- R. Saadati
^{1}, - S. M. Vaezpour
^{1}and - B. E. Rhoades
^{2}Email author

**2009**:393245

https://doi.org/10.1155/2009/393245

© R. Saadati et al. 2009

**Received: **16 February 2009

**Accepted: **26 August 2009

**Published: **28 September 2009

## Abstract

## 1. Introduction and Preliminaries

Let be a Banach space and a self-map of . Let be some iteration procedure. Suppose that , the fixed point set of , is nonempty and that converges to a point . Let and define . If implies that , then the iteration procedure is said to be -stable. Without loss of generality, we may assume that is bounded, for if is not bounded, then it cannot possibly converge. If these conditions hold for , that is, Picard's iteration, then we will say that Picard's iteration is -stable.

Theorem 1.1 (see [1]).

for all , where , . Suppose that has a fixed point . Then, is Picard -stable.

Various kinds of analytical methods and numerical methods [2–10] were used to solve integral equations. To illustrate the basic idea of the method, we consider the general nonlinear system:

where is a linear operator, is a nonlinear operator, and is a given continuous function. The basic character of the method is to construct a functional for the system, which reads

where is a Lagrange multiplier which can be identified optimally via variational theory, is the th approximate solution, and denotes a restricted variation; that is, .

Now, we consider the Fredholm integral equation of second kind in the general case, which reads

where is the kernel of the integral equation. There is a simple iteration formula for (1.4) in the form

Now, we show that the nonlinear mapping , defined by

First, we show that the nonlinear mapping has a fixed point. For we have

Therefore, if

then, the nonlinear mapping has a fixed point.

Second, we show that the nonlinear mapping satisfies (1.1). Let (1.6) hold. Putting and shows that (1.1) holds for the nonlinear mapping .

All of the conditions of Theorem 1.1 hold for the nonlinear mapping and hence it is -stable. As a result, we can state the following theorem.

Theorem 1.2.

and . Then the nonlinear mapping , in the norm of , is -stable.

Theorem 1.3 (see [11]).

and assume that . Then, if , the above iteration converges, in the norm of to the solution of (1.4).

Corollary 1.4.

and , then stability of the nonlinear mapping in the norm of is a coefficient condition for the above iteration to converge in the norm of , and to the solution of (1.4).

## 2. Test Examples

In this section we present some test examples to show that the stability of the iteration method is a coefficient condition for the convergence in the norm of to the solution of (1.4). In fact the stability interval is a subset of converges interval.

Example 2.1 (see [12]).

which implies that has a fixed point. Also, putting and shows that (1.1) holds for the nonlinear mapping . All of the conditions of Theorem 1.1 hold for the nonlinear mapping and hence it is -stable.

Example 2.2 (see [12]).

By (2.13), we have the following results:

Continuing this way ad infinitum, we obtain

The above sequence is convergent if , that is, and the exact solution is

which implies that has a fixed point. Also, putting and shows that (1.1) holds for the nonlinear mapping . All of conditions of Theorem 1.1 hold for the nonlinear mapping and hence it is -stable.

Example 2.3.

Continuing this way ad infinitum, we obtain

which implies that has a fixed point. Also, putting and shows that (1.1) holds for the nonlinear mapping . All of the conditions of Theorem 1.1 hold for the nonlinear mapping and hence it is -stable.

## Declarations

### Acknowledgments

The authors would like to thank referees and area editor Professor Nan-jing Huang for giving useful comments and suggestions for the improvement of this paper. This paper is dedicated to Professor Mehdi Dehghan

## Authors’ Affiliations

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