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A new approach to Hyers-Ulam stability of $$r$$-variable quadratic functional equations. (English) Zbl 1458.39023
Summary: In this paper, we investigate the general solution of a new quadratic functional equation of the form $\sum_{1\leq i<j<k \leq r} \phi (l_i+l_j+l_k)=(r-2) \sum_{i=1,i \neq j}^r \phi (l_i+l_j) + \tfrac{-r^2+3r-2}{2} \sum_{i=1}^r\phi (l_i).$ We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.
##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 47H10 Fixed-point theorems
##### Keywords:
quadratic mapping; fixed-point method; Ulam stability
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##### References:
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