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. 1. Introduction The stability problem of the functional equation was conjectured by Ulam [12] during the conference in the University of Wisconsin in 1940. In the next year, Hyers in [5] solved the problem of stability in the case of additive mapping. Since then it is called the Hyers-Ulam stability. In 1979, Baker et al. in [4] introduced the following: if f satisfies the inequality |E1(f) −E2(f)| ≤ϵ, then either f is bounded or E1(f) = E2(f). The stability of this type is called the superstability. The superstability of the cosine functional equation (also called the d’Alembert equation) f(x + y) + f(x −y) = 2f(x)f(y), was investigated by Baker [3]. Their results were improved by Badora [1] and Badora and Ger [2].