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A Fixed Point Approach to the Stability of an Integral Equation Related to the Wave Equation
We will apply the fixed point method for proving the generalized Hyers-Ulam stability of the integral equation which is strongly related to the wave equation.
In 1940, Ulam  gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms: Let be a group and let be a metric group with the metric . Given , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism with for all ?
The case of approximately additive functions was solved by Hyers  under the assumption that and are the Banach spaces. Indeed, he proved that each solution of the inequality , for all and , can be approximated by an exact solution, say an additive function. In this case, the Cauchy additive functional equation, , is said to have the Hyers-Ulam stability.
Rassias  attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: and proved the Hyers theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. (Aoki  has provided a proof of a special case of Rassias’ theorem just for the stability of the additive function. Aoki did not prove the stability of the linear function, which was implied by Rassias’ theorem.) Since then, the stability of several functional equations has been extensively investigated [5–12].
The terminologies generalized Hyers-Ulam stability, Hyers-Ulam-Rassias stability, and Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations.
Let and be fixed real numbers with . For any differentiable function , the function defined as is a solution of the wave equation as we see from which we know that satisfies the wave equation (3).
Conversely, we know that every solution of the wave equation (3) can be expressed by where are arbitrary twice differentiable functions. If these and satisfy for all , then expressed by (5) satisfies the integral equation (7). These facts imply that the integral equation (7) is strongly connected with the wave equation (3).
Cădariu and Radu  applied the fixed point method to the investigation of the Cauchy additive functional equation. Using such a clever idea, they could present another proof for the Hyers-Ulam stability of that equation [14–19].
In this paper, we introduce the integral equation: which may be considered as a special form of (2), and prove the generalized Hyers-Ulam stability of the integral equation (7) by using ideas from [13, 15, 19, 20]. More precisely, assume that is a given function and is an arbitrary and continuous function which satisfies the integral inequality: If there exist a function and a constant such that then we say that the integral equation (7) has the generalized Hyers-Ulam stability.
For a nonempty set , we introduce the definition of the generalized metric on . A function is called a generalized metric on if and only if satisfies () if and only if ; () for all ; () for all . We remark that the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include the infinity.
We now introduce one of fundamental results of fixed point theory. For the proof, we refer to . This theorem will play an important role in proving our main theorems.
Theorem 1. Let be a generalized complete metric space. Assume that is a strictly contractive operator with the Lipschitz constant . If there exists a nonnegative integer such that for some , then the following are true: (a)the sequence converges to a fixed point of ; (b) is the unique fixed point of in (c)if , then
3. The Generalized Hyers-Ulam Stability
In the following theorem, for given real numbers , , , and satisfying , , and , let , , and be finite intervals. Assume that and are positive constants with . Moreover, let be a continuous function satisfying for all and .
We denote by the set of all functions with the following properties: (a) is continuous for all and ; (b) for all and ; (c) for all and . Moreover, we introduce a generalized metric on as follows:
Theorem 2. If a function satisfies the integral inequality: for all and , then there exists a unique function which satisfies for all and .
Proof. First, we show that is complete. Let be a Cauchy sequence in . Then, for any there exists an integer such that for all . In view of (13), we have
If and are fixed, (17) implies that is a Cauchy sequence in . Since is complete, converges for any and . Thus, considering , we can define a function by
Since is bounded on , (17) implies that converges uniformly to in the usual topology of . Hence, is continuous and is bounded on with an upper bound ; that is, . (It has not been proved yet that converges to in .)
If we let increase to infinity, it follows from (17) that By considering (13), we get This implies that the Cauchy sequence converges to in . Hence, is complete.
We now define an operator by for all . Then, according to the fundamental theorem of calculus, is continuous on . Furthermore, it follows from (12), , and (21) that for any and . Hence, we conclude that .
We assert that is strictly contractive on . Given any , let be an arbitrary constant with . That is, for all and . Then, it follows from (12), (21), and (23) that for all and . That is, . Hence, we may conclude that for any and we note that .
We prove that the distance between the first two successive approximations of is finite. Let be given. By , , and (13) and from the fact that , we have for any and . Thus, (13) implies that
Therefore, it follows from Theorem 1 that there exists a such that in and .
In view of and (13), it is obvious that , where was chosen with the property (26). Now, Theorem 1 implies that is the unique element of which satisfies for any and .
Finally, Theorem 1, together with (13) and (14), implies that since (14) means that . In view of (13), we can conclude that (16) holds for all and .
Remark 3. Even though condition (12) seems to be strict, the condition can be satisfied provided that and are chosen so that is small enough and is a large number.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author would like to express his cordial thanks to the referees for useful remarks. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557).
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Copyright © 2013 Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.