On the stability of the orthogonally quartic functional equation (Q875718)

Let \(X\) be a normed space equipped with an abstract orthogonality relation \(\bot\) [cf. \textit{J.\ Rätz}, Aequationes Math. 28, 35--49 (1985; Zbl 0569.39006)] and let \(Y\) be a Banach space. For mappings \(f:X\to Y\), the author introduces the orthogonally quartic functional equation: \[ x\bot y\;\Rightarrow f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+24f(x)-6f(y). \] [For an unconditional quartic equation see \textit{S. H. Lee, S. M. Im} and \textit{I. S. Hwang}, J. Math. Anal. Appl. 307, 387--394 (2005; Zbl 1072.39024)]. Although the solution of the above conditional equation is not given, its stability is shown. Namely, if the norm of the difference of the both sides of the equation is bounded by \(\theta\left(\| x\| ^p+\| y\| ^p\right)\) for all \(x,y\in X\) such that \(x\bot y\), (with \(\theta\geq 0\), \(0\leq p\neq 4\)), then there exists a unique orthogonally quartic mapping \(T:X\to Y\) such that \[ \| f(x)-T(x)\| \leq \frac{\theta}{| 32-2^{p+1}| } \| x\| ^p,\quad x\in X. \]