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The stability of a cubic type functional equation with the fixed point alternative. (English) Zbl 1077.39026
The authors prove that a function \(f\) between real vector spaces satisfies the functional equation \[ \begin{split} f(x+y+2z)+f(x+y-2z)+f(2x)+f(2y)\\=2[f(x+y)+2f(x+z)+2f(x-z)+f(y+z)+2f(y-z)]\end{split}\tag{\(*\)} \] if and only if \(f\) is cubic, i.e. it satisfies \(f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)\).
Using the fixed point alternative theorem, they establish the generalized Hyers-Ulam-Rassias stability of the equation \((*)\). The same technique for establishing stability of other equations can be found in the papers of L. Cadariu and V. Radu [JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 4 (2003; Zbl 1043.39010)] and of K.-H. Park and Y.-S. Jung [Commun. Korean Math. Soc. 19, No. 2, 253–266 (2004)].

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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