/ / Article
Special Issue

## Recent Developments on Sequence Spaces and Compact Operators with Applications

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 943127 | 5 pages | https://doi.org/10.1155/2014/943127

# Applications of Normal S-Iterative Method to a Nonlinear Integral Equation

Accepted22 Jul 2014
Published11 Aug 2014

#### Abstract

It has been shown that a normal S-iterative method converges to the solution of a mixed type Volterra-Fredholm functional nonlinear integral equation. Furthermore, a data dependence result for the solution of this integral equation has been proven.

#### 1. Introduction

The scientists working in almost every field of science are faced with nonlinear problems, because nature itself is intrinsically nonlinear. Such problems can be modelled as nonlinear mathematical equations. Solving nonlinear equations is, of course, considered to be a matter of the uttermost importance in mathematics and its manifold applications. There are numerous systematic approaches which are classified as direct and iterative methods to solve such equations in the existing literature. Indeed, by using direct methods, finding solutions to a complicated nonlinear equation can be an almost insurmountable challenge. In this context, iterative methods have become very important mathematical tools for finding solutions to a nonlinear equation. For a comprehensive review and references to the extensive literature on the iterative methods, the interested reader may refer to some recent works .

Recently, Sahu  and Khan , who was probably unaware of Sahu’s work, introduced the following iterative process which has been called normal S-iterative method and Picard-Mann hybrid iterative process by Sahu and Khan, respectively, and hereinafter referred to as the “normal S-iterative method.”

Definition 1. Let be an ambient space and let be a self-map of . A normal S-iterative method is defined by where is a real sequence in satisfying certain control condition(s).

It has been shown both analytically and numerically in [9, 10] that iterative method (1) converges at a rate faster than all Picard , Mann , and Ishikawa  iterative processes in the sense of Berinde  for the class of contraction mappings.

This iterative method, due to its simplicity and fastness, has attracted the attention of many researchers and has been examined in various aspects; see .

In this paper, inspired by the performance and achievements of normal S-iterative method (1), we will give some of its applications. We will show that normal S-iterative method (1) converges strongly to the solution of the following mixed type Volterra-Fredholm functional nonlinear integral equation which was considered in : where is an interval in , , continuous functions, and .

Also we give a data dependence result for the solution of integral equation (2) with the help of normal S-iterative method (1).

We end this section with some known results which will be useful in proving our main results.

Theorem 2 (see ). We suppose that the following conditions are satisfied:), ;();()there exist nonnegative constants , , and such that for all , , , , ;()there exist nonnegative constants and such that for all , , , ;().
Then (2) has a unique solution .

Lemma 3 (see ). 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 inequality holds:

#### 2. Main Results

Theorem 4. One opines that all conditions ()–() in Theorem 2 are performed. Let be a real sequence satisfying . Then (2) has a unique solution, say , in and normal S-iterative method (1) converges to .

Proof. We consider the Banach space , where is Chebyshev’s norm. Let be an iterative sequence generated by normal S-iterative method (1) for the operator defined by We will show that as .
From (1), (2), and assumptions ()–(), we have that Combining (8) with (9), we obtain or, from assumption (), Thus, by induction, we get Since for all , assumption () yields Having regard to the fact that for all , we can rewrite (12) as which yields .

We now prove the data dependence of the solution for integral equation (2) with the help of the normal S-iterative method (1).

Let be as in the proof of Theorem 4 and , two operators defined by where , , , .

Theorem 5. Let , , and be defined as in Theorem 2 and let be an iterative sequence defined by normal S-iterative method (1) associated with . Let be an iterative sequence generated by where is defined as in the proof of Theorem 4 and is a real sequence in satisfying (i) , for all , and (ii) . One supposes further that (iii) there exist nonnegative constants and such that and , for all and for all .
If and are solutions of corresponding equations (15) and (16), respectively, then one has that

Proof. Using (1), (15), (16), (17), and assumptions ()–() and (iii), we obtain Combining (19) with (20) and using assumptions () and in the resulting inequality, we get Denote that It is clear that inequality (21) satisfies all conditions in Lemma 3, and hence it follows that

#### Conflict of Interests

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

#### Acknowledgment

The author would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

1. C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965, Springer, London, UK, 2009. View at: MathSciNet
2. F. Gürsoy and V. Karakaya, “Some convergence and stability results for two new Kirk type hybrid fixed point iterative algorithms,” Journal of Function Spaces, vol. 2014, Article ID 684191, 8 pages, 2014. View at: Publisher Site | Google Scholar
3. F. G{\"u}rsoy, V. Karakaya, and B. E. Rhoades, “Data dependence results of new multi-step and S-iterative schemes for contractive-like operators,” Fixed Point Theory and Applications, vol. 2013, artcile 76, 12 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
4. H. Kiziltunc and S. Temir, “Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces,” Computers & Mathematics with Applications, vol. 61, no. 9, pp. 2480–2489, 2011. View at: Publisher Site | Google Scholar
5. M. Basarir and A. Sahin, “On the strong and Δ—convergence of new multi-step and S-iteration processes in a CAT (0) space,” Journal of Inequalities and Applications, vol. 2013, article 482, 2013. View at: Publisher Site | Google Scholar
6. M. O. Olatinwo, “Convergence and stability results for some iterative schemes,” Acta Universitatis Apulensis, no. 26, pp. 225–236, 2011. View at: Google Scholar | MathSciNet
7. S. Almezel, Q. H. Ansari, and M. A. Khamsi, Eds., Topics in Fixed Point Theory, Springer, 2014.
8. V. Berinde, Iterative Approximation of Fixed Points, Springer, Berlin , Germany, 2007. View at: MathSciNet
9. D. R. Sahu, “Applications of the S-iteration 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
10. S. H. Khan, “A Picard-Mann hybrid iterative process,” Fixed Point Theory and Applications, vol. 2013, article 69, 2013. View at: Publisher Site | Google Scholar | MathSciNet
11. E. Picard, “Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives,” Journal de Matématiques Pures et Appliquées, vol. 6, pp. 145–210, 1890. View at: Google Scholar
12. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953.
13. 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 | MathSciNet
14. V. Berinde, “Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators,” Fixed Point Theory and Applications, vol. 2004, no. 2, pp. 97–105, 2004. View at: Publisher Site | Google Scholar | MathSciNet
15. 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
16. N. Hussain, V. Kumar, and M. A. Kutbi, “On rate of convergence of Jungck-type iterative schemes,” Abstract and Applied Analysis, vol. 2013, Article ID 132626, 15 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
17. S. H. Khan, “Fixed points of contractive-like operators by a faster iterative process,” WASET International Journal of Mathematical, Computational Science and Engineering, vol. 7, pp. 57–59, 2013. View at: Google Scholar
18. S. M. Kang, A. Rafiq, and S. Lee, “Convergence analysis of an iterative scheme for Lipschitzian hemicontractive mappings in Hilbert spaces,” Journal of Inequalities and Applications, vol. 2013, article 312, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
19. S. M. Kang, A. Rafiq, and S. Lee, “Strong convergence of an implicit S -iterative process for Lipschitzian hemicontractive mappings,” Abstract and Applied Analysis, vol. 2012, Article ID 804745, 7 pages, 2012. View at: Publisher Site | Google Scholar
20. V. Kumar, A. Latif, A. Rafiq, and N. Hussain, “$S$-iteration process for quasi-contractive mappings,” Journal of Inequalities and Applications, vol. 2013, article 206, 15 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
21. C. Crăciun and M. Şerban, “A nonlinear integral equation via Picard operators,” Fixed Point Theory, vol. 12, no. 1, pp. 57–70, 2011. View at: Google Scholar | MathSciNet
22. Ş. M. Şoltuz 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 Site | Google Scholar | MathSciNet

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. 