- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 381980, 7 pages

http://dx.doi.org/10.1155/2013/381980

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

^{1}Department of Mathematical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, 34210 Istanbul, Turkey^{2}Department 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: (z_{1}),
(z_{2}),
(z_{3}). 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, 16–27].

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, 24–26, 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.*

#### References

- E. Picard, “Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives,”
*Journal de Mathématiques Pures et Appliquées*, vol. 6, pp. 145–210, 1890. - W. R. Mann, “Mean value methods in iteration,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Krasnoselkij, “Two remarks on the method of successive approximations,”
*Uspekhi Matematicheskikh Nauk*, vol. 10, no. 1(63), pp. 123–127, 1955. View at MathSciNet - S. Ishikawa, “Fixed points by a new iteration method,”
*Proceedings of the American Mathematical Society*, vol. 44, pp. 147–150, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor, “New approximation schemes for general variational inequalities,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 217–229, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. E. Rhoades and S. M. Soltuz, “The equivalence between Mann-Ishikawa iterations and multistep iteration,”
*Nonlinear Analysis*, vol. 58, no. 1-2, pp. 219–228, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Chugh, V. Kumar, and S. Kumar, “Strong converge of a new three step iterative scheme in Banach spaces,”
*American Journal of Computational Mathematics*, vol. 2, pp. 345–357, 2012. View at Publisher · View at Google Scholar - R. Kannan, “Some results on fixed points,”
*Bulletin of the Calcutta Mathematical Society*, vol. 60, pp. 71–76, 1968. View at Zentralblatt MATH · View at MathSciNet - S. K. Chatterjea, “Fixed-point theorems,”
*Doklady Bolgarskoĭ Akademii Nauk*, vol. 25, pp. 727–730, 1972. View at Zentralblatt MATH · View at MathSciNet - T. Zamfirescu, “Fix point theorems in metric spaces,”
*Archiv der Mathematik*, vol. 23, pp. 292–298, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. E. Rhoades, “A comparison of various definitions of contractive mappings,”
*Transactions of the American Mathematical Society*, vol. 226, pp. 257–290, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Berinde, “On the convergence of the Ishikawa iteration in the class of quasi contractive operators,”
*Acta Mathematica Universitatis Comenianae*, vol. 73, no. 1, pp. 119–126, 2004. View at Zentralblatt MATH · View at MathSciNet - M. O. Osilike and A. Udomene, “Short proofs of stability results for fixed point iteration procedures for a class of contractive-type mappings,”
*Indian Journal of Pure and Applied Mathematics*, vol. 30, no. 12, pp. 1229–1234, 1999. View at Zentralblatt MATH · View at MathSciNet - C. O. Imoru and M. O. Olatinwo, “On the stability of Picard and Mann iteration processes,”
*Carpathian Journal of Mathematics*, vol. 19, no. 2, pp. 155–160, 2003. View at Zentralblatt MATH · View at MathSciNet - F. Gursoy, V. Karakaya, and B. E. Rhoades, “Data dependence results of new multistep and S-iterative schemes for contractive-like operators,”
*Fixed Point Theory and Applications*, vol. 2013, article 76, 2013. View at Publisher · View at Google Scholar - A. Rafiq, “On the convergence of the three-step iteration process in the class of quasi-contractive operators,”
*Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis*, vol. 22, no. 3, pp. 305–309, 2006. View at Zentralblatt MATH · View at MathSciNet - B. E. Rhoades, “Fixed point theorems and stability results for fixed point iteration procedures. II,”
*Indian Journal of Pure and Applied Mathematics*, vol. 24, no. 11, pp. 691–703, 1993. View at Zentralblatt MATH · View at MathSciNet - I. A. Rus,
*Generalized Contractions and Applications*, Cluj University Press, Cluj-Napoca, Romania, 2001. View at MathSciNet - İ. Yildirim, M. Özdemir, and H. Kiziltunç, “On the convergence of a new two-step iteration in the class of quasi-contractive operators,”
*International Journal of Mathematical Analysis*, vol. 3, no. 37–40, pp. 1881–1892, 2009. View at Zentralblatt MATH · View at MathSciNet - J. A. Park, “Mann-iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces,”
*Journal of the Korean Mathematical Society*, vol. 31, no. 3, pp. 333–337, 1994. View at Zentralblatt MATH · View at MathSciNet - M. Ertürk and V. Karakaya, “n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces,”
*Journal of Inequalities and Applications*, vol. 2013, article 196, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - W. Phuengrattana and S. Suantai, “On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval,”
*Journal of Computational and Applied Mathematics*, vol. 235, no. 9, pp. 3006–3014, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Chugh and V. Kumar, “Strong convergence of SP iterative scheme for quasi-contractive operators in Banach spaces,”
*International Journal of Computer Applications*, vol. 31, no. 5, pp. 21–27, 2011. - S. M. Soltuz, “Data dependence for Mann iteration,”
*Octogon Mathematical Magazine*, vol. 9, pp. 825–828, 2001. - S. M. Soltuz, “Data dependence for Ishikawa iteration,”
*Lecturas Matemáticas*, vol. 25, no. 2, pp. 149–155, 2004. View at MathSciNet - S. M. Soltuz and T. Grosan, “Data dependence for Ishikawa iteration when dealing with contractive-like operators,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 242916, 7 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. M. Soltuz, “The equivalence between Krasnoselskij, Mann, Ishikawa, Noor and multistep iterations,”
*Mathematical Communications*, vol. 12, no. 1, pp. 53–61, 2007. View at MathSciNet - F. Gürsoy, V. Karakaya, and B. E. Rhoades, “The equivalence among new multistep iteration, S-iteration and some other iterative schemes,” http://arxiv.org/abs/1211.5701.
- N. Hussain, R. Chugh, V. Kumar, and A. Rafiq, “On the rate of convergence of Kirk-type iterative schemes,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 526503, 22 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - V. Berinde,
*Iterative Approximation of Fixed Points*, Springer, Berlin, Germany, 2007. View at MathSciNet - R. Chugh and V. Kumar, “Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators,”
*International Journal of Computer Applications*, vol. 40, no. 15, pp. 41–46, 2011.