#### Abstract

We prove the generalized Hyers-Ulam stability of the one-dimensional wave equation, , in a class of twice continuously differentiable functions.

#### 1. Introduction

In 1940, Ulam [1] 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 [2] 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 [3] 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 [49].

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).

In this paper, using an idea from [10], we prove the generalized Hyers-Ulam stability of the (one-dimensional) wave equation (2).

#### 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.

#### Acknowledgments

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).