Hyperstability results for the general linear functional equation in non-Archimedean 2-Banach spaces

Abstract

Let X be a 2-normed space over R, ℝ Y be a non-Archimedean 2-Banach space over non-Archimedean field K, r, s ℝ \ {0} , and R, S ∈ K \ {0}. In this paper, a short preface on non- Archimedean 2-Banach spaces (Y, ||,··||) is given. Then, we reformulate the Brzdek fixed point theorem in non-Archimedean 2-Banach spaces. Using the Brzdek fixed point method, we prove hyperstability results of the general linear functional equation h(rx + sy) = Rh(x) + Sh(y), x, y, ∈ X, in non-Archimedean 2-Banach spaces. In fact, under some natural assumptions on control function Y: X × X × Y → [0, ∞) , we show that every map satisfying ||h(rx + sy) - Rh(x) - Sh(y), z||* = ≤ y(x, y, z), x, y ∈ z, ∈ Y, is hyperstable in the class of functions h: X → Y.