The Scientific World Journal

Volume 2014 (2014), Article ID 840504, 5 pages

http://dx.doi.org/10.1155/2014/840504

Research Article

Convergence Analysis for a Modified SP Iterative Method

Fatma Öztürk Çeliker

Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, 34220 Istanbul, Turkey

Received 13 June 2014; Accepted 9 July 2014; Published 10 August 2014

Academic Editor: Syed A. Mohiuddine

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.

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 fi…xed 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.

Table 1: Comparison speed of convergence among various iteration methods.

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

1
3.101431265
3.944094141
3.101431265
3.101431265

2
3.000970459
3.032885422
3.003472396
3.006099262

3
3.000010797
3.001099931
3.000191044
3.000474311

4
3.000000126
3.000033381
3.000012841
3.000040908

5
3.000000001
3.000000928
3.000000964
3.000003733

6
3.000000000
3.000000024
3.000000078
3.000000354

7
3.000000000
3.000000000
3.000000007
3.000000034

8
3.000000000
3.000000000
3.000000001
3.000000004

9
3.000000000
3.000000000
3.000000000
3.000000000

Table 2: Comparison speed of convergence among various iteration methods.

Number of iterations
CR
Normal S
S
Picard

1
3.101431265
3.944094141
3.944094141
14.45128320

2
3.004853706
3.056995075
3.079213170
3.944094141

3
3.000341967
3.004449310
3.007910488
3.101431265

4
3.000027911
3.000384457
3.000829879
3.011228065

5
3.000002459
3.000035123
3.000088928
3.001247045

6
3.000000227
3.000003324
3.000009637
3.000138554

7
3.000000022
3.000000323
3.000001051
3.000015395

8
3.000000003
3.000000032
3.000000115
3.000001710

9
3.000000000
3.000000003
3.000000013
3.000000190

10
3.000000000
3.000000000
3.000000001
3.000000021

11
3.000000000
3.000000000
3.000000000
3.000000002

12
3.000000000
3.000000000
3.000000000
3.000000000

Table 3: Comparison speed of convergence among various iteration methods.

Number of
SP
Noor
Ishikawa
Mann

iterations

1
3.101431265
3.101431265
3.944094141
14.45128320

2
3.017380074
3.053700718
3.500544608
9.197688670

3
3.006056041
3.037176288
3.346563527
7.322609407

4
3.002849358
3.028678163
3.267333303
6.346715746

5
3.001583841
3.023463545
3.218710750
5.744057924

6
3.000979045
3.019921623
3.185684999
5.333095485

7
3.000651430
3.017350921
3.161716071
5.034023149

8
3.000457519
3.015395770
3.143487338
4.806124994

9
3.000334906
3.013856108
3.129133091
4.626397579

10
3.000253300
3.012610550
3.117521325
4.480838008

11
3.000196722
3.011581068
3.107924338
4.360421594

12
3.000156165
3.010715155
3.099852480
4.259063398

13
3.000126272
3.009976148
3.092963854
4.172507477

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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