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We determine the solutions on various intervals in [ 0 , ∞ [ to the functional equation f (x m) = r f (x) for real r and positive m. Explicit formulas, involving periodic functions, are given for the set S of all solutions. The formulas for r < 0 are more complicated. An approach to S with the help of the axiom of choice is also given. A special attention is laid on solutions that are continuous on [ 0 , ∞ [ or on various open subintervals. We also describe solutions satisfying some asymptotic properties at the boundary of these intervals. [ABSTRACT FROM AUTHOR]

Let Y (+) be a group, D ⊆ ℤ2 where ℤ(+, ⩽) denotes the ordered group of all integers, and ℤ2 := ℤ×ℤ. We shall use the notations Dx := {u ∈ ℤ | ∃v ∈ X : (u, v) ∈ D}, Dy := {v ∈ ℤ | ∃u ∈ ℤ : (u, v) ∈ D}, Dx+y := {z ∈ ℤ | ∃(u, v) ∈ D : z = u + v}. The main purpose of the article is to find sets D ⊆ ℤ2 that the general solution of the functional equation f (x+y) = g(x)+h(y) for all (x, y) ∈ D with unknown functions f : Dx+y → Y, g : Dx → Y, h : Dy → Y is in the form of f (u) = a(u) + C1 + C2 for all u ∈ Dx+y, g(v) = a(v) + C1 for all v ∈ Dx, h(z) = a(z) + C2 for all z ∈ Dy where a : ℤ → Y is an additive function, C1, C2 ∈ Y are constants.

Functional equations involving certain generalized forms of the so-called 'laws of sciences' are considered. The resolution of these equations is linked to the concept of comparison meaningfulness that appears in measurement theory and dimension theory. The results obtained are stated without assuming any topological requirement. [ABSTRACT FROM AUTHOR]

The functional equation f (x + y) = g (x y) is investigated with unknown functions f : A + A → Y , g : A · A → Y in the following cases: A : = α , β ⊆ F + where F is an Archimedean ordered field; A is the set of all positive integers; A is the set of all positive dyadic rational numbers. The set Y is an arbitrarily fixed (infinite) set. The main result of the paper shows that there exists a set A ⊆ R + that is closed under addition and multiplication and there exist functions f, g : A → Y which satisfy the equation f (x + y) = g (x y) for all x , y ∈ A such that the range of the function f is infinite. Finally, some application of the above results is also given. [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]

Several authors studied the so called exponential polynomials, characterized as solutions to the equation f (x + y) = ∑ k = 1 n u i (x) v i (y). In the present paper we deal with a more general equation ∑ j = 1 M P j (x , y) f j (a j x + c j y) = ∑ k = 1 n u k (x) v k (y). Here all f j , u k , v k are assumed to be unknown scalar functions on R d , while P j are polynomials. We prove that f j are ratios of exponential polynomials and polynomials, or sums of exponential functions multiplied by rational functions. [ABSTRACT FROM AUTHOR]

Motivated by the well known fact that any nonzero solution of the fundamental Cauchy functional equation may arbitrarily be prescribed on a Hamel basis, we deal with the following problem: given a functional equation * E 1 (φ) = E 2 (φ) with the unknown function φ : X ⟶ Y , what must the set ∅ ≠ Z ⊂ X be like in order to ensure that an arbitrary function φ 0 : Z ⟶ Y admits a unique function φ : X ⟶ Y solving equation ( ∗ ) and such that φ | Z = φ 0 ; if such a set does exist it is termed to be a basic set. We discuss the problem of existence of basic sets for exponential functions, d'Alembert's functions, sine functions, Cuculière's functions and hyperbolic tangent type functions, among others. [ABSTRACT FROM AUTHOR]

The issue of distributivity of aggregation operators is crucial for many different areas such as fuzzy sets and fuzzy logic, pseudo-analysis and measure theory, and particulary in the decision making theory. The problem of distributivity of an operator form a special class of uni-nullnorms over a general uninorm is being addressed through this paper. The class in question consists of uni-nullnorms with continuous and Archimedean underlying t-norms and t-conorms, and the results presented here are a natural continuation and extension of some previous works with an emphasis on a much wider class of uninorms as inner operators. [ABSTRACT FROM AUTHOR]

Translativity of the beta-type function B f : I 2 → 0 , ∞ , B f x , y : = f x f y f x + y , where f is a single variable function defined either on I = R or I = [ 0 , ∞) , or I = 0 , ∞ , is considered. In each of these three cases a complete solution is given. [ABSTRACT FROM AUTHOR]

Given a nonincreasing null sequence T = (T j) j ⩾ 1 of nonnegative random variables satisfying some classical integrability assumptions and E (∑ j T j α) = 1 for some α > 0 , we characterize the solutions of the well-known functional equation f (t) = E ∏ j ⩾ 1 f (t T j) , t ⩾ 0 , related to the so-called smoothing transform and its min-type variant. In order to do so within the class of nonnegative and nonincreasing functions, we provide a new three-step method whose merits are that it simplifies earlier approaches in some relevant aspects; it works under weaker, close to optimal conditions in the so-called boundary case when E (∑ j ⩾ 1 T j α log T j) = 0 ; it can be expected to work as well in more general setups like random environment. At the end of this article, we also give a one-to-one correspondence between those solutions that are Laplace transforms and thus correspond to the fixed points of the smoothing transform and certain fractal random measures. The latter are defined on the boundary of a weighted tree related to an associated branching random walk. [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]

We show a simple example for ordered semigroup 𝕊 = 𝕊 (+,⩽) that 𝕊 ⊆ℝ (ℝ denotes the real line) and ]a, b[ + ]c, d[ = ]a + c, b + d[ for all a, b, c, d ∈ 𝕊 such that a < b and c < d, but the intervals are no translation invariant, that is, the equation c +]a, b[ = ]c + a, c + b[ is not always fulfilled for all elements a, b, c ∈ 𝕊 such that a < b.

The issue of distributivity of aggregation operations is crucial for many different areas such as utility theory and integration theory. Of special interest are aggregation operations with annihilator. This paper is focused on the problem of distributivity for some so called associative, commutative aggregation operations with annihilator a, known as associative a-CAOA, and uninorms. The full characterization of distributive pairs for T-uninorms, S-uninorms and bi-uninorms is given. [ABSTRACT FROM AUTHOR]

We axiomatically characterize the χ 2 dissimilarity measure. To this end, we solve a new generalization of a functional equation discussed in Aczel (Lectures on functional equations and their applications. Academic Press, 1966). [ABSTRACT FROM AUTHOR]

In this paper we deal with the product of two or three Cauchy differences equaled to zero. We show that in the case of two Cauchy differences, the condition of absolute continuity and differentiability of the two functions involved implies that one of them must be linear, i.e., we have a trivial solution. In the case of the product of three Cauchy differences the situation changes drastically: there exists non trivial C ∞ solutions, while in the case of real analytic functions we obtain that at least one of the functions involved must be linear. Some open problems are then presented. [ABSTRACT FROM AUTHOR]

This paper studies the conditional distributivity for T-uninorms over uninorms in the most general setting, transforming it into the (conditional) distributivity equation involving two uninorms. [ABSTRACT FROM AUTHOR]

Let K be either R or C and α ∈ R . We determine the solutions f : R 3 → K of the following new parametric functional equation: f (x 1 x 2 + α y 1 z 2 + α y 2 z 1 , x 1 y 2 + x 2 y 1 + α z 1 z 2 , x 1 z 2 + x 2 z 1 + y 1 y 2) = f (x 1 , y 1 , z 1) f (x 2 , y 2 , z 2) , (x 1 , y 1 , z 1) , (x 2 , y 2 , z 2) ∈ R 3 , which results from the product of two numbers in a cubic free field. We equip R 3 with a binary operation to show that the non-zero solutions of this equation are monoid homomorphisms and we investigate our results to introduce and find the solutions of d'Alembert's functional equations with endomorphisms. [ABSTRACT FROM AUTHOR]

Let (Ω , F , P) be a probability space and let α , β : F → R be random variables. We provide sufficient conditions under which every bounded continuous solution φ : R → R of the equation φ (x) = ∫ Ω φ α (ω) (x - β (ω)) P (d ω) is constant. We also show that any non-constant bounded continuous solution of the above equation has to be oscillating at infinity. [ABSTRACT FROM AUTHOR]

Applying medial limits we describe bounded solutions φ : S → R of the functional equation φ (x) = ∫ Ω g (ω) φ (f (x , ω)) d μ (ω) + G (x) , where (Ω , A , μ) is a measure space, S ⊂ R , f : S × Ω → S , g : Ω → R is integrable and G : S → R is bounded. The main purpose of this paper is to extend results obtained in Morawiec (Results Math 75(3):102, 2020) to the above general functional equation in wider classes of functions and under weaker assumptions. [ABSTRACT FROM AUTHOR]

Let I ⊆ R be a nonempty open subinterval. We say that a two-variable mean M : I × I → R enjoys the balancing property if, for all x , y ∈ I , the equality 1 M (M (x , M (x , y)) , M (M (x , y) , y)) = M (x , y) holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, that M is analytic, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizes regular quasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question are continuously differentiable. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class of Matkowski means. [ABSTRACT FROM AUTHOR]

Assume that (Ω , A , P) is a probability space, f : [ 0 , 1 ] × Ω → [ 0 , 1 ] is a function such that f (0 , ω) = 0 , f (1 , ω) = 1 for every ω ∈ Ω , g : [ 0 , 1 ] → R is a bounded function such that g (0) = g (1) = 0 , and a , b ∈ R . Applying medial limits we describe bounded solutions φ : [ 0 , 1 ] → R of the equation φ (x) = ∫ Ω φ (f (x , ω)) d P (ω) + g (x) satisfying the boundary conditions φ (0) = a and φ (1) = b . [ABSTRACT FROM AUTHOR]

Let S1 denote the set of all pairs (x, y) of real numbers that fulfill the condition x2-y2=1, and S2 denote the set of all pairs (x, y) of real numbers that fulfill the condition x2+y2=1. In this paper we consider quadratic real functions f that satisfy the additional equation y2f(x)=x2f(y) under the condition (x,y)∈Si(i=1,2). We prove that each of these conditions implies f(x)=f(1)x2 for all x∈R. [ABSTRACT FROM AUTHOR]

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]

We determine continuous bijections f, acting on a real interval into itself, whose k-fold iterate is the quasi-arithmetic mean of all its subsequent iterates from f0 up to fn (where 0⩽k⩽n). Namely, we prove that if at most one of the numbers k, n is odd, then such functions consist of at most three affine pieces. [ABSTRACT FROM AUTHOR]

Let C be an affine plane curve. We consider additive functions f:K→K for which f(x)f(y)=0, whenever (x,y)∈C. We show that if K=R and C is the hyperbola with defining equation xy=1, then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over Q or over Fp, the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero. [ABSTRACT FROM AUTHOR]

In 1990, Benz asked whether a real additive mapping satisfying $$xf(y)=yf(x)$$ for all points ( x, y) on the unit circle must be linear. In 2005, Boros and Erdei showed that it must be so. Here we generalize the problem to a pair of additive functions f, g related by the functional equation $$xf(y)=yg(x)$$ for all points ( x, y) on a specified curve. We find that for many (but not all) types of curves this forces f and g to be equal and linear. [ABSTRACT FROM AUTHOR]

In this note we determine the unique solution to the functional equation $$f(x + y) ( x- y) = \left( f(x) -f (y) \right) (x + y)$$ . We require no additional assumptions on the function $$f$$ . Moreover we solve this functional equation if $$f$$ is only defined on a finite interval. The interest in this type of functional equation is motivated by the study of symmetrizing measures for (the generator of) a Lévy-driven Ornstein-Uhlenbeck process. [ABSTRACT FROM AUTHOR]

We find all real-valued general solutions $f:S\rightarrow \mathbb{R}$ of the d’Alembert functional equation with involution $$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in S$, where $S$ is a commutative semigroup and $\unicode[STIX]{x1D70E}~:~S\rightarrow S$ is an involution. Also, we find the Lebesgue measurable solutions $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the above functional equation, where $\unicode[STIX]{x1D70E}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a Lebesgue measurable involution. As a direct consequence, we obtain the Lebesgue measurable solutions $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the classical d’Alembert functional equation $$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in \mathbb{R}^{n}$. We also exhibit the locally bounded solutions $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the above equations. [ABSTRACT FROM PUBLISHER]

In this paper, we give all the solutions $${g,h:\mathbb{R}\to\mathbb{R}}$$ (the reals) of the functional equation 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 Hosszú 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. [ABSTRACT FROM AUTHOR]

We extend previous results on generalized Cauchy difference equations (in which f = g = h) to functional equations of the form where L is a given function and f, g, h, k are unknown. We prove results assuming either that L is sufficiently smooth or that $${L(x,y)=\phi(x)+\phi(y)-\phi(x+y)}$$ for some function ϕ. We illustrate these results with applications to several examples, including the Olkin-Baker equation. [ABSTRACT FROM AUTHOR]

The equation belongs to the class of linear functional equations. The solutions form a linear space with respect to the usual pointwise operations. According to the classical results of the theory they must be generalized polynomials. New investigations have been started a few years ago. They clarified that the existence of non-trivial solutions depends on the algebraic properties of some related families of parameters. The problem is to find the necessary and sufficient conditions for the existence of non-trivial solutions in terms of these kinds of properties. One of the earliest results is due to Z. Daróczy []. It can be considered as the solution of the problem in case of n = 2. We are going to take more steps forward by solving the problem in case of n = 3. [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]

We determine nontrivial intervals $${I \subset(0,+\infty)}$$ , numbers $${\alpha\in\mathbb R}$$ and continuous bijections $${f \colon I \to I}$$ such that f( x) f( x) = x for every $${x\in I}$$ . [ABSTRACT FROM AUTHOR]

We characterize solutions $${f, g : \mathbb{R} \to \mathbb{R}}$$ of the functional equation f( x + g( x) y) = f( x) f( y) under the assumption that f is locally bounded above at each point $${x \in \mathbb{R}}$$. Our result refers to Goła̧b and Schinzel (Publ Math Debr 6:113-125, ) and Wołodźko (Aequationes Math 2:12-29, ). [ABSTRACT FROM AUTHOR]

We study the functional equation f( x) f( x) = x imposing no continuity assumptions on its bijective solutions defined on an interval. All the continuous bijections satisfying the equation were determined by the second author in (Aequat. Math. (in print), ) when solving the problem (Problem posed during the forty-ninth International Symposium on Functional Equations ) posed by Brillouët-Belluot. [ABSTRACT FROM AUTHOR]

We solve the problem posed by Nicole Brillouët-Belluot (During the Forty-ninth International Symposium on Functional Equations, ) of determining all continuous bijections f : I → I satisfying where I is an arbitrary subinterval of the real line. [ABSTRACT FROM AUTHOR]

Let $${I\subset\mathbb{R}}$$ be a nonempty open interval and let $${L:I^2\to I}$$ be a fixed strict mean. A function $${M:I^2\to I}$$ is said to be an L-conjugate mean on I if there exist $${p,q\in{]}0,1]}$$ and a strictly monotone and continuous function φ such thatfor all $${x,y\in I}$$ . Here L( x, y) is a fixed quasi-arithmetic mean. We will solve the equality problem in this class of means. [ABSTRACT FROM AUTHOR]

Given a random-valued function f : [0, 1] × Ω → [0, 1] on a probability space (Ω,A, P), we consider bounded solutions Ψ : [0, 1] → R of the inequality Ψ(x) ≤ ∫ Ψ(f(x,ω))dP(ω) and a uniqueness-type problem for bounded solutions φ of equations of the type φ(x) = h(x, φ º f(x,·)). Analogues for f : R × Ω → R of the form f(x, ω) = x + ξ(ω) are proved. Some particular cases are studied in more details, especially those where the probability space under considerations is simply the set of positive integers. [ABSTRACT FROM AUTHOR]

Summary. We present a collection of open problems connected with the functional equation of associativity that have arisen in the course of various investigations in the theory of probabilistic metric spaces, statistical dependence, models for fuzzy logic and related topics. [ABSTRACT FROM AUTHOR]

For every $${\alpha\in\mathbb R}$$ we determine all increasing bijections f : (0, + ∞)→ (0, + ∞) such that $${f(1)\neq1}$$ and f( x) f( x) = x for every $${x\in (0,+\infty)}$$ . [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a characterization of monomial functions with the aid of an operator $$\tilde D$$ introduced in Section 1. The present results were inspired by A. Dinghas [2] and A. Simon and P. Volkmann [5], who treated the characterization of polynomial functions by the so called Dinghas interval-derivatives. [ABSTRACT FROM AUTHOR]

Recent applications of functional equations to questions of allocation, aggregation, utility, taxation, theories of measurement and dimensional analysis are discussed and open problems formulated. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to solve the idempotency equation F(x, x) = x, x ∈ [0, 1] for a class of functions of the type 'convex linear combination of a t-norm and a t-conorm'. In the non-strict Archimedean case and for each k ∈ (0, 1), we obtain a unique solution F for this equation. While these functions F are not associative, we also prove that they satisfy the bisymmetry condition. [ABSTRACT FROM AUTHOR]