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On cubic derivations. (English) Zbl 1236.39027

Summary: We say a functional equation \((\xi)\) is stable if any function \(g\) satisfying the equation \((\xi)\) approximately is near to true solution of \((\xi)\). 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 \[ \begin{cases} f(xy)=x^3f(y)+f(x)y^3,\\ f(2x+y)+f(2x-y)=2f(x+y) +2f(x-y)+12f(x)\end{cases} \] on Banach algebras.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
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