A new type of approximation for the radical quintic functional equation in non-Archimedean (2, β)-Banach spaces

Let R be the set of real numbers. In this paper, we first introduce the notions of non-Archimedean (2,beta)-normed spaces (X, vertical bar vertical bar center dot,center dot vertical bar vertical bar*,(beta)) and we will reformulate the fixed point theorem [10, Theorem 1] in this space, after it, we introduce and solve the radical quintic functional equation f ((5)root x(5) + y(5)) = f(x) + f(y), x,y epsilon R. also under some weak natural assumptions on the function gamma : R x R x X -> [0,infinity), we show that this theorem is a very efficient and convenient tool for proving the hyperstability results when f : R -> X satisfy the following radical quintic inequality vertical bar vertical bar f ((5)root x(5) + y(5)) - f(x) - f(y), z vertical bar vertical bar*,(beta) <= gamma(x,y,z), x,y epsilon R backslash {0}, z epsilon X, with x NOTEQUAL; -y. (c) 2017 Elsevier Inc. All rights reserved.