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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: \[ \begin{gathered} f(2x+y)+f(2x-y)+4f(x)+f(y)+f(-y)=2f(x+y)+2f(x-y)+2f(2x)\quad\text{and}\\ f(ax+y)+f(ax-y)=af(x+y)+af(x-y)+2a(a^2-1)f(x) \end{gathered} \] (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 \(a\not\in\{-1,0,1\}\), \(b\neq 0\), \(a+b\neq 0\), \(a-b\neq 0\)) is given for functions mapping a topological vector space into a Banach space. Both real and complex case is considered.

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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