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On the stability of the quadratic mapping in Banach modules. (English) Zbl 1017.39010
Let \(B\) be a unital Banach algebra with norm \(|\cdot|\), \(B_1=\{a\in B:|a|=1\}\) and let \({}_BM_1\), \({}_BM_2\) be left Banach \(B\)-modules. A quadratic mapping \(Q:{}_BM_1\to{}_BM_2\) is called \(B\)-quadratic if \(Q(ax)=a^2Q(x)\) for all \(a\in B\), \(x\in{}_BM_1\). Let \(\varphi:{}_BM_1\times{}_BM_2\to[0,\infty)\) be a function such that one of the series \(\sum_{n=1}^{\infty}2^{-2n}\varphi(2^{n-1}x,2^{n-1}x)\) and \(\sum_{n=1}^{\infty}2^{2n-2}\varphi(2^{-n}x,2^{-n}x)\) converges for every \(x\in{}_BM_1\). Denote by \(\widetilde{\varphi}(x)\) the sum of the convergent series. The following theorem is proved:
Theorem. Let \(f:{}_BM_1\to{}_BM_2\) be a mapping such that \(f(0)=0\) and \[ \bigl\|f(ax+ay)+f(ax-ay)-2a^2f(x)-2a^2f(y)\bigr\|\leq\varphi(x,y) \] for all \(a\in B_1\) and all \(x,y\in{}_BM_1\). If \(f(tx)\) is continuous in \(t\in{\mathbb R}\) for each fixed \(x\in{}_BM_1\), then there exists a unique \(B\)-quadratic mapping \(Q:{}_BM_1\to{}_BM_2\) such that \(\bigl\|f(x)-Q(x)\bigr\|\leq\widetilde{\varphi}(x)\) for all \(x\in{}_BM_1\).
The similar results are obtained for the other functional equations: \[ \begin{aligned} f(ax+y)+f(ax-y)&=2a^2f(x)+2f(y),\\ a^2f(x+y)+a^2f(x-y)&=2f(ax)+2f(ay),\\ f(ax+ay)+f(ax-ay)&=2a^2g(x)+2a^2g(y),\\ a^2f(x+y)+a^2f(x-y)&=2g(ax)+2g(ay) \end{aligned} \] and for the classical quadratic functional equation.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
47B48 Linear operators on Banach algebras
47H99 Nonlinear operators and their properties
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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