On the Stability of Wave Equation

and Applied Analysis 3 for each r ∈ R. Then we have


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.
The case of approximately additive functions was solved by Hyers [2] under the assumption that  1 and  2 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.
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.

Main Results
For a given integer  ≥ 2,   denotes the th coordinate of any point  in R  ; that is  = ( 1 , . . .,   , . . .,   ), and || denotes the Euclidean distance between  and the origin; that is, Given a real number  > 0, assume that real numbers  and  2 satisfy  >  and 0 <  2 < ∞, and define We remark that (, ) ∈ × if and only if ||/ ∈ .Using an idea from [11], we define a class  of all twice continuously differentiable functions  :  ×  → R with the properties (i) (, ) = V(||/) for all  ∈  and  ∈  and for some V :  → R;

If we define
for all  1 ,  2 ∈  and  ∈ R, then  is a vector space over real numbers.That is,  is a large class such that it is a vector space.
for all  ∈  and  ∈ , then there exists a solution  0 :  ×  → R of the wave equation (2) which belongs to  and satisfies for all  ∈  and  ∈ .
Proof.Let V : R → R be a function which satisfies for all  ∈  and  ∈ .For any  ∈ {1, 2, . . ., }, 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 or for all  ∈ , where we set () := V  ().Set for each  ∈ .Then we have According to (18) and [13, Theorem 1], there exists a unique real number  such that for all  ∈ .
Hence, it follows from the last inequalities that for any  ∈ .
If we define a function  0 :  ×  → R by then we have for all  ∈  and  ∈ , which implies that  0 (, ) is a solution of the wave equation ( 2).
Assume now that  and  1 are given real numbers satisfying 0 <  <  and 0 <  1 < ∞.We then set and define a class   of all twice continuously differentiable functions  :   ×   → R with the properties (iii) (, ) = V(||/) for all  ∈   and  ∈   and for some V : It might be remarked that (, ) ∈   ×   if and only if ||/ ∈   .If we define for all  1 ,  2 ∈   and  ∈ R, then   is a vector space over real numbers.
for all  ∈   and  ∈   .
Proof.If V : R → R 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 () := V  ().Set for any  ∈   .Then we get According to (35) and [13,Corollary 2], there exists a unique real number  such that for all  ∈   .
From the last inequalities, it follows that for each  ∈   .On account of (iv), we have lim Then, a similar argument to the last part of the proof of Theorem 1 shows that  0 (, ) is a solution of the wave equation ( 2) and it belongs to   .Finally, the validity of (34) immediately follows from (41).

Remarks
Remark 1.The inequality (10) in Theorem 1 can be rewritten as for all  ∈  and  ∈ .If we further substitute sin  for / in the previous inequality, then we obtain for any  ∈  and  ∈ .
For the case of  = 3, the inequality (10) can be rewritten as for all  ∈  and  ∈ .

Remark 2. As in
for all  ∈   and  ∈   .