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Solution of the Ulam stability problem for cubic mappings. (English) Zbl 0984.39014
A functional equation \[ f(x+2y)+3f(x)=3f(x+y)+f(x-y)+6f(y)\tag{1} \] is introduced. The following theorem is proved. Let \(X\) be a normed space, let \(Y\) be a real Banach space and let \(c\geq 0\). If \(f:X\to Y\) fulfils the inequality \[ \|f(x+2y)+3f(x)-(3f(x+y)+f(x-y)+6f(y))\|\leq c \] for all \(x,y\in X\), then there exists the unique mapping \(C:X\to Y\) satisfying (1) for all \(x,y\in X\), such that \(\|f(x)-C(x)\|\leq {11\over 42}c\) for all \(x\in X\).

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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