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A functional equation having monomials as solutions. (English) Zbl 1191.39026
For vector spaces \(X,Y\) and \(f: X\to Y\) consider the functional equations
\[ f(2x+y)+f(2x-y)=2^{n-2}[f(x+y)+f(x-y)+6f(x)] \] for \(n=1,2,3\) and
\[ f(2x+y)+f(2x-y)+6f(y)=4[f(x+y)+f(x-y)+6f(x)]. \] General solutions of the above equation as well as their stability are investigated. The stability results are obtained using both the direct and the fixed point methods.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
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