A generalized mixed quadratic-quartic functional equation (Q432522)

The article deals with the following functional equation \[ f(kx + y) + f(kx - y) = g(x + y) + g(x - y) + h(x) + \widetilde{h}(y) \quad (x, y \in X), \] where \(f,g,h,\widetilde{h}:\;X \to Y\) are unknown functions, \(k\) is a fixed integer with \(k \neq 0,\pm 1\), \(X\) and \(Y\) are linear spaces. The main result gives the complete description of solutions to this equation. As special cases the following functional equations are also considered: \[ \begin{multlined} f(kx + y) + f(kx - y)\\ = k^2f(x + y) + k^2f(x - y) + 2(f(kx) - k^2f(x)) - 2(k^2 - 1)f(y) \quad (x, y \in X)\end{multlined} \] and \[ \begin{multlined} f(kx + y) + f(kx - y)\\ = k^2f(x + y) + k^2f(x - y) + 2k^2(k^2 - 1)f(x) - 2(k^2 - 1)f(y) \quad (x, y \in X).\end{multlined} \] In the end of the article the equation under consideration is studied in the case when linear spaces \(X\) and \(Y\) are changed to abelian groups \(G\) and \(S\) with some additional properties.