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

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.


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][2][3][4][5][6][7][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 Siterative method. " Definition 1. Let be an ambient space and let be a selfmap of . A normal S-iterative method is defined by where { } ∞ =0 is a real sequence in [0, 1] 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.
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]:

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The Scientific World Journal : 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 Lemma 3 (see [22]). Let { } ∞ =0 be a nonnegative sequence for which one assumes there exists 0 ∈ N, such that for all ≥ 0 one has satisfied the inequality where ∈ (0, 1), for all ∈ N, ∑ ∞ =0 = ∞, and ≥ 0, for all ∈ N. Then the following inequality holds:

Main Results
We will show that → * as → ∞.
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 If * and̃ * are solutions of corresponding equations (15) and (16), respectively, then one has that * −̃ * ≤ Proof. Using (1) Combining (19) with (20) and using assumptions (A 5 ) and 1/2 ≤ in the resulting inequality, we get It is clear that inequality (21)