Goła̧b–Schinzel equation on cylinders

We determine continuous solutions of the Go?a?b–Schinzel functional equation on cylinders.     

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).     

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.     

Bounded solutions of the Golab—Schinzel equation

Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)ⁿ y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) .     

On solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),     

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

On a conditional Gołab-Schinzel equation

Let We show that for every function satisfying the conditional equation

A generalization of the Egalitarian and the Kalai–Smorodinsky bargaining solutions

We characterize the class of weakly efficient n-person bargaining solutions that solely depend on the ratios of the players’ ideal payoffs. In the case of at least three players the ratio between the solution payoffs of any two players is a power of the ratio between their ideal payoffs. As special cases this class contains the Egalitarian and the Kalai–Smorodinsky bargaining solutions, which can be pinned down by imposing additional axioms.     

Two-soliton solutions of the complex short pulse equation via Riemann–Hilbert approach

Based on the basic conclusions of the Riemann–Hilbert method for solving the initial value problem of the complex short-pulse equations, the general form of the two-soliton solutions of the complex short pulse equation is given in this paper. Under the two different assumptions of the scattering coefficient, the expression of the two-soliton solutions of the equation is given specifically.     

A note of a modular equation of Ramanujan–Selberg continued fraction

The Ramanujan Journal - We show that if $$E/\mathbb {Q}$$ is an elliptic curve with a rational p-torsion for $$p=2$$ or 3, then there is a congruence relation between Ramanujan’s tau function...     

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

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

A note on solutions of the Korteweg–de Vries hierarchy

olutions of the Korteweg–de Vries hierarchy are discussed. It is shown that results by Wazwaz [Wazwaz AM. Multiple-soliton solutions of the perturbed KdV equation. Commun Nonlinear Sci Simul 2010;15911:3270–73] are the well-known consequences of the full integrability for the Korteweg–de Vries hierarchy.     

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].