A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations
Two Pexiderized versions of the quadratic functional equation are considered: \[ \begin{aligned} f(x\ast y)+f(x\ast y^{-1})&=2g(x)+2g(y),\\ f(x\ast y)+g(x\ast y^{-1})&=2h(x)+2k(y), \end{aligned} \] where \(f,g,h,k\) are functions mapping a group \((G,\ast)\) into a~real or complex topological vector space~\(X\). The main results are obtained for functions mapping a~vector space~\(V\) into a~Banach space~\(X\). It is shown that if the remainder \(f(x+y)+f(x-y)-2g(x)-2g(y)\) is bounded in some sense, then there exists exactly one quadratic function \(Q:V\to X\) such that \(f\) is close (in the same sense) to \(f(0)+Q\) and \(g\) is close to \(\frac{f(0)}{2}+Q\). It is also shown that if the remainder \(f(x+y)+g(x-y)-2h(x)-2k(y)\) is bounded (in some sense), then there exist exactly one quadratic function \(Q:V\to X\) and two unique additive functions \(T,T':V\to X\) such that \(f\) is close (in the same sense) to \(f(0)+Q+T\), \(g\) is close to \(g(0)+Q+T'\) and \(\frac{h+k}{2}\) is close to \(\frac{f(0)+g(0)}{4}+Q+\frac{T}{2}\).
scientific article; zbMATH DE number 2105429