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 FOR NONAUTONOMOUS NEUTRAL DYNAMIC DELAY EQUATIONS ON TIME SCALES

The article is concerned with oscillation of nonautonomous neutral dynamic delay equations on time scales. Sufficient conditions are established for the existence of bounded positive solutions and for oscillation of all solutions of this equation. Some results extend known results for difference equations when the time scale is the set Z^＋ of positive integers and for differential equations when the time scale is the set IR of real numbers.

LINEARIZED OSCILLATIONS OF DELAY DIFFERENTIAL EQUATIONS AND APPLICATIONS

The necessary and sufficient conditions for the oscillations of every solution of the nonlinear delay equation (t)+f(x(t-l))+g(x( t-k ))=0 are oblained.

SOME OSCILLATION CRITERIA FOR A CLASS OF HIGHER ORDER NONLINEAR DYNAMIC EQUATIONS WITH A DELAY ARGUMENT ON TIME SCALES

In this paper,we establish some oscillation criteria for higher order nonlinear delay dynamic equations of the form[rnφ(⋯r2(r1x^(Δ))^(Δ)⋯)^(Δ)]^(Δ)(t)+h(t)f(x(τ(t)))=0 on an arbitrary time scale T with supT=∞,where n≥2,φ(u)=|u|^(γ)sgn(u)forγ>0,ri(1≤i≤n)are positive rd-continuous functions and h∈C_(rd)(T,(0,∞)).The functionτ∈C_(rd)(T,T)satisfiesτ(t)≤t and lim_(t→∞)τ(t)=∞and f∈C(R,R).By using a generalized Riccati transformation,we give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.The obtained results are new for the corresponding higher order differential equations and difference equations.In the end,some applications and examples are provided to illustrate the importance of the main results.

Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays

In this paper, we study oscillatory properties of solutions for the nonlinear impulsive hyperbolic equations with several delays. We establish sufficient conditions for oscillation of all solutions.

LINEARIZED OSCILLATIONS FOR AUTONOMOUS DELAY DIFFERENTIAL EQUATIONS

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

RECENT DEVELOPMENTS IN THE OSCILLATION OF NEUTRAL DELAY DIFFERENCE EQUATIONS WITH SUMMABLE COEFFICIENTS

This paper surveys some new results on the oscillation of all solutions of the linear nentral delaydifference equation of the form △(yn - pnyn-k) + qnyn-l = 0, n = 0,1, 2…where { pn } and { qn } are twe real numbers sequences with qn≥0, and k and l are positive integers. These re-sults do not require the usual assumptionAlso, some interesting open problems on this topic am given.

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.

Time Delay Induced Oscillation: An Example on a Class of <i>n</i>Coupled Van Der Pol Oscillators Model with Delays

In this paper, a class of n coupled van der Pol oscillator model with delays is considered. By employing an analysis approach, some sufficient conditions to guarantee the existence of stability and oscillations for themodel are obtained. Examples are provided to demonstrate the results.

Boundedness and Oscillation of Third Order Neutral Differential Equations with Deviating Arguments

we consider the third-order neutral functional differential equations with deviating arguments. A new theorem is presented that improves a number of results reported in the literature. Examples are included to illustrate new results.

GENERIC OSCILLATIONS OF FIRST ORDER IMPULSIVE DELAY DIFFERENTIAL EQUATIONS

In this paper, we will study generic oscillation and generic nonoscillation of first order impulsive delay differential equations. A necessary and sufficient condition and some sufficient conditions are obtained for both phenomena based on the root of characteristic equation.

Oscillation of Third-order Delay Dynamic Equations on Time Scales

This paper is concerned with the oscillatory behavior of a class of third-order noonlinear variable delay neutral functional dynamic equations on time scale. By using the generalized Riccati transformation and inequality technique, we establish some new oscilla- tion criteria for the equations. Our results extend and improve some known results, but also unify the oscillation of third-order nonlinear variable delay functional differential equations and functional difference equations with a nonlinear neutral term. Some examples are given to illustrate the importance of our results.

Oscillation Theorem of Systems of Quasilinear Impulsive Delay Hyperbolic Equations

In this paper, oscillatory properties for solutions of the systems of certain quasilinear impulsive delay hyperbolic equations with nonlinear diffusion coefficient are investigated. A sufficient criterion for oscillations of such systems is obtained.

Oscillation of Nonlinear Impulsive Delay Hyperbolic Partial Differential Equations

In this paper,by making use of the calculous technique and some results of the impulsive differential inequality,oscillatory properties of the solutions of certain nonlinear impulsive delay hyperbolic partial differential equations with nonlinear diffusion coefficient are investigated.Sufficient conditions for oscillations of such equations are obtained.

Oscillation of Second-order Delay Differential Equation

By using some differential inequality, a second-order delay differential equation(r(t)x′(t))′ + p(t)x(q(t)) = 0has been investigated and some necessary condition for this equation has a nonoscillatorysolution and some sufficient condition which ensures that all of the solutions of the aboveequation are oscillatory are obtained.

Oscillation of Second Order Delay Differential Equations with Nonlinear Neutral Term

By using the averaging technique, we obtain new oscillation criteria for second order delay differential equation with nonlinear neutral term. These results generalize and improve some known results about neutral delay differential equation of second order.

NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATIONS OF NEUTRAL HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH DELAYS

This paper is concerned with the oscillations of neutral hyperbolic partial differential equations with delays. Necessary and sufficient, conditions are obtained for the oscillations of all solutions of the equations, and these results are illustrated by some examples.

Oscillation of Solutions for a Class of Nonlinear Neutral Partial Differential Equations with Continuous Distribution Delay

In this paper, some sufficient conditions are obtained for the oscillation of solutions for a class of second order nonlinear neutral partial differential equations with continuous distribution delay under Robin and Dirichlet＇s boundary value conditions.

Oscillation theorems of Higher Order Nonlinear Delay Differential Equations

This paper has given necessary and sufficient conditions for oscillation of a classof higher order nonlinear delay differential equations, and given some sufficientconditions or necessary conditions for oscillation of a forced delay differentialequation.

Oscillation in a Variable Delay Logistic Difference Equation

Consider the nonautonomous delay logistic equation △yn=pnyn（1-yn-ln/k）,n≥0, where {Pn}n≥0 is a sequence of nonnegative real numbers, {In}n≥0 is a sequence of positive integers satisfying n→∞lim（n-ln）=∞, and k is a positive constant. Only solutions which are positive for n ≥ 0 are considered. We obtain a new sufficient for all positive solutions of （1） to oscillate about k which contains the corresponding result in [2] when i = 1.