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The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

In the paper there are two kinds of functional symmetry considered, i.e., (1)-power symmetry and (- 1) -power symmetry, in a rich subfamily B of the family of all complex valued functions on a symmetric set G ⊂ C . The first one defines the values f (- z) as identical with f(z) and the other as the inverse of the f(z) in G . The announced notions allow a unique decomposition of functions f ∈ B into a product of two factors f 1 , f - 1 having the (1)- and the (- 1) -power symmetry property, respectively. In the paper there are also given examples of such partitions f = f 1 · f - 1 , for various f ∈ B , a solution of the problem of invariance of the above functional symmetries with respect to some one- and two-argument function operations and two applications of the mentioned function decomposition. [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]

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 its additive version, Bohr–Mollerup's remarkable theorem states that the unique (up to an additive constant) convex solution f(x) to the equation Δ f (x) = ln x on the open half-line (0 , ∞) is the log-gamma function f (x) = ln Γ (x) , where Δ denotes the classical difference operator and Γ (x) denotes the Euler gamma function. In a recently published open access book, the authors provided and illustrated a far-reaching generalization of Bohr–Mollerup's theorem by considering the functional equation Δ f (x) = g (x) , where g can be chosen from a wide and rich class of functions that have convexity or concavity properties of any order. They also showed that the solutions f(x) arising from this generalization satisfy counterparts of many properties of the log-gamma function (or equivalently, the gamma function), including analogues of Bohr–Mollerup's theorem itself, Burnside's formula, Euler's infinite product, Euler's reflection formula, Gauss' limit, Gauss' multiplication formula, Gautschi's inequality, Legendre's duplication formula, Raabe's formula, Stirling's formula, Wallis's product formula, Weierstrass' infinite product, and Wendel's inequality for the gamma function. In this paper, we review the main results of this new and intriguing theory and provide an illustrative application. [ABSTRACT FROM AUTHOR]

In this paper we determine the solutions (φ , f 1 , f 2) of the Pexider functional equation φ ( x + y 2) (f 1 (x) - f 2 (y)) = 0 , (x , y) ∈ I 1 × I 2 , where I 1 and I 2 are nonempty open subintervals. Special cases of the above equation regularly arise in problems with two-variable means. We show that, under a rather weak regularity condition, the coordinate-functions of a typical solution of the equation are constant over several subintervals of their domain. The regularity condition in question will be that the set of zeros of φ is closed. We also discuss particular solutions where this condition is not met. [ABSTRACT FROM AUTHOR]

This paper deals with functional equations in the form of f (x) + g (y) = h (x , y) where h is given and f and g are unknown. We will show that if h is a Borel measurable function associated with characterizations of the uniform or Cauchy distributions, then there is no measurable solutions of the equation. Our proof uses a characterization of the Dirac measure and it is also applicable to the arctan equation. [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]

In this research paper, we investigate a generalization of Vincze's type functional equations involving several (up to four) unknown functions in connection with the maximum functional equation as max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + ψ (y) , max { ψ (x y) , ψ (x y - 1) } = ψ (x) η (y) + χ (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) , max { ψ (x y) , ψ (x y - 1) } = ϕ (x) η (y) + χ (y) , where G is an arbitrary group, x , y ∈ G , and ψ , η , χ , ϕ : G → R are unknown functions. [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]

In this paper we give necessary conditions for quadratic functions f : R → R that satisfy the additional equation y 2 f (x) = x 2 f (y) under the condition x y = 1 . [ABSTRACT FROM AUTHOR]

On the occasion of his 80th birthday, Ludwig Reich, editor-in-chief of Aequationes Mathematicae from 1996 to 2008, gave an interview conducted via Zoom. This paper presents his answers to several questions regarding his education and academic life. [ABSTRACT FROM AUTHOR]

The author, together with M. Veerapazham, S. Wang and W. Zhang, considered the existence and uniqueness of continuous solutions to an iterative equation involving the multiplication of iterates in Gopalakrishna et al. (Sci. China Math., to appear). In this paper we continue to investigate this equation for differentiable solutions. Similarly to continuous solutions until Gopalakrishna et al. (Sci. China Math., to appear), no result on differentiable solutions of this equation on non-compact intervals of R has been obtained until now. Although our strategy here is to use the logarithmic conjugation to reduce the equation to the well-known polynomial-like iterative equation, all known results on differentiable solutions of the latter are given on compact intervals. We revisit polynomial-like iterative equations on the whole of R and use the Banach contraction principle to prove the existence and uniqueness of differentiable solutions to our equation on R + . We then technically extend our results on R + to R - and show that none of the pairs of solutions obtained on R + and R - can be combined at the origin to obtain even a continuous solution of the equation on the whole R . [ABSTRACT FROM AUTHOR]

The aim of this paper is to offer some hyperstability results for the following functional equation ∑ λ ∈ Λ f (x λ. y) = L f (x) + ∑ λ ∈ Λ f (λ. y) (x , y ∈ S) , where S is a semigroup, Λ is a finite subgroup of the group of endomorphisms of S, L is the cardinality of Λ (i.e. L = c a r d (Λ) ) and f : S → G such that (G , +) is a L-cancellative abelian group with a metric d. Moreover, we discuss some remarks concerning particular cases of the considered equation and the inhomogeneous equation ∑ λ ∈ Λ f (x λ. y) = L f (x) + ∑ λ ∈ Λ f (λ. y) + F (x , y) (x , y ∈ S) , where F : S × S → G . [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a probabilistic (martingale) characterization to the general measurable solution of the Lobachevsky functional equation. We show that finding the general solution of this equation is equivalent to establishing that a space-transformation of a Brownian Motion by a suitable function is a martingale. This method can be applied for Cauchy's, Jensen's and some other functional equations. [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]

In this survey paper we present some recent results in the iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embedddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in $${{\mathbb{R}}^n}$$ space; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers-Ulam stability of the translation equation. Most of the results presented here was obtained by the means of functional equations. We indicate the relations between the iteration theory and functional equations. [ABSTRACT FROM AUTHOR]

In this paper we will get a class of functional equations involving a countable set of terms, summed by the well known Bochner–Fejér summation procedure, which are closely associated with the set of almost periodic functions. We will show that the zeros of a prefixed almost periodic function determine analytic solutions of such a functional equation associated with it, and we will obtain other solutions which are analytic or meromorphic on a certain domain. [ABSTRACT FROM AUTHOR]

In this paper we attempt to define axiomatic measures of non-compactness for Sobolev spaces of integer order W n , p (Ω) , where Ω ⊂ R d (which is equivalent to Ω being any set of infinite measure). We consider two cases, one with Ω being an open subset with finite measure, and another when Ω = R d , and discuss basic features of the measure of non-compactness in each case. Next we define a partial measure of non-compactness on space L p (Ω ; B) , where B is a Banach space. Furthermore, we give some application to solve the functional integro-differential equation in W 1 , p (Ω). [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 we prove the following result: Let R be a prime ring with c h a r (R) ≠ 2 , 3 , 5 and let T : R → R be an additive mapping satisfying the relation T (x 4) = x T (x 2) x for all x ∈ R . In this case T is of the form T (x) = λ x for all x ∈ R and some fixed element λ ∈ C , where C is the extended centroid of R. [ABSTRACT FROM AUTHOR]

In this paper we introduce the concept of translation invariant functions: considering an arbitrary set ∅ ≠ S ⊂ R n , the function F : S ⟶ R is translation invariant if F (x) = F (y) implies F (x + t) = F (y + t) for any vectors x , y , t ∈ R n such that x , y , x + t , y + t ∈ S . In our main results we shall consider an open, connected set ∅ ≠ D ⊂ R n . We prove that if F : D ⟶ R is a translation invariant, continuous function, then there exists a vector a = (a 1 , ⋯ , a n) ∈ R n and a strictly monotone, continuous function f such that F (x 1 , ⋯ , x n) = f (a 1 x 1 + ⋯ + a n x n) holds for all (x 1 , ⋯ , x n) ∈ D . Using this result we also show that continuous solutions F : D ⟶ R of the system of functional equations F (x 1 , ⋯ , x j + t j , ⋯ , x n) = Ψ j (F (x 1 , ⋯ , x j , ⋯ , x n) , t j) (j = 1 , ⋯ , n) can be represented as the composition of a strictly monotone, continuous function and a linear functional as well. Applying the latter theorem, we give a characterization of Cobb–Douglas type utility functions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate maps on sets of positive operators which are induced by the continuous functional calculus and transform a Kubo–Ando mean σ into another τ . We establish that under quite mild conditions, a mapping ϕ can have this property only in the trivial case, i.e. when σ and τ are nontrivial weighted harmonic means and ϕ stems from a function which is a constant multiple of the generating function of such a mean. In the setting where exactly one of σ and τ is a weighted arithmetic mean, we show that under fairly weak assumptions, the mentioned transformer property never holds. Finally, when both of σ and τ are such a mean, it turns out that the latter property is only satisfied in the trivial case, i.e. for maps induced by affine functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. Firstly, we study the non-existence of finite order meromorphic solutions to the Cubic type of Fermat functional equation f (z) 3 - 3 τ f (z) f (z + c) + f (z + c) 3 = 1 . In addition, the paper is concerned with the description of finite order entire solutions of the Quadratic type of Fermat functional equation f (z) 2 - 2 μ f (z) f (z + c) + f (z + c) 2 = 1 . [ABSTRACT FROM AUTHOR]

We consider measures of intrinsic uncertainty, or fuzziness. Two earlier papers introduced the axiom of localization and the direct product composition law and determined explicit forms for measures of fuzziness with these properties. In addition, justification was provided for choosing “Prod” over “Min” for the fuzzy direct product. In the present paper we continue to pursue the issue of the most appropriate fuzzy direct product. Indeed, supposing only that the fuzzy direct product is a t-norm, we find all possible localizable composable measures of fuzziness and their corresponding fuzzy direct products. [ABSTRACT FROM AUTHOR]

The main result of the present paper is about the solutions of the functional equation F ( x + y 2) + f 1 (x) + f 2 (y) = G (g 1 (x) + g 2 (y)) , x , y ∈ I , derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where F , f 1 , f 2 , g 1 , g 2 : I → R and G : g 1 (I) + g 2 (I) → R are the unknown functions assumed to be continuously differentiable with 0 ∉ g 1 ′ (I) ∪ g 2 ′ (I) , and the set I stands for a nonempty open subinterval of R . In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that F is affine on I and g 1 and g 2 are continuous functions strictly monotone in the same sense, and secondly that g 1 and g 2 are invertible affine functions with a common additive part. [ABSTRACT FROM AUTHOR]

The following MW-problem was posed independently by Janusz Matkowski and Jacek Wesołowski in different forms in 1985 and 2009, respectively: Are there increasing and continuous functions φ:[0,1]→[0,1]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi :[0,1]\rightarrow [0,1]$$\end{document}, distinct from the identity on [0, 1], such that φ(0)=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi (0)=0$$\end{document}, φ(1)=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi (1)=1$$\end{document} and φ(x)=φ(x2)+φ(x+12)-φ(12)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varphi (x)=\varphi (\frac{x}{2})+\varphi (\frac{x+1}{2})-\varphi (\frac{1}{2})$$\end{document} for every x∈[0,1]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x\in [0,1]$$\end{document}? By now, it is known that each of the de Rham functions Rp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_p$$\end{document}, where p∈(0,1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$p\in (0,1)$$\end{document}, is a solution of the MW-problem, and for any Borel probability measure μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu $$\end{document} concentrated on (0, 1) the formula ϕμ(x)=∫(0,1)Rp(x)dμ(p)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi _\mu (x)=\int _{(0,1)}R_p(x)\, d\mu (p)$$\end{document} defines a solution ϕμ:[0,1]→[0,1]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\phi _\mu :[0,1]\rightarrow [0,1]$$\end{document} of this problem as well. In this paper, we give a new family of solutions of the MW-problem consisting of Cantor-type functions. We also prove that there are strictly increasing solutions of the MW-problem that are not of the above integral form with any Borel probability measure μ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu $$\end{document}. [ABSTRACT FROM AUTHOR]

In this paper it is proved that, for a function f : X → E mapping from a normed linear space X into an inner product space E, the functional inequality ‖ 2 f (x) + 2 f (y) - f (x - y) ‖ ⩽ ‖ f (x + y) ‖ , ‖ x ‖ + ‖ y ‖ ⩾ d for some d > 0 , implies f is quadratic. Other types of functional inequalities related to the quadratic functional equation have also been investigated. Besides we establish the Hyers–Ulam stability on restricted domains, and we improve the bounds and thus the stability results obtained in Jung (J Math Anal Appl 222:126–137, 1998) and Rassias (J Math Anal Appl 276: 747–762, 2002). Finally we apply our recent results to the asymptotic behavior of quadratic functional equations of different types. [ABSTRACT FROM AUTHOR]

We continue our study started in Morawiec and Zürcher (Aequ Math 92(4):601-615, 2018) of the functional equation φ(x)=∑n=0Nφ(fn(x))-∑n=0Nφ(fn(0))and its increasing and continuous solutions φ:[0,1]→[0,1] such that φ(0)=0 and φ(1)=1. In this paper we assume that f0,...,fN:[0,1]→[0,1] are strictly increasing contractions such that 0≤f0(0)

Two years ago, during the 21st European Conference on Iteration Theory, Gregory Derfel asked: "Does there exist a non-trivial bounded continuous solution of the equation 2f(x)=f(x-1)+f(-2x)?" He repeated the question during the 55th International Symposium on Functional Equations. In this paper we present a partial solution of a more general problem, connected to the functional equation f(x)=M(f(x+t1),f(x+t2),...,f(x+tn-1),f(ax)), where n∈N,t1,t2,...,tn-1∈R\{0},a∈(-∞,0) and M is a given function in n variables satisfying some additional properties. In particular, M can be a weighted quasi-arithmetic mean in n variables. [ABSTRACT FROM AUTHOR]

The paper deals with the mean value theorem of differential and integral calculus due to Flett (Math Gazette 42:38-39, ) and its various extensions. Since this theorem is a source of interest in various fields of mathematics (including functional equations), we aim to provide a detailed study of various (known as well as new) sufficient conditions for its validity, their geometric interpretations, and a comparison of the corresponding classes of functions. Moreover, our approach enables many new alternative proofs of known results, for instance, Pawlikowska's extension of Flett's theorem for higher-order derivatives. Some interesting open problems are also formulated. [ABSTRACT FROM AUTHOR]

The study of quantum integers and their operations is closely related to the studies of symmetries of roots of polynomials and of fundamental questions of decompositions in Additive Number Theory. In his papers on quantum arithmetics, Melvyn Nathanson raises the question of classifying solutions of functional equations arising from the multiplication of quantum integers, starting with polynomial solutions and then generalizing to rational function solutions. The classification of polynomial solutions with fields of coefficients of characteristic zero and support base P has been completed. In a paper concerning the Grothendieck group associated to the collection of polynomial solutions, Nathanson poses a problem which asks whether the set of rational function solutions strictly contains the set of ratios of polynomial solutions. It is now known that there are infinitely many rational function solutions Γ with fields of coefficients of characteristic zero not constructible as ratios of polynomial solutions, even in the purely cyclotomic case, which is the case most similar to the polynomial solution case. The classification of polynomial solutions is thus not sufficient, in essential ways, to resolve the classification problem of all rational function solutions with fields of coefficients of characteristic zero. In this paper we study symmetries of roots of rational functions and the classification of the important class-the last and main obstruction to the classification problem-of rational function solutions, the purely cyclotomic, purely nonrational primitive solutions with fields of coefficients of characteristic zero and support base P, which allows us to complete the classification problem raised by Nathanson. [ABSTRACT FROM AUTHOR]

It is known that if a bivariate mean K is symmetric, continuous and strictly increasing in each variable, then for every mean M there is a unique mean N such that K is invariant with respect to the mean-type mapping M , N , which means that K ∘ M , N = K and N is called a K-complementary mean for M (Matkowski in Aequ Math 57(1):87–107, 1999). This paper extends this result for a large class of nonsymmetric means. As an application, the limits of the sequences of iterates of the related mean-type mappings are determined, which allows us to find all continuous solutions of some functional equations. [ABSTRACT FROM AUTHOR]

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function. [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]

This paper examines various aspects related to the Cauchy functional equation $$f(x+y)=f(x)+f(y)$$ , a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation. [ABSTRACT FROM AUTHOR]

In this paper we generalize the fundamental equation of information to the symmetric cone domain and find the general solution under the assumption of continuity of the respective functions. [ABSTRACT FROM AUTHOR]

In this paper, we prove that continuous non-PM functions with non-monotonicity height equal to 1 need not be strictly monotone on its range, unlike PM functions. An existence theorem is obtained for the iterative roots of such functions. We also discuss the Hyers–Ulam stability for the functional equation of the iterative root problem. [ABSTRACT FROM AUTHOR]

Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability p, then the probability of a correct verdict tends to one as the jury size tends to infinity (Marquis de Condorcet in Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix, Imprim. Royale, Paris, 1785). Recently, Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) developed a model where jurors sequentially receive independent signals from an interval according to a distribution which depends on the state of Nature and on the juror's "ability", and vote sequentially. This paper shows that, to mimic Condorcet's binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio α (t) of the probability that a mean-zero random variable satisfies X > t given that | X | > t . In particular, we show that under natural symmetry assumptions the tail-balances α (t) uniquely determine the signal distribution and so the distributions assumed in Alpern and Chen (Eur J Oper Res 258:1072–1081, 2017, Theory Decis 83:259–282, 2017) are uniquely determined for α (t) linear. [ABSTRACT FROM AUTHOR]

In this paper we consider a generalization of d'Alembert's equation and Wilson's equation on commutative semigroups using only the semigroup operation, ie. we consider the functional equation h (x + 2 y) + h (x) = 2 f (y) h (x + y) , x , y ∈ S , where f , h : S → K , (S , +) is a commutative semigroup, K is a quadratically closed field, char K ≠ 2 . [ABSTRACT FROM AUTHOR]

Several results concerning the homomorphism functional equation f x ∨ y = f x ∨ f y , in the class of semilattices, as well as other similar equations, are presented in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we present some characterizations of the q-gamma function by some functional equations using E. Artin's technique. [ABSTRACT FROM AUTHOR]

Recently, Dong (Aequ Math 86:269-277, ) has proved the generalized stability of the functional equation $${\|f(x + y)\| = \|f(x) + f(y)\|}$$ under the assumption that X, the domain of f, is an Abelian group. In this paper, we prove a generalization of this result by removing the commutativity assumption of X. [ABSTRACT FROM AUTHOR]

In this paper we study the functional equation where a, b, c are fixed complex numbers and $${f \colon \mathbb{C} \to \mathbb{C}}$$ is the unknown function. We show, that if there is i such that $${b_i / c_i \neq b_j /c_j}$$ holds for any $${1 \leq j \leq n,\ j \neq i}$$ , the functional equation has a nonconstant solution if and only if there are field automorphisms $${\phi_1, \ldots, \phi_k}$$ of $${\mathbb{C}}$$ such that $${\phi_1 \cdots \phi_k}$$ is a solution of the equation. [ABSTRACT FROM AUTHOR]