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We discuss the set-valued dynamics related to the theory of functional equations. We look for selections of convex set-valued functions satisfying set-valued Euler-Lagrange inclusions. We improve and extend upon some of the results in [13, 20], but under weaker assumptions. Some applications of our results are also provided.

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.

Let G be an Abelian group with a metric d and E be a normed space. For any f : G → E we define the Drygas difference of the function f by the formula Λ f ( x , y ) : = 2 f ( x ) + f ( y ) + f ( - y ) - f ( x + y ) - f ( x - y ) $$\Lambda {\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right): = 2{\rm{f}}\left( {\rm{x}} \right) + {\rm{f}}\left( {\rm{y}} \right) + {\rm{f}}\left( {{\rm{ - y}}} \right) - {\rm{f}}\left( {{\rm{x + y}}} \right) - {\rm{f}}\left( {{\rm{x - y}}} \right)$$ for all x, y ∈ G. In this article, we prove that if ˄f is Lipschitz, then there exists a Drygas function D : G → E such that f − D is Lipschitz with the same constant. Moreover, some results concerning the approximation of the Drygas functional equation in the Lipschitz norms are presented.

Let $${\mathscr {H}}$$ be a complex Hilbert space and $${\mathscr {B}}({\mathscr {H}})$$ the algebra of all bounded linear operators on $${\mathscr {H}}$$. We give the concrete forms of surjective continue unital linear maps from $${\mathscr {B}}({\mathscr {H}})$$ onto itself that preserves G-quasi-isometric operators.

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 $${\mathbb {C}}$$ be the set of complex numbers and $$\sigma $$ be a continuous automorphism and $$\tau $$ be a continuous anti-automorphism such that $$\sigma ^{2}=\tau ^{2}=id.$$ The purpose of this paper is to generalize the small-dimension lemma [20, Small Dimension Lemma] and by help of it we find on any compact group G the non-zero continuous solutions $$f:G\rightarrow {\mathbb {C}}$$ of the functional equation $$\begin{aligned} f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y), \ \ \ x,y \in G, \end{aligned}$$ in terms of continuous characters of G.

In this article, we obtain a characterizations and representations of set-valued solutions defined on an Abelian group G with values in a Hausdorff topological vector space of the following generalized bi-quadratic functional equation: $$\begin{aligned}&F(x+y,z+w)+F(x+y,z-w)+F(x-y,z+w)+F(x-y,z-w) \\&\quad =aF(x,z)+bF(x,w)+cF(y,z)+dF(y,w), \end{aligned}$$ for some nonnegative real numbers a, b, c, and d.

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.

Let $$\mathbb {R}$$ denote the set of real numbers. The purpose of the present paper is first to introduce and solve the radical quintic functional equation $$\begin{aligned} f\left( \root 5 \of {x^5+y^5}\right) = f(x)+f(y),\;\;\;x,y\in \mathbb {R}, \end{aligned}$$ for f a mapping from $$\mathbb {R}$$ into a vector space. We also establish stability results in quasi- $$\beta $$ -Banach spaces, and then the stability by using subadditive and subquadratic functions in ( $$\beta , p$$ )-Banach spaces for this functional equation. In addition, we also present a counterexample that does not satisfy the stability based on Ulam’s question.

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.

Let N be the set of all positive integers, G an Abelian group with a metric d and E a normed space. For any f : G → E we define the k-quadratic difference of the function f by the formula Qk ƒ(x; y) := 2ƒ(x) + 2k2ƒ(y) - f(x + ky) - f(x - ky) for x; y ∈ G and k ∈ N. Under some assumptions about f and Qkƒ we prove that if Qkƒ is Lipschitz, then there exists a quadratic function K : G → E such that ƒ -K is Lipschitz with the same constant. Moreover, some results concerning the stability of the k-quadratic functional equation in the Lipschitz norms are presented. Mathematics Subject Classification (2010): Primary 39B82, 39B52.Key words: k-quadratic functional equation, stability, Lipschitz space.

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].

The object of this paper is to solve the generalized radical cubic functional equation, and discuss the stability problem in quasi-$\beta$-Banach spaces and then the stability by using subadditive and subquadratic functions in ($\beta, p$)-Banach spaces for the generalized radical cubic functional equation. Also certain non-stability results are investigated via specific counterexamples. Our results are generalization of the main results which are established by Z. Alizadeh and A. G. Ghazanfari in 2016.

Let \({\mathbb {C}}\) be the set of complex numbers, X be a normed space and Y be a Banach space. We investigate the Hyers-Ulam stability theorem when \(f:{\mathbb {C}}\rightarrow Y\) satisfy the following \(\sigma -\)quadratic inequality $$\begin{aligned} \Vert f(x+y)+f(x+\sigma (y))-2f(x)-2f(y)\Vert \le \epsilon \end{aligned}$$ in a set \(\Omega \subset {\mathbb {C}}^2\) of Lebesgue measure \(m(\Omega )=0\), where \(\sigma :X\rightarrow X\) is an involution.

We determine the Solutions f : S → H of the generalized Jensen’s functional equation f( x + σ(y)) + f( x + τ(y)) = 2f(x), x , y∈ S and the solutions f : S → H of the generalized quadratic functional equation f ( x + σ(y)) + f (x + τ(y)) = 2f (x) + 2f (y), x, y ∈ S, where S is a commutative semigroup, H is an abelian group (2-torsion free in the first equation and uniquely 2-divisible in the second) and σ, τ are two involutions of S.

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 .

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 establish the Hyers-Ulam-Rassias stability of the mixed type additive-cubic functional equation $$f(2x+y)+f(2x-y)-f(4x)=2[f(x+y)+f(x-y)]-8f(2x)+10f(x)-2f(-x),$$ with $x\bot y,$ where $\bot $ is the orthogonality in the sense of Ratz in modular 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.