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Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of an additive functional equation in several variables. (English) Zbl 1148.39027
Let \(n>1\) be an integer and \(X,Y\) be real vector spaces, \(\delta,\theta\geq 0\) and \(p>0\), \(p\neq 1\) with \(\delta=0\) if \(p>1\). Based on Hyers’ method, the author investigates the stability problem of a functional equation in the following settings: \[ \Bigl\|nf\Bigl(\sum_{i=1}^nx_i\Bigr)-\sum_{i=1}^nf(x_i)- \sum_{1\leq i<j\leq n}f(x_i+x_j)\Bigr\|\leq \delta+\theta\sum_{i=1}^n\|x_i\|^p \] \[ \Bigl\|nf\Bigl(\sum_{i=1}^nx_i\Bigr)-\sum_{i=1}^nf(x_i)- \sum_{1\leq i<j\leq n}f(x_i+x_j)\Bigr\|\leq \delta+\theta\sum_{1\leq i<j\leq n}\|x_i\|^{p/2}\|x_j\|^{p/2} \] The main results state, that, in each cases, the solutions of the inequalities are “close” to a unique additive mapping. In particular, the solutions of the corresponding functional equations are additive.
Consult also the papers T. Aoki [J. Math. Soc. Japan 2, 64–66 (1950; Zbl 0040.35501)] and L. Maligranda [Aequationes Math. 75, No. 3, 289–296 (2008; Zbl 1158.39019)] for interesting and important historical remarks.

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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