Hyperstability Results for the General Linear Functional Equation in Non-Archimedean 2-Banach Spaces

Abstract

Let 𝑋 be a 2-normed space over 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 𝛾: 𝑋 × 𝑋 × 𝑌 → [0, ∞) , we show that every map satisfying ‖ℎ(𝑟𝑥 + 𝑠𝑦) − 𝑅ℎ(𝑥) − 𝑆ℎ(𝑦), 𝑧‖_∗ ≤ 𝛾(𝑥, 𝑦, 𝑧), 𝑥, 𝑦 ∈ 𝑋, 𝑧 ∈ 𝑌, is hyperstable in the class of functions ℎ: 𝑋 → 𝑌.