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On cubic derivations. (English) Zbl 1236.39027

Summary: We say a functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to true solution of (ξ). Also, we say that a functional equation is superstable if every approximately solution is an exact solution of it. In this paper, we investigate the stability and superstability of the system of functional equations

f(xy)=x 3 f(y)+f(x)y 3 ,f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)

on Banach algebras.

MSC:
39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
46L57Derivations, dissipations and positive semigroups in C * -algebras