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Solution of the Ulam stability problem for an Euler type quadratic functional equation. (English) Zbl 1017.39011
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: $\begin{split} 2\bigl[Q(x_0-x_1)+Q(x_1-x_2)+Q(x_2-x_3)+Q(x_3-x_0)\bigr]=\\ =Q(x_0-x_1-x_2+x_3)+Q(x_0+x_1-x_2-x_3)+2Q(x_0-x_1+x_2-x_3). \end{split}\tag{1}$ 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 $\begin{split} \|2[f(x_0-x_1)+f(x_1-x_2)+ f(x_2-x_3)+f(x_3-x_0)]-\\ -[f(x_0-x_1-x_2+x_3)+ f(x_0+x_1-x_2-x_3)+ 2f(x_0-x_1+x_2-x_3)]\|\leq c \end{split}$ 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 $$x\in X$$.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 39B62 Functional inequalities, including subadditivity, convexity, etc.
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