A fixed point application for the stability and hyperstability of multi-Jensen-quadratic mappings (Q2288064)

The concept of stability for a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. In \(1940\) Ulam posed the first stability problem. In the next year \textit{D. H. Hyers} [Proc. Natl. Acad. Sci. USA 27, 222--224 (1941; Zbl 0061.26403)] gave a partial 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. \textit{K. Ciepliński} [J. Math. Anal. Appl. 363, No. 1, 249--254 (2010; Zbl 1211.39017)] proved the stability of multi-Jensen mappings in normed spaces. In the present work, the authors reduce a system of \(n\) equations, defining the multi-Jensen-quadratic mappings, to a single functional equation. Then they prove the generalized Hyers-Ulam stability for multi-Jensen-quadratic functional equations by using the fixed point method.