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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
##### Keywords:
Hyers-Ulam stability; cubic mapping; normed space; Banach space