On Stability and Hyperstability of an Equation Characterizing Multi-Cauchy–Jensen Mappings

Recently, functions of several variables satisfying, with respect to each variable, some functional equation (among them Cauchy’s, Jensen’s, quadratic and other ones) have been studied. We give a new characterization of multi-Cauchy–Jensen mappings, which states that a function fulfilling some equation on a restricted domain is multi-Cauchy–Jensen. Next, using a fixed point theorem, it is 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. This result is a tool for obtaining a generalized Hyers–Ulam stability or hyperstability of this equation for particular control functions, which is presented in several examples.