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In this paper we present a complete asymptotic expansion of the Archimedean compound of two symmetric homogeneous means and derive recursive algorithms for coefficients in this expansion. We also show some examples and obtain explicit expansions for the Archimedean compounds of the arithmetic, geometric and harmonic means. [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]

We investigate the functional equation f (p (x)) = q (f (x)) where p and q are given real functions. In the paper "On solvability of f (p (x)) = q (f (x)) for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494", we solved the problem of the solvability of f (p (x)) = q (f (x)) under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output 'solvable/not solvable'. [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]

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]

For a given p-variable mean M:Ip→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M :I^p \rightarrow I$$\end{document} (I is a subinterval of R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}$$\end{document}), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its (p+1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(p+1)$$\end{document}-variable β\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\beta $$\end{document}-invariant extension as the unique solution K:Ip+1→I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$K :I^{p+1} \rightarrow I$$\end{document} of the functional equation K(M(x2,⋯,xp+1),M(x1,x3,⋯,xp+1),⋯,M(x1,⋯,xp))=K(x1,⋯,xp+1),for allx1,⋯,xp+1∈I\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned}&K\big (M(x_2,\dots ,x_{p+1}),M(x_1,x_3,\dots ,x_{p+1}),\dots ,M(x_1,\dots ,x_p)\big )\\&\quad =K(x_1,\dots ,x_{p+1}), \text { for all }x_1,\dots ,x_{p+1} \in I \end{aligned}$$\end{document}in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions. [ABSTRACT FROM AUTHOR]

The conditional distributivity between two different aggregation operators, which has received wide attention from the researchers, is vital for many fields, for example, utility theory, integration theory and so on. In some existing generalization, the restrictive but not completely justified condition that the values of the inner operator are less than 1. However, for a more general and reasonable setting, the values of the inner operator should be strictly bounded between 0 and 1. Therefore, the aim of this paper is to introduce and fully characterize this kind of conditional distributivity of a semi-t-operator over a uninorm. In comparison with the corresponding results obtained, there are many new solutions. [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]

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 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]

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]

The equation x + g (y + f (x)) = y + g (x + f (y)) was introduced by Marcin E. Kuczma in connection with his research on compatible means. Kuczma determined the analytic solutions of the equation in order to prove that compatible homogeneous analytic means are necessarily power means. Kuczma's result was improved by J. Sikorska, who determined the twice differentiable solutions, and then by N. Brillouët-Belluot, who found all differentiable solutions. In this paper we determine all continuous solutions. As a corollary we find that compatible continuous homogeneous means are necessarily power 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 Borowiecka et al. (Bernoulli 21(4):2513-2551, 2015) the authors show that every generalized convolution can be used to define a Markov process, which can be treated as a Lévy process in the sense of this convolution. The Bessel process is the best known example here. In this paper we present new classes of regular generalized convolutions enlarging the class of such Markov processes. We give here a full characterization of such generalized convolutions ⋄ for which δx⋄δ1, x∈[0,1], is a convex linear combination of n=3 fixed measures and only the coefficients of the linear combination depend on x. For n=2 it was shown in Jasiulis-Goldyn and Misiewicz (J Theor Probab 24(3):746-755, 2011) that such a convolution is unique (up to the scale and power parameters). We show also that characterizing such convolutions for n⩾3 is equivalent to solving the Levi-Civita functional equation in the class of continuous generalized characteristic functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the solution (f, g) of the equation [f(x)-f(y)]g′(αx+(1-α)y)=[g(x)-g(y)]f′(αx+(1-α)y),x,y∈I,where I⊂R is an open interval, f,g:I→R are differentiable, α is a fixed number from (0, 1). [ABSTRACT FROM AUTHOR]

In this paper, we give the solution of a problem formulated in Kominek and Sikorska (Aequationes Math 90:107-121, 2016) in connection with the functional equation f(xy)-f(x)-f(y)=g(x+y)-g(x)g(y).Our result can also be interpreted in the way that, under some additional condition, the logarithmic and the exponential Cauchy equations are strongly alien. [ABSTRACT FROM AUTHOR]

Given a continuous strictly monotone function $$\varphi $$ defined on an open real interval I and a probability measure $$\mu $$ on the Borel subsets of [0, 1], the Makó-Páles mean is defined by Under some conditions on the functions $$\varphi $$ and $$\psi $$ defined on I, the quotient mean is given by In this paper, we study some invariance of the quotient mean with respect to Makó-Páles means. [ABSTRACT FROM AUTHOR]

This paper aims to study the cross-migrative property for uninorms. We only consider the most usual classes of uninorms as follows: uninorms in $${\mathcal{U}_{min}}$$ and $${\mathcal{U}_{max}}$$ , representable uninorms, idempotent uninorms and uninorms continuous in the open unit square, and limit the research to those uninorms which have the same neutral element. This study shows that there is no cross-migrativity between representable uninorms and other classes of uninorms. The relationship is the same between conjunctive uninorms and disjunctive uninorms. We give the sufficient and necessary conditions for the cross-migrativity equation to hold for all of the other possible combinations of uninorms. [ABSTRACT FROM AUTHOR]

In this paper, we present an extention of Hyers-Ulam stability of Sahoo-Riedel's points for real-valued differentiable functions on [ a, b] and then we obtain stability results of Flett's points for functions in the class of continuously differentiable functions on [ a, b] with f′( a) = f′( b). [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize the distributivity equations between quasi-arithmetic means and some binary operators, such as, nullnorms, semi-nullnorms and semi-t-operators. It is shown that the distributivity equations between nullnorms (semi-nullnorms) and quasi-arithmetic means degenerate into the distributivity equations between t-norms or t-conorms (semi-t-norms or semi-t-conorms) and quasi-arithmetic means. But for semi-t-operators, the distributive equations have new solutions. These new results extend the previous ones about the distributivity between t-norms (t-conorms) and quasi-arithmetic means. [ABSTRACT FROM AUTHOR]

We study the stability properties of the equation which is motivated by numerical integration. In Szostok and Wa̧sowicz (Appl Math Lett 24(4):541-544, ) the stability of the simplest equation of the type (0.1) was investigated thus the inequality was studied. In the current paper we present a somewhat different approach to the problem of stability of (0.1). Namely, we deal with the inequality [ABSTRACT FROM AUTHOR]

In this paper we investigate geometrically convex solutions $${g : R^{+} \rightarrow R^{+}}$$ to the generalized gamma functional equation with initial condition given by where $${f : R^{+} \rightarrow R^{+}}$$ is a given function. We prove that if f satisfies some appropriate conditions, then (*) has a unique eventually geometrically convex solution g, determined by the formulae Some known results in the literature are generalized and improved. [ABSTRACT FROM AUTHOR]

The functional equation of composite type arose in the course of the studies on the problem of extension and restriction of the number of arguments of a mean M performed by G. Aumann in the third decade of the past century. A solution to (1) in the analytic case was ulteriorly obtained by Aumann himself and remained as a noteworthy characterization of analytic quasiarithmetic means. An ample generalization of Eq. (1) which involves general weighting operators is considered in this paper. Under mild conditions on the regularity of the involved means, the general solution to this generalized equation is obtained for a particularly tractable class of weighting operators. [ABSTRACT FROM AUTHOR]

The diagonalization of a two-variable functional equation (related to competition) leads to the iterative equation It was shown in Kahlig (Appl Math 39:293-303, ) that if a function $${f:{\mathbb{R}}\rightarrow {\mathbb{R}}}$$ , such that f(0) = 0, satisfies this equation for all $${x\in (-1,1),}$$ and is twice differentiable at the point 0, then $${f=\tanh \circ (p\,\tan ^{-1}) }$$ for some real p. In this paper we prove the following stronger result. A function $${f:{\mathbb{R}} \rightarrow {\mathbb{R}},\;f(0) =0}$$ , differentiable at the point 0, satisfies this functional equation if, and only if, there is a real p such that $${f=\tanh \circ (p\,\tan ^{-1})}$$ . We also show that the assumption of the differentiability of f at 0 cannot be replaced by the continuity of f. The corresponding result for the iterative equation coming from a three- respectively four-variable competition equation is also proved. Our conjecture is that analogous results hold true for the diagonalization of any n-variable competition equation $${(n=5, 6, 7, \ldots)}$$ . [ABSTRACT FROM AUTHOR]

We consider continuous solutions $${f : \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} = (0, \infty)}$$ of the functional equation $${f(xf(x)) = \varphi (f(x))}$$ where $${\varphi}$$ is a given continuous map $${\mathbb{R}_{+} \rightarrow \mathbb{R}_{+}}$$ . A solution f is singular if there are $${0 < a \leq b< \infty}$$ such that $${f|_{(0,a)} > 1, f|_{[a,b]} \equiv 1}$$ , and $${f|_{(b,\infty)} < 1}$$ ; other solutions are regular. It is known that the range R of a singular solution can contain periodic orbits of $${\varphi}$$ of all periods. In this paper we show that the range of a regular solution f contains no periodic point of $${\varphi}$$ of period different from $${2^n, n \in \mathbb{N}}$$ so that $${\varphi|_{R_f}}$$ has zero topological entropy. It follows, that the regular solutions are just the solutions f satisfying one of the conditions: (i) $${R_{f} \subseteq (0,1]}$$ , (ii) $${R_{f} \subseteq [1, \infty)}$$ , (iii) there are $${0 1}$$ . [ABSTRACT FROM AUTHOR]

The functional equation related to competition ([]) for y = cx with a fixed c > 0, leads to the equation The case c = 1 (a first order iterative functional equation) was treated in []. In this paper we consider the case c ≠ 1 (when the equation is of the second order). We show that a function $${f:\mathbb{R} \rightarrow \mathbb{R},\,f\left( 0\right) =0}$$ , differentiable at the point 0 satisfies this functional equation iff there is a real p such that $${f=\tanh \circ \left( p\tan ^{-1} \right) }$$ which extends the main result of []. [ABSTRACT FROM AUTHOR]

In this paper we determine the general solution of some special cases of the generalized Hosszú functional equation on intervals [0,1] and (0,1). [ABSTRACT FROM AUTHOR]

Given rational vector spaces V, W a mapping $${f \colon V \to W}$$ is called a generalized polynomial of degree at most n, if there are homogeneous generalized polynomials f of degree i such that $${f = \sum_{i = 0}^n f_i}$$ . Homogeneous generalized polynomials f of degree i are mappings of the form $${f_i (x) = f_i^*(x, x, \ldots , x)}$$ with $${f_i^* \colon V^i \to W i}$$ -linear. In the literature one may find quite a lot of functional equations such that their general solution is of the form f or $${f_n + f_{n - 1}}$$ where n is a small positive integer ( ≤ 6 or ≤ 4 respectively). In this paper, given an arbitrary positive integer n and an arbitrary subset $${L \subseteq \{0, 1, \ldots, n\}}$$ such that $${n \in L}$$ , a method is described to find (many) functional equations, such that their general solution is given by $${\sum_{i \in L} f_i}$$ . For the cases $${L = \{n\}}$$ and $${L = \{n - 1, n\}}$$ additional equations are given. [ABSTRACT FROM AUTHOR]