On approximate homomorphisms: a fixed point approach

Consider the functional equation Ȝ_1(f)= Ȝ_2(f)(̃Ȝ) in a certain general setting. A function g is an approximate solution of (Ȝ) if Ȝ1(g) and 2(g) are close in some sense. The Ulam stability problem asks whether or not there is a true solution of (Ȝ) near g. A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each m = 1, 2, 3, 4, we will find out the general solution of the functional equation f (ax + y) + f (ax − y) = am−2[ f (x + y) + f (x − y)]+2(a2 − 1)[ am−2f (x) + (m − 2)(1 − (m − 2)2) 6 f (y)] for any fixed integer a with a = 0±1. Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions.