- Review Article
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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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ1_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ4_HTML.gif)
where is the kernel of the integral equation. There is a simple iteration formula for (1.4) in the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ5_HTML.gif)
Now, we show that the nonlinear mapping , defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ6_HTML.gif)
is -stable in
.
First, we show that the nonlinear mapping has a fixed point. For
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ7_HTML.gif)
Therefore, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ9_HTML.gif)
for to construct a sequence of successive iterations
to the solution of (1.4). In addition, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ10_HTML.gif)
and
. Then the nonlinear mapping
, in the norm of
, is
-stable.
Theorem 1.3 (see [11]).
Use the iteration scheme
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ11_HTML.gif)
for to construct a sequence of successive iteration
to the solution of (1.4). In addition, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ13_HTML.gif)
for If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ15_HTML.gif)
The iteration formula reads
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ16_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ17_HTML.gif)
Substituting (2.3) into (2.2), we have the following results:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ18_HTML.gif)
Continuing this way ad infinitum, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ19_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ20_HTML.gif)
The above sequence is convergent if , and the exact solution is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ21_HTML.gif)
On the other hand we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ22_HTML.gif)
Then if for mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ23_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ25_HTML.gif)
its iteration formula reads
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ26_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ27_HTML.gif)
By (2.13), we have the following results:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ28_HTML.gif)
Continuing this way ad infinitum, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ29_HTML.gif)
The above sequence is convergent if , that is,
and the exact solution is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ30_HTML.gif)
On the other hand we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ31_HTML.gif)
Then if , for mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ32_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ34_HTML.gif)
its iteration formula reads
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ35_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ36_HTML.gif)
Substituting (2.22) into (2.21), we have the following results:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ37_HTML.gif)
Continuing this way ad infinitum, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ38_HTML.gif)
The above sequence is convergent if ; that is,
, and the exact solution is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ39_HTML.gif)
On the other hand we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ40_HTML.gif)
Then if , for mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ41_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F393245/MediaObjects/13663_2009_Article_1138_Equ42_HTML.gif)
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