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 [1–8].
Recently, Sahu [9] and Khan [10], 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 [11], Mann [12], and Ishikawa [13] iterative processes in the sense of Berinde [14] 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 [15–20].
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 [21]: 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 [21]). 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 [22]). 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.