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On the Hyers-Ulam stability of an Euler-Lagrange type cubic functional equation. (English) Zbl 1087.39029
The paper consists of two parts. In the first one the general solutions of two functional equations of cubic type are given: $$ \gather f(2x+y)+f(2x-y)+4f(x)+f(y)+f(-y)=2f(x+y)+2f(x-y)+2f(2x)\quad\text{and}\\ f(ax+y)+f(ax-y)=af(x+y)+af(x-y)+2a(a^2-1)f(x) \endgather $$ (where $a\not\in\{-1,0,1\}$ is a fixed integer). In the second part the stability in the spirit of Hyers, Ulam, Rassias and Găvruta of the functional equation $$ f(ax+by)+f(ax-by)=ab^2f(x+y)+ab^2f(x-y)+2a(a^2-b^2)f(x) $$ (where $a,b$ are fixed integers such that $a\not\in\{-1,0,1\}$, $b\ne 0$, $a+b\ne 0$, $a-b\ne 0$) is given for functions mapping a topological vector space into a Banach space. Both real and complex case is considered.

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges