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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