On the Global Character of the Difference Equation X n + 1 =

We investigate the global stability, the periodic character, and the boundedness nature of solutions of the difference equation x n +1 = f + n x n m (2 k +1) + i x n m 2 l A + x n m 2 l , n =0,1,… where k and l are non-negative integers, the parameters f , n , i , A are non-negative real numbers with f + n + i >0, and the initial conditions are non-negative real numbers. We show that the solutions exhibit a trichotomy character depending upon the parameters n , i and A .     

Asymptotic Behavior of a Class of Nonlinear Delay Difference Equations

In this paper, we shall study the asymptotic behavior of solutions of difference equations of the form x n +1 = x n p f ( x n m k 1 , x n m k 2 ,…, x n m k r ), n =0,1,…, where p is a positive constant and k 1 ,…, k r are (fixed) nonnegative integers. In particular, permanence and global attractivity will be discussed.     

A General Comparison Result for Higher Order Nonlinear Difference Equations With Deviating Arguments

The authors consider m -th order nonlinear difference equations of the form D m p x n + i h j ( n , x s j ( n ) )=0, j =1,2,( E j ) where m S 1, n ] N 0 ={0,1,2,…}, D 0 p x n = x n , D i p x n = p n i j ( D i m 1 p x n ), i =1,2,…, m , j x n = x n +1 m x n , { p n 1 },…,{ p n m } are real sequences, p n i >0, and p n m L 1. In Eq. ( E 1 ) , p = a and p n i = a n i , and in Eq. ( E 2 ) , p = A and p n i = A n i , i =1,2,…, m . Here, { s j ( n )} are sequences of nonnegative integers with s j ( n ) M X as n M X , and h j : N 0 2 R M R is continuous with uh j ( n , u )>0 for u p 0. They prove a comparison result on the oscillation of solutions and the asymptotic behavior of nonoscillatory solutions of Eq. ( E j ) for j =1,2. Examples illustrating the results are also included.     

一般退化时滞差分系统解的存在性及通解

本文研究退化时滞差分系统Ex(k+ 1)= Ax(k)+ ∑li= 1Bix(k- i)+ f(k) (k= 0,1,2,…),x(k)= φ(k) (k= 0,- 1,- 2,…,- l),其中E、A、Bi∈Rm ×n,x(k)∈Rn,f(k)∈Rm ,rank(E)< n.给出了上述系统解的存在性条件及通解表达式.     

A Note on the Difference Equation x n +1 = ∑ i =0 k α i x n − i p i

In this note we improve Theorem 2 in Ref. [3] , about the difference equation x n +1 = ~ i =0 k f i x n m i p i , n =0,1,2,..., where k is a positive integer, f i , p i ] (0, X ) for i =0,..., k , and the initial conditions x m k , x m k +1 ,..., x 0 are arbitrary positive numbers.     

Existence and Comparison Results for First Order Periodic Implicit Difference Equations with Maxima

In this paper, we establish comparison results (maximum principles) which allow us to use the monotone method and the method of upper and lower solutions in order to build convergent sequences to the solutions of difference equations of the type j u k = f k , u k +1 , max l ] { k m h +1,…, k +1} u l , k ] I , u 0 = u T , with j u k = u k +1 m u k , I ={0,1,…, T m 1} and f ] C ( I 2 R 2 R , R ).     

Буняковский不等式之推广及其对于积分方程与希尔伯特空间之应用

<正> 不等式■(1) 通常称为布湼可夫斯基不等式,或席瓦耳智不等式,在本文中,作者推广此不等式为这里我们用 det u_(ij)(i,j=1,2,…,n)表第i列j行之元为 u_(ij)之n列行列式,f_i,g_j(i,j=1,2,…,n)表任一希尔伯特空间之任意二组之元,(f_i,g_j)表f_i与g_j二元之内乘积.     

Asymptotic Behavior of Solutions of Some Linear Difference Equations with Oscillatory Decreasing Coefficients

We investigate the asymptotic behavior of solutions of the system x ( n +1)=[ A + B ( n ) V ( n )+ R ( n )] x ( n ), n S n 0 , where A is an invertible m 2 m matrix with real eigenvalues, B ( n )= ~ j =1 r B j e i u j n , u j are real and u j p ~ (1+2 M ) for any M ] Z , B j are constant m 2 m matrices, the matrix V ( n ) satisfies V ( n ) M 0 as n M X , ~ n =0 X Á V ( n +1) m V ( n ) Á < X , ~ n =0 X Á V ( n ) Á 2 < X , and ~ n =0 X Á R ( n ) Á < X . If AV ( n )= V ( n ) A , then we show that the original system is asymptotically equivalent to a system x ( n +1)=[ A + B 0 V ( n )+ R 1 ( n )] x ( n ), where B 0 is a constant matrix and ~ n =0 X Á R 1 ( n ) Á < X . From this, it is possible to deduce the asymptotic behavior of solutions as n M X . We illustrate our method by investigating the asymptotic behavior of solutions of x 1 ( n +2) m 2(cos f 1 ) x 1 ( n +1)+ x 1 ( n )+ a sin n f n g x 2 ( n )=0 x 2 ( n +2) m 2(cos f 2 ) x 2 ( n +1)+ x 2 ( n )+ b sin n f n g x 1 ( n )=0 , where 0< f 1 , f 2 < ~ , 1/2< g h 1, f 1 p f 2 , and 0< f <2 ~ .     

具正規核的積分方程

<正> §1.引言 凡合條件即是說凡合條件kk[x,y]=kk[x,y](1.1)的核k(x,y)叫做正規核(normal kernel).這種核顯然包括實對稱核、實畸對稱核、艾氏核及畸艾氏核等為特例。在本文中,我們將討論具此種核之積分方程之性質及解法尤其是關於此種核之特值及奇值(即希米特(E.Schmidt)的特值)之性質     

除數問题(Ⅲ)

<正> 設d_k(n)是n分解為k個因子的數目,設 R_k(x)=(a_(k,0)+a_(k,1)ln x +…+a_(k,k-1)ln~(k-1)x)x(x>0)是ξ~k(s)x~s/s在s=1的留數.定義△_k(x)=D_k(x)-R_k(x).設σ_k真是使估計式     

On Globally Periodic Solutions of the Difference Equation x n + 1 = f ( x n )/ x n − 1

We study the second-order difference equation x n +1 = f ( x n ) x n m 1 where f ] C 1 ([0, X ),[0, X )) and x n ] (0, X ) for all n ] Z . For the cases p h 5, we find necessary and sufficient conditions on f for all solutions to be periodic with period p . We answer some questions and conjectures of Kulenovi ' and Ladas.     

n阶时滞差分方程的边值问题

本文研究ｎ阶时滞差分方程的边值问题：ｘ（ｋ＋ｎ）＝ｆ（ｋ，ｘｋ（），ｘ（ｋ），ｘ（ｋ＋１），…，ｘ（ｋ＋ｎ－１）），ｋ∈ＩＴ，ｘ（ｍ）＝φ（ｍ），ｍ∈Ｉ－ｒ，ｘ（１）＝ａ１，ｘ（２）＝ａ２，…，ｘ（ｎ－２）＝ａｎ－２，ｘ（Ｔ）＝Ａ，｛得到了解的存在性和唯一性的结果．     

一类共振条件下的高阶p-Laplacian算子多点边值问题

主要讨论了下列n阶带p-Laplacian算子多点边值问题在共振条件下解的存在性.(Φp(x(n-1)))′+f(t,x,x′,…,x(n-2))=0,0相似文献   

紧支撑正交插值的多小波和多尺度函数

本文给出一类伸缩因子为α的紧支撑正交插值多尺度函数和多小波的构造方法.设{Vj}是尺度函数Φ(x)=[φ1(x),φ2(x),…,φa(x)]T生成的多分辨分析,Vj(?)L2(R)是{a-j/2φ(?)(ajx-k),k∈Z,(?)=1,2,…,a)线性扩张构成的子空间,其插值性是指φ1(x),φ2(x),…,φa(x)满足φj(k+(?)/a)=δk,0δj,e,j,(?)∈{1,2,…,a).当Φ(x)是正交插值的,则多分辨分析的分解或重构系数能用采样点表示而不需要用计算内积的方法产生.基于此,我们建立多小波采样定理,即如果一个连续信号f(x)∈VN,则f(x)=∑i=0a-1∑k∈Zf(k/aN+i/aN+1)φi+1(aNx-k),并给出对应多小波的显式构造公式.更进一步,证明了本文构造的多小波也有插值性.最后,还给出一个构造算例.     

条件Erlang分布双参数加法定理的推广

X(m)和Y(k)服从参数(m,λ)和(k,μ)的Erlang分布且相互独立.证明了在X(m)相似文献   

一类连续体上连续映射的周期点

设X是个阶有限的遗传可分解可链连续体, f:X→X是X上的连续自映射, On(x,f)={fi(x):0≤i≤n)是f的一个返回轨道, inf(On(x,f))相似文献   

条件经验随机过程的渐近性质

设(Xi,Yi)(i=1,2,…,n)是来自总体(X,Y)的样本(独立同分布),其中X∈R1,Y∈Rq.M(x y)是Y=y时X的条件分布,Mnkn(x y)为M(x y)的第kn个最近邻域的经验分布估计量,讨论条件经验过程Sn(t,x,y)=kn12(Mnkn(x y)-M(x y))的渐近性质,得出在适当条件下,对固定的y,Sn(t,x,y)(x,t为参数)弱收敛于某一G aussian过程S(.).     

Hammerstein型非线性积分方程正解的个数

<正> 本文是作者工作[8]、[9]的继续.在[9]中作者利用Leray-Schauder拓扑度理论研究了多项式型Hammerstein非线性积分方程的固有值,即设     

二阶非线性脉冲微分方程的振动性

考虑二阶脉冲微分方程(r(t)(x′(t))σ)′+f(t,x(t),x′(t))=0,t t0,t≠tk,k=1,2,…x(tk+)=gk(x(tk)),x′(tk+)=hk(x′(tk)),k=1,2,…(E)其中0 t0相似文献   

On the Periodic Character of Some Difference Equations

We consider positive solutions of the following difference equation x n =max A x n m k , B x n m m , n =0,1,…, where A , B are any positive real numbers and k , m are any positive integers. We prove that every positive solution is eventually periodic and determine the period in terms of the parameters A , B , k , and m .