On the superstability of generalized d’Alembert harmonic functions

Abstract

The aim of this paper is to study the superstability problem of the d’Alembert type functional equation f(x + y + z) + f(x + y + σ(z)) + f(x + σ(y) + z) + f(σ(x) + y + z) = 4f(x)f(y)f(z) for all x, y, z ∈G, where G is an abelian group and σ: G →G is an endomorphism such that σ(σ(x)) = x for an unknown function f from G into C or into a commutative semisimple Banach algebra. 1. Introduction In 1940, Ulam [19] gave a wide ranging talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of important unsolved problems and among those the following question concerning the stability of homomorphisms Let G1 be a group and let (G2, d) be a metric group. Given δ > 0, does there exist ϵ > 0 such that if a mapping h: G1 →G2 satisfies the inequality d(h(xy), h(x)h(y)) ≤δ for all x, y ∈G1, then there is a homomorphism a: G1 →G2 with d(h(x), a(x)) ≤ϵ for all x ∈G1?