using similar arguments as in the proof of Theorem 5, we get Making use of the fact that is a continuous map we have Hence an application of Lemma 4 to (44) leads to

Corollary 7. Since the Mann (2), Ishikawa (3), and Noor (4) iterative processes are special cases of the multistep iterative scheme (5), the data dependence results of these iterative processes can be obtained similarly.

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