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.

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.

NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATIONS OF HYPERBOLIC AND ELLIPTIC TYPE PARTIAL DIFFERENCE EQUATIONS

This paper is concerned with hyperbolic type delay partial difference equation and elliptic type equation where a, b,p, qi are real numbers, hi and h are nonnegative integers, a, b,p ≠ 0 and not all of qi are zero for 1 ≤ i ≤ u. Sufficient and necessary conditions for all solutions of the equation mentioned above to be oscillatory are obtained.

Oscillation of Nonlinear Partial Difference Equations with Continuous Variables

这份报纸与连续变量和相应双方程涉及非线性的部分差别方程的摆动。几个足够的条件为这些方程的摆动被获得。

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 Delay Difference Equations

In this paper we study the Oscillatory behaviour of the second order delay differenceequation.（1）△（rn△An）+PnAn-k=0,n=n0,n0+1……where{Pn}（?）is a nonnegative Sequenceof real number,（?）is a positive sequence of real number with sum from n=n0 to +∞（1/rn）=+∞,K is a positive integer and △An=An+1-An we prove that each one of following conditions.imples that al solutions of Eq（1）oscillate,where Rn=sum from i=n0 to n（1/ri

Comparison theorems for the oscillation of difference equations with continuous arguments and applications

Zhang and Yan in ref. [1] gave some sufficient conditions for the oscillation of eqs.(1) and (2) by using the methods as in difference equations with discrete arguments, and thereby revealed certain oscillation relation between difference equations with continuous arguments and discrete ones. However, the oscillation results in ref. [1] need the hypothesis liminf p_i(t)】0, which is an essential condition.t→∞ In this note, we compare eq. (1) with certain delay differential equation, and thereby establish some new sufficient conditions for the oscillation of eqs. (1) and (2). These conditions are integral conditions which do not need the hypothesis liminf p_i(t)】0.

A NEW OSCILLATION CRITERION FOR FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION

In this paper, a new sufficient condition for the oscillation of all solutions of first order neutral delay differential equations is obtained. Secondly, the result can also be extended to a general neutral differential equation, and many known results in the literatures are improved.

A NECESSARY AND SUFFICIENT CONDITION FOR THE OSCILLATION-OF NEUTRAL EQUATIONS

The problem of oscillation of neutral delay differential equations is of theoretical as well as practical interest, and the oscillation theory of such equations has been extensively developed during the past few years. However, almost all the known results deal with the equations only with positive coefficients.

OSCILLATION THEOREMS FOR A NEUTRAL DELAY DIFFERENCE EQUATION

Consider the neutral delay difference equation △[p(t)x(t)-q(t)x(t-τ)] + r(t)x(t-δ(t)) =0 (*)where t ∈{a, a + 1, a + 2,…} and p(t) > 0, q(t) ≥ 0, r(t) ≥ 0, δ(t) ∈ {0, 1, 2,…} with,lim(t-δ(t)) =∞ and τ∈ { 1, 2, 3,…}. Note that the delay of the difference equation may vary, thus the equation may not be of constant order. We obtain some sufficient conditions for the oscillation of Equation (*) and the second order self-adjoint difference equation △[p(t-1)△y(t-1)]+r(t)y(t) = 0.And the work in Timothy Peil [5] is improved.AMS (1991) No. 39A10, 39A12

OSCILLATION THEOREM TO SYSTEMS OF IMPULSIVE NEUTRAL DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we study the oscillation of solutions to the systems of impulsive neutral delay parabolic partial differential equations. Under two different boundary conditions, we obtain some sufficient conditions for oscillation of all solutions to the systems.

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.

OSCILLATION OF NONLINEAR DELAY DIFFERENCE EQUATIONS

This paper deals with the oscillatory properties of a class of nonlinear difference equations with several delays. Sufficient criteria in the form of infinite sum for the equations to be oscillatory are obtained.

OSCILLATION OF SYSTEMS OF IMPULSIVE DELAY PARABOLIC EQUATIONS ABOUT BOUNDARY VALUE PROBLEMS

In this paper, oscillation of solutions to a class of impulsive delay parabolic partial differential equations system with higher order Laplace operator is studied. Under two different boundary value conditions, we establish some sufficient criteria with respect to the oscillations of such systems, employing first-order impulsive delay differential inequalities. The results fully reflect the influence action of impulsive and delay in oscillation.

OSCILLATION CRITERIA FOR SECOND ORDER NEUTRAL DIFFERENCE EQUATIONS

By using the Riccati transformation and the Cauchy mean inequality, we shall derive some oscillatory criteria for the second order neutral delay difference equationΔ [a nΔ (x n+p nx g(n) )]+q nf(x σ(n) )=0. These results generalize and improve some known results about both neutral and delay difference equations.

OSCILLATION CRITERIA FOR CERTAIN NONLINEAR DIFFERENCE EQUATIONS

In this paper, we present new oscillation criteria for certain class of second-order difference equation obtained by using a technique similar to the integral averaging technique. Our theorem complement some known oscillation criteria.

On a Delay Reaction-Diffusion Difference Equation of Neutral Type

In this paper, we first consider a delay difference equation of neutral type of the form: Δ(y_n+py_(n-k))+q_ny_(n-)=0 for n∈Z^+(0) (1*) and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241 248) to guarantee that every non-oscillatory solution of (1~*) with p=1 tends to zero as n→∞ Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form: Δ_1(u_(n,m)+pu_(n-k,m)+q_(n,m)u_(n-m)=a^2Δ_2~2u_(n+1,m-1) for (n,m)∈Z^+(0)×Ω. (2*) study various casks of p in the neutral term and obtain that if p≥-1 then every non-oscillatory solution of (2~*) tends uniformly in m∈Ω to zero as n→∞: if p=-1 then every solution of (2~*) oscillates and if p<-1 then every non-oscillatory solution of (2~*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses.

ASYMPTOTIC STABILITY OF THIRD ORDER DELAY DIFFERENCE EQUATIONS WITH THREE PARAMETERS

In this paper, we obtain a necessary and sufficient condition for the asymptotical stability of the zero solution to the third order delay difference equations.

THE ASYMPTOTIC STABILITY REGION TO A CLASS OF HIGHER ORDER DELAY LINEAR DIFFERENCE EQUATIONS

In this paper, we establish necessary and sufficient conditions for the zero solution to a class of higher order delay linear difference equations to be asymptotically stable, which are easy to be verified and to be applied.

OSCILLATION FOR A CLASS OF LOGISTIC EQUATIONS

The oscillation of a delay difference equationis studied and the necessary and sufficient condition for all positive solutions of (E) to oscillate about its positive equilibrium is obtained.