Stability of exponential equations in Schwartz distributions

We reformulate the superstability of exponential equation and cosine functional equation J.A. Baker, The stability of cosine equation, Proc. Amer. Math. Soc. 80 (1980) 411–416] in some spaces of generalized functions such as the Schwartz distributions, Sato hyperfunctions, and Gelfand generalized functions, which completes the previous results of partial generalizations of the stability problems J. Chung, A distributional version of functional equations and their stabilities, Nonlinear Anal. 62 (2005) 1037–1051; J. Chung, S.Y. Chung, D. Kim, The stability of Cauchy equations in the space of Schwartz distributions, J. Math. Anal. Appl. 295 (2004) 107–114].