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

Let X be a normed space over F is an element of {R, C}, Y be a non-Archimedean Banach space over a non-Archimedean non-trivial field K and c, d, C, D be constants such that, c, d is an element of F \ {0} and C, D is an element of K \ {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 -> Y. In fact, by exerting some natural assumptions on control function gamma : X-2 \ {0} -> R+, we show that the map h : X -> Y that satisfies the inequality ||h(cx + dy) - Ch(x) - Dh(y)||(*) <= gamma(x, y), is a solution to general linear functional equation for every x, y is an element of X \ {0}. Finally, this paper concludes with some consequences of the results.