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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.

MSC:
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
WorldCat.org
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References:
[1] Bonsall, F.; Duncan, J.: Complete normed algebras. (1973) · Zbl 0271.46039
[2] Borelli, C.; Forti, G.: On a general Hyers--Ulam stability result. Internat. J. Math. math. Sci. 18, 229-236 (1995) · Zbl 0826.39009
[3] Cholewa, P. W.: Remarks on the stability of functional equations. Aequationes math. 27, 76-86 (1984) · Zbl 0549.39006
[4] Czerwik, S.: On the stability of the quadratic mapping in the normed space. Abh. math. Sem. Hamburg 62, 59-64 (1992) · Zbl 0779.39003
[5] Hyers, D. H.: On the stability of the linear functional equation. Proc. natl. Acad. sci. USA 27, 222-224 (1941) · Zbl 0061.26403
[6] Hyers, D. H.; Isac, G.; Rassias, Th.M.: Stability of functional equations in several variables. (1998) · Zbl 0907.39025
[7] Jun, K.; Lee, Y.: On the Hyers--Ulam--rassias stability of a pexiderized quadratic inequality. Math. ineq. Appl. 4, 93-118 (2001) · Zbl 0976.39031
[8] Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. amer. Math. soc. 72, 297-300 (1978) · Zbl 0398.47040
[9] Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. math. Anal. appl. 251, 264-284 (2000) · Zbl 0964.39026
[10] Schröder, H.: K-theory for real c\ast-algebras and applications. Pitman res. Notes math. Ser. 290 (1993)
[11] Skof, F.: Proprietà locali e approssimazione di operatori. Rend. sem. Mat. fis. Milano 53, 113-129 (1983)
[12] Ulam, S. M.: Problems in modern mathematics. (1960) · Zbl 0086.24101