-Stability Approach to Variational Iteration Method for Solving Integral Equations

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 [2–10] 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.