Weierstrass preparation of quadratic differentials

Special cases or variants of the following play an important role in the asymptotic analysis of ordinary differential equations with turning points. Theorem.Let a(t, x) be a smooth complex-valued function germ at the origin in C × R ^m which is holomorphic in t. Suppose that a(t, 0) does not vanish identically. Then there is a smooth change of variable t = g(s, x), holomorphic in s, such that a(t, x)dt ² =P(s, x)ds ² where P is a monic polynomial in s.     

Nonconvex and nonmonotone perturbations of evolution inclusions of subdifferential type

In this paper we prove the existence of strong solutions for evolution inclusions of the form – (t) (x(t))+F(t,x)) defined in a separable Hilbert space, where (·) denotes the subdifferential of a proper, closed, convex function (·) andF(t,x) is a multivalued nonconvex, nonmonotone perturbation satisfying a general growth condition.Research supported by N.S.F. Grant D.M.S.-8802688.     

Stability and instability criteria for Kaplan–Yorke solutions

We derive sufficient conditions for the stability and instability of periodic solutions of Kaplan–Yorke type to the equation where f is even in the first and odd in the second argument. The criteria are based on the monotonicity of the coefficient in a transformed version of the variational equation. For the special case of cubic f, we show that this monotonicity property is satisfied if and only if the set is contained in a region E defined by a quadratic form (bounded by an an ellipse or a hyperbola). The coefficients of this quadratic form are expressible in terms of the Taylor coefficients of f. Further, the parameter α in the equation and the amplitude z of the periodic solution are related by an elliptic integral. Using the relation between this integral and the arithmeticgeometric mean, we obtain upper and lower estimates on this relation, and on the inverse function. Combining these estimates with the inequality that defines the region E, we obtain stability criteria explicit in terms of the Taylor coefficients of f. These criteria go well beyond local stability analysis, as examples show. This research was supported by the Alexander von Humboldt Foundation (Germany) Received: March 14, 2005; revised: August 16, 2005     

Note on some s-shaped bifurcation curves

In this article we consider asymptotic behavior of some bifurcation curves of the two-point boundary value problem -u′ (x) =λf(u(x)) for 0 < x < 1; u(0) = u(1) = 0. Infact we prove that λ grows linearly with respective to p(p = u(1/2)) for p large     

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation where p > 0 is a constant. These results are extensions of the earlier results of Hille, Wintner, Opial, Yan for second order linear differential equations and include the recent results of Li and Yeh, Kusano and Yoshida, Yang and Lo for half-linear differential equations. Authors’ address: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China     

Frequency Domain Methods and Decoupling of Linear Infinite Dimensional Differential Algebraic Systems

We discuss the analysis of linear constant coefficient differential algebraic equations on infinite dimensional Hilbert spaces. We give solution concepts and discuss solvability criteria which are mainly based on Laplace transform. Furthermore, we investigate the decoupling of these systems motivated by the Kronecker normal form for the finite dimensional case. Applications are given by the analysis of mixed systems of ordinary differential, partial differential and differential algebraic equations.     

Oscillating solutions of scalar delay-differential equations with state dependence

This paper is concerned with the oscillatory behavior of the delay-differential equation X'(t)=F(t,x_t) including the equations x'(t)=-a(t)x(t-r(t,x(t))), [display math001] as special cases.We give conditions for the existence of a nonoscillatory solution of (1) and criteria for the oscillation of all solutions of (1), aiming at extending or generalizing to (1) some of the recent oscillation and nonoscillation results for delay equations of the form x'(t)=-a(t)x(t-p)).     

Analytic integrability and characterization of centers for nilpotent singular points

A method which provides necessary conditions to obtain a local analytic first integral in a neighborhood of a nilpotent singular point is developed. As an application we provide sufficient conditions in order that systems of the form where P_n and Q_n are homogeneous polynomials of degree n = 2, 3, 4, 5 have a local analytic first integral of the form H=y²+F(x, y), where F starts with terms of order higher than 2. We remark that, in general, the existence of such integral is only guaranteed when the singular point is a nilpotent center and the system has a formal first integral, see [6]. Therefore, we characterize the nilpotent centers of systems which have a local analytic first integral.     

A note on homoclinic orbits for second order Hamiltonian system

Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a（t）|s|^2∫^t0f（t,s）ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a（t）dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too.     

On a closed orbit of some constrained system

We study on what one calls a constrained system of ODEs on It consists of two ordinary differential equations and an algebraic equation with respect to three unknown functions. We seek closed orbits of such a system. A necessary and sufficient condition for the system to have non-trivial closed orbits is given. Elementary tools such as Lyapunov functions and Poincaré’s index theory are used in the proof of the result. As an application we consider a constrained system associated with a non-constraint system of ODEs called the modified Bonhöffer-van der Pol system.     

一类具分布时滞的微分方程的正周期解的存在性与多解性

讨论具分布时滞的微分方程x′(t)=-a(t,x)x(t)+∫-0τf(t,r,x(t+r))dr,x′(t)=a(t,x)x(t)-∫0-τf(t,r,x(t+r))drx′(t)=-g(t,x(t))+∫0-τf(t,r,x(t+r))dr,x′(t)=g(t,x(t))-∫0-τf(t,r,x(t+r))dr正周期解问题,利用锥不动点定理,获得了这类问题正解存在性和多重性的充分条件,推广了已有文献的相关结果.     

Continuous transformations of differential equations with delays

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer's transformationz(t)=g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of thenth order into an equation of the same type. Further, some forms of these equations having certain subspaces of solutions are considered.     

On the existence of periodic solutions for the third-order nonlinear ordinary differential equations

In this paper,the existence of periodic solution for the third-order nonlinear ordinarydifferential equation of the form -^x f(-^x) g(-^x) h(x)=p(t) is consldered,where f,g,h andp are the continuous functlons, and p (t T) =p (t). By using the Leray-Schauder degreemethod,some sufficient conditions to guarantee the existence of T periodic solution for the equation are obtained. Some previous results are extended and improved.     

Exponentially Dichotomous Operators and Exponential Dichotomy of Evolution Equations on the Half-Line

To an evolution family on the half-line of bounded operators on a Banach space X we associate operators I_X and I_Z related to the integral equation and a closed subspace Z of X. We characterize the exponential dichotomy of by the exponential dichotomy and the quasi-exponential dichotomy of the operators X we associate operators I_X and I_Z, respectively.     

具p-Laplacian算子型奇异方程组边值问题正解的存在性

本文讨论了一类具p-Laplacian算子型奇导方程组边值问题(φp(x'))'+α1(t),f(x(t),y(t))=0,(φp(y'))'+α2(t)g(x(t),y(t))=0,x(0)-β1x'(0)=0,x(1)+δ1x'(1)=0,y(0)-β2Y'(0)=0,y(1)+δ2y'(1)=0正解的存在性,其中φp(x)=|x|p-2x,p>1.通过使用不动点指数定理,在适当的条件下,建立了这类奇异方程组边值问题存在一个或者多个正解的充分条件.这些结果能用来研究椭圆型方程组边值问题径向对称解的存在性.     

二阶拟线性微分方程组边值问题的三个对称正解

本文讨论二阶拟线性微分方程组边值问题( p(x'))'+a(t),(t,x,y)=0,( q(y'))'+b(t)g(t,x,y)=0,x(0)-B0(x'(0))=x(1)+Bo(x'(1))=0,y(0)-B1(y'(0))=y(1)+B1(y'(1))=0,其中f,g是非负连续的函数.利用五个泛函的不动点定理,赋予f和g一些增长条件保证至少三个对称正解的存在性.     

中立型标量积分微分方程的周期解

本文考虑中立型标量方程x′(t)=a(t)x(t)+∫     

具时滞的高维周期系统周期解的存在性与唯一性

本文考虑了具时滞的高维周期系统x’(t)=A(t,x(t))x(t)+f(t,x(t-r)),其中(t,x)∈R×R~n,A(t,x)是n×n连续矩阵,f(t,x)是n维连续向量,且A(t+T,x)=A(T,x),f(t十T,x)=f(t,x).利用不动点方法,建立了保证其T周期解的存在性及唯一性的充分条件.所得结果推广、改进了文[1-3]的主要结果.     

一类具有偏差变元的高阶中立型Rayleigh方程周期解的存在性

利用重合度理论和不等式分析技巧,获得了一类具有偏差变元的高阶中立型Rayleigh方程(x(t)-cx(t-σ))~((m))+f(x'(t))+β(t)g(x(t-τ(t)))=p(t)周期解存在性的新的充分条件,有意义的是函数f(x)和非线性项前的系数β(t)可以变号.     

二阶中立型微分方程非振动解的存在性

考虑了一类变系数的具有强迫项的二阶中立型微分方程(x(t)+R(t)x(h(t)))″+P(t)x(g_1(t))-Q(t)x(g_2(t))=f(t)非振动解的存在性问题.通过Banach压缩映像原理,分别得到了方程存在满足■|x(t)|>0的非振动解x(t)的充分条件与必要条件,推广了一阶变系数方程的相应结果.