Positive Periodic Solutions of Systems of First Order Ordinary Differential Equations

Consider the n-dimensional nonautonomous system ?(t) = A(t)G(x(t)) ? B(t)F(x(t ? τ(t))) Let u = (u ₁,…,u _n ), $f^{i}_{0}={\rm lim}_{\|{\rm u}\|\rightarrow 0}{f^{i}(\rm u)\over \|u\|}$ , $f^{i}_{\infty}={\rm lim}_{\|{\rm u}\|\rightarrow \infty}{f^{i}(\rm u)\over \|u\|}$ , i = l,…,n, F = (f ¹…,f ⁿ ), ${\rm F_{0}}={\rm max}_{i=1,\ldots,n}{f^{i}_{0}}$ and ${\rm F_{\infty}}={\rm max}_{i=1,\ldots,n}{f^{i}_{\infty}}$ . Under some quite general conditions, we prove that either F₀ = 0 and F_∞ = ∞, or F₀ = ∞ and F_∞ = 0, guarantee the existence of positive periodic solutions for the system for all λ > 0. Furthermore, we show that F₀ = F_∞ = 0, or F_∞ = F_∞ = ∞ guarantee the multiplicity of positive periodic solutions for the system for sufficiently large, or small λ, respectively. We also establish the nonexistence of the system when either F₀ and F_∞ > 0, or F₀ and F_∞, < for sufficiently large, or small λ, respectively. We shall use fixed point theorems in a cone.     

Painlevé Classification of Binomial Type Ordinary Differential Equations of the Arbitrary Order

The complete Painlevé classification of the binomial ordinary differential equations of the arbitrary order n ≥ 4 is built. Six classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

Binomial-Type Ordinary Differential Equations of the Third Order

The complete Painlevé classification of the binomial ordinary differential equations of the third order is built. Eight classes of equations with Painlevé property are obtained. All of these equations are solved in terms of elementary functions and known Painlevé transcendents.     

On the Explicit Solution of Certain First Order Systems of Partial Differential Equations

We consider two classes of systems of partial differential equations of first order. One consists of generalized Stokes-Beltrami equations $ Aw_z = w^*_z $ , $ \lambda Bw_{\bar z} -w^*_{\bar z} $ with square matrices A and B and a scalar factor u . The other may be written in matrix notation as $ v_{\bar z} = c{\bar v} $ where c denotes a square matrix. This system is known as a Pascali system. Both systems are in close connections to certain systems of second order for which the solutions can be represented using particular differential operators. On the basis of these relations we give the solutions of the first order systems explicitly.     

一类二阶非线性常微分方程的初值问题

本文建立了一类二阶非线性常微分方程初值问题的一个定理,给出了它的关于解的周期性、振动性和估计式的三个推论及对应于它们的例子,指出由Thomas、DeSpantz和Lerman、Klamkin和Reid及Stare等人所考察过的一些二阶非线性常微分方程都是本文方程的特例。     

Movable Singularities of a Class of Nonlinear Ordinary Differential Equations of Arbitrary Order

In this paper we consider nonlinear ordinary differential equations   y ^{( n )}= F ( y ', y , x )  of arbitrary order   n ≥ 3  , where F is algebraic in   y , y '  and locally analytic in x . We prove that for   n > 3  these equations always admit movable branch points. In the case   n = 3  these equations admit movable branch points unless they are of the known class   y '= a ( x )( y ')²+ ( b ₂( x ) y ²+ b ₁( x ) y + b ₀( x )) y '+ ( c ₄( x ) y ⁴+ c ₃( x ) y ³+ c ₂( x ) y ²+ c ₁( x ) y + c ₀( x ))  , where   a ,  b_j ,  c_j   are locally analytic in x .     

Oscillation and Nonoscillation Criteria for Two-Dimensional Systems of First Order Linear Ordinary Differential Equations

Sufficient conditions are established for the oscillation and nonoscillation of the system     

Classification of Discrete Symmetries of Ordinary Differential Equations

A simple method for determining all discrete point symmetries of a given differential equation has been developed recently. The method uses constant matrices that represent inequivalent automorphisms of the Lie algebra spanned by the Lie point symmetry generators. It may be difficult to obtain these matrices if there are three or more independent generators, because the matrix elements are determined by a large system of algebraic equations. This paper contains a classification of the automorphisms that can occur in the calculation of discrete symmetries of scalar ordinary differential equations, up to equivalence under real point transformations. (The results are also applicable to many partial differential equations.) Where these automorphisms can be realized as point transformations, we list all inequivalent realizations. By using this classification as a look-up table, readers can calculate the discrete point symmetries of a given ordinary differential equation with very little effort.     

一类二阶微分方程解的振动性质

利用积分平均技巧研究二阶微分方程(r(t)(x(t) )x′(t) )′ q(t) f(x(t) ) g(x′(t) ) =0 .解的振动性质 ,得到了一些保证此方程所有解振动的充分条件 .特别 ,本文的结果改进了文 [1 ]的主要结果 .     

A Class of Multistep Method Containing Second Order Derivatives for Solving Stiff Ordinary Differential Equations

In this paper a general k-step k-order multistep method containing derivatives of second order is given. In particular, a class of k-step (k+1)th-order stiff stable multistep methods for k=3-9 is constructed. Under the same accuracy, these methods are possessed of a larger absolute stability region than those of Gear's [1] and Enright's [2]. Hence they are suitable for solving stiff initial value problems in ordinary differential equations.     

泛函微分方程多个正周期解的存在性

考虑一类一阶非线性泛函微分方程,利用锥中的不动点理论给出存在多个正周期解的一些新的充分条件.     

Linear Second Order Ordinary Differential Equations with Nonsmooth Coefficients

This paper deals with a Dirichlet boundary value problem for a linear second order ordinary differential operator, whose coefficients belong to certainL^p-spaces. Its solution is to be understood in the sense of Sobolev, so that the Fredholm alternative holds. The main purpose of this paper is, in case of unique solvability, to introduce a Green's function by means of which the solution can be given explicitly by integrals. We give the precise definition of the Green's function via Riesz' Representation Theorem and establish some of its basic properties. As a preliminary tool the Cauchy initial value problem is considered.     

一类二阶非线性常微分方程零解的全局稳定性

本文用变量分离法研究了二阶非线性微分方程ｘ＋ φ（ｘ）ｐ（ｘ ）＋ ｇ（ｘ）ｆ （ｘ）＝ ０ 零解的全局稳定性     

某些二阶微分方程解的渐近性

Two criteria for asymptotic behavior of certain second order differential equations are obtained. The results extend and improve Feng‘s theorem.     

Contact Classification of Linear Ordinary Differential Equations: I

It is known that a linear ordinary differential equation of order n3 can be transformed to the Laguerre–Forsyth form y ⁽ⁿ⁾= _i=3 ⁿ a _n–i (x)y ^(n–i) by a point transformation of variables. The classification of equations of this form in a neighborhood of a regular point up to a contact transformation is given.