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.

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.

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.

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.

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.

Surface crack imaging based on delayed temperature difference at symmetric points by laser spot thermography

Laser spot thermography is a novel technique for the detection of surface cracks with a laser to heat sample locally and with an IR camera to record the surface temperature distribution. Common methods to characterize cracks are only suitable for the situation that the laser scanning path is vertical to the crack. But due to the randomness of cracks,when the scanning path is parallel to the crack,surface cracks cannot be detected by these methods. To tackle this problem,a method is presented which is suitable for the situation that the scanning path is parallel to crack. The main idea is to evaluate the crack-caused asymmetries of the surface temperature distribution. The effect of temperature gradient and the maximum scanning interval are analyzed by a 2D simulation. A new crack imaging technique is presented that is based on delayed temperature difference at symmetric points to characterize the crack in the thermal image. Compared well with those obtained by the spatial first derivative method,experimental results are shown to efficiently prove this method.

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.

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 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

Study on the influence of side-blown airflow velocities on plasma and combustion wave generated from fused silica induced by combined pulse laser

This study examines the impact of variations in side-blowing airflow velocity on plasma generation,combustion wave propagation mechanisms,and surface damage in fused silica induced by a combined millisecond-nanosecond pulsed laser.The airflow rate and pulse delay are the main experimental variables.The evolution of plasma motion was recorded using ultrafast time-resolved optical shadowing.The experimental results demonstrate that the expansion velocities of the plasma and combustion wave are influenced differently by the sideblowing airflow at different airflow rates(0.2 Ma,0.4 Ma,and 0.6 Ma).As the flow rate of the sideblow air stream increases,the initial expansion velocities of the plasma and combustion wave gradually decrease,and the side-blow air stream increasingly suppresses the plasma.It is important to note that the target vapor is always formed and ionized into plasma during the combined pulse laser action.Therefore,the side-blown airflow alone cannot completely clear the plasma.Depending on the delay conditions,the pressure of the side-blowing airflow,the influence of inverse Bremsstrahlung radiation absorption and target surface absorption mechanisms can lead to a phenomenon known as the double combustion waves when using a nanosecond pulse laser.Both simulation and experimental results are consistent,indicating the potential for further exploration of fused silica targets in the laser field.

渔业资源的评估方法和模型研究进展

渔业资源评估模型主要有产量模型(production models)、Delay-difference模型、消耗模型(depletion models)和年龄结构模型(age-structured models)。产量模型在渔业资源评估中应用很广泛,但不能精确反映统计误差;Delay-difference模型增加了生物参数、并考虑生物过程中的时间延迟,它是扩充了的产量模型,但当可获得的数据不足时,不能精确估算模型中的参数,甚至可能产生错误的估算值;此时,消耗模型相对于Delay-difference模型是有效的,该模型研究渔获量如何影响残存鱼的相对丰度;年龄结构模型能更好地显示种群的动态变化,能为配置合理的捕捞强度提供科学的依据,但需要很多数据,尤其要求精确的渔获物年龄数据。我国渔业资源的评估多采用产量模型和年龄结构模型,对于Delay-difference模型和消耗模型的应用甚为少见。

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].

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.