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The goal of this paper is to study the general solution of the following general radical quintic functional equation f ( a x 5 + b y 5 5 ) = r f ( x ) + s f ( y ) for f a mapping from the field of real numbers into a vector space, where a , b , r , s are fixed nonzero reals. Also, we prove the generalized hyperstability results for the general radical quintic functional equation by using the fixed point theorem (cf. Dung and Hang (2018) [15] , Theorem 2.1) in quasi-β-Banach spaces. Namely, we show, under some weak natural assumptions, functions satisfying the above equation approximately (in some sense) must be actually solutions to it.

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 .

We present results on approximate solutions to the biadditive equation f ( x + y , z - w ) + f ( x - y , z + w ) = 2 f ( x , z ) - 2 f ( y , w ) on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. In this way we obtain inequalities characterizing biadditive mappings and inner product spaces. Our outcomes are connected with the well known issues of Ulam stability and hyperstability.

The aim of this paper is first to reformulate the fixed point theorem [4, Theorem 1] in 2-Banach spaces, after it, we introduce and solve the radical quartic functional equation f ( x 4 + y 4 4 ) = f ( x ) + f ( y ) . We also show that the fixed point methods allow to investigate Ulam's type stability of radical quartic functional equation in 2-Banach spaces.

In this paper, we prove some hyperstability results of the following Cauchy–Jensen functional equation f ( x ) + f ( y ) + α f ( z ) = α f ( x + y α + z ) , in Banach spaces, for any fixed positive integer α ≥ 2 .