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# Hyers-Ulam Stability of Nonlinear Integral Equation

*Fixed Point Theory and Applications*
**volumeÂ 2010**, ArticleÂ number:Â 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. [3â€“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 [8â€“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?"

## 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:**222â€“224. 10.1073/pnas.27.4.222Jung S-M:

**Hyersâ€”Ulam stability of differential equation .***Journal of Inequalities and Applications*2010,**2010:**-12.Takahasi S-E, Miura T, Miyajima S:

**On the Hyersâ€”Ulam stability of the Banach space-valued differential equation .***Bulletin of the Korean Mathematical Society*2002,**39**(2):309â€“315. 10.4134/BKMS.2002.39.2.309Wang G, Zhou M, Sun L:

**Hyersâ€”Ulam stability of linear differential equations of first order.***Applied Mathematics Letters*2008,**21**(10):1024â€“1028. 10.1016/j.aml.2007.10.020Li Y, Hua L:

**Hyersâ€”Ulam stability of a polynomial equation.***Banach Journal of Mathematical Analysis*2009,**3**(2):86â€“90.Kuczma M:

*An Introduction to the Theory of Functional Equations and Inequalities*. PMN, Warsaw, Poland; 1985.BrzdÄ™k J:

**On a method of proving the Hyersâ€”Ulam stability of functional equations on restricted domains.***The Australian Journal of Mathematical Analysis and Applications*2009,**6**(1, article 4):1â€“10.CiepliÅ„ski K:

**Stability of the multi-Jensen equation.***Journal of Mathematical Analysis and Applications*2010,**363**(1):249â€“254. 10.1016/j.jmaa.2009.08.021Moszner Z:

**On the stability of functional equations.***Aequationes Mathematicae*2009,**77**(1â€“2):33â€“88. 10.1007/s00010-008-2945-7Paneah B:

**A new approach to the stability of linear functional operators.***Aequationes Mathematicae*2009,**78**(1â€“2):45â€“61. 10.1007/s00010-009-2956-zPrager W, Schwaiger J:

**Stability of the multi-Jensen equation.***Bulletin of the Korean Mathematical Society*2008,**45**(1):133â€“142. 10.4134/BKMS.2008.45.1.133Jung 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):19â€“25.Folland GB:

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

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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