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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 910565, 6 pages
On the Stability of Wave Equation
Mathematics Section, College of Science and Technology, Hongik University, Sejong 339-701, Republic of Korea
Received 30 September 2013; Accepted 11 November 2013
Academic Editor: Junesang Choi
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.
We prove the generalized Hyers-Ulam stability of the wave equation, , in a class of twice continuously differentiable functions under some conditions.
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 the Hyers’ theorem. That is, Rassias proved the generalized Hyers-Ulam stability (or the Hyers-Ulam-Rassias stability) of the Cauchy additive functional equation. Since then, the stability of several functional equations has been extensively investigated [4–10].
The terminologies, the generalized Hyers-Ulam stability and the Hyers-Ulam stability, can also be applied to the case of other functional equations, of differential equations, and of various integral equations.
Given a real number , the partial differential equation is called the wave equation, where and denote the second time derivative and the Laplacian of , respectively.
For an integer , assume that and are open (connected) subsets of and , respectively. Let be a function. If, for each twice continuously differentiable function satisfying there exist a solution of the 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. Main Results
For a given integer , denotes the th coordinate of any point in ; that is , and denotes the Euclidean distance between and the origin; that is,
Given a real number , assume that real numbers and satisfy and , and define We remark that if and only if . Using an idea from , we define a class of all twice continuously differentiable functions with the properties (i) for all and and for some ;(ii) .
If we define for all and , then is a vector space over real numbers. That is, is a large class such that it is a vector space.
Theorem 1. Let a function be given such that there exists a positive real number with If a satisfies the inequality for all and , then there exists a solution of the wave equation (2) which belongs to and satisfies for all and .
Proof. Let be a function which satisfies
for all and . For any , we differentiate with respect to to get
Similarly, we obtain the second partial derivative of with respect to as follows:
Hence, we have
By a similar way, we further get the second derivative of with respect to as follows:
Therefore, it follows from (14) and (15) that
for any , , and , and it follows from (8) and (9) that
for all , where we set .
Set for each . Then we have According to (18) and [13, Theorem 1], there exists a unique real number such that or for all .
Hence, it follows from the last inequalities that for any .
Due to , it holds that . Replacing with in the last inequalities, we get for all and .
If we define a function by then we have for all and , which implies that is a solution of the wave equation (2).
It is now to show that . Let be a function with the property Then we have which implies that can be expressed as , where . Moreover, we get which verifies that . Finally, by (24), the inequality (10) holds true.
Assume now that and are given real numbers satisfying and . We then set and define a class of all twice continuously differentiable functions with the properties (iii) for all and and for some ;(iv).
It might be remarked that if and only if . If we define for all and , then is a vector space over real numbers.
Theorem 2. Let a function be given such that there exists a positive real number with If a satisfies the inequality for all and , then there exists a solution of the wave equation (2) which belongs to and satisfies for all and .
Proof. If is given by (11), then we can simply follow the lines in the first part of the proof of Theorem 1 to obtain
for all , where .
Set for any . Then we get According to (35) and [13, Corollary 2], there exists a unique real number such that or for all .
From the last inequalities, it follows that for each .
On account of , we have . Replacing with in the last inequalities, we obtain for alland .
Let us define a function by Then, a similar argument to the last part of the proof of Theorem 1 shows that is a solution of the wave equation (2) and it belongs to . Finally, the validity of (34) immediately follows from (41).
Remark 1. The inequality (10) in Theorem 1 can be rewritten as
for all and . If we further substitute for in the previous inequality, then we obtain
for any and .
For the case of , the inequality (10) can be rewritten as for all and .
Remark 2. As in Remark 1, the inequality (34) in Theorem 2 can be rewritten as
for all and . If we substitute for in the previous inequality, then we get
for any and .
For the case of , the inequality (34) can be rewritten as for all and .
Remark 3. It is an open problem whether the wave equation (2) has the generalized Hyers-Ulam stability for the case of either and or and or and .
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). This work was also supported by the 2011 Hongik University Research Fund.
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