LAUSR

Abstract Let R be the set of real numbers. In this paper, we first introduce the notions of non-Archimedean ( 2 , β ) -normed spaces ( X , ‖ ⋅ , ⋅ ‖ ⁎ , β ) 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 ( x 5 + y 5 5 ) = f ( x ) + f ( y ) , x , y ∈ R . Also, under some weak natural assumptions on the function γ : R × R × X → [ 0 , ∞ ) , 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 ‖ f ( x 5 + y 5 5 ) − f ( x ) − f ( y ) , z ‖ ⁎ , β ≤ γ ( x , y , z ) , x , y ∈ R ∖ { 0 } , z ∈ X , with x ≠ − y .