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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 $$\bigl\|f(ax+ay)+f(ax-ay)-2a^2f(x)-2a^2f(y)\bigr\|\le\varphi(x,y)$$ for all $a\in B_1$ and all $x,y\in{}_BM_1$. If $f(tx)$ is continuous in $t\in{\Bbb R}$ for each fixed $x\in{}_BM_1$, then there exists a unique $B$-quadratic mapping $Q:{}_BM_1\to{}_BM_2$ such that $\bigl\|f(x)-Q(x)\bigr\|\le\widetilde{\varphi}(x)$ for all $x\in{}_BM_1$. The similar results are obtained for the other functional equations: $$ \align f(ax+y)+f(ax-y)&=2a^2f(x)+2f(y),\\ a^2f(x+y)+a^2f(x-y)&=2f(ax)+2f(ay),\\ f(ax+ay)+f(ax-ay)&=2a^2g(x)+2a^2g(y),\\ a^2f(x+y)+a^2f(x-y)&=2g(ax)+2g(ay) \endalign $$ and for the classical quadratic functional equation.

39B82Stability, separation, extension, and related topics
39B52Functional equations for functions with more general domains and/or ranges
47B48Operators on Banach algebras
47H99Nonlinear operators
46H25Normed modules and Banach modules, topological modules
Full Text: DOI
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