Abstract
We will apply the successive approximation method for proving the Hyers-Ulam stability of a nonlinear integral equation.
Similar content being viewed by others
1. Introduction
We say a functional equation is stable if, for every approximate solution, there exists an exact solution near it. In 1940, Ulam posed the following problem concerning the stability of functional equations [1]: we are given a group and a metric group
with metric
Given
does there exist a
such that if
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ1_HTML.gif)
for all then a homomorphism
exists with
for all
The problem for the case of the approximately additive mappings was solved by Hyers [2] when
and
are Banach space. Since then, the stability problems of functional equations have been extensively investigated by several mathematicians (cf. [3–5]). Recently, Y. Li and L. Hua proved the stability of Banach's fixed point theorem [6]. The interested reader can also find further details in the book of Kuczma (see [7, chapter XVII]). Examples of some recent developments, discussions, and critiques of that idea of stability can be found, for example, in [8–12].
In this paper, we study the Hyers-Ulam stability for the nonlinear Volterra integral equation of second kind. Jung was the author who investigated the Hyers-Ulam stability of Volterra integral equation on any compact interval. In 2007, he proved the following [13].
Given and
, let
denote a closed interval
and let
be a continuous function which satisfies a Lipschitz condition
for all
and
, where
is a constant with
. If a continuous function
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ2_HTML.gif)
for all and for some
, where
is a complex number, then there exists a unique continuous function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ3_HTML.gif)
for all .
The purpose of this paper is to discuss the Hyers-Ulam stability of the following nonhomogeneous nonlinear Volterra integral equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ4_HTML.gif)
where . We will use the successive approximation method, to prove that (1.4) has the Hyers-Ulam stability under some appropriate conditions. The method of this paper is distinctive. This new technique is simpler and clearer than methods which are used in some papers, (cf. [13, 14]). On the other hand, Hyers-Ulam stability constant obtained in our paper is different to the other works, [13].
2. Basic Concepts
Consider the nonhomogeneous nonlinear Volterra integral equation (1.4). We assume that is continuous on the interval
and
is continuous with respect to the three variables
,
, and
on the domain
; and
is Lipschitz with respect to
. In this paper, we consider the complete metric space
and assume that
is a bounded linear transformation on
.
Note that, the linear mapping is called bounded, if there exists
such that
, for all
. In this case, we define
. Thus
is bounded if and only if
, [15].
One says that (1.4) has the Hyers-Ulam stability if there exists a constant with the following property: for every
,
, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ5_HTML.gif)
then there exists some satisfying
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ6_HTML.gif)
We call such a Hyers-Ulam stability constant for (1.4).
3. Existence of the Solution of Nonlinear Integral Equations
Consider the iterative scheme
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ7_HTML.gif)
Since is assumed Lipschitz, we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ8_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ9_HTML.gif)
in which , for all
. So, we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ10_HTML.gif)
Therefore, since is complete metric space, if
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ11_HTML.gif)
is absolutely and uniformly convergent by Weirstrass's M-test theorem. On the other hand, can be written as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ12_HTML.gif)
So there exists a unique solution such that
. Now by taking the limit of both sides of (3.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ13_HTML.gif)
So, there exists a unique solution such that
.
4. Main Results
In this section, we prove that the nonlinear integral equation in (1.4) has the Hyers-Ulam stability.
Theorem 4.1.
The equation , where
is defined by (1.4), has the Hyers-Ulam stability; that is, for every
and
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ14_HTML.gif)
there exists a unique such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ15_HTML.gif)
for some .
Proof.
Let ,
, and
. In the previous section we have proved that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ16_HTML.gif)
is an exact solution of the equation . Clearly there is
with
, because
is uniformly convergent to
as
. Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ17_HTML.gif)
where . This completes the proof.
Corollary 4.2.
For infinite interval, Theorem 4.1 is not true necessarily. For example, the exact solution of the integral equation ,
, is
. By choosing
and
,
is obtained, so
,
. Hence, there exists no Hyers-Ulam stability constant
such that the relation
is true.
Corollary 4.3.
Theorem 4.1 holds for every finite interval ,
,
, and
, when
.
Corollary 4.4.
If one applies the successive approximation method for solving (1.4) and for some
, then
, such that
is the exact solution of (1.4).
Example 4.5.
If we put and
(
is constant), (1.4) will be a linear Volterra integral equation of second kind in the following form:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F927640/MediaObjects/13663_2010_Article_1367_Equ18_HTML.gif)
In this example, if on square
, then
satisfies in the Lipschitz condition, where
is the Lipschitz constant. Also
; therefore, if
, the Volterra equation (4.5) has the Hyers-Ulam stability.
5. Conclusions
Let be a finite interval, and let
and
be integral equations in which
is a nonlinear integral map. In this paper, we showed that
has the Hyers-Ulam stability; that is, if
is an approximate solution of the integral equation and
for all
and
, then
, in which
is the exact solution and
is positive constant.
6. Ideas
In this paper, we proved that (1.4) has the Hyers-Ulam stability. In (1.4), is a linear transformation. It is an open problem that raises the following question: "What can we say about the Hyers-Ulam stability of the general nonlinear Volterra integral equation (1.4) when
is not necessarily linear?"
References
Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1960:xvii+150.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
Jung S-M: Hyers—Ulam stability of differential equation . Journal of Inequalities and Applications 2010, 2010:-12.
Takahasi S-E, Miura T, Miyajima S: On the Hyers—Ulam stability of the Banach space-valued differential equation . Bulletin of the Korean Mathematical Society 2002,39(2):309–315. 10.4134/BKMS.2002.39.2.309
Wang G, Zhou M, Sun L: Hyers—Ulam stability of linear differential equations of first order. Applied Mathematics Letters 2008,21(10):1024–1028. 10.1016/j.aml.2007.10.020
Li Y, Hua L: Hyers—Ulam stability of a polynomial equation. Banach Journal of Mathematical Analysis 2009,3(2):86–90.
Kuczma M: An Introduction to the Theory of Functional Equations and Inequalities. PMN, Warsaw, Poland; 1985.
Brzdęk J: On a method of proving the Hyers—Ulam stability of functional equations on restricted domains. The Australian Journal of Mathematical Analysis and Applications 2009,6(1, article 4):1–10.
Ciepliński K: Stability of the multi-Jensen equation. Journal of Mathematical Analysis and Applications 2010,363(1):249–254. 10.1016/j.jmaa.2009.08.021
Moszner Z: On the stability of functional equations. Aequationes Mathematicae 2009,77(1–2):33–88. 10.1007/s00010-008-2945-7
Paneah B: A new approach to the stability of linear functional operators. Aequationes Mathematicae 2009,78(1–2):45–61. 10.1007/s00010-009-2956-z
Prager W, Schwaiger J: Stability of the multi-Jensen equation. Bulletin of the Korean Mathematical Society 2008,45(1):133–142. 10.4134/BKMS.2008.45.1.133
Jung S-M: A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory and Applications 2007, 2007:-9.
Gachpazan M, Baghani O: HyersUlam stability of Volterra integral equation. Journal of Nonlinear Analysis and Its Applications 2010, (2):19–25.
Folland GB: Real Analysis Modern Techniques and Their Application. University of Washington, Seattle, Wash, USA; 1984:xiv+350.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Gachpazan, M., Baghani, O. Hyers-Ulam Stability of Nonlinear Integral Equation. Fixed Point Theory Appl 2010, 927640 (2010). https://doi.org/10.1155/2010/927640
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/927640