概周期解的存在性、唯一性与稳定性

本文给出了一些保证微分方程的周期解和概周期解的存在性、唯一性、稳定性与不稳定性的充分性条件及周期解的存在范围估计式.所得结果推广[1]的主要结果及[2-6]的有关结果.     

Positive solutions of a nonlinear impulsive equation with periodic boundary conditions

In this paper, we investigate nonlinear second order differential equations subject to linear impulse conditions and periodic boundary conditions. Sign properties of an associated Green’s function are exploited to get existence results for positive solutions of the nonlinear boundary value problem with impulse. Upper and lower bounds for positive solutions are also given. The results obtained yield periodic positive solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.     

The existence of periodic solutions of higher order nonlinear periodic difference equations

Sufficient conditions for the existence of at least one periodic solution of two classes of nonlinear higher order periodic difference equations are established, respectively. The results show us that sufficient conditions for the existence of T ? periodic solutions of difference equation are different from those ones for the existence of T ? periodic solutions of differential equation. Copyright © 2012 John Wiley & Sons, Ltd.     

The necessary and sufficient conditions for the existence of periodic orbits in a Lotka-Volterra system

By extending Darboux method to three dimension, we present necessary and sufficient conditions for the existence of periodic orbits in three species Lotka-Volterra systems with the same intrinsic growth rates. Therefore, all the published sufficient or necessary conditions for the existence of periodic orbits of the system are included in our results. Furthermore, we prove the stability of periodic orbits. Hopf bifurcation is shown for the emergence of periodic orbits and new phenomenon is presented: at critical values, each equilibrium are surrounded by either equilibria or periodic orbits.     

On the solutions of a second order nonlinear system with almost periodic forcing

A second order almost periodic perturbed nonlinear system with small parameter is discussed in this paper. Based on some analytical techniques and the method of averaging, some sufficient conditions are obtained for existence of one almost periodic solution. Also some suitable conditions are given for existence of two almost periodic solutions. The obtained new results generalized the known results in Seifert [1], [3] and He [2], [4]. Finally, some applications are presented to illustrate that our results are a good generalization of the known results.     

New results of existence and stability of periodic solution for a delay multispecies Logarithmic population model

In this Letter, for a general class of delayed periodic multispecies Logarithmic population model, we prove some new results on the existence of positive periodic solutions by contraction principle. The global exponential stability of positive periodic solutions is discussed further, and conditions for exponential convergence are given. The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases.     

Periodic solutions of Weyl fractional order integral systems†

In this paper, we show the existence and uniqueness results for periodic solutions of Weyl fractional order integral systems. A numerical example is given to illustrate our theoretical results. Our results show that periodic orbits can be obtained by putting the periodic conditions to some certain fractional order integral systems. Copyright © 2016 John Wiley & Sons, Ltd.     

On a theorem of Favard

We obtain sufficient conditions for the existence of almost periodic solutions of almost periodic linear differential equations thereby extending Favard's classical theorem. These results are meant to complement previous results of the authors who have shown by means of a counterexample that the boundedness of all solutions is not, by itself, sufficient to guarantee the existence of an almost periodic solution to a linear almost periodic differential equation.

     

Dynamics of a periodic Watt-type predator–prey system with impulsive effect

In this paper, an impulsive periodic predator–prey system with Watt-type functional response is investigated. By using the Floquet theory of linear periodic impulsive equation, the stability conditions for the prey-eradication positive periodic solution are given, and the boundedness of the system is proved. By the method of coincidence degree, the sufficient conditions for the existence of at least one strictly positive periodic solution are obtained. Furthermore, we give numerical analysis to confirm our theoretical results. It will be useful for ecosystem control.     

Second order nonlinear differential equations with linear impulse and periodic boundary conditions

In this study, we establish existence and uniqueness theorems for solutions of second order nonlinear differential equations on a finite interval subject to linear impulse conditions and periodic boundary conditions. The results obtained yield periodic solutions of the corresponding periodic impulsive nonlinear differential equation on the whole real axis.     

Almost periodic solutions for undamped nonhomogeneous delay-differential equations

We first establish a result giving conditions that certain undamped delay differential equations with almost periodic time dependence have unique almost periodic solutions. Using this result we obtain conditions that a second order scalar nonlinear delay differential equation with almost periodic forcing will have a unique almost periodic solution having saddle-type stability properties. These results use the method of averaging.

     

Existence of solutions for sub-linear or super-linear operator equations

We investigate solutions to superlinear or sublinear operator equations and obtain some abstract existence results by minimax methods. These results apply to superlinear or sublinear Hamiltonian systems satisfying several boundary value conditions including Sturm-Liouville boundary value conditions and generalized periodic boundary value conditions, and yield some new theorems concerning existence of solutions or nontrivial solutions. In particular, some famous results about periodic solutions to superlinear or sublinear Hamiltonian systems by Rabinowitz or Benci and Rabinowitz are special cases of the theorems.     

Periodic jordan rings and order structure

The aim of this paper is to obtain information about a periodic Jordan ring by using only properties of its idempotent elements. Osborn proves that a power-associative periodic ring having only one nonzero idempotent element is a division ring. so associative. He also proves that a periodic Jordan ring is a subdirect product of simple periodic Jordan rings and that a simple periodic Jordan ring is either a periodic field or a Jordan ring of capacity 2. Using these results we obtain some necessary and suficient conditions for a periodic Jordan ring to be associative, and these conditions are only given in terms of the idempotent elements. We also characterize the periodic Jordan ring which are a direct product of periodic fields and simple periodic Jordan rings of capacity two.     

Almost periodic solutions of discrete Volterra equations

For discrete Volterra equations with or without delay, we obtain several results concerning almost periodic solutions and asymptotically almost periodic solutions under certain conditions. We also investigate the relations among solutions of equations discussed and give an example to illustrate our results.     

Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations

In this paper, we investigate a classical periodic Lotka–Volterra competing system with impulsive perturbations. The conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are given by applying Floquet theory of linear periodic impulsive equation, and we also give the conditions for the global stability of these solutions as a consequence of some abstract monotone iterative schemes introduced in this paper, which will be also used to get some sufficient conditions for persistence. By using the method of coincidence degree, the conditions for the existence of at least one strictly positive (componentwise) periodic solution are derived. The theoretical results are confirmed by a specific example and numerical simulations. It shows that the dynamic behaviors of the system we consider are quite different from the corresponding system without pulses.     

具有强迫项Lienard类型方程周期解及概周期解的存在性

本文利用指数型二分性理论,给出具有强迫项的Ｌｉｅｎａｒｄ类型方程周期解及概周期解存在的充分条件,这些条件是由ＦｉｎｋＡ．Ｍ．,林发兴在文［１,２］对Ｌｉｅｎａｒｄ方程关于同样问题所获结果的自然扩展与推广     

Pseudo almost periodic solutions for a class of nonlinear Duffing system with a deviating argument

This paper is concerned with a nonlinear Duffing system with pseudo almost periodic coefficients and delay. Under proper conditions, by using theory of exponential dichotomies and contraction mapping principle, some sufficient conditions are established to ensure the existence and uniqueness of pseudo almost periodic solutions, which improve and extend some known relevant results. Moreover, an example is given to illustrate the theoretical results.     

Permanence,extinction and periodic solution of the predator–prey system with Beddington–DeAngelis functional response and stage structure for prey

In this paper, we study the permanence, extinction and periodic solution of the periodic predator–prey system with Beddington–DeAngelis functional response and stage structure for prey. A set of sufficient and necessary conditions which guarantee the predator and prey species to be permanent are obtained. In addition, sufficient conditions are derived for the existence of positive periodic solutions to the system. Numeric simulations show the feasibility of the main results.     

含参数泛函微分方程概周期正解的存在性

研究了一类含参数泛函微分方程概周期正解的存在性问题.结合有界性及渐近概周期性获得了系统存在概周期正解的几组充分条件,并将结果应用于几类种群动力学模型,分别获得了系统在概周期环境下存在概周期解的一组充分条件.     

On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions

We prove the existence of positive solutions of second-order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions. Obtained results yield positive periodic solutions of the equation on the whole real axis, provided that the coefficients are periodic.