Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces

Let \( X \) be a normed space over \( \mathbb{F} \in\{ \mathbb{R}, \mathbb{C}\} \) , \( Y \) be a non-Archimedean Banach space over a non-Archimedean non-trivial field \(\mathbb{K}\) and \(c,d,C,D\) be constants such that, \( c, d \in \mathbb{F}\setminus\{0\} \) and \( C, D \in \mathbb{K}\setminus\{0\} \) . In this paper, some preliminaries on non-Archimedean Banach spaces and the concept of hyperstability are presented. Next, the well-known fixed point method [7, Theorem1] is reformulated in non-Archimedean Banach spaces. Using this method, we prove that the general linear functional equation \( h(cx+dy)= Ch(x)+Dh(y) \) is hyperstable in the class of functions \( h:X\rightarrow Y \) . In fact, by exerting some natural assumptions on control function \( \gamma:X^{2}\setminus\{0\}\rightarrow \mathbb{R}_{+} \) , we show that the map \( h:X\rightarrow Y \) that satisfies the inequality \( \lVert h(cx+dy)- Ch(x)-Dh(y)\rVert_{\ast}\leq \gamma(x,y) \) , is a solution to general linear functional equation for every \( x, y \in X\setminus\{0\} \) . Finally, this paper concludes with some consequences of the results.