Open Access

-Stability Approach to Variational Iteration Method for Solving Integral Equations

Fixed Point Theory and Applications20092009:393245

DOI: 10.1155/2009/393245

Received: 16 February 2009

Accepted: 26 August 2009

Published: 28 September 2009

Abstract

We consider -stability definition according to Y. Qing and B. E. Rhoades (2008) and we show that the variational iteration method for solving integral equations is -stable. Finally, we present some text examples to illustrate our result.

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]).

Let be a Banach space and a self-map of satisfying
(1.1)

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

Various kinds of analytical methods and numerical methods [210] were used to solve integral equations. To illustrate the basic idea of the method, we consider the general nonlinear system:

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

Now, we show that the nonlinear mapping , defined by

(1.6)

is -stable in .

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

(1.7)

Therefore, if

(1.8)

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.

Use the iteration scheme
(1.9)
for to construct a sequence of successive iterations to the solution of (1.4). In addition, if
(1.10)

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

Theorem 1.3 (see [11]).

Use the iteration scheme
(1.11)
for to construct a sequence of successive iteration to the solution of (1.4). In addition, let
(1.12)

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

Corollary 1.4.

Consider the iteration scheme
(1.13)
for If
(1.14)

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]).

Consider the integral equation
(2.1)
The iteration formula reads
(2.2)
(2.3)
Substituting (2.3) into (2.2), we have the following results:
(2.4)
Continuing this way ad infinitum, we obtain
(2.5)
then
(2.6)
The above sequence is convergent if , and the exact solution is
(2.7)
On the other hand we have
(2.8)
Then if for mapping
(2.9)
we have
(2.10)

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]).

Consider the integral equation
(2.11)
its iteration formula reads
(2.12)
Then we have
(2.13)

By (2.13), we have the following results:

(2.14)

Continuing this way ad infinitum, we obtain

(2.15)

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

(2.16)
On the other hand we have
(2.17)
Then if , for mapping
(2.18)
we have
(2.19)

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.

Consider the integral equation
(2.20)
its iteration formula reads
(2.21)
(2.22)
Substituting (2.22) into (2.21), we have the following results:
(2.23)

Continuing this way ad infinitum, we obtain

(2.24)
The above sequence is convergent if ; that is, , and the exact solution is
(2.25)
On the other hand we have
(2.26)
Then if , for mapping
(2.27)
we have
(2.28)

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

(1)
Department of Mathematics and Computer Science, Amirkabir University of Technology
(2)
Department of Mathematics, Indiana University

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Copyright

© R. Saadati et al. 2009

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