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On the stability of the orthogonally quartic functional equation. (English) Zbl 1117.39020

Let \(X\) be a normed space equipped with an abstract orthogonality relation \(\bot\) [cf. 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 S. H. Lee, S. M. Im and 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 \[ \| f(x)-T(x)\| \leq \frac{\theta}{| 32-2^{p+1}| } \| x\| ^p,\quad x\in X. \]

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
39B55 Orthogonal additivity and other conditional functional equations
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