Approximation of an additive-quadratic functional equation in RN-spaces (Q2912650)

The authors consider functions \(f\colon X\to Y\) where \(X\) is a real vector space and \(Y\) a random normed (RN) space. Given positive reals \(a,b,c\) and assuming that \(f\) satisfies approximately (in a certain sense described in the paper) the equation NEWLINENEWLINE\[NEWLINE\begin{multlined} af(b^{-1}(x+y+z))+af(b^{-1}(x-y+z))+af(b^{-1}(x+y-z))+af(b^{-1}(-x+y+z))\\NEWLINE=cf(x)+cf(y)+cf(z) \,\, (x,y,z\in X)\end{multlined}NEWLINE\]NEWLINE they prove that \(f\) is (in a certain sense) close to some additive function \(A\) if \(f\) is odd. They also prove that \(f\) is close to some quadratic function \(Q\) if \(f\) is even and satisfies \(f(0)=0\). Also the case where (only) \(f(0)=0\) is satisfied is discussed.NEWLINENEWLINEIn doing so they construct odd functions \(A\) and even functions \(Q\) with \(Q(0)=0\) satisfying the equation above exactly. They do not say why then \(A\) is additive and \(Q\) is quadratic. Also some remarks about the general solution of the equation are missing. Moreover nothing is mentioned about the approximative case in general. Thus the case of general \(f\) (\(f(0)\) arbitrary) remains open.