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Let 𝑆 be a commutative semigroup, 𝐾 a quadratically closed commutative field of characteristic different from 2, 𝐺 a 2-cancellative abelian group and 𝐻 an abelian group uniquely divisible by 2. The goal of this paper is to find the general non-zero solution f : S 2 → K f\colon S^{2}\to K of the d’Alembert type equation f ⁢ ( x + y , z + w ) + f ⁢ ( x + σ ⁢ ( y ) , z + τ ⁢ ( w ) ) = 2 ⁢ f ⁢ ( x , z ) ⁢ f ⁢ ( y , w ) , x , y , z , w ∈ S , f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z)f(y,w),\quad x,y,z,w\in S, the general non-zero solution f : S 2 → G f\colon S^{2}\to G of the Jensen type equation f ⁢ ( x + y , z + w ) + f ⁢ ( x + σ ⁢ ( y ) , z + τ ⁢ ( w ) ) = 2 ⁢ f ⁢ ( x , z ) , x , y , z , w ∈ S , f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z),\quad x,y,z,w\in S, the general non-zero solution f : S 2 → H f\colon S^{2}\to H of the quadratic type equation f ⁢ ( x + y , z + w ) + f ⁢ ( x + σ ⁢ ( y ) , z + τ ⁢ ( w ) ) = 2 ⁢ f ⁢ ( x , z ) + 2 ⁢ f ⁢ ( y , w ) , x , y , z , w ∈ S , f(x+y,z+w)+f(x+\sigma(y),z+\tau(w))=2f(x,z)+2f(y,w),\quad x,y,z,w\in S, where σ , τ : S → S \sigma,\tau\colon S\to S are two involutions.

In this work, we give some characterizations or representations of set-valued solutions defined on a commutative monoid $$(M,+)$$ with values in a Hausdorff topological vector space of the following two-variable functional equation with involutions: $$\begin{aligned} F(x+y,z+w)+F(x+\sigma (y),z+\tau (w)) =\alpha F(x,z)+\beta F(y,w), \end{aligned}$$ where $$\alpha ,\;\beta $$ are fixed nonnegative real numbers and $$\sigma ,\tau : M\rightarrow M$$ are involutions (i.e., $$\sigma (x+y)=\sigma (x)+\sigma (y)$$ and $$\sigma \circ \sigma (x)=x$$ for all $$x,y\in M$$ ).

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 X be a normed space, Y be a Banach space and $$f,g: X\rightarrow Y$$. In this paper, we investigate the Hyers–Ulam stability theorem for the generalized quadratic functional equation $$\begin{aligned} f(kx+y)+f(kx-y)=2k^2g(x)+2f(y) \end{aligned}$$in a set $$\Omega \subset X\times X$$, where k is a positive integer. By the Baire category theorem, we derive some consequences of our main result.

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 $$\mathbb {R}$$ be the set of real numbers and Y be a Banach space. We investigate the Hyers-Ulam stability theorem when $$f,g:\mathbb {R}\rightarrow Y$$ satisfy the following Pexider quadratic inequality $$\begin{aligned} \Vert f(x+y)+f(x-y)-2g(x)-2f(y)\Vert \le \epsilon , \end{aligned}$$ in a set $$\Omega \subset \mathbb {R}^2$$ of Lebesgue measure $$m(\Omega )=0$$ .

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.

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 .

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.

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.

All paper is related with the non-zero continuous solutions f : G → ℂ of the functional equation f ( x σ ( y ) ) + f ( τ ( y ) x ) = 2 f ( x ) f ( y ) , x , y ∈ G , $${\rm{f}}({\rm{x}}\sigma ({\rm{y}})) + {\rm{f}}(\tau ({\rm{y}}){\rm{x}}) = 2{\rm{f}}({\rm{x}}){\rm{f}}({\rm{y}}),\;\;\;\;\;{\rm{x}},{\rm{y}} \in {\rm{G}},$$ where σ; τ are continuous automorphism or continuous anti-automorphism defined on a compact group G and possibly non-abelian, such that σ2 = τ2 = id: The solutions are given in terms of unitary characters of G

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].

Let S be a semi-group, and let ?, ? ? Antihom(S,S) satisfy ? ? ? = ? ? ? = id: We show that any solutions f : S ? C of the functional equation f (x?(y)) + f(?(y)x) = 2 f(x)f(y), x, y ? S; has the form f = (m + m ? ? ? ?)=2, where m is a multiplicative function on S.

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic difference of the function f by the formulaQkf(x, y) := f(x + ky) + f(x - ky) - f(x + y) - f(x - y) - 2(k2 - 1) f(y)for all x, y ∈ G and for any integer k with k ≠ 1, -1. In this paper, we achieve the general solution of equation Qkf(x, y) = 0; after it, we show that if Qkf is Lipschitz, then there exists a quadratic function K : G → E such that f - K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].Mathematics Subject Classification (2010): Primary 39B82, 39B52.Keywords: Generalized quadratic functional equation, stability, Lipschitz space

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.

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.

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 .

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 $$\begin{aligned} \sum _{i=1}^{n} p_{i}=1. \end{aligned}$$

Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in Banach algebras for the following $\alpha$-Cauchy-Jensen functional equation:$$f\left(\frac{x+y}{\alpha}+z\right)+f\left(\frac{x-y}{\alpha}+z\right)=\frac{2}{\alpha} f(x)+2 f(z),$$where $\alpha \in \mathbb{N}_{\geq 2}$.

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.

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.

espanolLet S be a monoid, C be the set of complex numbers, and let σ,τ ∈ Antihom(S,S) satisfy τ ○ τ =σ ○ σ= id. The aim of this paper is to describe the solution ⨍,g: S → C of the functional equation in terms of multiplicative and additive functions. EnglishLet S be a monoid, C be the set of complex numbers, and let σ,τ ∈ Antihom(S,S) satisfy τ ○ τ =σ ○ σ= id. The aim of this paper is to describe the solution ⨍,g: S → C of the functional equation ⨍(xσ(y)) + ⨍(τ(y)x) = 2f(x)g(y), x, y ∈ S, in terms of multiplicative and additive functions.

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.

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