Open Access

Hyers-Ulam Stability of Nonlinear Integral Equation

Fixed Point Theory and Applications20102010:927640

Received: 8 April 2010

Accepted: 13 August 2010

Published: 19 August 2010


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. [35]). 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 [812].

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].

Definition 2.1 (cf. [5, 13]).

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
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 .


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?"

Authors’ Affiliations

Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad


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© Mortaza Gachpazan and Omid Baghani. 2010

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