Stability for quadratic functional equation in the spaces of generalized functions

Abstract

In this paper, we consider the general solution of quadratic functional equation f ( a x + y ) + f ( a x − y ) = f ( x + y ) + f ( x − y ) + 2 ( a 2 − 1 ) f ( x ) for any integer a with a ≠ − 1 , 0 , 1 . Moreover we reformulate and prove the Hyers–Ulam–Rassias stability theorem of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. The generalized Hyers–Ulam stability originated from the Th.M. Rassias's stability theorem that appeared in his paper [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300].