Recent Developments on Sequence Spaces and Compact Operators with Applications
View this Special IssueResearch Article  Open Access
Convergence Analysis for a Modified SP Iterative Method
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 PicardS 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 PicardS 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 PicardS 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 wellknown PicardBanach 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 wellknown 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 PicardS 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 PicardS 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 threestep iteration method due to Abbas and Nazir [17].
We are now able to establish the following data dependence result.



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.
References
 N. Kadioglu and I. Yildirim, “Approximating fixed points of nonexpansive mappings by a faster iteration process,” http://arxiv.org/abs/1402.6530. View at: Google Scholar
 F. Gürsoy and V. Karakaya, “A PicardS hybrid type iteration method for solving a differential equation with retarded argument,” http://arxiv.org/abs/1403.2546. View at: Google Scholar
 W. Phuengrattana and S. Suantai, “On the rate of convergence of Mann, Ishikawa, Noor and SPiterations for continuous functions on an arbitrary interval,” Journal of Computational and Applied Mathematics, vol. 235, no. 9, pp. 3006–3014, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 E. Picard, “Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives,” Journal de Mathématiques pures et appliquées, vol. 6, pp. 145–210, 1890. View at: Google Scholar
 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at: Publisher Site  Google Scholar  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 Site  Google Scholar  Zentralblatt MATH  MathSciNet
 R. Agarwal, D. O Regan, and D. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Journal of Nonlinear and Convex Analysis, vol. 8, pp. 61–79, 2007. View at: Google Scholar
 V. Berinde, “Picard iteration converges faster than Mann iteration for a class of quasicontractive operators,” Fixed Point Theory and Applications, no. 2, pp. 97–105, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin, Germany, 2007. View at: MathSciNet
 X. Weng, “Fixed point iteration for local strictly pseudocontractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Ş. M. Şoltuz and T. Grosan, “Data dependence for Ishikawa iteration when dealing with contractivelike operators,” Fixed Point Theory and Applications, vol. 2008, Article ID 242916, 2008. View at: Publisher Site  Google Scholar
 D. R. Sahu, “Applications of the Siteration process to constrained minimization problems and split feasibility problems,” Fixed Point Theory, vol. 12, no. 1, pp. 187–204, 2011. View at: Google Scholar  MathSciNet
 R. Chugh, V. Kumar, and S. Kumar, “Strong convergence of a new three step iterative scheme in Banach spaces,” The American Journal of Computational Mathematics, vol. 2, pp. 345–357, 2012. View at: Publisher Site  Google Scholar
 I. Karahan and M. Ozdemir, “A general iterative method for approximation of fixed points and their applications,” Advances in Fixed Point Theory, vol. 3, no. 3, pp. 510–526, 2013. View at: Google Scholar
 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 Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. R. Sahu and A. Petruşel, “Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 17, pp. 6012–6023, 2011. View at: Publisher Site  Google Scholar
 M. Abbas and T. Nazir, “A new faster iteration process applied to constrained minimization and feasibility problems,” Matematicki Vesnik, vol. 66, no. 2, p. 223, 2014. View at: Google Scholar
Copyright
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.