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On stability and hyperstability of an equation characterizing multi-Cauchy-Jensen mappings. (English) Zbl 1404.39026
The concept of stability for a functional equation arises when one replaces a functional equation by inequality, which acts as a perturbation of the equation. In 1940 S. M. Ulam posed the first stability problem. In 1941 D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222–224 (1941; Zbl 0061.26403)] gave a partial affirmative answer to the question of Ulam. Hyers’ Theorem was generalized by T. Aoki [J. Math. Soc. Japan 2, 64–66 (1950; Zbl 0040.35501)] for additive mappings and by T. M. Rassias [Proc. Am. Math. Soc. 72, 297–300 (1978; Zbl 0398.47040)] for linear mappings by considering an unbounded Cauchy difference.
In the present work, the authors establish a new characterization of multi-Cauchy-Jensen mappings, which states that a function fulfilling some equation on a restricted domain is multi-Cauchy-Jensen on the whole space. Moreover, they proved that a function which approximately satisfies on restricted domain the equation characterizing such functions is close in some sense to the solution of the equation.

MSC:
39B52 Functional equations for functions with more general domains and/or ranges
39B82 Stability, separation, extension, and related topics for functional equations
39B72 Systems of functional equations and inequalities
47H10 Fixed-point theorems
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