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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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