Oscillation Theorems Research Articles

LaSalle stationary oscillation theorem for affine periodic dynamic systems on time scales

In this paper, for affine periodic systems on time scales, we establish LaSalle stationary oscillation theorem to obtain the existence and asymptotic stability of affine periodic solutions on time scales. As applications, we present the existence and asymptotic stability of affine periodic solutions on time scales via Lyapunov’s method.

Complex oscillation of non-homogeneous differential equations in the unit disc

In this paper, we investigate the relationship between solutions and their derivatives of non homogeneous linear differential equation for k ≥ 2 and small functions, where are finite order meromorphic functions in the unit disc . We obtain some oscillation theorems for , where f (z) is a solution and ϕ(z) is a small function.

Applications of a Class of Herglotz Operator Pencils

We identify a class of operator pencils, arising in a number of applications, which have only real eigenvalues. In the one-dimensional case we prove a novel version of the Sturm oscillation theorem: if the dependence on the eigenvalue parameter is of degree $k$, then the real axis can be partitioned into a union of $k$ disjoint intervals, each of which enjoys a Sturm oscillation theorem: on each interval there is an increasing sequence of eigenvalues that are indexed by the number of roots of the associated eigenfunction. One consequence of this is that it guarantees that the spectra of these operator pencils have finite accumulation points, implying that the operators do not have compact resolvents. As an application we apply this theory to an epidemic model and several species dispersal models arising in biology.

New oscillation theorems for second order quasi-linear difference equations with sub-linear neutral term

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the second order neutral difference equation \t\t\tΔ(an(Δzn)β)+qnxn−ℓγ=0,n≥n0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\Delta \\bigl( a_{n} ( \\Delta z_{n} )^{\\beta} \\bigr) + q_{n}x_{n - \\ell}^{\\gamma} = 0,\\quad n \\ge n_{0}, $$\\end{document} where z_{n} = x_{n} + p_{n}x_{n - k}^{alpha} . The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the importance of the main results.

Oscillation theorems for nonlinear fractional difference equations

In this study, we discuss some theorems related to the oscillatory behavior of nonlinear fractional difference equations equipped with well-known fractional Riemann–Liouville difference operator. Then we give an example for the illustration of the results obtained.

Oscillation theorems for linear Hamiltonian systems with nonlinear dependence on the spectral parameter and the comparative index

We consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. In our consideration we do not impose any controllability and strict normality assumptions and omit the Legendre condition for the Hamiltonian. We adapt the theory developed by A.A. Abramov for the Hamiltonian spectral problems based on piecewise constant transformations of their conjoined bases using the notion of the comparative index. We introduce the concept of oscillation numbers which generalize the notion of proper focal points and prove the oscillation theorem relating the number of finite eigenvalues in the given interval with the values of the oscillation numbers at the end points of this interval.

Discrete oscillation theorems for symplectic eigenvalue problems with general boundary conditions depending nonlinearly on spectral parameter

In this paper we establish new oscillation theorems for discrete symplectic eigenvalue problems with general boundary conditions. We suppose that the symplectic coefficient matrix of the system and the boundary conditions are nonlinear functions of the spectral parameter and that they satisfy certain natural monotonicity assumptions. In our new theory we admit possible oscillations in the coefficients of the symplectic system and the boundary conditions by incorporating their nonconstant rank with respect to the spectral parameter. We also prove necessary and sufficient conditions for boundedness of the real part of spectrum of these eigenvalue problems.

Oscillation problems for Hill's equation with periodic damping

This paper deals with the second-order linear differential equation x″+a(t)x′+b(t)x=0, where a and b are periodic coefficients. The main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory. Our nonoscillation theorem and oscillation theorem are proved by using the Riccati technique. In our theorem, the composite function of an indefinite integral of b and a suitable multiple-valued continuously differentiable function are focused, and the composite function of them plays an important role. The results obtained here include a result by Kwong and Wong [15] and a result by Sugie and Matsumura [26]. An application to a equation of Whittaker–Hill type is given to show the usefulness of our results. Finally, simulations are also attached to illustrate that our oscillation criterion is sharp.

Oscillation theorems for certain delay difference inequalities

Oscillation theorems for certain delay difference inequalities

Oscillation Criteria of Second-Order Mixed Neutral Delay Dynamic Equations on Time Scales

In this artical, we consider a second-order neutral dynamic equation on a time scales. A number of oscillation theorems are shown that supplement and extend some known results in the eassay.

Oscillation theorems for second order neutral difference equations with “Maxima”

Oscillation theorems for second order neutral difference equations with “Maxima”

English

English

Integral condition for oscillation of half-linear differential equations with damping

The purpose of this paper is to provide an oscillation theorem that can be applied to half-linear differential equations with time-varying coefficients. A parametric curve by the coefficients is focused in order to obtain our theorem. This parametric curve is a generalization of the curve given by the characteristic equation of the second-order linear differential equation with constant coefficients. The obtained theorem is proved by transforming the half-linear differential equation to a standard polar coordinates system and using phase plane analysis carefully.

Oscillation theorems for second-order nonlinear delay differential equations of neutral type

In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form \[\frac{d}{dt}\bigg[r(t)\frac{d}{dt}[x(t)+p(t)x(\tau(t))]\bigg]+q(t)G\bigl(x(\sigma(t))\bigr)=0\: for\: t\geq t_{0},\] under the assumptions $\int^{\infty}\frac{1}{r(\eta)}d\eta=\infty$ and $\int^{\infty}\frac{1}{r(\eta)}d\eta<\infty$ for various ranges of the bounded neutral coefficient $p$. Unlike most of the previous results, $\tau^{\prime}$ is allowed to be oscillatory. Further, some illustrative examples showing applicability of the new results are included.

Oscillation theorems for fourth‐order delay differential equations with a negative middle term

This paper deals with the oscillation of the fourth‐order linear delay differential equation with a negative middle term under the assumption that all solutions of the auxiliary third‐order differential equation are nonoscillatory. Examples are included to illustrate the importance of results obtained.

Oscillation criteria for second-order superlinear Emden–Fowler neutral differential equations

We study oscillatory behavior of solutions to a class of second-order superlinear Emden–Fowler neutral differential equations. New oscillation theorems are presented and their efficiency is illustrated.

Oscillation criteria for second order superlinear dynamic equations with oscillating coefficients

This paper is concerned with oscillation of second order superlinear dynamic equations with oscillating coefficients. By using generalized Riccati transformations, oscillation theorems are obtained on an arbitrary time scale. We illustrate the versatility of our results by means of examples.

Wong’s Oscillation Theorem for the Second-Order Delay Differential Equations

Wong’s Oscillation Theorem for the Second-Order Delay Differential Equations

Oscillation theorems for second-order non-linear differential equation with a non-linear neutral term

In this paper, the authors present sufficient conditions for the oscillation of all solutions of the second order nonlinear neutral differential equation with sub-linear neutral term. The obtained results generalize and improve those reported in the literature. Examples are provided to illustrate the importance of the main results.

Oscillation theorems for impulsive parabolic differential system of neutral type

In this paper, oscillatory properties of solutions to a nonlinear impulsive parabolic differential system of neutral type are investigated. A series of sufficient conditions are established for problems with Robin and Dirichlet boundary conditions. Examples are provided to confirm the validity of the analysis.