Hyers-Ulam Stability of Nonlinear Integral Equation

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

(1.1)

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

(1.2)

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

(1.3)

for all .

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

(1.4)

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

(2.1)

then there exists some satisfying such that

(2.2)

We call such a Hyers-Ulam stability constant for (1.4).

3. Existence of the Solution of Nonlinear Integral Equations

Consider the iterative scheme

(3.1)

Since is assumed Lipschitz, we can write

(3.2)

Hence,

(3.3)

in which , for all . So, we can write

(3.4)

Therefore, since is complete metric space, if , then

(3.5)

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

(3.6)

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

(3.7)

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

(4.1)

there exists a unique such that

(4.2)

for some .

Proof.

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

(4.3)

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

(4.4)

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:

(4.5)

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?"