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Ulam stability problem for generalized $$A$$-quadratic mappings. (English) Zbl 1069.39030
Let $$U$$ and $$V$$ be real vector spaces and let $$n>1$$ be a given natural number. The authors consider the functional equation $\sum_{i=1}^{n}Q\left(\sum_{j\neq i}x_j-(n-1)x_i\right)+nQ\left(\sum_{i=1}^{n}x_i\right)=n^2\sum_{i=1}^{n}Q(x_i)\tag $$*$$$ for all $$x_1,\ldots,x_n\in U$$, where $$Q:U\to V$$. It is easily seen that for $$n=2$$ the equation ($$*$$) is equivalent to the quadratic functional equation $Q(x+y)+Q(x-y)=2Q(x)+2Q(y),\;\;\;x,y\in U.$ It is shown that this equivalence holds also for $$n>2$$. Then, a stability of the equation ($$*$$) is proved. Let $$Df(x_1,\ldots,x_n)$$ denote the difference between the left and right hand sides of ($$*$$) with $$Q$$ replaced by $$f:U\to V$$. Assume that $$V$$ is a Banach space and $\| Df(x_1,\ldots,x_n)\| \leq\varphi(x_1,\ldots,x_n),\quad x_1,\ldots,x_n\in U$ where $$\varphi : U^n\to \mathbb{R}_{+}$$ satisfies appropriate conditions. Then there exists a unique mapping $$Q:U\to V$$ – a solution of ($$*$$) which is close (in some sense) to $$f$$. In particular, this result can be applied to $$\varphi(x_1,\ldots,x_n):=\theta+\varepsilon(\| x_1\| ^p+\cdots+\| x_n\| ^p)$$ with $$\varepsilon\geq 0$$, $$\theta\geq 0$$, $$p\neq 2$$ and $$\theta=0$$ if $$p>2$$. The results are also generalized to the case of mappings between Banach left modules over a unital Banach $$*$$-algebra.

MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges
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References:
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