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The Scientific World Journal / 2014 / Article
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Recent Developments on Sequence Spaces and Compact Operators with Applications

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Research Article | Open Access

Volume 2014 |Article ID 840504 | 5 pages | https://doi.org/10.1155/2014/840504

Convergence Analysis for a Modified SP Iterative Method

Academic Editor: Syed A. Mohiuddine
Received13 Jun 2014
Accepted09 Jul 2014
Published10 Aug 2014

Abstract

We consider a new iterative method due to Kadioglu and Yildirim (2014) for further investigation. We study convergence analysis of this iterative method when applied to class of contraction mappings. Furthermore, we give a data dependence result for fixed point of contraction mappings with the help of the new iteration method.

1. Introduction

Recent progress in nonlinear science reveals that iterative methods are most powerful tools which are used to approximate solutions of nonlinear problems whose solutions are inaccessible analytically. Therefore, in recent years, an intensive interest has been devoted to developing faster and more effective iterative methods for solving nonlinear problems arising from diverse branches in science and engineering.

Very recently the following iterative methods are introduced in [1] and [2], respectively: where is a nonempty convex subset of a Banach space , is a self map of , and , are real sequences in .

While the iterative method (1) fails to be named in [1], the iterative method (2) is called Picard-S iteration method in [2]. Since iterative method (1) is a special case of SP iterative method of Phuengrattana and Suantai [3], we will call it here Modified SP iterative method.

It was shown in [1] that Modified SP iterative method (1) is faster than all Picard [4], Mann [5], Ishikawa [6], and [7] iterative methods in the sense of Definitions 1 and 2 given below for the class of contraction mappings satisfying Using the same class of contraction mappings (3), Gürsoy and Karakaya [2] showed that Picard-S iteration method (2) is also faster than all Picard [4], Mann [5], Ishikawa [6], [7], and some other iterative methods in the existing literature.

In this paper, we show that Modified SP iterative method converges to the fixed point of contraction mappings (3). Also, we establish an equivalence between convergence of iterative methods (1) and (2). For the sake of completness, we give a comparison result between the rate of convergences of iterative methods (1) and (2), and it thus will be shown that Picard-S iteration method is still the fastest method. Finally, a data dependence result for the fixed point of the contraction mappings (3) is proven.

The following definitions and lemmas will be needed in order to obtain the main results of this paper.

Definition 1 (see [8]). Let and be two sequences of real numbers with limits and , respectively. Suppose that exists.(i)If , then we say that converges faster to than to .(ii)If , then we say that and have the same rate of convergence.

Definition 2 (see [8]). Assume that for two fixed point iteration processes and both converging to the same fixed point , the following error estimates, are available where and are two sequences of positive numbers (converging to zero). If converges faster than , then converges faster than to .

Definition 3 (see [9]). Let be two operators. We say that is an approximate operator of if for all and for a fixed we have

Lemma 4 (see [10]). Let and be nonnegative real sequences and suppose that for all , , , and as holds. Then .

Lemma 5 (see [11]). Let be a nonnegative sequence such that there exists , for all ; the following inequality holds. Consider where , for all , and , . Then

2. Main Results

Theorem 6. Let be a nonempty closed convex subset of a Banach space and a contraction map satisfying condition (3). Let be an iterative sequence generated by (1) with real sequences , in satisfying . Then converges to a unique fixed point of , say .

Proof. The well-known Picard-Banach theorem guarantees the existence and uniqueness of . We will show that as . From (3) and (1) we have By induction on the inequality (10), we derive Since , taking the limit of both sides of inequality (11) yields ; that is, as .

Theorem 7. Let , , and with fixed point be as in Theorem 6. Let , be two iterative sequences defined by (1) and (2), respectively, with real sequences , in satisfying . Then the following are equivalent:(i) converges to ;(ii) converges to .

Proof. We will prove (i)(ii). Now by using (1), (2), and condition (3), we have
Define Since and , which implies as . Since also , for all hence the assumption leads to Thus all conditions of Lemma 4 are fulfilled by (12), and so . Since
Using the same argument as above one can easily show the implication (ii)(i); thus it is omitted here.

Theorem 8. Let , , and with fixed point be as in Theorem 6. Let , be real sequences in satisfying(i).
For given , consider iterative sequences and defined by (1) and (2), respectively. Then converges to faster than does.

Proof. The following inequality comes from inequality (10) of Theorem 6:
The following inequality is due to ([2], inequality of Theorem 1):
Define
Since Therefore, taking into account assumption (i), we obtain It thus follows from well-known ratio test that . Hence, we have which implies that is faster than .

In order to support analytical proof of Theorem 8 and to illustrate the efficiency of Picard-S iteration method (2), we will use a numerical example provided by Sahu [12] for the sake of consistent comparison.

Example 9. Let and . Let be a mapping and for all . is a contraction with contractivity factor and ; see [12]. Take with initial value . Tables 1, 2, and 3 show that Picard-S iteration method (2) converges faster than all SP [3], Picard [4], Mann [5], Ishikawa [6], [7], CR [13], [14], Noor [15], and Normal- [16] iteration methods including a new three-step iteration method due to Abbas and Nazir [17].
We are now able to establish the following data dependence result.


Number of iterations Picard-S Abbas and Nazir Modified SP S *

13.1014312653.9440941413.1014312653.101431265
23.0009704593.0328854223.0034723963.006099262
33.0000107973.0010999313.0001910443.000474311
43.0000001263.0000333813.0000128413.000040908
53.0000000013.0000009283.0000009643.000003733
63.0000000003.0000000243.0000000783.000000354
73.0000000003.0000000003.0000000073.000000034
83.0000000003.0000000003.0000000013.000000004
93.0000000003.0000000003.0000000003.000000000


Number of iterationsCRNormal SSPicard

13.1014312653.9440941413.94409414114.45128320
23.0048537063.0569950753.0792131703.944094141
33.0003419673.0044493103.0079104883.101431265
43.0000279113.0003844573.0008298793.011228065
53.0000024593.0000351233.0000889283.001247045
63.0000002273.0000033243.0000096373.000138554
73.0000000223.0000003233.0000010513.000015395
83.0000000033.0000000323.0000001153.000001710
93.0000000003.0000000033.0000000133.000000190
103.0000000003.0000000003.0000000013.000000021
113.0000000003.0000000003.0000000003.000000002
123.0000000003.0000000003.0000000003.000000000


Number of SP Noor Ishikawa Mann
iterations

13.1014312653.1014312653.94409414114.45128320
23.0173800743.0537007183.5005446089.197688670
33.0060560413.0371762883.3465635277.322609407
43.0028493583.0286781633.2673333036.346715746
53.0015838413.0234635453.2187107505.744057924
63.0009790453.0199216233.1856849995.333095485
73.0006514303.0173509213.1617160715.034023149
83.0004575193.0153957703.1434873384.806124994
93.0003349063.0138561083.1291330914.626397579
103.0002533003.0126105503.1175213254.480838008
113.0001967223.0115810683.1079243384.360421594
123.0001561653.0107151553.0998524804.259063398
133.0001262723.0099761483.0929638544.172507477

Theorem 10. Let be an approximate operator of satisfying condition (3). Let be an iterative sequence generated by (1) for and define an iterative sequence as follows: where , are real sequences in satisfying (i) , (ii) for all , and (iii) . If and such that as , then we have where is a fixed number and .

Proof. It follows from (1), (3), (22), and assumption (ii) that
From assumption (i) we have and thus, inequality (24) becomes
Denote that It follows from Lemma 5 that From Theorem 6 we know that . Thus, using this fact together with the assumption we obtain

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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Copyright © 2014 Fatma Öztürk Çeliker. 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.

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