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.