具有时滞的细胞神经网络周期解与稳定性

本文研究了具有时滞的细胞神经网络周期解存在性和平凡解的稳定性问题。利用Ｌｙａｐｕｎｏｖ函数法并结合不等式分析技巧,我们首先证明了时滞细胞网络的解是有界的,然后建立了时滞细胞神经物周期解的存在准则,最后在时滞细胞神经网络有平衡点时,给出了神经网络系统的平衡点指数稳定的充分条件。其结果推广了文「７,８」的相应结果。     

具有时滞的周期Lotka-Volterra型系统的全局渐近稳定性

考虑一般具有时间依赖时滞和连续分布时滞的Ｎ－种群周期Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ型系统。通过使用Ｌｉａｐｕｎｏｖ函数方法得到了关于正周期解的存在性和全局渐近稳定性的充分条件。这些条件改进和推广了最近被Ｗａｎｇ,Ｃｈｅｎ,Ｌｕ「２」和Ａｈｌｉｐ,Ｋｉｎｇ「４」得到的相应结果。     

一类非线性时滞差分方程解的若干性质

本文研究了一类非线性时滞差分方程解的有界持久性、全局渐近稳定性以及无界解的存在性,所得结果解决及部分解决了文献「４」提出的两个ｏｐｅｎ问题,包含了文献「６」中相应的结果。     

一类n维非自治系统的周期解

本文应用Ｓｃｈａｕｄｅｒ和Ｒｏｔｈ不动点定理，给出了周期系统存在周期解的一组充分性条件，推广了文「１」「２」的结果。     

Banach空间中—u‘’∈F（t,u（t）,u（t））的解

本文给出了弱序列完备Ｂａｎａｃｈ空间中二阶微分包含的解的存在性定理，推广了「１」中的结果到集值情形，同时由于利用已有的定理「２」，简化了类似「１」宫的证明。     

正定型超曲面上的闭曲线与非线性振荡的周期

本文在Ｒ＾ｎ＋１中定义了一类超曲面并讨论了其上闭测地线上度的下界以及Ｈａｍｉｌｔｏｎ系统的周期解的最小周期，这类超曲面比星形超曲面广泛，因而本文结论包括了文献「１」、「２」、「３」中的相应结论。     

高阶非线性立型微分方程的周期解

利用ｋ－集压缩延拓理论,研究了一类高阶非线性中型微分方程周期解的存在性,推广了文「１」,「２」的相应结果。     

广义Lienard方程非平凡周期解的存在性

本文给出了广义Ｌｉｅｎａｒｄ方程ｘ＋ｆ（ｘ）（ｘ）ｘ＋ｇ（ｘ）＝０存在非平凡周期解的两个充分条件,推广了文「４,５」中的结果,并且指出文「１」中的一个疏漏。     

一类中立型泛函微分方程周期解问题

本文利用Ｆｏｕｒｉｅｒ级数理论和不动点原理研究了一类中立型泛函微分方程（ＮＦＤＥＳ）：（ｄ／ｄｔ）∫「ｄＤ（ｓ）」ｘ（ｔ＋ｓ）＝∫ｒ「ＤＬ（ｓ）」ｘ（ｔ＋ｓ）＋Ｎｘ（ｔ）的周期解问题,得到了解的存在性和唯一性的新结果,所得结果推广和改进了文「５」的工作。     

一类时滞种群模型的周期正解

利用一些分析技巧和重合度理论,研究了一类时滞种群模型：Ｎ’（ｔ）＝Ｎ（ｔ）「ｂ（ｔ）－∞／∑／ｊ＝１ａｊｌｎＮ（ｔ＝ｒｊ）」周期正解的存在性,得到了一些新的判据,同时也改进了一些相关文献的结果。     

一类时滞Logistic方程正周期解的存在性

In the present paper,we consider the existence of positive periodic solu-tions for a kind of delay Logistic equations.By using a fixed point theorem in cones,we give some new existence results of single and multiple positive periodic solutionsfor a kind of delay Logistic equations.Some biomathematical models are presentedto illustrate our results.     

Existence of positive periodic solutions of nonlinear neutral differential systems with variable delays

In this work we use some mixed techniques of the Mawhin coincidence degree theory and fixed point theorem to prove the existence of positive periodic solutions of delay systems. As a consequence, we offer existence criteria and sufficient conditions for existence of periodic solutions to the systems with feedback control. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved.     

Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state‐dependent delay

In this paper, we study the dynamics of a Nicholson's blowflies equation with state‐dependent delay. For the constant delay, it is known that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and global existence of periodic solutions has been established. Here, we consider the state‐dependent delay instead of the constant delay and generalize the results on the existence of slowly oscillating periodic solutions under a set of mild conditions on the parameters and the delay function. In particular, when the positive equilibrium gets unstable, a global unstable manifold connects the positive equilibrium to a slowly oscillating periodic orbit. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence of Global and Periodic Solutions for Delay Equation with Quasilinear Perturbation

Delay parabolic problems have been studied by many authors. Some authors investigated more general delay problem (refer to [1], [2]), some investigated concrete delay partial differential equations. Recently, we have done some work on delay parabolic problem. We discussed semilinear parabolic delay problem and obtained some results on the existence of solutions. In particular the results on existence of periodic solutions are characteristic (see [3], [4], [5], [6]). The purpose of this paper is to study delay equation with quasilinear perturbation. We present the existence of global and periodic solutions of abstract evolution equations in Section 2. The abstract results are used to obtain the existence of global and periodic solutions of delay parabolic problem with quasilinear perturbation in Section 3. We make preparation for our investigation and give a generalization of Gronwall inequality (Lemma 1.3) which is used in next section.     

Periodic solutions of abstract functional differential equations with state‐dependent delay

In this paper, we are concerned with the existence of solutions of systems determined by abstract functional differential equations with infinite and state‐dependent delay. We establish the existence of mild solutions and the existence of periodic solutions. Our results are based on local Lipschitz conditions of the involved functions. We apply our results to study the existence of periodic solutions of a partial differential equation with infinite and state‐dependent delay. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamical analysis of toxin producing Phytoplankton–Zooplankton interactions

In the present paper we consider a toxin producing phytoplankton–zooplankton model in which the toxin liberation by phytoplankton species follows a discrete time variation. Firstly we consider the elementary dynamical properties of the toxic-phytoplankton–zooplankton interacting model system in absence of time delay. Then we establish the existence of local Hopf-bifurcation as the time delay crosses a threshold value and also prove the existence of stability switching phenomena. Explicit results are derived for stability and direction of the bifurcating periodic orbit by using normal form theory and center manifold arguments. Global existence of periodic orbits is also established by using a global Hopf-bifurcation theorem. Finally, the basic outcomes are mentioned along with numerical results to provide some support to the analytical findings.     

退化时滞微分方程的周期解问题

本文讨论退化时滞微分方程的周期解问题．特别地，我们给出二维退化滞后微分方程的周期解的存在性问题，并在最后举例说明其应用。     

Positive periodic solution for a neutral delay competitive system

In this paper we study the existence of positive periodic solutions for a class of neutral type competitive system with delay. Applying the continuation theorem of coincidence degree theory, a general criteria on the existence of positive periodic solutions are established. Some the related results are generalized and improved. It is shown that the arguments delays have no effect on the existence positive periodic solution of system.     

Global dynamics behaviors for a nonautonomous Lotka–Volterra almost periodic dispersal system with delays

A nonautonomous Lotka–Volterra dispersal system with continuous delays and discrete delays is considered. By using a comparison theorem and delay differential equation basic theory, we obtain sufficient conditions for the permanence of the population in every patch. By constructing a suitable Lyapunov functional, we prove that the system is globally asymptotically stable under some appropriate conditions. Using almost periodic functional hull theory, we get sufficient conditions for the existence, uniqueness and globally asymptotical stability for an almost periodic solution. This implies that the population in every patch exhibits stable almost periodic fluctuation. Furthermore, the results show that the permanence and global stability of system, and the existence and uniqueness of a positive almost periodic solution, depend on the delay; then we call it “profitless”.     

On the existence,uniqueness and stability of periodic solutions of large scale systems with unbounded delay

In this paper, we consider the periodic solution problems for the systems with unbounded delay, and the existence, uniqueness and stability of the periodic solution are dealt with unitedly. First we establish the suitable delay-differential inequality, then study seperately the problems of periodic solution for the systems with bounded delay, with unbounded delay and the Volterra integral-differential systems with infinite delay by using the character of matrix measure and the asymptotic fixed point theorem of poincaré's periodic operator in the dfferent phase spaces. A series of simple criteria for the existence, uniqueness and stability of these systems are obtained.