Erratum to: Quartic functional equations in Lipschitz spaces

Abstract

In [1] we proved the stability of quartic functional equations by using 2-variable quadratic functional equations inLipschitz spaces. In this paper,we show that the results presented in [1] only hold for quadratic functional equations and sowe correct our results and approximate the quartic functional equations by using biquadratic functional equations. Indeed, the equality K (2x, 2y) = 24K (x, y), used in [1, Theorem 2.1], does not hold for the 2-variable quadratic functional equation, that is, the function Q defined in [1, Theorem 2.1] is quadratic and it is not quartic. We need to introduce the notion of symmetric left invariant mean. Let G be an abelian group, E a vector space, and S(E) a family of subsets of E . By B(G, S(E))we denote the family of all functions f : G −→ E such that Im f ⊂ A for some A ∈ S(E). We say that B(G, S(E)) admits a symmetric left invariant mean (briefly SLIM), if the family S(E) is linearly invariant and there exists a linear operatorM : B(G, S(E)) −→ E such that (i) if fx,y ∈ B(G, S(E)) and (x, y) ∈ G × G, then M[ fx,y] = M[ fy,x ],