Solution of the Ulam stability problem for an Euler type quadratic functional equation (Q1864530)

Let \(ABCD\) be a~quadrilateral and let \(A',B',C',D',M,N\) be the midpoints of the sides \(AB\), \(BC\), \(CD,\) \(DA\) and of the diagonals \(AC,BD\), respectively. Then \[ |AB|^2+|BC|^2+|CD|^2+|DA|^2= 2|A'C'|^2+2|B'D'|^2+4|MN|^2. \] This well known geometric identity leads to the following functional equation: \[ \begin{multlined} 2\bigl[Q(x_0-x_1)+Q(x_1-x_2)+Q(x_2-x_3)+Q(x_3-x_0)\bigr]=\\ =Q(x_0-x_1-x_2+x_3)+Q(x_0+x_1-x_2-x_3)+2Q(x_0-x_1+x_2-x_3). \end{multlined}\tag{1} \] The following stability result is proved: Theorem. Let \(X\) be a normed space and let \(Y\) be a Banach space. Assume that \(f:X\to Y\) fulfils the functional inequality \[ \begin{multlined} \|2[f(x_0-x_1)+f(x_1-x_2)+ f(x_2-x_3)+f(x_3-x_0)]-\\ -[f(x_0-x_1-x_2+x_3)+ f(x_0+x_1-x_2-x_3)+ 2f(x_0-x_1+x_2-x_3)]\|\leq c \end{multlined} \] for all \(x_0,x_1,x_2,x_3\in X\) and for some constant~\(c\geq 0\) independent of \(x_0,x_1,x_2,x_3\). Then the limit \(Q(x)=\lim_{n\to\infty}2^{-2n}f(2^nx)\) exists for all \(x\in X\) and \(Q:X\to Y\) is the unique mapping satisfying~(1) and such that \(\bigl\|f(x)-Q(x)\bigr\|\leq{11\over 12}c\) for all \(x\in X\).