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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 $|·|$, ${B}_{1}=\left\{a\in B:|a|=1\right\}$ and let ${}_{B}{M}_{1}$, ${}_{B}{M}_{2}$ be left Banach $B$-modules. A quadratic mapping $Q:{}_{B}{M}_{1}\to {}_{B}{M}_{2}$ is called $B$-quadratic if $Q\left(ax\right)={a}^{2}Q\left(x\right)$ for all $a\in B$, $x\in {}_{B}{M}_{1}$. Let $\varphi :{}_{B}{M}_{1}×{}_{B}{M}_{2}\to \left[0,\infty \right)$ be a function such that one of the series ${\sum }_{n=1}^{\infty }{2}^{-2n}\varphi \left({2}^{n-1}x,{2}^{n-1}x\right)$ and ${\sum }_{n=1}^{\infty }{2}^{2n-2}\varphi \left({2}^{-n}x,{2}^{-n}x\right)$ converges for every $x\in {}_{B}{M}_{1}$. Denote by $\stackrel{˜}{\varphi }\left(x\right)$ the sum of the convergent series. The following theorem is proved:

Theorem. Let $f:{}_{B}{M}_{1}\to {}_{B}{M}_{2}$ be a mapping such that $f\left(0\right)=0$ and

$∥f\left(ax+ay\right)+f\left(ax-ay\right)-2{a}^{2}f\left(x\right)-2{a}^{2}f\left(y\right)∥\le \varphi \left(x,y\right)$

for all $a\in {B}_{1}$ and all $x,y\in {}_{B}{M}_{1}$. If $f\left(tx\right)$ is continuous in $t\in ℝ$ for each fixed $x\in {}_{B}{M}_{1}$, then there exists a unique $B$-quadratic mapping $Q:{}_{B}{M}_{1}\to {}_{B}{M}_{2}$ such that $∥f\left(x\right)-Q\left(x\right)∥\le \stackrel{˜}{\varphi }\left(x\right)$ for all $x\in {}_{B}{M}_{1}$.

The similar results are obtained for the other functional equations:

$\begin{array}{cc}\hfill f\left(ax+y\right)+f\left(ax-y\right)& =2{a}^{2}f\left(x\right)+2f\left(y\right),\hfill \\ \hfill {a}^{2}f\left(x+y\right)+{a}^{2}f\left(x-y\right)& =2f\left(ax\right)+2f\left(ay\right),\hfill \\ \hfill f\left(ax+ay\right)+f\left(ax-ay\right)& =2{a}^{2}g\left(x\right)+2{a}^{2}g\left(y\right),\hfill \\ \hfill {a}^{2}f\left(x+y\right)+{a}^{2}f\left(x-y\right)& =2g\left(ax\right)+2g\left(ay\right)\hfill \end{array}$

and for the classical quadratic functional equation.

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges 47B48 Operators on Banach algebras 47H99 Nonlinear operators 46H25 Normed modules and Banach modules, topological modules
##### Keywords:
Banach module; quadratic functional equation; stability