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

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

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),     

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.     

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.     

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 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).     

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

Nonoscillatory solutions of a second-order difference equation of Poincaré type

Poincaré’s classical theorem about the convergence of ratios of successive values of solutions applies if the characteristic roots of the associated limiting equation are simple and have different moduli. In this work, it is shown that for the nonoscillatory solutions the conclusion of Poincaré’s theorem is also true in the case where the limiting equation has a double positive characteristic root.     

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.     

Hölder continuity of solutions of an elliptic equation uniformly degenerating on part of the domain

We consider a second-order divergence elliptic equation in a domain D divided by a hyperplane in two parts. The equation is uniformly elliptic in one of these parts and is uniformly degenerate with respect to a small parameter ? in the other. We show that each solution is Hölder continuous in D with Hölder exponent independent of ?.     

A Note on the analytic solutions of the Camassa–Holm equation

In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to $H^s(\mathbb{R})$ with $s>3/2$, $6u_0{6}_{L^1}<\infty$ and $u_0?u_{0xx}$ does not change sign, we prove that the solution stays analytic globally in time. To cite this article: M.C. Lombardo et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Local solvability and a priori estimates for classical solutions to an equation of Benjamin-Bona-Mahony-Bürgers type

We establish the local (in time) solvability in the classical sense for the Cauchy problem and first and second boundary-value problems on the half-line for a nonlinear equation similar to Benjamin-Bona-Mahony-Bürgers-type equation. We also derive an a priori estimate that implies sufficient blow-up conditions for the second boundary-value problem. We obtain analytically an upper bound of the blow-up time and refine it numerically using Richardson effective accuracy order technique.     

Asymptotics and existence of bounded solutions of a nonlinear Schrödinger equation on the half-line

We study the asymptotics and existence of nonzero bounded solutions of the Schrödinger equation on the half-line with potential that implicitly depends on the wave function via a nonlinear second-order ordinary differential equation. We prove the existence of countably many nonzero bounded solutions on the half-line and derive asymptotic formulas at infinity for these solutions.     

Asymptotic behavior of the solutions of the p-Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p→∞ is studied.