On the Stability of One-Dimensional Wave Equation
We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation, , in a class of twice continuously differentiable functions.
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 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 Hyers’ theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4–9].
The terminologies, the generalized Hyers-Ulam stability, and the Hyers-Ulam stability can also be applied to the case of other functional equations, differential equations, and various integral equations.
Given a real number , the partial differential equation is called the (one-dimensional) wave equation, where and denote the second time derivative and the second space derivative of , respectively.
Let be a function. If, for each twice continuously differentiable function satisfying there exist a solution of the (one-dimensional) wave equation (2) and a function such that where is independent of and , then we say that the wave equation (2) has the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability).
2. Generalized Hyers-Ulam Stability
In the following theorem, using the d’Alembert method (method of characteristic coordinates), we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2).
Theorem 1. Let a function be given such that the double integral exists for all . If a twice continuously differentiable function satisfies the inequality for all , then there exists a solution of the wave equation (2) which satisfies for all .
Proof. Let us define a function by
If we set and , then we have and
for all . Hence, we have
for any . Thus, it follows from inequality (6) that
for any .
Therefore, we get or equivalently for all .
On account of (8), we get Hence, it follows from (13) and the last equalities that for all .
If we set and in the last inequality, then we obtain for all , where we set
By some tedious calculations, we get for all . Hence, we know that for any ; that is, is a solution of the wave equation (2).
Corollary 2. Given a constant , let a function be given as If a twice continuously differentiable function satisfies inequality (6), for all , then there exists a solution of the wave equation (2) which satisfies for all .
Proof. Since for all , in view of Theorem 1, we conclude that the statement of this corollary is true.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557).
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