Stability of a Cauchy-Jensen additive mapping in various normed spaces (Q1716124)

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, S. M. Ulam posed the first stability problem. In the following year \textit{D. H. Hyers} [Proc. Natl. Acad. Sci. USA 27, 222--224 (1941; Zbl 0061.26403; JFM 67.0424.01)] 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 use a fixed point method and a direct method to prove the Hyers-Ulam stability of the Cauchy-Jensen additive functional equation in various normed spaces. For the entire collection see [Zbl 1402.47001].