On stability and hyperstability of an equation characterizing multi-Cauchy-Jensen mappings
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 \textit{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 \textit{T. Aoki} [J. Math. Soc. Japan 2, 64--66 (1950; Zbl 0040.35501)] for additive mappings and by \textit{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.
scientific article; zbMATH DE number 6909674
scientific article