THEORY AND APPLICATIONS OF MONOTONE SEMIFLOWS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

The theory of monotone semiflows has been widely applied to functional differential equations (FDEs). The studies on the theory and applications of monotone semiflows for FDEs are very important and interesting. A brief des-cription of our recent works are as follows.By using general monotone semiflow theory, several results of positively invariant sets, monotone solutions and contracting rectangles of retarded functional differential equations(RFDEs) with infinite delay are gained under the assumption of quasimonotonicity; sufficient conditions for the existence, un-iqueness and global attractivity of periodic solutions are also established by combining the theory of monotone semiflows for neutral functional differential equations(NFDEs) and Krasnoselskii's fixed point theorem.     

Existence and global attractivity of unique positive almost periodic solution for a model of hematopoiesis

Sufficient conditions are obtained which guarantee the uniform persistence and global attractivity of solutions for the model of hematopoiesis. Then some criteria are established for the existence, uniqueness and global attractivity of almost periodic solutions of almost periodic system.     

ATTRACTIVE BEHAVIOR OF SOLUTIONS OF A NONLINEAR HIGHER ORDER DIFFERENCE EQUATIONS

In this paper, we study the global attractivity of solutions of higher order nonlinear difference equation xn+1=f(xn+1-1,xn+1-2,…,xn,xn-1,)n=0,1,2,…, where f is a continuous, positive, decreasing function on [0,+∞)1+1\{(0,…,0)}. We extend some known global behavior of the solutions of nonlinear higher order difference equations.     

EXISTENCE OF POSITIVE SOLUTIONS AND 1／c-GLOBAL ATTRACTIVITY OF A NONLINEAR DELAY DIFFERENCE EQUATION WITH VARIABLE COEFFICIENTS

Abstract. The existence of positive solutions and the global attractivity of the difference equa-tion     

DYNAMIC BEHAVIORS OF A NONAUTONOMOUS RATIO-DEPENDENT LESLIE SYSTEM INCORPORATING A PREY REFUGE

A nonautonomous ratio-dependent Leslie system incorporating a prey refuge is studied in this paper. By applying the comparison theorem of diferential equations and constructing a suitable Lyapunov function, a set of sufcient conditions which guarantee the persistent property and global attractivity of the system is obtained. Also, by applying the comparison theorem of diferential equations and Fluctuation Lemma, a set of sufcient conditions which ensure the extinction of the prey species and the global attractivity of predator species is obtained. This result shows that for the Lotka-Volterra type predator-prey system, when the value of prey refuge increases, predator species will be driven to extinction due to the lack of food. Our study shows that the alternative food resource predator species is always permanent, which means that prey refuge has no infuence on the permanence of predator species. However, refuge plays an important role in the persistent property of the prey species: large enough prey refuge could keep the persistent property of the prey species, while small enough refuge could lead to the extinction of prey species. Numerical simulations show the feasibility of the main results.     

GLOBAL ATTRACTIVITY OF THE HOPFIELD NEURAL NETWORKS

1 IntroductionHopfield neural networks is an associative memory model, it is also a neuraloptimization networks. This model has been studied extensively (see [1--12]).But in most of the literature, the signal function gi(x) is differentiable and isa sigmoid curve, and the sufficient conditions about the global attractivity ofthe Hopfield neural networks have no relevance to the delay. The aim of thispaper is to give a delayed sufficient condition of the global attractivity of theHopfield neur…     

GLOBAL ATTRACTIVITY OF A DELAYED RATIO-DEPENDENT COOPERATIVE SYSTEM WITH STAGE STRUCTURE

An autonomous stage-structured ratio-dependent cooperative system, which was proposed by Muhammadhaji, Teng and Abdurahman, is revisited in this paper. By introducing a new lemma and using the iterative method, a set of sufficient conditions which guarantee the global attractivity of the positive equilibrium is obtained. It is shown that the conditions which guarantee the permanence of the system are enough to ensure the global attractivity of the system. Our result not only complements but also supplements one of the main results of Muhammadhaji, Teng and Abdurahman （Permanence and extinction analysis for a delayed ratio-dependent cooperative system with stage structure, Afr. Mat.（2013）DOI 10.1007/s13370-013-0162-6）.     

Asymptotic Behavior on a Competition Model Arising from an Unstirred Chemostat

This paper is concerned with the unstirred chemostat model with two-species and one non-reproducingresource.The global attractivity of the positive steady-state solutions of the original system is established.Moreover,the effects of the growth rate on the unique positive equilibrium of the single population model arestudied.     

正负系数中立型差分方程的全局吸引性

A neutral difference equation with positive and negative coefficients△(xn-cnxn-k) pnxn-l-qnxn-r=0,n=0,1,2.…，is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.     

Attractivity in a Delayed Three—species Ratio—dependent Predator—prey System without Dominating Instantaneous Negative Feedback

A delayed three-species ratio-dependent predator-prey food-chain model without dominating in-stantaneous negative feedback is investigated. It is shown that the system is permanent under some appropriate conditions, and sufficient conditions are derived for the global attractivity of the positive equilibrium of the system.     

高阶非线性差分方程的导数收敛及其应用

首先研究高阶线性差分方程的整体收敛性，并证明了高阶非线性差分方程各阶导数的整体收敛；进而得到了关于高阶非线性差分方程整体收敛的一个定理，最后利用这个定理部分解决了Ladas提出的一个猜测。     

Stability and attractivity of periodic solutions of parabolic systems with time delays

This paper is concerned with the existence, stability, and global attractivity of time-periodic solutions for a class of coupled parabolic equations in a bounded domain. The problem under consideration includes coupled system of parabolic and ordinary differential equations, and time delays may appear in the nonlinear reaction functions. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement using Schauder fixed point theorem, while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme. The results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system. Applications are also given to three Lotka-Volterra reaction diffusion model problems, and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions.     

具有可变时滞的非线性非自治差分方程的线性化全局渐近稳定性及其应用

证明了在一定条件下,具有可变时滞的非线性非自治差分方程的全局渐近稳定性可由某种线性差分方程的渐近稳定性确定,给出了这类差分方程全局渐近稳定的充分条件.作为实例,获得了具有可变时滞的离散型非自治广义Log istic方程的全局吸收性判别准则.     

Global attractivity in monotone and subhomogeneous almost periodic systems

A global attractivity theorem is first proved for a class of skew-product semiflows. Then this result is applied to monotone and subhomogeneous almost periodic reaction-diffusion equations, ordinary differential systems and delay differential equations for their global dynamics.     

一类时滞反应扩散方程的稳态解的全局吸引性

本文研究了一类带扩散项造血生物模型当非线性项不具有单调性时,Dirichlet边值条件下稳态解的全局吸引性.     

Asymptotic behavior of recursions via fixed point theory

In this paper we provide a formulation of initial value problems for (explicit and implicit) difference equations in terms of abstract equations in sequence spaces. They will be solved using appropriate fixed point theorems and we obtain quantitative attractivity properties.     

Global attractivity of a family of nonautonomous max-type difference equations

This paper studies a family of nonautonomous max-type difference equations with several delays. Using some transformations we prove global attractivity of positive solutions to these equations under some conditions. Some of our results considerably extend related ones in the literature. Two examples are given to illustrate the main results.     

Stability of attractivity regions for autonomous functional differential equations

We consider attractivity regions of the zero solutions of autonomous retarded functional differential equations (FDEs). It is shown that the attractivity region remains unchanged if the FDE undergoes a small perturbation in certain classes of FDEs. Especially we obtain the openness of the set of parameters α∈(0,π/2) such that the attractivity region of the equation \(\dot x(t) = - \alpha x(t - 1)[1 + x(t)]\) is given by the set of continuous functions ?:[?1,0] → R with ?(0)>?1.     

一类时滞差分方程的全局吸引性

本文得到一类时滞差分方程解的全局吸引性的几个新的充分条件，包含或改进了一些已知结果，把G.Ladas的一个猜想的研究向前推进了。     

中立型泛函微分方程的周期解

对于中立型泛函微分方程,证明了解的毕竟有界性蕴含周期解的存在性,把常微分方程中著名的Yoshizawa周期解存在定理推广到中立型泛函微分方程,然后利用所得结果给出一类产生于电力系统的中立型时滞泛函微分方程周期解存在惟一与吸引的条件。