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.


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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