STABILITY AND EXISTENCE OF PERIODIC SOLUTIONS AND ALMOST PERIODIC SOLUTIONS ON LIENARD SYSTEMS

1MainResultsConsidersystem11~.x f(x)x' g(x)~0(1)wheref(x)islocallyintegrable,g(x)isdifferentiablealldg(0)=0.Theroem1Thezerosolutionofsystem(1)isuniformlyasymptoticallystableifbyequivalenttransf'Ormu=xov=X' F(x).DefineW[t,(uif\v)]j6ug(s)ds Iv',thenwisaposi…     

Lienard方程周期解、概周期解的存在性

本文考虑Ｌｉｅｎａｒｄ方程ｘ”十ｆ（ｘ）ｘ’＋ｇ（ｘ）＝ｅ（ｔ），我们得到：当且时，对于任意周期或概周期。数ｅ（ｔ），它有周期或概周期解．而对于Ｌｉｅｎａｒｄ方程ｘ”＋ｆ（ｘ）ｘ’＋ｃｘ＝ｅ（ｔ），我们得到：当ｃ＞０且时，对于任意周期、或概周期函数ｅ（ｔ），它有周期或概周期解．     

Logistic型方程周期解和概周期解的存在性

In this paper, we study the Logistic type equation x= a（t）x -b（t）x^2＋ e（t）. Under the assumptions that e（t） is small enough and a（t）, b（t） are contained in some positive intervals, we prove that this equation has a positive bounded solution which is stable. Moreover, this solution is a periodic solution if a（t）, b（t） and e（t） are periodic functions, and this solution is an almost periodic solution if a（t）, b（t） and e（t） are almost periodic functions.     

Lienard方程周期解、概周期解的存在性

本文考虑 Lienard方程 x″+f (x) x′+g(x) =e(t) ,我们得到 :当 -∞ 0且 0 相似文献   

THE PERTURBATION OF ALMOST PERIODIC SOLUTION OF ALMOST PERIODIC SYSTEM

By using the exponential dichotomy and the averaging method,a perturbation theoryis established for the almost periodic solutions of an almost differential system.Suppose that the almost periodic differential system(dx)/(dt)=f(x,t) ε~2g(x,t,ε)(1)has an almost periodic solution x=x_0(t,M)for ε=0,where M=(m_1,…,m_k)is theparameter vector.The author discusses the conditions under which(1)has an almostperiodic solution x=x(t,ε)such that x(t,ε)=x_0(t,M)holds uniformly.The results obtained are quite complete.     

On the Stability of Periodic Impulsive Systems

In this paper, we consider periodic systems of ordinary differential equations with impulse perturbations at fixed points of time. It is assumed that the system possesses the trivial solution. We show that if the trivial solution of the system is stable or asymptotically stable, then it is uniformly stable or uniformly asymptotically stable, respectively. By using the method of Lyapunov functions, we establish criteria of uniform asymptotical stability and instability.     

具有反馈控制的离散两种群竞争系统的概周期解

对于具有反馈控制的离散概周期两种群竞争系统的概周期解问题,进行了深入的讨论,得到了该系统存在唯一的一致渐近稳定概周期解的充分条件.     

Periodic Solutions for Nonlinear Diferential Equations

ＰｅｒｉｏｄｉｃＳｏｌｕｔｉｏｎｓｆｏｒＮｏｎｌｉｎｅａｒＤｉｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎｓＣｏｎｇＦｕｚｈｏｎｇ（从福仲）（ＯｆｆｉｃｅｏｆＭａｔｈｅｍａｔｉｃｓ，８６００３Ｕｎｉｔ，Ｃｈａｎｇｃｈｕｎ，１３００２２）ＭａｏＤｏｎｇｍｉｎｇ（...     

一类广义Liénard方程的概周期解

研究广义 Liénard方程: x'=φ(y)-F(x), y'=-g(x)+p(t), 利用 Amerio的结果证明方程的解部分变元的最终有界性意味着概周期解的存在性, 推广了Cieutat^[1]的主要结果.     

STABILITY AND ALMOST PERIODIC SOLUTIONS OF NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

Itiswellknownthattheexistenceofalmostperiodicsolutionsiscloselyrelatedtothestabilityofsolutions.Forfunctionaldifferentialequationswithinfinitedelay,Y.Hin.[5'6]studiedtheproblemsontheexistenceofalmostperiodicsolutionsandthestability.However,therearefewpapersll2]dealingwithneutralfunctionaldifferentialequationswithinfinitedelay.Inthepresentpaper,forneutralfunctionaldifferentialequationswithinfinitedelay,weprovetheinherencetheoremfortheuniformlystableoperatorD(t),definethestabilitywithrespecttot…     

SINGULAR PERTURBATIONS FOR QUASILINEAR HYPERBOLIC EQUATIONS

This paper deals with the following mixed problem for Quasilinear hyperbolic equationsThe M order uniformly valid asymptotic solutions are obtained and there errors areestimated.     

Chemotaxis-fluild 系统的全局弱解

We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.     

一类具无穷时滞泛函微分方程的周期解

讨论具有无穷时滞中立型泛函微分方程$ \frac{\rm d}{{\rm d}t}\left(x(t)-\int_{-\infty}^{0}g(s,x(t+s)){\rm d}s\right) =A(t,x(t))x(t)+f(t,x_t)$的周期解问题,利用重合度理论中的延拓定理得到了周期解的存在性和唯一性条件;特别地,当$g(s,x)\equiv 0, A(t,x)=A(t)$时, 给出了存在唯一稳定周期解的条件.     

Some new refined Hardy type inequalities with general kernels and measures

We state and prove some new refined Hardy type inequalities using the notation of superquadratic and subquadratic functions with an integral operator A _k defined by $$ A_kf(x):=\frac{1}{K(x)} \int\limits_{\Omega_2} k(x,y)f(y)d\mu_2(y), $$ where ${k: \Omega_1 \times \Omega_2 \to \mathbb{R}}$ is a general nonnegative kernel, (Ω₁, μ ₁) and (Ω₂, μ ₂) are measure spaces and $$ K(x):=\int\limits_{\Omega_2} k(x,y)d\mu_2(y), \, x \in \Omega_1. $$ The relations to other results of this type are discussed and, in particular, some new integral identities of independent interest are obtained.     

Periodic Solutions to Porous Media Equations of Parabolic-elliptic Type

This paper is concerned with a equation, which is a model of filtration in partially saturated porous media, with mixed boundary condition of Dirichlet-Neumann type {∂_tb(u) - ∇ • a [∇u + k(b(u))] = f \qquad in \quad (0, ∞) × Ω u = h(t, x) \qquad on \quad (0, ∞) × Γ_0 v • a [∇u + k(b(u))] = g(t, x) \qquad on \quad (0, ∞) × Γ_1 We have proved that there exists one and only one periodic solution of the problem under the data f, g and h with same period. Moreover, we have proved that the unique periodic solution ω is asymptotically statble in the sense that for any solution u of the problem b(u(t)) - b(ω(t)) → 0\qquad in L²(Ω) as t → ∞.     

一类具有脉冲效应的浮游生物模型的持久性以及概周期解

利用脉冲微分方程的对比定理以及李雅普诺夫函数法,我们研究了一类具有脉冲效应的浮游生物模型的持久性以及概周期解.文中所得结论改进了以往的研究成果.文中所用的研究方法可以用来研究其他带有脉冲的生物数学模型的持久性以及概周期解.最后,我们总结阐述了脉冲如何影响模型的持久性,概周期解以及一致渐进稳定性.     

The principal eigenvalue of a space–time periodic parabolic operator

This paper deals with the generalized principal eigenvalue of the parabolic operator , where the coefficients are periodic in t and x. We give the definition of this eigenvalue and we prove that it can be approximated by a sequence of principal eigenvalues associated to the same operator in a bounded domain, with periodicity in time and Dirichlet boundary conditions in space. Next, we define a family of periodic principal eigenvalues associated with the operator and use it to give a characterization of the generalized principal eigenvalue. Finally, we study the dependence of all these eigenvalues with respect to the coefficients.      

一个描述血吸虫病的数学模型的周期解

研究一个描述血吸虫病的周期微分方程模型dx/dt=-rx+A/S(t)y,dy/dt=-δ(t)y+B(S(t)-y) x2/1+ x.数值计算发现该系统同时具有渐近稳定的零解和一个正周期解.通过证明该系统解的有界性,并在一个函数空间上构造单调有界序列,进而证明了在一定条件下正周期解的存在性.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

UNIQUENESS OF SOLUTIONS FOR HIGHER DIMENSIONAL QUASILINEAR DEGENERATE PARABOLIC EQUATION

This paper studies the uniqueness of generalized solutions for the first boundary value problem of the form where A_u≥0.It is proved that if u∈BV_x(Q_T) and A(u, x, t) is strictly increasing with respect to u, the solution is unique.