Ulam stability problem for quadratic mappings of Euler--Lagrange
The authors consider a generalization of the \textit{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 \textit{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.
scientific article; zbMATH DE number 2173620