We investigate the solvability in continuous functions of the Abel equation φ(Fx) - φ(x) = 1 where F is a given continuous mapping of a topological space X. This property depends on the dynamics generated by F. The solvability of all linear equations P(x)ψ(Fx) + Q(x)ψ(x) = γ(x) follows from solvability of the Abel equation in case F is a homeomorphism. If F is noninvertible but X is locally compact then such a total solvability is determined by the same property of the cohomological equation φ(Fx)...

We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given increasing homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. In a series of papers by P. Kahlig and J. Smítal it was proved that the range of any non-constant solution is an interval whose end-points are fixed under $\varphi$ and which contains in its interior no fixed point except for $1$. They also provide a characterization of the class of monotone solutions and prove a necessary and sufficient condition for any solution...

We consider the functional equation $f\left(xf\right(x\left)\right)=\varphi \left(f\right(x\left)\right)$ where $\varphi J\to J$ is a given homeomorphism of an open interval $J\subset (0,\infty )$ and $f(0,\infty )\to J$ is an unknown continuous function. A characterization of the class $𝒮(J,\varphi )$ of continuous solutions $f$ is given in a series of papers by Kahlig and Smítal 1998–2002, and in a recent paper by Reich et al. 2004, in the case when $\varphi$ is increasing. In the present paper we solve the converse problem, for which continuous maps $f(0,\infty )\to J$, where $J$ is an interval, there is an increasing homeomorphism $\varphi$ of $J$ such that $f\in 𝒮(J,\varphi )$. We...

We deal with the linear functional equation (E) $g\left(x\right)={\sum }_{i=1}^r{p}_{i}g\left(c_{i}x\right)$, where g:(0,∞) → (0,∞) is unknown, $(p₁,...,p_r)$ is a probability distribution, and $c_i$’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

For the Abel equation on a real-analytic manifold a dynamical criterion of solvability in real-analytic functions is proved.

Let (Ω,,P) be a probability space and let τ: ℝ×Ω → ℝ be a function which is strictly increasing and continuous with respect to the first variable, measurable with respect to the second variable. Given the set of all continuous probability distribution solutions of the equation $F\left(x\right)={\int }_{\Omega }F\left(\tau (x,\omega )\right)dP\left(\omega \right)$ we determine the set of all its probability distribution solutions.

The paper describes the general form of an ordinary differential equation of an order $n+1$$(n\ge 1)$ which allows a nontrivial global transformation consisting of the...

In this paper we consider the Aleksandrov equation f(L + x) = f(L) + f(x) where L is contained in Rn and f: L --> R and we describe the class of solutions bounded from below, with zeros and assuming on the boundary of the set of zeros only values multiple of a fixed a > 0. This class is the natural generalization of that described by Aleksandrov itself in the one-dimensional case.

Ce texte est consacré à une famille de distributions statistiques — qui comprend les distributions de V. Pareto, celles du type exponentiel et celles que l'on appellera ici «contra-paretiennes» (ou «anti-paretiennes») — dont l'unité tient à ce qu'elles vérifient toutes une même relation fonctionnelle. Celle-ci est d'ailleurs interprétable en termes d'inégalité des distributions ; elle fournit en outre une méthode simple et efficace d'ajustement des distributions de la famille à des «données» observées....