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In this work, we prove a simple fixed point theorem in non-Archimedean $$(n,\beta )$$ -Banach spaces, by applying this fixed point theorem, we will study the stability and the hyperstability of the kth radical-type functional equation: $$\begin{aligned} f\left( \root k \of {x^k+y^k}\right) = f(x)+f(y), \end{aligned}$$ where f is a mapping on the set of real numbers and k is a fixed positive integer. Furthermore, we give some important consequences from our main results.

In this paper, using the Brzdek’s fixed point theorem [9, Theorem 1] in non-Archimedean (2,β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation $$f\left( {\left( {\sum\limits_{i = 1}^k {a_i x_i^p } } \right)^{1/p} } \right) = \sum\limits_{i = 1}^k {a_i f\left( {x_i } \right)} ,$$ where p ∈ {1, ⋯, 5}, a1, ⋯, ak are fixed nonzero reals when p ∊ {1, 3, 5} and are fixed positive reals when p ∈ {2, 4}.

Let X be a normed space, U ⊂ X ∖ { 0 } {U\subset X\setminus\{0\}} a non-empty subset, and ( G , + ) {(G,+)} a commutative group equipped with a complete ultrametric d that is invariant (i.e., d ( x + z , y + z ) = d ( x , y {d(x+z,y+z)=d(x,y} ) for x , y , z ∈ G {x,y,z\in G} ). Under some weak natural assumptions on U and on the function γ : U 3 → [ 0 , ∞ ) {\gamma\colon U^{3}\to[0,\infty)} , we study the new generalized hyperstability results when f : U → G {f\colon U\to G} satisfies the inequality d ⁢ ( α ⁢ f ⁢ ( x + y α + z ) , α ⁢ f ⁢ ( z ) + f ⁢ ( y ) + f ⁢ ( x ) ) ≤ γ ⁢ ( x , y , z ) d\biggl{(}\alpha f\biggl{(}\frac{x+y}{\alpha}+z\biggr{)},\alpha f(z)+f(y)+f(x)% \biggr{)}\leq\gamma(x,y,z) for all x , y , z ∈ U {x,y,z\in U} , where x + y α + z ∈ U {\frac{x+y}{\alpha}+z\in U} and α ≥ 2 {\alpha\geq 2} is a fixed positive integer. The method is based on a quite recent fixed point theorem (Theorem 1 in [J. Brzdȩk and K. Ciepliński, A fixed point approach to the stability of functional equations in non-Archimedean metric spaces, Nonlinear Anal. 74 2011, 18, 6861–6867]) (cf. [8, Theorem 1]) in some functions spaces.

We have proved theHyers-Ulam stability and the hyperstability of the quadratic functional equation f (x + y + z) + f (x + y − z) + f (x − y + z) + f (−x + y + z) = 4[f (x) + f (y) + f (z)] in the class of functions from an abelian group G into a Banach space.

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.

The purpose of this paper is first to reformulate the fixed point theorem (see Theorem 1 of [J. Brzdȩk, J. Chudziak and Z. Páles, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 2011, 17, 6728–6732]) in β-Banach spaces. We also show that this theorem is a very efficient and convenient tool for proving the hyperstability results of the general linear equation in β-Banach 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.

Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation $$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$ for functions $f$ and $h$ mapping a semigroup $(G,\cdot )$ into a commutative semigroup $(E,+)$, where the map $\unicode[STIX]{x1D70E}:G\rightarrow G$ is an endomorphism of $G$ with $\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$ for all $x\in G$. We derive from these results some characterisations of inner product spaces. We also obtain a description of solutions to the equation and hyperstability results for the $\unicode[STIX]{x1D70E}$-quadratic and $\unicode[STIX]{x1D70E}$-Drygas equations.

Using the fixed point theorem [12, Theorem 1] in (2,?)-Banach spaces, we prove the generalized hyperstability results of the bi-Jensen functional equation 4f(x + z/2; y + w/2) = f (x,y) + f (x,w) + f (z,y) + f (y,w). Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. The method we use here can be applied to various similar equations in many variables.

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.

Motivated by the notion of Ulam stability, we investigate the generalized hyperstability results for the Drygas functional equation $$\begin{aligned} f(x+y)+f(x-y)=2f(x)+f(y)+f(-y), \end{aligned}$$ on a restricted domain. The method is based on a quite recent fixed point theorem (cf. [6, Theorem 1]) in some functions spaces. We derive from them some characterizations of inner product spaces. Our results are improvements and generalizations of the main results of Piszczek and Szczawinska [29].

We present a method that allows to study approximate solutions to the two-variable Jensen functional equation $$\displaystyle 2f\Big (\frac {x+z}{2},\frac {y+w}{2}\Big )=f(x,y)+f(z,w) $$ on a restricted domain. Namely, we show that (under some weak natural assumptions) functions that satisfy the equation approximately (in some sense) must be actually solutions to it. The method is based on a quite recent fixed point theorem in some functions spaces and can be applied to various similar equations in many variables. Our outcomes are connected with the well-known issues of Ulam stability and hyperstability.

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 .

In this paper, we will investigate some hyperstability results of an n-dimensional Jensen type functional equation $$\begin{aligned} \sum _{i=1}^{n} p_{i} f(x_{i})=f\left( \sum _{i=1}^{n} p_{i} x_{i}\right) , \end{aligned}$$ where \(n>1\) is an integer, and \(p_{1},\ldots ,p_{n}\) are positive rational numbers with $$\begin{aligned} \sum _{i=1}^{n} p_{i}=1. \end{aligned}$$

Let \((G,+)\) be an abelian group equipped with a complete ultrametric d that is invariant (i.e., \(d(x + z, y + z)= d(x, y\)) for \(x, y, z \in G\)), X be a normed space and \(U\subset X\setminus \{0\} \) be a nonempty subset. Under some weak natural assumptions on U and on the function \(\chi :U^3\rightarrow [0,\infty )\), we study new hyperstability results when \(f:U\rightarrow G\) satisfy the following Apollonius inequality $$\begin{aligned}d\;x-z,y-z,x-y,z-\frac{x+y}{2}\in U. \end{aligned}$$ Moreover, we derive some consequences from our main results.

In this paper, we prove the hyperstability of the following mixed additive-quadratic-Jensen functional equation 2 f ( x + y 2 ) + f ( x − y 2 ) + f ( y − x 2 ) = f ( x ) + f ( y ) $$2f({{x + y} \over 2}) + f({{x - y} \over 2}) + f({{y - x} \over 2}) = f(x) + f(y)$$ in the class of functions from an 2-divisible abelian group G into a Banach space.

In this paper, we establish some hyperstability results of the following Cauchy-Jensen functional equation in Banach spaces.

Let $$\mathbb {R}$$ be the set of real numbers, $$(G,+)$$ be a commutative group and d be a complete ultrametric on G that is invariant (i.e., $$d(x + z, y + z)= d(x, y$$ ) for $$x, y, z \in G$$ ). Under some weak natural assumptions on the function $$\gamma :{\mathbb {R}}^2\rightarrow [0,\infty )$$ , we study the generalised hyperstability results when $$f:\mathbb {R}\rightarrow G$$ satisfy the following radical cubic inequality $$\begin{aligned} d\big (f(\root 3 \of {x^3+y^3}),f(x)+f(y)\big ) \le \gamma (x,y), \quad x,y\in \mathbb {R}{\setminus }\{0\}, \end{aligned}$$ with $$x\ne -y$$ . The method is based on a quite recent fixed point theorem (cf. Brzdek and Cieplinnski in Nonlinear Anal 74:6861–6867, 2011, Theorem 1) in some functions spaces.