Asymptotic equivalence of alternately advanced and delayed differential systems with piecewise constant generalized arguments

In this paper, we investigate the existence, uniqueness and the asymptotic equivalence of a linear system and a perturbed system of differential equations with piecewise alternately advanced and retarded argument of generalized type (DEPCAG). This is based in the study of an equivalent integral equation with Cauchy and Green matrices type and in a solution of a DEPCAG integral inequality of Gronwall type. Several examples are also given to show the feasibility of results.     

Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments

We study scalar advanced and delayed differential equations with piecewise constant generalized arguments, in short DEPCAG of mixed type, that is, the arguments are general step functions. It is shown that the argument deviation generates, under certain conditions, oscillations of the solutions, which is an impossible phenomenon for the corresponding equation without the argument deviations. Criteria for existence of periodic solutions of such equations are discussed. New criteria extend and improve related results reported in the literature. The efficiency of our criteria is illustrated via several numerical examples and simulations.     

Stability of oscillatory solutions of differential equations with general piecewise constant arguments of mixed type

We examine scalar differential equations with general piecewise constant arguments of mixed type, in short DEPCAG of mixed type, that is, the arguments are general step functions. Criteria of existence of the oscillatory and nonoscillatory solutions of such equations are proposed. Necessary and sufficient conditions for stability of the zero solution are obtained. Our results are new, extend and improve earlier publications. Several numerical examples and simulations are also given to show the feasibility of our results.     

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.     

YoshiZawa型周期解定理和Massera型周期解定理研究进展简介

微分方程解的有界性和周期解的存在性是檄分方程理论研究中的两个重要课题，二者之间有着紧密的联系．在解的有界性与周期解的存在性的研究中，Yoshizawa周期解定理和Massera周期解定理是非常重要的结果，具有重要的理论意义和应用价值．本文以Yoshizawa型周期解定理和Massera型周期解定理的研究为主，简要介绍泛函微分方程周期解理论研究方面的一些新进展。     

具扰动项的时滞Logistic方程的周期解

本文首先讨论一类时滞微分方程的周期解的存在性，然后将本文结果应用到具扰动项的 时滞Ｌｏｇｉｓｔｉｃ方程得出其正周期解的存在性，本文结果推广和改进了「１」和「６」的相应结果。     

Positive periodic solutions of competitive kolmogorov diffusion systems with interference constants

In this paper, we consider the positive periodic solutions of multispecies patches connected by discrete diffusion with a within-patch dynamics of periodic Kolmogorov type. A new criterion for existence and globally asymptotic stability of positive periodic solution is established. The results in [1–4] are improved and extended.     

Linearly implicit Liénard systems

The presence of nonlinearities in the capacitance and the inductance in van der Pol type electrical circuits defines a linearly implicit (or quasilinear) counterpart of the classical Liénard systems. When the reactances remain positive, the existence of a unique attracting periodic solution follows, with minor modifications, as in the classical setting. Novel results are obtained when the values of reactances may vanish at certain points of the state space; these points yield singularities of the model, and the existence of an attracting periodic solution can be characterized in terms of the behavior of certain smooth solutions crossing the singular manifold through so-called I-singularities.     

Existence and uniqueness of periodic solution for forced Rayleigh type equations

By using the coincidence degree theory and some analysis skill, some new sufficient conditions for the existence and uniqueness of periodic solution for a kind of forced Rayleigh type equation are established, which are complement of previously known results.     

An impulsive periodic predator-prey Lotka–Volterra type dispersal system with mixed functional responses

An impulsive two species periodic predator-prey Lotka–Volterra type dispersal system with mixed functional responses is presented and studied in this paper. Conditions for the permanence and extinction of the predator-prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results.     

Existence and global attractivity of a positive periodic solution to a Lotka–Volterra model with mutual interference and Holling III type functional response

In this paper, a Volterra model with mutual interference and Holling III type functional response is investigated, some sufficient conditions which guarantee the existence and global attractivity of a positive periodic solution for the system are obtained and two examples are given to verify the results by using MatLab. Furthermore, our results improve the main results in the literature.     

Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

This paper is concerned with a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects. By using the fixed point theory based on monotone operator, some simple conditions are obtained for the existence of at least one positive periodic solution of the model. The existence result of this paper implies that the functional response on prey does not influence the existence of positive periodic solution of the model, which completes some results given in recent years. Further, by applying the comparison theorem in impulsive differential equations and constructing a suitable Lyapunov functional, the permanence and global attractivity of the model are also investigated. The main results in this paper extend, complement, and improve the previously known result. And some examples and numerical simulations are given to illustrate the feasibility and effectiveness of the main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Banach空间中常微分方程的周期解

本文研究Ｂａｎａｃｈ空间中常微分方程的周期解的存在性，在耗散型条件下得到一系列结果。同时，还给出一无穷维微分方程存在周期解的例子。     

Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independent

In this paper, we study two species time-delayed predator-prey Lotka-Volterra type dispersal systems with periodic coefficients, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of patches and cannot disperse. Sufficient conditions on the boundedness, permanence and existence of positive periodic solution for this systems are established. The theoretical results are confirmed by a special example and numerical simulations.     

Dynamic Behavior of a Logistic Type Equation with Infinite Delay

A non-autonomous Logistic type equation with infinite delay is investigated.For general nonau-tonomous case,sufficient conditions which guarantee the uniform persistence and globally attractivity of thesystem are obtained;For almost periodic case,by means of a suitable Lyapunov functional,sufficient conditionsare derived for the existence and uniqueness of almost periodic solution of the system.Some new results areobtained.     

Permanence for a class of periodic time-dependent predator–prey system with dispersal in a patchy-environment

In this paper, we study two species predator–prey Lotka–Volterra type dispersal system with periodic coefficients in two patches, in which both the prey and predator species can disperse between two patches. By utilizing analytic method, sufficient and realistic conditions on permanence and the existence of periodic solution are established. The theoretical results are confirmed by a special example and numerical simulations.     

Permanence and periodicity of a delayed ratio-dependent predator–prey model with Holling type functional response and stage structure

A periodic and delayed ratio-dependent predator–prey system with Holling type III functional response and stage structure for both prey and predator is investigated. It is assumed that immature predator and mature individuals of each species are divided by a fixed age, and immature predator do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solution of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

Rate of decay of stable periodic solutions of Duffing equations

In this paper, we consider the second-order equations of Duffing type. Bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution. Furthermore, the stability of the unique periodic solution is analyzed; the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Dynamic analysis of a turbidostat model with the feedback control

In this paper, a mathematical model with impulsive state feedback control is proposed for turbidostat system. The sufficient conditions of existence of positive order one periodic solution are obtained by using the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentration of microorganism and substrate. By investigating the periodic solution, the period and the initial point of the periodic solution are given. The results show that turbidostat with impulsive state feedback control tends to an order one periodic solution.