Global Attractivity of a Kind of Delay Difference Equations

Several new sufficient conditions are given for the global attractivity of solutions of a kind of delay difference equations. They either include or improve some known results and put the study of Ladas' conjecture forward.

Asymptotic Behavior and Nonexistence of Positive Solutions of Delay Difference Equations *L

Aim To obtain new criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations. Methods By means of Hlder inequality and a method of direct analysis, some interesting Lemmas were offered. Results and Conclusion New criteria for asymptotic behavior and nonexistence of positive solutions of nonlinear neutral delay difference equations are established, which extend and improve the results obtained in the literature. Some interesting examples illustrating the importance of our results are also included.

GLOBAL ASYMPTOTIC STABILITY FOR A NONLINEAR DELAY DIFFERENCE EQUATION

In this paper,a sufficient condition for the global asymptotic stability of the solutions of the following nonlinear delay difference equation is obtained, xn+ 1=xn+xn- 1xn- 2 +a xnxn- 1+xn- 2 +a, n =0 ,1 ,..., where a∈ [0 ,∞ ) and the initial values x- 2 ,x- 1,x0 ∈ (0 ,∞ ) .As a special case,a conjecture by Ladas is confirmed.

GLOBAL ATTRACTIVITY FOR A CLASS OF NONLINEARDELAY DIFFERENCE EQUATIONS

The global attractivity of the delay difference equation Deltax(n) + a(n)x(n) + f(n, Sigma(s = -k)(0)(q) over bar (s), (n) x(s + n)) = 0, which includes the discrete type of many mathematical ecological equations, was discussed. The sufficient conditions that guarantee every solution to converge to zero are obtained. Many known results are improved and generated.

BOUNDEDNESS AND PERSISTENCE AND GLOBAL ASYMPTOTIC STABILITY FOR A CLASS OF DELAY DIFFERENCE EQUATIONS WITH HIGHER ORDER

Some sufficient, conditions for boundedness and persistence and global asymptotic stability of solutions for a class of delay difference equations with higher order are obtained, which partly solve G. Ladas' two open problems and extend some known results.

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

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.

Global Attractivity in a Delay Difference Equation

A new sufficient condition is given for the global attractivity of solutions ofthe delay difference equation xn+1= xnf(xn, zn-1), n = 0,1…,As an application, ourresults partly confirm a conjecture of G. Ladas.

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

Linearized Oscillations in Nonlinear Delay Difference Equations

Consider the nonlinear delay difference equation xn+1-xn+sum j=1 to m pjfj（xn-kj）=0. We establish a linearized oscillation result of this equation,which is the extension of the result in the paper [1].

Two Positive Periodic Solutions of Delay Functional Difference System with Feedback Control

By using a fixed point theorem on a cone to investigate the existence of two positive periodic solutions for the following delay difference system with feedback control argument of the form {△x（n）=-b（n）x（n）＋f（n,x（n-τ1（n））,…,x（n-τm（n））,u（n-δ（n）））,△u（n）=-η（n）u（n）＋a（n）x（n-σ（n））,n∈Z.

Stability of Difference Systems with Finite Delay

In this paper, the authors establish some theorems that can ascertain the zero solutions of systemsx(n+1)=f(n,x n)(1)are uniformly stable,asymptotically stable or uniformly asymptotically stable. In the obtained theorems, ΔV is not required to be always negative, where ΔV(n,x n)≡V(n+1,x(n+1)) -V(n,x(n))=V(n+1,f(n,x n))-V(n,x(n)), especially, in Theorem 1, ΔV may be even positive, which greatly improve the known results and are more convenient to use.

GLOBAL ATTRACTIVITY IN A DELAY LOGISTIC DIFFERENCE EQUATION

This paper studies the global attractivity of the positive equilibrium 1 of the delay logistic difference equation Δ y n=p ny n(1-y τ(n) ), n=0,1,2,...,(*)where {p n} is a sequence of positive real numbers, {τ(n)} is a nondecreasing sequence of integers, τ(n)<n and lim n→∞τ(n)=∞ .It is proved that ifnj=τ(n)p j≤54 for sufficiently large n and ∞j=0p j=∞,then all positive solutions of Eq.(*) tend to 1 as n→∞ .The results improve the existing results in literature.

Existence of positive solutions in a delay logistic difference equation

The author studied the existence of positive solutions of the delay logistic difference equation Δ y n=p ny n(1-y τ(n) ), n =0,1,2,.... where { p n } is a sequence of positive real numbers, { τ(n) } is a nondecreasing sequence of integers, τ(n)<n and lim n →∞ τ(n) =∞. A sufficient condition for the existence of positive solutions of the equation was given.

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.

ON UNIFORM ASYMPTOTIC STABILITY OF INFINITE DELAY DIFFERENCE EQUATIONS

For the infinite delay difference equations of the general form, two new uniform asymptotic stability criteria are established in terms of the discrete Liapunov functionals.

BOUNDEDNESS OF INFINITE DELAY DIFFERENCE SYSTEMS IN TERMS OF TWO MEASURES

In this paper, by means of Liapunov functionals, we establish the criteria of uniform boundedness and uniform ultimate boundedness for infinite delay difference systems in terms of two measures. Moreover, an example is given to illustrate the obtained results.

MULTIPLE POSITIVE PERIODIC SOLUTIONS OF A CLASS OF DELAY DIFFERENCE EQUATIONS

Based on the Leggett-Williams fixed point theorem for a Banach space, we establish the existence of three positive periodic solutions for a class of delay difference equations.

A THEOREM OF UNIFORMLY ASYMPTOTIC STABILITY FOR DELAY DIFFERENCE SYSTEM

In this paper, we establish a criterion of unformly asymptotic stability for finite delay difference systems in terms of two measures by employing Lyapunov functionals method.

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.