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-Stability Approach to Variational Iteration Method for Solving Integral Equations
Fixed Point Theory and Applications volume 2009, Article number: 393245 (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
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
is -stable in .
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.
Use the iteration scheme
for to construct a sequence of successive iterations to the solution of (1.4). In addition, if
and . Then the nonlinear mapping , in the norm of , is -stable.
Theorem 1.3 (see [11]).
Use the iteration scheme
for to construct a sequence of successive iteration to the solution of (1.4). In addition, let
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
for If
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
The iteration formula reads
Substituting (2.3) into (2.2), we have the following results:
Continuing this way ad infinitum, we obtain
then
The above sequence is convergent if , and the exact solution is
On the other hand we have
Then if for mapping
we have
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
its iteration formula reads
Then we have
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
On the other hand we have
Then if , for mapping
we have
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
its iteration formula reads
Substituting (2.22) into (2.21), 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
On the other hand we have
Then if , for mapping
we have
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.
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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
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Saadati, R., Vaezpour, S.M. & Rhoades, B.E. -Stability Approach to Variational Iteration Method for Solving Integral Equations. Fixed Point Theory Appl 2009, 393245 (2009). https://doi.org/10.1155/2009/393245
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DOI: https://doi.org/10.1155/2009/393245