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Ulam stability problem for quadratic mappings of Euler–Lagrange. (English) Zbl 1074.39027
The authors consider a generalization of the quadratic functional equation. Let $${\mathcal A}$$ and $${\mathcal B}$$ be linear spaces and $$a,b\in\mathbb{N}$$. A mapping $$Q:{\mathcal A}\to {\mathcal B}$$ satisfying the equation $(a+b)aQ(x)+(a+b)bQ(y)=Q(ax+by)+abQ(x-y),\;\;\;x,y\in {\mathcal A}$ is called the {quadratic mapping of Euler–Lagrange}.
The aim of the paper is to prove the stability of the above equation. It is shown that, with $${\mathcal B}$$ being a normed space, for $$f:{\mathcal A}\to {\mathcal B}$$ satisfying the inequality $\| (a+b)aQ(x)+(a+b)bQ(y)-Q(ax+by)-abQ(x-y)\| \leq\varphi(x,y),\;\;\;x,y\in {\mathcal A}$ with the approximate reminder $$\varphi$$ satisfying some assumptions, there exists a unique quadratic mapping of Euler-Lagrange $$Q:{\mathcal A}\to{\mathcal B}$$ which is, in some sense, close to $$f$$. Additionally, similar results are given for mappings between Banach modules.

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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References:
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