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 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.

New Oscillatory Theorems for Third-Order Nonlinear Delay Dynamic Equations on Time Scales

This paper is concerned with the oscillatory properties of the third-order nonlinear delay dynamic equations of the form??on time scales , where ?is a quotient of odd positive integers. Applying the inequality technique we present two new sufficient conditions which ensure that every solution of equations is oscillatory or converges to zero. The results obtained improve and complement some known results in the literature.

Oscillation Criteria for Third-order NonlinearNeutral Dynamic Equations on Time Scales

Many practical problems, such as those from electronic engineering, mechanicalengineering, ecological engineering, aerospace engineering and so on, need to bedescribed by dynamic equations on time scales, so it is important in theory andpractical significance to study these equations. In this paper, the oscillation andasymptotic behavior of third-order nonlinear neutral delay dynamic equations ontime scales are studied by using generalized Riccati transformation technique, integralaveraging methods and comparison theorems. The main purpose of this paperis to establish some new oscillation criteria for such dynamic equations. The newKamenev criteria and Philos criteria are given, and an example is considered toillustrate our main results.

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.

Nonlinear delay difference equations for housing dynamics assuming heterogeneous backward-looking expectations

China＇s first interest rate hike during the last decade, aiming to cool down the seemingly overheated real estate market, had aroused more caution on housing market. This paper aims to analyze the housing price dynamics after an unanticipated economic shock, which was believed to have similar properties with the backward-looking expecta- tion models. The analysis of the housing price dynamics is based on the cobweb model with a simple user cost affected demand and a stock-flow supply assumption. Several nth- order delay rational difference equations are set up to illustrate the properties of housing dynamics phenomena, such as the equilibrium or oscillations, overshoot or undershoot and convergent or divergent, for a kind of heterogeneous backward-looking expectation models. The results show that demand elasticity is less than supply elasticity is not a necessary condition for the occurrence of oscillation. The housing price dynamics will vary substantially with the heterogeneous backward-looking expectation assumption and some other endogenous factors.

Oscillation of Second-order Nonlinear Dynamic Equation on Time Scales

： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many knownresults for second order dynamic equations. Some examples are given to illustrate the main results of this article.

A NOTE ON THE JULIA SETS OF ENTIRE SOLUTIONS TO DELAY DIFFERENTIAL EQUATIONS

Let f be an entire solution of the Tumura-Clunie type non-linear delay differential equation.We mainly investigate the dynamical properties of Julia sets of f,and the lower bound estimates of the measure of related limiting directions is verified.

Oscillation of third-order functional dynamic equations on time scales

We consider the nonlinear functional dynamic equation （p（t）[（r（t）x^△（t））^△]γ）^△＋q（t）f（x（τ（t））） =0, for t≥t0,on a time scale T, where γ〉 0 is the quotient of odd positive integers, p, r, τ- and q are positive rd-continuous functions defined on the time scale 1F, and lirut→∞ τ（t） = ∞. The main aim of this paper is to establish some new sufficient conditions which guarantee that the equation has oscillatory solutions or the solutions tend to zero as →∞ τ. The main investigation depends on the Riccati substitution and the analysis of the associated Riccati dynamic inequality. Our results extend, complement and improve some previously obtained ones. In particular, the results provided substantial improvement for those obtained by Yu and Wang [J Comput Appl Math, 225 （2009）, 531-540]. Some examples illustrating the main results are given.

On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

In this paper, we study certain non-autonomous third order delay differential equations with continuous deviating argument and established sufficient conditions for the stability and boundedness of solutions of the equations. The conditions stated complement previously known results. Example is also given to illustrate the correctness and significance of the result obtained.

Oscillation of fourth-order delay dynamic equations

This paper is concerned with oscillatory behavior of a class of fourth-order delay dynamic equations on a time scale.In the general time scales case,four oscillation theorems are presented that can be used in cases where known results fail to apply.The results obtained can be applied to an equation which is referred to as Swift-Hohenberg delay equation on a time scale.These criteria improve a number of related contributions to the subject.Some illustrative examples are provided.

OSCILLATION OF THIRD-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

In this paper, we give some new criteria for the asymptotic behavior and oscillation of third-order delay differential equation. The oscillation of the studied equation is studied under two conditions, and our results improve some ones in D?urina et al.(2018). Some examples are given to illustrate the main results with Euler-type differential equations.

Oscillation Theorems for Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

By employing the generalized Riccati transformation technique,we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation [r(t)[y(t)+p(t)y(■(t))]~Δ]~Δ+q(t)f(y((δ(t)))=0 on a time scale■.The results improve some oscillation results for neutral delay dynamic equations and in the special case when■our results cover and improve the oscillation results for second- order neutral delay differential equations established by Li and Liu[Canad.J.Math.,48(1996), 871 886].When■,our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh[Comp.Math.Appl.,36(1998),123-132].When ■ ■our results are essentially new.Some examples illustrating our main results are given.

Pullback Dynamics of 2D Non-autonomous Navier-Stokes Equations with Klein-Voight Damping and Multi-delays

This paper is concerned with the pullback dynamics of 2D non-autonomous Navier-Stokes-Voigt equations with continuous and distributed delays on bounded domain. Under some regular assumptions on initial and delay data, the existence of evolutionary process and the family of pullback attractors for this fluid flow model with Klein-Voight damping are derived. The regular assumption of external force is less than [1].

Oscillation Criteria of Second Order Half Linear Delay Dynamic Equations on Time Scales

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation （r（t）g（x△（t）））△＋p（t）f（x（τ（t）））=0 on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=R, difference equations when T=N but can be applied on different types of time scales such as when T=N for q〉1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

STABILITY OF SOLUTIONS FOR CERTAIN THIRD-ORDER NONLINEAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

In this paper, we investigate stochastic asymptotic stability of the zero solution for certain third-order nonlinear stochastic delay differential equations by constructing Lyapunov functionals.

OSCILLATION CRITERIA FOR SECOND-ORDER NEUTRAL DELAY DYNAMIC EQUATIONS ON TIME SCALES

In this paper, we consider a class of second-order neutral delay dynamic equations on a time scale T. By means of Riccati transformation technique, we establish some new oscillation criteria in two different conditions. The obtained results enrich the well-known oscillation results for some dynamic equations.

Kamenev-type criteria for nonlinear damped dynamic equations

We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves related contributions to the subject. An example is provided to illustrate assumptions in our theorem are less restrictive.

EFFECTS OF TIME DELAYED VELOCITY FEEDBACKS ON SELF-SUSTAINED OSCILLATOR WITH EXCITATION

Both the primary resonant solutions and their bifurcations due to time delayed velocity feedbacks used in a self-sustained oscillator with excitation were further investigated. A model was proposed by adding linear and nonlinear time delayed feedbacks to a representative non-autonomous system( with external forcing). The stability condition of the linearized system at trivial equilibrium was discussed, which leads to a critical stability boundary where periodic solutions may occur. The main attention was focused on bifurcations from the primary resonant solutions. It is found that the stable primary resonant solution may appear periodically in the time delay. Meanwhile, the unstable regions for such solutions are also obtained, predicting the occurrence of quasi-periodic motions.

ON THE BOUNDEDNESS OF A CLASS OF NONLINEAR DYNAMIC EQUATIONS OF THE THIRD ORDER

In this paper, a modified nonlinear dynamic inequality on time scales is used to study the boundedness of a class of nonlinear third-order dynamic equations on time scales. These theorems contain as special cases results for dynamic differential equations, difference equations and q-difference equations.