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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:

f(2x+y)+f(2x-y)+4f(x)+f(y)+f(-y)=2f(x+y)+2f(x-y)+2f(2x)andf(ax+y)+f(ax-y)=af(x+y)+af(x-y)+2a(a 2 -1)f(x)

(where a{-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 2 f(x+y)+ab 2 f(x-y)+2a(a 2 -b 2 )f(x)

(where a,b are fixed integers such that a{-1,0,1}, b0, a+b0, a-b0) is given for functions mapping a topological vector space into a Banach space. Both real and complex case is considered.


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