Non-Archimedean hyperstability of Cauchy-Jensen functional equations on a restricted domain

Let X be a normed space, U subset of X \ {0} a non-empty subset, and (G, +) a commutative group equipped with a complete ultrametric d that is invariant (i.e., d(x + z, y + z) = d(x, y) for x, y, z is an element of G). Under some weak natural assumptions on U and on the function y: U-3 -> [0, infinity), we study the new generalized hyperstability results when f : U -> G satisfies the inequality d(alpha f(x + y/alpha + z),alpha f(z) + f(y) + f(x)) <= gamma(x, y, z) for all x, y, z is an element of U, where x+y/alpha + z is an element of U and alpha >= 2 is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdek and K. Cieplinski, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 (2011), no. 18, 6861-6867]) in some functions spaces.