On the superstability of the cosine and sine type functional equations

Abstract

In this paper, we study the superstablity problem of the cosine and sine type functional equations: $f(xσ(y)a) + f(xya) = 2f(x)f(y)$ and $f(xσ(y)a) − f(xya) = 2f(x)f(y)$, where $f : S → C$ is a complex valued function; $S$ is a semigroup; $σ$ is an involution of $S$ and a is a fixed element in the center of $S$.