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Nearly quadratic mappings over \(p\)-adic fields. (English) Zbl 1250.39013

Summary: We establish some stability results over \(p\)-adic fields for the generalized quadratic functional equation \[ \sum^n_{k=2} \sum^k_{i_{1}=2} \sum^{k+1}_{i_{2}=i_{1}+1} \cdots \sum^n_{i_{n-k+1}=i_{n-k}+1} f\left(\sum^n_{_{\substack{ i=1\\ i \neq i_1, \dots, i_{n-k+1}}}} x_i - \sum^{n-k+1}_{r=1} x_{i_r}\right) + f\left(\sum^n_{i=1} x_i\right) = 2^{n-1} \sum^n_{i=1} f(x_i), \] where \(n\in \mathbb N\) and \(n\geq 2\).

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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