A generalization of the Gołąb-Schinzel functional equation

For every fixed real p, the continuous real valued functions f defined on a linear topological space and satisfying the functional equation $$f\left( p[f(y)x+y]+(1-p)[f(x)y+x]\right) =f(x)f(y)$$ are determined. For p = 0 or p = 1 this equation coincides with the classical Go??b-Schinzel equation.     

Subgroups of the groupZ n and a generalization of the Gołąb-Schinzel functional equation

Letn be a positive integer and letX be a linear space over a commutative fieldK. In the set = (K\{0}) × X we define a binary operation ·: × by

Continuous on rays solutions of an equation of the Go?a?b-Schinzel type

Let X be a real linear space. We characterize continuous on rays solutions f,g:X→R of the equation f(x+g(x)y)=f(x)f(y). Our result refers to papers of J. Chudziak (2006) [14] and J. Brzd?k (2003) [11].     

Semigroup-valued solutions of the Gołąb-Schinzel type functional equation

Let (S, o) be a semigroup. We determine all solutions of the functional equation

The pexiderized Go?a?b-Schinzel functional equation

Let X be a linear space over a commutative field K. We characterize a general solution f,g,h,k:X→K of the pexiderized Go?a?b-Schinzel equation f(x+g(x)y)=h(x)k(y), as well as real continuous solutions of the equation.     

On locally bounded above solutions of an equation of the Goła̧b–Schinzel type

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 Go?a?b and Schinzel (Publ Math Debr 6:113–125, 1959) and Wo?od?ko (Aequationes Math 2:12–29, 1968).     

On a relationship between solutions of the third Painlevé equation

The direct and inverse Bäcklund transformations for the third Painlevé equation in the case O is used to obtain a nonlinear functional relationship connecting the solutions of this equation for different values of the parameters that occur in it.Belarus State University of Information Technology and Electronics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 364–366, March, 1995.     

Stability problem for the Go?a?b-Schinzel type functional equations

Let X be a vector space over a field K of real or complex numbers, n∈N and λ∈K?{0}. We study the stability problem for the Go?a?b-Schinzel type functional equations

f(x+fn(x)y)=λf(x)f(y)     

On a conditional Gołab-Schinzel equation

Let We show that for every function satisfying the conditional equation

Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

We construct new elliptic solutions of the restricted Toda chain. These solutions give rise to a new explicit class of orthogonal polynomials, which can be considered as a generalization of the Stieltjes–Carlitz elliptic polynomials. Relations between characteristic (i.e., positive definite) functions, Toda chain, and orthogonal polynomials are developed in order to derive the main properties of these polynomials. Explicit expressions are found for the recurrence coefficients and the weight function for these polynomials. In the degenerate cases of the elliptic functions, the modified Meixner polynomials and the Krall–Laguerre polynomials appear.     

Integrability,existence of global solutions,and wave breaking criteria for a generalization of the Camassa–Holm equation

Recent generalizations of the Camassa–Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well-posedness in Sobolev spaces for the family under consideration. Moreover, we prove that wave breaking phenomena occurs under certain mild hypothesis. Based on the machinery developed by Dubrovin [Commun. Math. Phys. 267, 117–139 (2006)] regarding bi-Hamiltonian deformations, we introduce the notion of quasi-integrability and prove that there exists a unique bi-Hamiltonian structure for the equation only when it is reduced to the Dullin–Gotwald–Holm equation. Our results suggest that a recent shallow water model incorporating Coriollis effects is integrable only in specific situations. Finally, to finish the scheme of geometric integrability of the family of equations initiated in a previous work, we prove that the Dullin–Gotwald–Holm equation describes pseudo-spherical surfaces.     

On bounded solutions of a nonlinear Schrödinger equation on the half-line

We consider the problem on nonzero solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear ordinary differential equation of the second order under zero boundary conditions for the wave function and the condition that the potential is zero at the beginning of the interval and its derivative is zero at infinity. The problem is reduced to the analysis and investigation of solutions of the Cauchy problem for a system of two nonlinear second-order ordinary differential equations with initial conditions depending on two parameters. We show that if the solution of the Cauchy problem for some parameter values can be extended to the entire half-line, then there exists a nonzero solution of the original problem with finitely many zeros.     

On dominant contractions and a generalization of the zero–two law

Zaharopol proved the following result: let \({T,S:L^1(X,{\mathcal{F}},\mu)\to L^1(X, {\mathcal{F}},\mu)}\) be two positive contractions such that T ≤ S. If \({\|S-T\| <1 }\) then \({\left\|S^n-T^n\right\| <1 }\) for all \({n\in\mathbb{N}}\). In the present paper we generalize this result to multi-parameter contractions acting on L ¹. As an application of that result we prove a generalization of the “zero–two” law.