Search

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S−1, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we present the central solutions f , g : G × G → C of the functional Eq. f (x 1 y 1 , x 2 y 2) + g (x 1 y 1 - 1 , x 2) + f (x 1 , x 2 y 2 - 1) = f (x 1 y 1 - 1 , x 2 y 2 - 1) + g (x 1 y 1 , x 2) + f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a 2-divisible group. [ABSTRACT FROM AUTHOR]

The analysis of stability in functional equations (FEs) within neutrosophic normed spaces is a significant challenge due to the inherent uncertainties and complexities involved. This paper proposes a novel approach to address this challenge, offering a comprehensive framework for investigating stability properties in such contexts. Neutrosophic normed spaces are a generalization of traditional normed spaces that incorporate neutrosophic logic. By providing a systematic methodology for addressing stability concerns in neutrosophic normed spaces, our approach facilitates enhanced understanding and control of complex systems characterized by indeterminacy and uncertainty. The primary focus of this research is to propose a novel class of Euler-Lagrange additive FE and investigate its Ulam-Hyers stability in neutrosophic normed spaces. Direct and fixed point techniques are utilized to achieve the required results. [ABSTRACT FROM AUTHOR]

We analyze in our paper questions of the theory of relativity. We approach this theory from the point of view of velocities and their composition. This is where the functional equations appear. Solving them leads to a world where velocities are bounded from above, the upper bound being exactly the "speed of light". [ABSTRACT FROM AUTHOR]

In this paper, we solve the system of functional equations { f (x + y) + g (y − x) = 2 f (x) , g (x + y) − f (y − x) = 2 g (y) and we investigate the stability of g-derivations in Banach algebras. [ABSTRACT FROM AUTHOR]

We consider a concept of angle in normed spaces which preserves isosceles orthogonality and approximate isosceles orthogonality and then we investigate linear mappings which approximately preserve such an angle. We show that such mappings are approximate similarities thus they are continuous and bounded below. We also state some results related to conformality and approximate conformality in normed spaces. In particular, an equivalent definition for conformality of mappings defined between inner product spaces is obtained from the methods developed in the paper. Finally, we pay attention to modular mappings defined between inner product C ∗ -modules over standard algebras with the property of approximately orthogonality preserving and study approximate conformality for such mappings. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

Let S be a monoid (= semigroup with identity), and let σ : S → S be a homomorphism such that σ ∘ σ = i d . In an earlier paper we solved the Pexiderized d'Alembert functional equation (PDFE) f (x y) + g (σ (y) x) = h (x) k (y) for unknown f , g , h , k : S → C , assuming that S is either regular or generated by its squares and that one of the unknown functions is central. The present paper has two main results. The first describes the solutions of PDFE on a general monoid in terms of multiplicative functions, solutions of a special case of the sine subtraction law, and solutions of other functional equations with just one unknown function. The second main result uses the first one to give a more detailed solution of PDFE on a larger class of monoids than has been treated previously. We also find the continuous solutions on topological monoids. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain some characterizations (or representations) of set-valued solutions defined on an Abelian monoid with values in a Hausdorff topological vector space of a new class of functional equations. [ABSTRACT FROM AUTHOR]

In this paper, we continue the discussion about relations between exponential polynomials and generalized moment functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite-dimensional variety is spanned by moment functions? Let m be an exponential on X. In our former paper, we have proved that if the linear space of all m-sine functions in the variety of an m-exponential monomial is (at most) one-dimensional, then this variety is spanned by moment functions generated by m. In this paper, we show that this may happen also in cases where the m-sine functions span a more than one-dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in d variables, describe exponentials on it and give the characterization of the so-called m-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in d variables is the polynomial ring in d variables. Finally, using the Ehrenpreis–Palamodov Theorem, we show that every exponential polynomial on the polynomial hypergroup in d variables is a linear combination of moment functions contained in its variety. [ABSTRACT FROM AUTHOR]

Let S be a semigroup, and let φ, ψ: S → S be two endomorphisms (which are not necessarily involutive). Our main goal in this paper is to solve the following generalized variant of d'Alembert's functional equation f (x ϕ (y)) + f (ψ (y) x) = 2 f (x) f (y) , x , y ∈ S , where f : S → ℂ is the unknown function by expressing its solutions in terms of multiplicative functions. Some consequences of this result are presented. [ABSTRACT FROM AUTHOR]

Let S be a semigroup and α, β ∈ ℝ. The purpose of this paper is to determine the general solution f : ℝ2 → S of the following parametric functional equation f ( x 1 + x 2 + α y 1 y 2 , x 1 y 2 + x 2 y 1 + β y 1 y 2) = f ( x 1 , y 1) f ( x 2 , y 2) , for all (x1, y1), (x2, y2) ∈ ℝ2, that generalizes some functional equations arising from number theory and is connected with the characterizations of the determinant of matrices. [ABSTRACT FROM AUTHOR]

In this paper, we determine the central solutions f : G × G → C of the functional equation f (x 1 σ y 1 , x 2 τ y 2) - f (x 1 σ y 1 , x 2) - f (x 1 , x 2 τ y 2) = f (x 1 y 1 , x 2 y 2) - f (x 1 y 1 , x 2) - f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a group, σ , τ : G → G are involutions. [ABSTRACT FROM AUTHOR]

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation f g - 1 ( f (x) + f (y) g (x) + g (y) ) = h k - 1 ( h (x) + h (y) k (x) + k (y) ) (x , y ∈ I) , where I is a nonempty open real interval, f , g , h , k : I → R are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability. [ABSTRACT FROM AUTHOR]

We deal with an alienation problem for an Euler–Lagrange type functional equation f (α x + β y) + f (α x - β y) = 2 α 2 f (x) + 2 β 2 f (y) assumed for fixed nonzero real numbers α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation g (x + y) + g (x - y) = 2 g (x) + 2 g (y). We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case g = γ f was examined. [ABSTRACT FROM AUTHOR]

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability in the sense of Hyers–Ulam–Rassias for the two types of generalized fractional differential systems by the ρ -Laplace transform method. Several examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Let X be an Abelian group, Y be a commutative monoid, K ⊂Y be a submonoid and F : X → 2Y \ {∅} be a set-valued map. Under some additional assumptions on ideals ℐ1 in X and ℐ2 in X2, we prove that if F is ℐ2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G : X → 2Y \{∅} such that F = G ℐ1-almost everywhere in X. Our considerations refers to the well known de Bruijn's result [1]. [ABSTRACT FROM AUTHOR]

In the present paper we consider the problem of characterization of maps which can be expressed as an affine difference i.e. a three-place map of the form t f (x) + (1 - t) f (y) - f (t x + (1 - t) y). We give a general solution of a functional equation associated with this problem. [ABSTRACT FROM AUTHOR]

In this paper, Hyers–Ulam stability of the diffusion equation on the half space R + n + 1 with n ≥ 2 is investigated: ∂ u ∂ t (x , y , t) =△ u (x , y , t) where x ∈ R n , y>0, and t>0. Using integral transforms, precisely, Fourier transform with respect to the tangential space x and Laplace transform for time t, we obtain the generalized Hyers–Ulam stability of diffusion equation in the half space R + n + 1 involving special functions such as gamma and error functions. [ABSTRACT FROM AUTHOR]

Let (S , +) be an abelian semigroup, let (H , +) be a uniquely 2-divisible, abelian group and let σ , τ be two endomorphisms of S. In this paper, we find the solutions f : S → H of the following Drygas type functional Eq. 1 f (x + σ (y)) + f (x + τ (y)) = 2 f (x) + f (σ (y)) + f (τ (y)) , x , y ∈ S , in terms of additive and bi-additive maps. Further, we solve a partly Pexiderized version of the above Eq and we use its solutions to determine the solutions of some related functional Eqs. In all results it is assumed that at least one of the endomorphisms σ and τ is surjective. [ABSTRACT FROM AUTHOR]

We consider the Levi–Civita equation f (x y) = g 1 (x) h 1 (y) + g 2 (x) h 2 (y) for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids. [ABSTRACT FROM AUTHOR]

In this paper we find the solutions of the functional equation f (x y) = g (x) h (y) + ∑ j = 1 n g j (x) h j (y) , x , y ∈ M , where M is a monoid, n ≥ 2 , and g j (for j = 1 , ... , n ) are linear combinations of at least 2 distinct nonzero multiplicative functions. [ABSTRACT FROM AUTHOR]

In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most n + m . If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly n + m . As in the previous paper, the results are proved using Leibniz difference operators. [ABSTRACT FROM AUTHOR]

Let 푆 be a commutative semigroup, 퐾 a quadratically closed commutative field of characteristic different from 2, 퐺 a 2-cancellative abelian group and 퐻 an abelian group uniquely divisible by 2. The goal of this paper is to find the general non-zero solution f : S 2 → 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 , the general non-zero solution f : S 2 → 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 , the general non-zero solution f : S 2 → 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 , where σ , τ : S → S are two involutions. [ABSTRACT FROM AUTHOR]

In this paper we study the blow-up analysis for some mean field equation on point vortices which is derived by C. Neri under a stochastic assumption. In particular, we derive some estimate which is the asymptotic behavior of blow-up solutions near the blow-up points. To obtain this result we shall employ the new scaling argument for blow-up solutions of Neri's mean field equation. Moreover, we also study the power type elliptic equation in higher dimension case related to mean field equation and derive such estimate. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form g (x + y 2 + z) + g (x + y 2 − z) + g (x − y 2 + z) + g (x − y 2 − z) = g (x) + g (y) + 4 g (z) , in 2-Banach spaces by using the Hyers direct method. [ABSTRACT FROM AUTHOR]

Given two functions f , g : I → R and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean M f , g ; μ : I 2 → I is defined by M f , g ; μ (x , y) : = (f g ) - 1 ∫ 0 1 f (t x + (1 - t) y) d μ (t) ∫ 0 1 g (t x + (1 - t) y) d μ (t) (x , y ∈ I). This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which M f , g ; μ (x , y) = M F , G ; μ (x , y) (x , y ∈ I) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means. [ABSTRACT FROM AUTHOR]

In the present paper we are concerned with the problem of characterization of maps which can be expressed as an affine difference i.e. a map of the form t f (x) + (1 - t) f (y) - f (t x + (1 - t) y) , where t ∈ (0 , 1) is a given number. We give a general solution of the functional equation associated with this problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new quadratic functional equation and, motivated by this equation, we investigate n-variables mappings which are quadratic in each variable. We show that such mappings can be unified as an equation, namely, multi-quadratic functional equation. We also apply a fixed point technique to study the stability for the multi-quadratic functional equations. Furthermore, we present an example and a few corollaries corresponding to the stability and hyperstability outcomes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the superstability of the following functional equations: f (P (x , y)) = g (x) h (y) , f (x + y) = g (x) h (y). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the solutions of Fréchet's functional equation in the context of Lie groups. In particular, we give the explicit right-abelian solutions of this equation for connected Lie groups. We also extend this result to homogeneous spaces and deal with some classical examples. [ABSTRACT FROM AUTHOR]