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Zannier proved that for a Laurent series f(x) satisfying the functional equation of type f(x m ) = P(x, f(x)), where \( P(x, y) \in {\Bbb C}(x)[y] \), if f(x) is not rational the set of such m consists of the powers of a single integer. He mentioned that the case f(x) = P(x, f(x m )) should be proved in a similar way. In this paper we first verify this statement and second we show a theorem which is useful for proving the transcendence of a Laurent series satisfying a certain functional equation. This theorem is a generalization of the result that a Laurent series which satisfiesf(x m ) = P(x, f(x)), where\( P(x, y) \in {\Bbb C}(x, y),\,m \geq 2 \)cannot represent an algebraic function unless it is rational (Ke. Nishioka [1], Zannier [8], Section 3).

A recent note of Ih-Ching Hsu poses an unsolved problem, to wit, the general solution of the functional equation g(x1, x2) + g(φ1(x1), φ2(x2)) = g(x1, φ2(x2)) + g(φ1(x1),x2), where the φi are given functions. This short paper obtains the general solution. It gives conditions which imply that anycontinuous solution has form g1(x1) + g2(x2).

The aim of this paper is to fill in the gaps in the formulation and the proof of a theorem contained in the paper by K.Ozeki (Aequ Math 25:247–252, 1982) published in this journal. We also give a short proof of this theorem and use it to obtain certain information about the factorization of polynomials of the form $$f(x)-f(y)$$ .

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 $$f(z)^3-3\tau f(z)f(z+c)+f(z+c)^3=1$$. In addition, the paper is concerned with the description of finite order entire solutions of the Quadratic type of Fermat functional equation $$f(z)^2-2\mu f(z)f(z+c)+f(z+c)^2=1$$.

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.

In the paper Brillouet-Belluot and Ebanks (Aequationes Math 60:233–242, 2000), the authors found all continuous functions f: [0, 1] → [0, + ∞) which verify f(0) = f(1) = 0 and the functional equation $$f(xy +c f(x) f(y)) = x f(y) + y f(x) +d \, f(x) f(y)$$ where c and d are given real numbers with c ≠ 0. In the present paper we obtain all continuous solutions \({f: \mathbb{R} \rightarrow \mathbb{R}}\) of the functional equation (1).

Two natural extensions of Jensen’s functional equation on the real line are the equations f(xy) + f(xy−1) = 2f(x) and f(xy) + f(y−1x) = 2f(x), where f is a map from a multiplicative group G into an abelian additive group H. In a series of papers (see Ng in Aequationes Math 39:85–99, 1990; Ng in Aequationes Math 58:311–320, 1999; Ng in Aequationes Math 62:143–159, 2001), Ng solved these functional equations for the case where G is a free group and the linear group \({{GL_n(R), R=\mathbb{Z},\mathbb{R}}}\) , is a quadratically closed field or a finite field. He also mentioned, without a detailed proof, in the above papers and in (see Ng in Aequationes Math 70:131–153, 2005) that when G is the symmetric group Sn, the group of all solutions of these functional equations coincides with the group of all homomorphisms from (Sn, ·) to (H, + ). The aim of this paper is to give an elementary and direct proof of this fact.

Two inverse operations are considered in this paper: solving a given equation, i.e. finding its solution space, and constructing an equation from a given set, the elements of which being supposed as its solutions, i.e. finding a suitable representation for a given solution space. These equations and representations need not be unique, they may depend on our requirements, e.g. on smoothness or discreteness. These considerations are explained on differential and functional equations. This paper offers a new interpretation of results taken from the area of difference, differential and functional equations.

In the paper On the mutual noncompatibility of homogeneous analytic non-power means (Aequationes Math. 45 (1993)) M. E. Kuczma considered analytic solutions of the functional equation $ x + g(y + f(x)) = y + g(x + f(y)) $ on the real line. Solutions in the class of twice differentiable functions are given in the author’s paper Differentiable solutions of a functional equation related to the non-power means (Aequationes Math. 55 (1998)). During the 38th International Symposium on Functional Equations N. Brillouet-Belluot presented the proof that differentiable solutions of this equation have the same form as in the previous cases. We present solutions of the equation in the class of monotonic Jensen convex or Jensen concave functions on the real line. This time we get already some new families of solutions.

In 1990 Cambridge University Press published the by now well-known monograph Iterative Functional Equations by Marek Kuczma, Bogdan Choczewski and Roman Ger. In the paper we will pay attention mainly on, we think, important papers published in the last decade or not discussed in this book. Of course, the selection we made reflects our personal preferences only.¶The topics of this paper are: 1. A view of the theory developed up to now; 2. Iterative roots; 3. Functional equations with superpositions of the unknown function; 4. Some linear and convolution equations; 5. Schilling's problem; 6. Daroczy's equation; 7. Simultaneous functional equations; 8. Schroder's equation; 9. Abel's equation; 10. Miscellaneous results and problems.

In this paper, the author discusses a generalization of Schroder’s and Abel’s functional equation of the form $$\sum _{n}\alpha _n(x)f\big (u_n(x)\big ) = g(x).$$ In fact, using the theory of operators and infinite matrices, we show under certain conditions this equation has only a unique bounded solution $$f:X\rightarrow {\mathbb {R}}.$$

We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ 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 $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ 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 $$g = \gamma f$$ g = γ f was examined.

We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential equations. Further we also give an affirmative answer to a conjecture of H.\'Swiatak. A part of the results was announced in the comprehensive paper [8] on series of our joint works. To prove the regularity of solutions of functional equation is in general one of central problem in the theory of functional equations., Comment: 9 pages, LaTeX

The subject of this paper is the investigation of the linear two variable functional equation (*) $$h_0 (x,y)f_0 (g_0 (x,y)) + \cdots + h_n (x,y)f_n (g_n (x,y)) = F(x,y),$$ whereg 0, ⋯,g n,h 0, ⋯,h n andF are given real valued functions on an open setΩ ⊂R 2, furtherf 0, ⋯,f n are unknown real functions. Assuming differentiability of sufficiently large order, we construct a linear partial differential operator $$D = \sum\limits_{i,j \geqslant 0,i + j \leqslant k} {\alpha _{ij} (x,y)} \partial _x^i \partial _y^i ,$$ whereα ij is defined onΩ, so that $$D[h_i (x,y)\varphi (g_i (x,y))] = 0, (x,y) \in \Omega , i = 1,...,n$$ for all sufficiently smooth functionsϕ defined on $$ \cup _{l = 1}^n g_i (\Omega )$$ . Then, applyingD to (*), we obtain $$D[h_0 (x,y)f_0 (g_0 (x,y))] = DF(x,y), (x,y) \in \Omega ,$$ , which is ak-th order linear differential-functional equation for the unknown functionf 0. Using Jarai's regularity theorems (see [3], [4], [5]) one can see that, if the given functionsg t,h t,F, are differentiable up to an orderk (1 ⩽k ⩽ ∞), then the measurability of the unknown functionsf t imply their differentiability up to the same order. In this paper we prove an analogous result, namely that the analyticity of the given functions implies that the unknown functions are also analytic provided that they are measurable.

The aim of this paper is to offer some hyperstability results for the following functional equation $$\begin{aligned} \sum _{\lambda \in \Lambda }f(x\lambda .y)=Lf(x)+\sum _{\lambda \in \Lambda }f(\lambda .y)\;\;\;\; (x,y\in S), \end{aligned}$$ where S is a semigroup, $$\Lambda $$ is a finite subgroup of the group of endomorphisms of S, L is the cardinality of $$\Lambda $$ (i.e. $$L=card(\Lambda )$$ ) and $$f:S\rightarrow G$$ 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 $$\begin{aligned} \sum _{\lambda \in \Lambda }f(x\lambda .y)=Lf(x)+\sum _{\lambda \in \Lambda }f(\lambda .y)+F(x,y)\;\;\; (x,y \in S), \end{aligned}$$ where $$F:S\times S \rightarrow G$$ .

The functional equationϕ(x) + ϕ(y) = ψ(xf(y) + yf(x)) (1) for the unknown functionsf, ϕ andψ mapping reals into reals appears in the title of N. H. Abel's paper [1] from 1827 and its differentiable solutions are given there. In 1900 D. Hilbert pointed to (1), and to other functional equations considered by Abel, in the second part of his fifth problem. He asked if these equations could be solved without, for instance, assumption of differentiability of given and unknown functions. Hilbert's question was recalled by J. Aczel in 1987, during the 25th International Symposium on Functional Equations in Hamburg-Rissen. In particular Aczel asked for all continuous solutions of (1). An answer to his question is contained in our paper. We determine all continuous functionsf: I → ℝ,ψ: Af(I × I) → ℝ andϕ: I → ℝ that satisfy (1). HereI denotes a real interval containing 0 andAf(x,y) := xf(y) + yf(x), x, y ∈ I. The list contains not only the differentiable solutions, implicitly described by Abel, but also some nondifferentiable ones.

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 $$\alpha (t)$$ α ( t ) of the probability that a mean-zero random variable satisfies X$$>t$$ > t given that $$|X|>t$$ | X | > t . In particular, we show that under natural symmetry assumptions the tail-balances $$\alpha (t)$$ 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 $$\alpha (t)$$ α ( t ) linear.

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.

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.

Several results concerning the homomorphism functional equation $$\begin{aligned} f\left( x\vee y\right) =f\left( x\right) \vee f\left( y\right) , \end{aligned}$$ in the class of semilattices, as well as other similar equations, are presented in the paper.

In the present paper we prove that the complex functional equation $$F(z)+F(2z)+\cdots +F(nz)=0$$ , $$n\ge 2$$ , $$z\in {\mathbb {C}}{\setminus }( -\infty ,0] $$ , is stable in the generalized Hyers–Ulam sense.

In this paper, we determine the complex-valued solutions of the functional equation $$\begin{aligned} f(x\alpha \sigma (y))+h(\tau (y)x)=2f(x)k(y) \end{aligned}$$for all $$x,y\in G$$, where G is a group, $$\alpha \in G$$ is a fixed element and $$\sigma ,\tau :G\rightarrow G$$ are involutions.

In this paper we find the solutions of the functional equation $$\begin{aligned} f(xy)=g(x)h(y)+\sum _{j=1}^{n}g_j(x)h_j(y),\;x,y \in M, \end{aligned}$$where M is a monoid, $$n\ge 2$$, and $$g_j$$ (for $$j=1,\ldots ,n$$) are linear combinations of at least 2 distinct nonzero multiplicative functions.

The general solution of the multiplicative symmetric functional equation $$f(f(x)y)=f(xf(y))$$, in abelian groups, was given by Dhombres. But it is still an open problem in non-abelian groups and abelian quasi-groups. In this paper, by using a new approach introduced by the author, we give a vast class of its solutions in arbitrary semigroups. It also works for non-abelian groups, and gives its general solution in abelian groups with a different representation from Dhombres’s solution. Also, we introduce some related equations and closed relationships between decomposer type and multiplicative symmetric equations. It is worth noting that the new approach and methods can be used for studying the equation on magmas (groupoids) as the most general algebraic structures for the topic. At last we pose a conjecture that guesses the general solution in arbitrary groups (and possibly semigroups).

Let $$(S,+)$$ be an abelian semigroup, let $$\sigma $$ be an involution of S, let X be a linear space over the field $${\mathbb {K}}\in \{{\mathbb {R}},{\mathbb {C}}\}$$ and let $$\mu $$,$$\nu $$ be linear combinations of Dirac measures. In the present paper, we find the general solution $$f:S\rightarrow X$$ of the following functional equation $$\begin{aligned} \int _{S}f(x+y+t)d\mu (t)+\int _{S}f(x+\sigma (y)+t)d\nu (t)=f(x)+f(y), \ \ \ x,y \in S, \end{aligned}$$in terms of additive and bi-additive maps. Many consequences of this result are presented.

The aim of this paper is to give an answer to a question posed by Alsina, Sikorska and Tomas. Namely, we show that, under suitable assumptions, a function $$f:X\!\rightarrow \!Y$$ from a normed space X into a normed space Y, satisfying the functional equation $$\begin{aligned} f\left( y-\frac{\rho '_+(x,y)}{\Vert x\Vert ^2}x\right) =f(y)-\frac{\rho '_+(f(x),f(y))}{\Vert f(x)\Vert ^2}f(x),\quad x,y\!\in \! X \end{aligned}$$ has to be a linear similarity (scalar multiple of a linear isometry).

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 $$f(x) = M \big ( f(x+t_{1}), f(x + t_{2} ),\ldots , f(x + t_{n-1} ), \,f(ax) \big ),$$ where $$n \in \mathbb {N}, \,t_{1},t_{2},\ldots ,t_{n-1} \in \mathbb {R} {\setminus } \{ 0\}, \,a \in (-\infty , 0)$$ 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.

The aim of the present paper is to describe some properties of functions with finite dimensional difference spaces by means of spectral analysis and spectral synthesis. We are going to apply these results to a version of a Levi-Civita functional equation, which has been recently studied by J. M. Almira and E. Shulman.

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 Levy 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 $$\diamond $$ for which $$\delta _x \diamond \delta _1$$ , $$x \in [0,1]$$ , 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 $$n \geqslant 3$$ is equivalent to solving the Levi-Civita functional equation in the class of continuous generalized characteristic functions.

The aim of this paper is to describe the solution (f, g) of the equation $$\begin{aligned}{}[f(x)-f(y)]g'(\alpha x+(1-\alpha )y)= [g(x)-g(y)]f'(\alpha x+(1-\alpha )y),\ x,y\in I, \end{aligned}$$ where $$I\subset \mathbb {R}$$ is an open interval, $$f,g:I\rightarrow \mathbb {R}$$ are differentiable, $$\alpha $$ is a fixed number from (0, 1).

In Schmitz (Aequ Math 91:373–389, 2017), the first author defines an “inverse ambiguous function” on a group G to be a bijective function $$f:G \rightarrow G$$ satisfying the functional equation $$f^{-1}(x) = (f(x))^{-1}$$ for all $$x \in G$$ . Using a simple criterion involving the number of elements in G not equal to their own inverse, the classification of finite abelian groups admitting inverse ambiguous functions is achieved. In this paper we aim to extend the results from (2017) to determine the existence of inverse ambiguous functions on members of certain families of non-abelian groups, namely the symmetric groups $$S_n$$ , the alternating groups $$A_n$$ , and the general linear groups GL(2, q) over a finite field $$\mathbb {F}_q$$ .

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 Mako–Pales mean is defined by $$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$ Under some conditions on the functions $$\varphi $$ and $$\psi $$ defined on I, the quotient mean is given by $$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$ In this paper, we study some invariance of the quotient mean with respect to Mako–Pales means.

In this paper, we utilize some fixed point theorems of contractive type to present a few existence and uniqueness theorems for a functional equation arising in dynamic programming of continuous multistage decision processes.

In this paper, we present hyperstability results of Jensen functional equations in ultrametric Banach spaces.

If \(f\,{:}\,G \rightarrow G\) is a bijection from a group G to itself, then the superscript \(-1\) written in proximity of f could, depending on its placement, indicate the inverse function of f or the function whose outputs are the inverses (in G) of the corresponding outputs of f. In this paper we investigate which groups G admit functions f where these two interpretations lead to the same outcome, that is, when \(f^{-1}(x) = (f(x))^{-1}\) for every \(x \in G\). We call such functions inverse ambiguous. After deriving some preliminary results, we turn our attention to the existence of inverse ambiguous functions defined on various fields F with respect to the underlying additive structure and multiplicative structure. Our study of this notational curiosity will lead to a surprising finding: namely, in many of the standard cases, there exist either continuous inverse ambiguous functions on F with respect to addition or on \(F^\times \) with respect to multiplication, but not both.

In this paper we study the solutions of the integral Van Vleck’s functional equation for the sine $$\begin{aligned} \int _{S}f(x\tau (y)t)d\mu (t)-\int _{S}f(xyt)d\mu (t) =2f(x)f(y),\; x,y\in S \end{aligned}$$ and the integral Kannappan’s functional equation $$\begin{aligned} \int _{S}f(xyt)d\mu (t)+\int _{S}f(x\tau (y)t)d\mu (t) =2f(x)f(y),\; x,y\in S, \end{aligned}$$ where S is a semigroup, $$\tau $$ is an involution of S and $$\mu $$ is a measure that is a linear combination of Dirac measures $$(\delta _{z_{i}})_{i\in I}$$ , such that for all $$i\in I$$ , $$z_{i}$$ is contained in the center of S. We express the solutions of the first equation by means of multiplicative functions on S, and we prove that the solutions of the second equation are closely related to the solutions of d’Alembert’s classic functional equation with involution.

In the present paper, we determine the complex-valued solutions (f, g) of the functional equation $$f(x\sigma(y))+f(\tau(y)x)=2f(x)g(y),$$ in the setting of groups and monoids that need not be abelian, where \({\sigma,\tau}\) are involutive automorphisms. We prove that their solutions can be expressed in terms of multiplicative and additive functions.

In this paper, we obtain hyperstability results for the $${\sigma}$$ -Drygas functional equation and the inhomogeneous $${\sigma}$$ -Drygas functional equation on semigroups. Namely, we show that a function satisfying the $${\sigma}$$ -Drygas equation approximately must be exactly the solution of it.

In the first section of this paper we consider some functional equations which are closely connected to derivations (i.e. additive mappings with the propertyD(ab) = aD(b) + D(a)b) on Banach algebras. IfD is a derivation on some algebraA, then the equationD(a) = − aD(a−1)a holds for all invertible elementsa ∈A. It seems natural to ask whether this functional equation characterizes derivations among all additive mappings. It is too much to expect an affirmative answer to this question in arbitrary algebras, since it may happen that even in normed algebras the group of all invertible elements contains only scalar multiples of the identity. We try to answer the question above in Banach algebras, since in Banach algebras invertible elements exist in abundance. In the second section of the paper we prove some results concerning representability of quadratic forms by bilinear forms.

1. In this paper we are concerned with the functional equation q~(x, t)=H (x, q9 [f (x), t], t), (1) where tp (x, t) is an unknown function and f (x), H (x, y, t) are known real functions of real variables and t is a real parameter. In paper [3] J. Matkowski has proved under assumptions which guarantee the existence and uniqueness of' solutions ~o,~ C' of equations qgn(X)=nn(X, (~n [fn(X)]), n=0, 1, 2, ..., (2) that tp, tends to q~o with derivatives up to order r, uniformly on every compact set. We shall prove that under some assumptions, solution ~0 (x, t) of equation (1) is of class C p with respect to the parameter t. For the linear equation q~ [f (x), t]=g(x, t)¢p(x, t )+ F (x, t)

The problem of determining all branching measures of inset information on open domains leads to the functional equation (*) $$\phi _1 (p,q) + \phi _2 (p + q,r) = \phi _3 (p,r) + \phi _4 (p + r,q)$$ for (p, q, r) inD 3 and real-valued mapsφ 1 (i = 1,⋯, 4) onD 2. Here, we letJ = ]0, 1[ m for some (fixed) positive integerm, and $$D_n = \left\{ {(p_1 ,p_2 , \ldots ,p_n )|p_i \in Jforalli,\sum\limits_{i = 1}^n {p_i \in J} } \right\}$$ forn = 2, 3,⋯. The main result of the paper is the general solution of equation (*). The general solution of (*) in the special case when the four unknown functions are equal and symmetric follows from a 1974 paper of Ng (Representation for measures of information with the branching property, Inform. and Control25, 45–56). The reduction of (*) to this special case is not easy, however, because of the absence of zeros from the domain. The first step is the known reduction of (*) to the two equations $$\begin{gathered} h(p,q) + k(p + q,r) = h(p,r) + k(p + r,q), \hfill \\ f(p,q) + g(p + q,r) + f(p,r) + g(p + r,q) = 0, \hfill \\ \end{gathered} $$ for (p, q, r) ∈ D 3 and real-valued mapsf, g, h, k onD 2. Equation (A) is eventually reduced to the special case with which Ng dealt. Some new methods developed include the general solution of some Sincov-type equations on restricted domains, and the general solution of $$U(x,y) = \Phi (x + y,q) - \Phi (x,q) - \Phi (y,q) - m(q)$$ for (x, y, q) ∈ D 3 and real-valued mapsm (onJ),U (onD 2), and skew-symmetric Ф (onD 2).

The purpose of this paper is to prove that for a general type of functional equation every solution which is measurable on a set of positive measure or satisfies the Baire condition on a Baire set of the second category is continuous. The results solve the problems of W. Sander raised in his paper "Verallgemeinerte CauchyFunktionalgleichungen", Aequationes Math. 18 (1978), 357-369. A typical, but not the most general, result is the following.

In this paper, we give all the solutions \({g,h:\mathbb{R}\to\mathbb{R}}\) (the reals) of the functional equation $$g(x)g(y)-g(x+y)=h(x+y-xy)-h(x)-h(y)+h(xy) \quad(x,y\in\mathbb{R}),$$ supposing additionally that h is continuous. This result is in connection with the alienation of the exponential Cauchy equation g(x + y) = g(x)g(y) and the Hosszu equation h(x + y−xy) + h(xy) = h(x) + h(y), namely it turns out that these equations are alien provided that h is continuous.

The functional equation of composite type M(M(x; M(x; y)); M(M(x; y); y)) = M(x; y) 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 at 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 equation (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. Fil: Berrone, Lucio Renato. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Cientifico Tecnológico Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingenieria y Agrimensura. Escuela de Ingenieria Electrónica. Laboratorio de Acústica y Electroacústica; Argentina

In this note we provide the solution to a problem posed by the first author in a previous paper. In particular, we prove a result relating the number of nonzero coefficients of a certain functional equation to the order of any derivation satisfying that equation.

In this paper local polynomials on Abelian groups are characterized by a “local” Frechet-type functional equation. We apply our result to generalize Montel’s Theorem and to obtain Montel-type theorems on commutative groups.

This paper treats a functional equation of d’Alembert, namely f(x + y) − f(x − y) = g(x)h(y), in the setting of groups and monoids that need not be abelian. Of the several possibilities for generalizing x − y to the non-abelian case, here we replace it by σ(y)x where σ is an involutive automorphism, and find the complex-valued solutions (f, g, h) of f(xy) − f(σ(y)x) = g(x)h(y) in terms of multiplicative and additive functions.

In this paper we investigate geometrically convex solutions \({g : R^{+} \rightarrow R^{+}}\) to the generalized gamma functional equation with initial condition given by $$g(x + 1) = f(x)g(x) \qquad{\rm for}\,x > 0 \,{\rm and}\, g(1) = 1,\qquad\qquad\qquad(*)$$ 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 $$g(x) = \lim\limits_{n\rightarrow \infty}\frac{f(n) . . . f(1)}{f(n + x) . . . f(x)} . \Big(f(n)\Big)^{\frac{{\rm ln}(n+1+x)-{\rm ln} (n+1)}{{\rm ln}(n+1) - {\rm ln} n}}$$ $$= \lim _{n\rightarrow \infty} \frac{f(n) . . . f(1)[f(n)]^{x}}{f(n + x) . . . f(x)} \quad {\rm for}\, x > 0.$$ Some known results in the literature are generalized and improved.

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

In this paper, we present some characterizations of the q-gamma function by some functional equations using E. Artin’s technique.