周期系数的平面Hamilton系统的平衡解的稳定性

本文考虑周期系数的平面Hamilton系统H(x,y,t)=H2(x,y,t)+H4(x,y,t)+d(x,y,t)的平衡解的稳定性。其中H2(x,y,t)=1/2[a(t)x2+y2],H4(x,y,t)=b4(t)x4+b2(t)(xy)2+b0(t)y4以及a(t),b0(t),b2(t),b4(t)是连续的T-周期函数,d(x,y,t)关于时间也是T-周期,在原点附近其阶为(x2+y2)3.     

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 .     

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.     

Positive periodic solutions for a nonlinear differential system with two parameters

In this article, we investigate a nonlinear system of differential equations with two parameters $$\left\{ \begin{array}{l} x"(t)=a(t)x(t)-\lambda f(t, x(t), y(t)),\y"(t)=-b(t)y(t)+\mu g(t, x(t), y(t)),\end{array}\right.$$ where $a,b \in C(\textbf{R},\textbf{R}_+)$ are $\omega-$periodic for some period $\omega > 0$, $a,b \not\equiv 0$, $f,g \in C(\textbf{R} \times \textbf{R}_+ \times \textbf{R}_+ ,\textbf{R}_+)$ are $\omega-$periodic functions in $t$, $\lambda$ and $\mu$ are positive parameters. Based upon a new fixed point theorem, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions to this system for any fixed $\lambda,\mu>0$. Finally, we give a simple example to illustrate our main result.     

某些二阶系统正解的存在性

考察如下边值问题正解的存在性x″(t) λa(t) f (x(t) ,y(t) ) =0y″(t) λb(t) g(x(t) ,y(t) ) =0x(0 ) =x(1 ) =y(0 ) =y(1 ) =0其中 f ,g:R × R R ;a,b:[0 ,1 ] R .所有的函数都被假定是连续的 ,此外 f ,g满足某些增长性条件 .本文得到了一些正解的存在性结果 .     

Relations de dépendance entre les équations fonctionnelles de Cauchy

Résumé Afin d'examiner les relations entre les différentes équations de Cauchy, nous résolvons, sans aucune hypothèse de régularité, l'équation fonctionnellea f(xy) + b f(x)f(y) + c f(x + y) + d (f(x) + f(y)) = 0, pour des fonctionsf, définies sur un anneau unifère divisible par deux et prenant leurs valeurs dans un corps, Les coefficientsa, b, c, etd appartiennent au centre de ce corps. Entre autres applications, nous en déduisons qu'une seule équation, à savoirf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), caractérise les endomorphismes des corps dont la caractéristique est différente de 2. En introduisant la notion d'équations fonctionnelles étrangères et d'équations fonctionnelles fortement étrangères, nous concluons à l'indépendance, au sens de cette notion, des équations classiques de Cauchy.

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Summary In order to study the inter-relations between the four Cauchy functional equations, we solve the functional equationa f(xy) + b f(x) f(y) + c f(x + y) + d(f(x) + f(y)) = 0. The functionf is defined over a ring which is divisible by 2 and which possesses a unit, while the values off are in a(skew)-field. The constantsa, b, c andd belong to this field and commute with all elements of thes-field. No regularity assumption is made onf. Among other applications, we show that the single equationf(xy) + f(x + y) = f(x)f(y) + f(x) + f(y), is enough to characterize field endormophisms in fields of characteristic different from 2. We introduce the notion of alien functional equations and that of strongly alien functional equations, to conclude that for such notions, Cauchy equations are indeed largely independent.

Dédié avec nos meilleurs voeux à Monsieur le Professeur Otto Haupt à l'occasion de son centenaire     

指数Diophantine方程(bn)~x+(2n)~y=((b+2)n)~z的例外解

设b是大于3的正奇数.运用初等方法讨论了方程(bn)^x+(2n)x+(2n)^y=((b+2)n)y=((b+2)n)^z适合(x,y,z)≠(1,1,1)的正整数解(x,y,z,n).证明了:i)对于任何给定的正整数N,存在无穷多个b可使该方程有满足min{x,y,z}≥N的正整数解(x,y,z,n);ii)对于任何给定的b,该方程仅有有限多组正整数解(x,y,z,n)满足y>z=x.     

GLOBAL EXISTENCE OF THE SOLUTIONS TO NONLINEAR HYPERBOLIC EQUATIONS IN EXTERIOR DOMAINS

This paper deals with the following IBV problem of nonlinear hyperbolic equations u_(tt)- sum from i, j=1 to n a_(jj)(u, Du)u_(x_ix_j)=b(u, Du), t>0, x∈Ω, u(O, x) =u~0(x), u_t(O, x) =u~1(v), x∈Ω, u(t, x)=O t>O, x∈()Ω,where Ωis the exterior domain of a compact set in R~n, and |a_(ij)(y)-δ_(ij)|= O(|y|~k), |b(y)|=O(|y|~(k+1)), near y=O. It is proved that under suitable assumptions on the smoothness,compatibility conditions and the shape of Ω, the above problem has a unique global smoothsolution for small initial data, in the case that k=1 add n≥7 or that k=2 and n≥4.Moreover, the solution ham some decay properties as t→ + ∞.     

二阶微分系统奇异正定超线性周期边值问题的多重正解

主要研究了二阶微分系统具有奇异正定超线性周期边值问题多重正解的存在性问题,利用Leray-Schauder抉择定理和锥不动点定理给出了奇异正定超线性周期边值问题-(p(t)x′)′+q1(t)x=f1(t,x,y),t∈I=[0,1]-(p(t)y′)′+q2(t)y=f2(t,x,y)x(0)=x(1),x[1](0)=x[1](1)y(0)=y(1),y[1](0)=y[1](1)(1.1)的多重正解的存在性,其中非线性项fi(t,x,y)(i=1,2)在x=∞,y=∞点处超线性,在(x,y)=(0,0)处具有奇性.这里定义x[1](t)=p(t)x′(t),y[1](t)=p(t)y′(t)为准导数,其中系数p(t),qi(t)(i=1,2)是定义在[0,1]上的可测函数,且p(t)>0,qi(t)>0(i=1,2),a.e[0,1],fi(t,x,y)∈C(I×R×R,R+),R+=(0,+∞).     

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.     

一类有滞量的微分方程的解的一种表达式

本文给出了方程 ａｎｘｎ（ｔ）＋ …＋ ａ０ｘ（ｔ）＋ ｂ１ｘ（ｔ－ τ１）＋ ｂ２ｘ （ｔ－ τ２ ）＝ ｔｋ 的解的一种表达式     

常微分方程的比较定理及其应用

<正> 本文§1讨论由驻定二阶系统(1.1)确定非驻定二阶系统(1.2)解的各种性态的比较定理.由于对(1.2)的定性研究,不能使用象Poincare-Bendixson理论那样的有力工具,显得办法很少,这种比较定理的意义之一在于,通过它可以将更多的工具间接用于(1.2).文[1]首先给出了许多这种比较定理,伹尚未得到使用,而且存在错误,本节将予纠正然     

THE GLOBAL SOLUTION OF INITIALBOUNDARY VALUE PROBLEM OF HIGHER-ORDER MULTIDIMENSIONAL EQUATION OF CHANGING TYPE

In practical problems there appears the higher-order equations of changing type. But,there is only a few of papers, which studied the problems for this kind of equations. In this paper a kind of the higher-order m     

一类可逆生化反应模型的定性分析

本文研究一类可逆生化反应的数学模型 :dxdt=a-( b+1 ) x+x2 y-cx3,dydt=bx-x2 y+cx3.应用微分方程定性理论 ,完整地解决了该系统极限环的存在性、唯一性和不存在性等问题 .     

Partial Difference Equations Analogous to the Cauchy-Riemann Equations and Related Functional Equations On Rings and Fields

Let (S, #, *) be an algebraic structure where # and * are binary operations with identities on the set S. Let (G, +) be an abelian group. We consider the functional equation (i) $$f(x * t, y)+ g(x, y\ \sharp\ t) = h(x, y)\ {\rm for\ all}\ x, y, t \in S,$$ where ?,g,h :S × S → G. As an application of (i) we solve $$f(x + t, y)- f(x, y) = -b(f(x, y+t)- f(x,y))\ {\rm for\ all}\ x, y, t \in S,$$ where ? :S × S → K (a field), and b ∈ K is a constant and b ≠ 0, ±1. If b = i, the pure imaginary unit, S = R and K = C, then the above equation may be considered as a discrete analogue of the Cauchy-Riemann equations. When (R, +, ?) is a commutative ring with 1, the functional equation (ii) $$\phi(y+xt)-\phi(xy+xt)=\phi(y+x)-\phi(xy+x)$$ for all x,y,t ∈ R, where ? : R → G, is basic to the general solutions of (i). We solve (ii) on certain rings and fields.     

带有阻尼项的偏泛函微分方程解的振动性

本文研究带有阻尼项的双曲型时滞偏微分方程 2 t2 u(x,t) +m(t) u t=a(t)△ u(x,t) +b(t)△ u(x,ρ(t) ) -q(t) f (u(x,σ(t) ) ,(x,t)∈ G≡Ω× R+ (1 )其中 ,R+=[0 ,+∞ ) ,Ω是一个具有逐段光滑边界的有界区域 .利用平均法和微分不等式方法得到方程 (1 )的若干新的振动准则 .     

Banach空间中的平均非扩张映象:不动点的存在定理

<正> 设X是Banach空间,E是X中的集合,T是映集合E到自身的映象.若T满足条件(称为平均非扩张条件)其中x,y∈E,a,b,c≥0且a+2b+2c≤1,则称T是平均非扩张映象. 文[1]概括了近年来研究关于平均非扩张映象不动点的一些主要结果.本文进     

On solution properties of some types of <Emphasis Type="Italic">p</Emphasis>-adic kinetic equations of the form reaction-diffusion

In this paper, solutions for two types of ultrametric kinetic equations of the form reaction-diffusion are obtained and properties of these solutions are investigated. General method to find the solution of equation of the form

Degenerate Convergence of Semigroups Related to a Model of Stochastic Gene Expression

We consider the Cauchy problem related to the system of equations:

关于线性偏微分方程的Cauchy问题的唯一性

<正> Cauchy问题的唯一性是偏微分方程的基本问题之一.经典的Cauchy-Kowalewski定理断言,解析方程或方程组的解析解是唯一的.1901年,Holmgren证明了,线性的解析方程或方程组的光滑解的唯一性.在取消关于系数的解析性的假设这个方向上的第一个结果是由Carleman在1939年给出的,他证明了两个自变量的相应结果,其中假设方程的主部的系数是实的,以及特征根是单重的,因而特征根的虚部如果不恒为零则总不为零.