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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)
##### Keywords:
Banach module; quadratic functional equation; stability
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##### References:
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