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On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations. (English) Zbl 1148.39026
Let $$n>2$$ be an integer, and let $$X,Y$$, $$f:X\to Y$$ be vector spaces. One of the corollaries of the main result is that the functional equation below is stable in the sense of Hyers and Ulam: $f\biggl(\sum_{i=1}^nx_i\biggr)+(n-2)\sum_{i=1}^nf(x_i)= \sum_{1\leq i<j\leq n}f(x_i+x_j)\qquad (x_1,\ldots,x_n\in X).$ The author also gives its general solutions, concluding that the even part of $$f$$ satisfies the classical quadratic equation, while the odd part of $$f$$ satisfies the Cauchy functional equation. The main tool of the proofs is based on Hyers’ method.
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
##### Keywords:
Hyers-Ulam stability
Full Text:
##### References:
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