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

EL-Fassi I.

Journal of Applied Analysis

2018

Vol. 24, nr 2

155--165

Let X be a normed space, U ⊂ 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 ∈ G). Under some weak natural assumptions on U and on the function γ : U3 → [0, ∞), we study the new generalized hyperstability results when f : U → G satisfies the inequality d(αf( x + y / α + z), αf(z) + f(y) + f(x)) ≤ γ(x, y, z) for all x, y, z ∈ U, where x+y α + z ∈ U and α ≥ 2 is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdęk and K. Ciepliński, 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.