# 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. [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

(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?"

## References

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13. Jung S-M: A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory and Applications 2007, 2007:-9.

14. Gachpazan M, Baghani O: HyersUlam stability of Volterra integral equation. Journal of Nonlinear Analysis and Its Applications 2010, (2):19–25.

15. Folland GB: Real Analysis Modern Techniques and Their Application. University of Washington, Seattle, Wash, USA; 1984:xiv+350.

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Correspondence to Mortaza Gachpazan.

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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

### Keywords

• Integral Equation
• Functional Equation
• Point Theorem
• Stability Constant
• Fixed Point Theorem