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Hyers-Ulam Stability of Nonlinear Integral Equation
Fixed Point Theory and Applications volume 2010, Article number: 927640 (2010)
Abstract
We will apply the successive approximation method for proving the Hyers-Ulam stability of a nonlinear integral equation.
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

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

for all and for some
, where
is a complex number, then there exists a unique continuous function
such that

for all .
The purpose of this paper is to discuss the Hyers-Ulam stability of the following nonhomogeneous nonlinear Volterra integral equation:

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

then there exists some satisfying
such that

We call such a Hyers-Ulam stability constant for (1.4).
3. Existence of the Solution of Nonlinear Integral Equations
Consider the iterative scheme

Since is assumed Lipschitz, we can write

Hence,

in which , for all
. So, we can write

Therefore, since is complete metric space, if
, then

is absolutely and uniformly convergent by Weirstrass's M-test theorem. On the other hand, can be written as follows:

So there exists a unique solution such that
. Now by taking the limit of both sides of (3.1), we have

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

there exists a unique such that

for some .
Proof.
Let ,
, and
. In the previous section we have proved that

is an exact solution of the equation . Clearly there is
with
, because
is uniformly convergent to
as
. Thus

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:

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?"
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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
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DOI: https://doi.org/10.1155/2010/927640