两个空间之间复合映射的不动点集和Reidemeister数

设X,Y为拓扑空间,f:X→Y,g:y→X．该文证明了下列结论:对每一自然数n, (1)f(Fix((g o,f)n))=Fix((f o g)n),g(Fix((f og )n))=Fix(g o f)n),且#Fix((g o f)n)= #Fix((f o g)n);(2)R((g o f)n)=R((f o g)n)．     

非线性二阶微分系统正解的存在性

考虑二阶微分系统边值问题[JB({]x″(t)+λ f(t,x(t),y(t))=0,\=y″(t)+μ g(t,x(t),y(t))=0,\ 00, f, g:［0,1］×［0,∞)×［0,∞)→R连续. 突破了以往文献要求非线性项 f, g非负的限制，运用锥上的一个不动点定理,在半正的情形下建立了问题正解的存在性     

一类二阶四点边值问题正解的存在性

讨论二阶四点微分方程组边值问题u″+p(t)f(t,u(t),v(t))=0,0 t 1,v″+q(t)g(t,u(t),v(t))=0,0 t 1,u(0)=a1x(ξ1),u(1)=b1x(η1)v(0)=a2x(ξ2),v(1)=b2x(η2)如果函数f,g:[0,1]×[0,∞)×[0,∞)→[0,∞)是连续的,并赋予f、g一定的增长条件,利用Leggett-Williama不动点定理,证明了上述边值问题至少存在三对正解.     

一个定理的再推广

定理若函数f-1(x),g-1(x)分别是函数f(x),g(x)的反函数(下同),且g-1(x)=g(x),则方程f(x)=g(x)有根a方程f-1(x)=g(x)有根g(a).证由f(a)=g(a),可得f-1(g(a))=a=g-1(g(a))=g(g(a)),即方程f-1(x)=g(x)有根g(a).由f-1(g(a))=g(g(a))=g-1(g(a))=a得f(a)=g(a),即方程f(x)=g(x)有根a.推     

本刊英文版Vol.30(2014),No.9论文摘要(英文)

<正>Least Number of Periodic Points of Self-maps of Lie Groups Jerzy JEZIERSKI Abstract There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M→M of a compact smooth manifold of dimension at least 3:NF_n(f)=min{#Fix(g~n);g~f;g is continuous}and NJD_n(f)=min{#Fix(g~n);g~f;g is smooth}.In general,NJD_n(f)may be much greater than NF_n(f).If M is a torus,then the invariants are     

RESEARCH ANNOUNCEMENTS FIXED POINTS AND BRAIDS Ⅱ

Let X be a connected compact polyhedron and let f:X→X be a map.Then theNielsen number N(f) is always a lower bound to MF[f]:=Min{#Fix(g)|g≈f:X→X},the least number of fixed points in the homotopy class.(See [2] or [4].)It is known [1] that if X has no local cut points and X is not a surface ofnegative Euler characteristic,then N(f)=MF[f] for all maps f:X→X.We now     

具有共振的边值问题解的存在性

利用Mawhin的重合度理论,研究具有共振的n-阶m-点边值问题x~((n))(t)=f(t,x(t),x′(t),…,x~((n-1))(t)),t∈(0,1)x(0)=x(η),x′(0)=x″(0)=…=x~((n-2))(0)=0,x~((n-1))(1)=α_ix~((n-1))(ξ_i)解的存在性,其中n≥2,m≥3,f:[0,1]×R~n→R将有界集映为有界集,且当x(t)∈C~(n-1)[0,1]时,f(t,x(t),x′(t),…,x~((n-1))(t))∈L~1[0,1],0<ξ_1<ξ_2<…<ξ_(m-2)<1,0<η<1,α_i∈R.在这里并不要求f具有连续性.     

关于Polya—de Bruijn定理的三个局限性问题的评注

设X、R为两个有限集合,有限群G作用在X上。又设R~x为从X到R的映射的全体。群G作用在R~x上通过:fg(d)=f(g(d)),g∈G,d∈X,f∈R~x。设ω为从R到环Q(包含有理数环在内的可换环)的映射,给f∈R~x赋权为W(f)=Π_(d∈x)W(f(d)),容易知W(f)=W(fg),g∈G。因而,可以给G—等价类集中的元F赋权为W(F)=W(f)f∈F。Plya[1]给出的计数多项式为:     

一类迭代函数方程组的C^m解的存在性和唯一性

讨论了较为广泛的一类迭代函数方程组G（x,f(x),…,f^n(x),g(x),…,g^n(x))=0 H(x,g(x),…,g^n(x),f(x),…,f^n(x)=0对任x∈J,其中J为实数轴R的连通闭子集,G,H∈C^m(J^2n 1,R),n≥2,对任一个整数m≥0,本在较弱的条件下证明了该方程组的C^m解的存在性和唯一性。     

一维p-拉普拉斯三个正解的存在性

考虑边值问题:(p(x'(t)))'+q(t)f(t,x(t),x'(t))=0,P>1,t∈[0,1],边值条件为x(0)=x(1)=0或x(0)=x'(1)=0．借助于一个新的不动点定理我们获得了存在至少三个正解的充分条件．问题的关键是非线性项f依赖于未知函数的一阶导数．最后,给出一个具体的例子．     

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 .     

Coincidence Wecken Property for Nilmanifolds

Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

Strong Convergence of the Kernel Estimates of Nonparametric Regression Functions

Let (X, Y), (X_1, Y_1),\cdots, (X_n, Y_n) be i. i. d. random vectors taking values in R_d\times R with E(|Y|)<\infinity, To estimate the regression function m(x)=E(Y|X=x), we use the kernel estimate $m_n(x)=[\sum\limits_{i = 1}^n {K(\frac{{{X_i} - x}}{{{h_n}}}){Y_i}/} \sum\limits_{i = 1}^n {K(\frac{{{X_j} - x}}{{{h_n}}})} \]$ where K(x) is a kernel function and h_n a window width. In this paper, we establish the strong consistency of m_n(x) when E(|Y|^p)<\infinity for some p>l or E{exp(t|Y|^\lambda)}<\infinity for some \lambda>0 and t>0. It is remakable that other conditions imposed here are independent of the distribution of (X, Y).     

具有四个有穷的IM公共小函数的整函数

本文证明了两个非常数整函数若具有四个有穷的ＩＭ公共小函数，则它们必恒等．从而在整函数的情形下，Ｎｅｖａｎｌｉｎｎａ五值定理中的五个ＩＭ公共值能否更换为五个ＩＭ公共小函数（含∞）的问题的回答是肯定的．     

一类三阶常微分方程非线性三点边值问题解的存在性

In this paper, existence of solutions of third-order differential equation
y′″（t）=f（t,y（t）,y′（t）,y″（t））
with nonlinear three-point boundary condition
{g（y（a）,y′（a）,y″（a））=0,
h（y（b）,y′（b））=0,
I（y（c）,y′（c）,y″（c））=0
is obtained by embedding Leray-Schauder degree theory in upper and lower solutions method,where a, b, c∈ R,a〈 b〈 c; f ： [a,c]×R^3→R,g：R^3→R,h：R^2→R and I：R^3→R are continuous functions. The existence result is obtained by defining the suitable upper and lower solutions and introducing an appropriate auxiliary boundary value problem. As an application, an example with an explicit solution is given to demonstrate the validity of the results in this paper.     

依赖于一阶导数的二阶脉冲微分方程边值问题的正解(英)

本文研究一类二阶脉冲微分方程：■的正解存在性．其中,0＜η＜1,0＜α＜1,f：[0,1]×[0,∞)×R→[0,∞),I_i：[0,∞)×R→R,J_i：[0,∞)×R→R,(i=1,2,…,k)均为连续函数．本文所用方法是文献[5]推广的Krasnoselskii不动点定理,此定理为解决依赖于一阶导数的边值问题提供了理论依据．基于此定理,获得了问题正解存在性定理．特别地,我们获得此类问题的Green函数,使问题的解决更直观和简单．