Sergei Yakovenko
We give a simplified proof and an improvement of a recent theorem by A. Grigoriev, placing an upper bound for the number of roots of linear combinations of solutions to systems of linear equations with polynomial or rational coefficients.
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In this note we show that a linear ordinary differential equation with polynomial coefficients is globally non-oscillating in $\mathbb{C} P^1$ if and only if it is Fuchsian, and at every its singular point any two distinct characteristic exponents have distinct real parts. As a byproduct of our study, we obtain a new explicit upper bound for the number of zeros of exponential polynomials in a horizontal strip.
The Riccati equation method is used to establish an oscillatory and a non oscillatory criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the obtained oscillatory criterion is a generalization of J. S. W. Wong's oscillatory criterion.
The Riccati equation method is used to establish Kamenev type conditions for the existence of oscillatory solutions to third order linear ordinary differential equations. Three oscillatory theorems are proved, which generalize the Lazer's oscillation criterion.
The Riccati equation method and an approach of the use of unknown factors is used to establish oscillation, suboscillation and nonoscillation criteria for linear systems of ordinary differential equations. A necessary condition for Lyapunov (asymptotic) stability for these systems is obtained.
In this paper we use the Riccati equation method with other ones to establish global solvability, stability and oscillation criteria for a class of two dimensional nonlinear systems of ordinary differential equations, which is a generalization of wide classes of second order nonlinear ordinary differential equations, studied by many authors. The applicability of some of these criteria are illustrated by examples.
Some properties of global solution of scalar Riccati equation are studied. On the basis of these properties using the Whiburn's and Leighton - Nehary's theorems some oscillatory and criteria are proved for second order linear systems of ordinary differential equations.
The Riccati equation method is used to establish oscillation and non-oscillation criteria for second order linear nonhomogeneous functional-differential equations.We show that the obtained oscillation criterion is a generalization of J. S. W. Wong's oscillation criterion for second order linear nonhomogeneous ordinary differential equations. Two examples, demonstrating the aptitude of the obtained criteria, are presented.
The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler equation and to the Van der Pol type equation.
In this paper we have considered higher order two dimensional coupled system of non-linear ordinary differential equations. We have given necessary and sufficient conditions on the non-linear functions such that the solutions pair oscilla
Forced oscillations for generalized $Φ$-Laplacian equations with Carathéodory perturbations
本文用拓扑方法和Brouwer度研究含广义Φ-Laplacian项的隐式常微分方程在Carathéodory条件下的受迫振荡全局分支结构，并给出分支集合的可视化示例。
In this paper the linearly damped oscillator equation is considered with the damping term generalized to a Caputo fractional derivative. The order of the derivative being considered is 0 less than or equal to nu which is less than or equal to 1 . At the lower end, nu = 0, the equation represents an un-damped oscillator and at the upper end, nu = 1, the ordinary linearly damped oscillator equation is recovered. A solution is found analytically and a comparison with the ordinary linearly damped oscillator is made. It is found that there are nine distinct cases as opposed to the usual three for t…
We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a $k$-dimensional differentiable manifold $M \subseteq \mathbb{R}^m$. We assume that $M$ is globally defined as the zero set of a smooth map and, as a first step, we determine a formula which reduces the computation of the degree of a tangent vector field on $M$ to the Brouwer degree of a suitable map in $\mathbb{R}^m$. As further applications, we study the set of harmonic solutions to periodic semi-esplicit differential-algebraic equations…
In this paper we consider the linear ordinary equation of the second order $$ L x(t)\equiv \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=f(t), \eqno{(1)} $$ and the corresponding homogeneous equation $$ \ddot{x}(t) +a(t)\dot{x}(t)+b(t)x(t)=0. \eqno{(2)} $$ Note that $[α,β]$ is called a nonoscillation interval if every nontrivial solution has at most one zero on this interval. Many investigations which seem to have no connection such as differential inequalities, the Polia-Mammana decomposition (i.e. representation of the operator $L$ in the form of products of the first order differential operators), u…
New necessary and sufficient conditions are given for the quantization of a class of periodic second order non-homogeneous ordinary differential equations in the complex plane in this paper. The problem is studied from the viewpoint of complex oscillation theory first developed by Bank and Laine (1982, 1983) and Gundersen and Steinbart (1994). We show that when a solution is complex non-oscillatory (finite exponent of convergence of zeros) then the solution, which can be written as special functions, must degenerate. This gives a necessary and sufficient condition when the Lommel function has …
We discuss the occurrence of oscillatory solutions which decay to 0 as $s\to+\infty$ for a class of perturbed second order ordinary differential equations. As opposed to other results in the recent literature, the perturbation is as small as desired in terms of its improper integrals and it is independent of the coefficients of the non-oscillatory unperturbed equation. This class of equations reveals thus a new pathology in the theory of perturbed oscillations.
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