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Generalized quadratic mappings in several variables. (English) Zbl 1058.39024
The author defines the generalized quadratic mapping as an even solution \(f:X\to Y\) (\(X\) and \(Y\) are given vector spaces) of the functional equation: \[ {(2 _{d-2}C_{l-1} - _{d-2}C_l - _{d-2}C_{l-2} )f\left(\sum_{j=1}^{d}x_j\right)+\sum_{\begin{matrix} \iota(j)=0,1\\ \sum_{j=1}^{d}\iota(j)=l\end{matrix}} f\left(\sum_{j=1}^{d}(-1)^{\iota(j)}x_j\right)} \]
\[ { =( _{d-1}C_{l} + _{d-1}C_{l-1} + 2 _{d-2}C_{l-1} - _{d-2}C_l - _{d-2}C_{l-2})\sum_{j=1}^{d}f(x_j)}, x_1,\ldots,x_d\in X \] and satisfying \(f(0)=0\). \(d\) and \(l\) are fixed integers such that \(1<l<d\) and \(_kC_j\) stands for the binomial coefficient \(\left(\begin{matrix} k\\ j\end{matrix}\right)\). Indeed, all even solutions of (1) are quadratic, i.e., they satisfy the functional equation \[ f(x+y)+f(x-y)=2f(x)+2f(y). \] Next, the stability of the considered equation is proved, provided that \(X\) is a normed space and \(Y\) is a Banach space. Let \(Df(x_1,\ldots,x_d)\) denotes the difference between the left and the right hand sides of (1). Assuming that \[ \| Df(x_1,\ldots,x_d)\| \leq\varphi(x_1,\ldots,x_d) \] where \(\varphi:X^d\to [0,\infty)\) satisfies some additional assumptions, it is proved that \(f\) can be suitably approximated by a unique quadratic mapping \(Q\). In particular, the results may be applied to the case: \[ \| Df(x_1,\ldots,x_d)\| \leq\Theta\sum_{j=1}^{d}\| x_j\| ^p,\;\;\;x_1,\ldots,x_d\in X \] with \(p\neq 2\).

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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