THE ASYMPTOTIC BEHAVIOR AND OSCILLATION FOR A CLASS OF THIRD-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS

In this paper,we consider a class of third-order nonlinear delay dynamic equations.First,we establish a Kiguradze-type lemma and some useful estimates.Second,we give a sufficient and necessary condition for the existence of eventually positive solutions having upper bounds and tending to zero.Third,we obtain new oscillation criteria by employing the Potzsche chain rule.Then,using the generalized Riccati transformation technique and averaging method,we establish the Philos-type oscillation criteria.Surprisingly,the integral value of the Philos-type oscillation criteria,which guarantees that all unbounded solutions oscillate,is greater than θ_(4)(t_(1),T).The results of Theorem 3.5 and Remark 3.6 are novel.Finally,we offer four examples to illustrate our results.

OSCILLATION AND NONOSCILLATION PROPERTIES OF TWO-DIMENSIONAL DELAY DIRECT CONTROL SYSTEMS

In a process of control, the oscillation property may influence the property of a system. However, there are very few results about oscillation of control systems.In this note, we consider a delay direct control system:

Nonoscillation for a Second Order Linear Delay Differential Equation with Impulses

A group of necessary and sufficient conditions for the nonoscillation of a second order linear delay equation with impulses

A NONOSCILLATION RESULT FOR DELAY DIFFERENTIAL EQUATIONS WITH IMPULSES

Sufficient conditions are obtained for the nonscillation of the impulsive delay differential equation

AN OSCILLATION CRITERIA FOR SECOND ORDERFUNCTIONAL EQUATIONS

This paper is concerned with the oscillation of second order linear functional equations of the form x(g(t)) = p(t)x(t) + Q(t)X(g(2)(t)), Where p, Q, g : [t(0), infinity) --> R+ = [0, infinity) are given real valued functions such that g(t) not equivalent to t, lim(t-->infinity) g(t) = infinity. It is proved here that when 0 less than or equal to m := lim inf(t-->infinity) Q(t)P(g(t)) less than or equal to 1/4 all solutions of this equation oscillate if the condition lim(t-->infinity) sup Q(t)P(g(t)) > (1 + root1 -4m/2)(2) (*) is satisfied. It should be emphasized that the condition (*) can not be improved in some sense.

LINEARIZED OSCILLATIONS FOR AUTONOMOUS DELAY DIFFERENTIAL EQUATIONS

The linearized oscillations of the nonlinear autonomous delay differential equation [GRAPHICS] are studied.

CONVERGENCE OF SOLUTIONS FOR RLC-NONLINEAR NETWORKS WITH TIME-VARYING ELEMENTS

This paper studies the asymptotic behavior of solutions for nonlinear RLCnetwoiks which have the following form The function p i,v,t),called tile mired potential function,call be used to construct Liapunov functions to prove the convergence of solutions under certain conditiolts. Under the assumption that every element value involving voltage source is asymptotically constallt, we establish four creteria for all solutiolls of such a system to converge to the set of equilibria of its limiting equations via LaSalle invariant principle.We also present two theorems on the existence of periodic solutions for periodically excited uonliltear circuits.This results generalize those of Brayton and Moser[1,2].

Oscillatory and Asymptotic Behavior for Second Order Quasilinear Difference Equations

This paper is concerned with the oscillatory (and nonoscillatory) behavior of solutions of second oder quasilinear difference equations of the type Some necessary and sufficient conditions are given for the equation to admit oscillatory and nonoscillatory solutions with special asymptotic properties. These results generalize and improve some known results.

OSCILLATION OF FIRST ORDER DELAY DIFFERENTIAL EQUATIONS WITH OSCILLATING COEFFICIENTS

In this paper,some sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formx′(t)+p(t)x(t-τ)=0are established,which improve and generalize some of the known results in the literature.

Linearized Oscillation of the First-Order Nonlinear Differential Equation

1 Introduction and Main Result Kulenovic, Ladas and Meimaridou obtained a result that the

Oscillation of High Order Neutral Delay Difference Equations with Continuous Arguments

In this paper, we study oscillation of solutions for a class of high order neutral delay difference equations with variable coefficients -τm [x（t） - c（t）x（t - τ）] = （-1）mp（t）x（t - σ）, t ≥ t0 〉 0. Some sufficient conditions are obtained for bounded oscillation of the solutions.

FORCED OSCILLATION FOR GENERAL DIFFERENTIAL EQUATIONS

New oscillation criteria for general differential equations of the form x^（n）（t）＋pn-1 （t）x^（n-1）（t）＋…＋P1（t）x′（t）＋p0（t）x（t）＋q1（t）x^μ （t） = q2 （t）x^λ （t）＋e（t） where λ,μ are the ratios of positive odd integers, 0 〈 μ 〈 1 and λ 〉 1 are established.

OSCILLATION OF A CLASS OF SECOND ORDER NEUTRAL DIFFERENCE EQUATIONS WITH CONTINUOUS ARGUMENTS

In this paper,we investigate oscillation of solutions to a class of second order neutral delay difference equations with continuous arguments. We then obtain some sufficient conditions for bounded oscillation of the solutions.

OSCILLATION THEOREMS FOR SECOND ORDER NEUTRAL DIFFERENCE EQUATIONS

The aim of this paper is to study the oscillation of second order neutral difference equations.Our results are based on the new comparison theorems,that reduce the problem of the oscillation of the second order equation to that of the first order equation.The comparison principles obtained essentially simplify the examination of the equations.