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 ~ .     

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.     

Existence of Positive Solutions for N-term Non-autonomous Fractional Differential Equations

Existence of positive solutions for the nonlinear fractional differential equation D ^αu = f(x,u), 0 < α < 1 has been given (S. Zhang. J. Math. Anal. Appl. 252 (2000), 804–812) where D ^α denotes Riemann–Liouville fractional derivative. In the present work we extend this analysis for n-term non autonomous fractional differential equations. We investigate existence of positive solutions for the following initial value problem

一类差分方程解的全局性分析

研究差分方程……的全局性质,其中A,B,β∈(O,+∞),p,q∈N<'+>={1,2,…},α=max{p,q),γ<,1>,γ<,2>,…,γ<,p>,C<,1>,C<,2>,…,C<,q>∈[0,1],满足∑γ<,i>=∑C<,J>=1,初始值x<,0>,x<-1>,…,x<,α>∈(0,∞).得到该方程的每个正...     

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 .     

Existence of Bounded Solutions for Quasilinear Subelliptic Dirichlet Problems

This paper proves the existence of solution for the following quasilinear subelliptic Dirichlet problem: {Σ^m_{j=1}X^∗_ja_j(X, v, Xv)+ a_o(x, v, Xv) + H(x,v, Xv) = 0 v ∈ M^{1,p}_0(Ω) ∩ L^∞(Ω) Here X = {X_1 , …, X_m} is a system of vector fields defined in an open domain M of R^n, n ≥ 2, Ω ⊂ ⊂ M, and X satisfies the so-called Hormander's condition at the order of r > 1 on M. M_{1,p}_0(Ω) is the weighted Sobolev's space associated with the system X . The Hamiltonian H grows at most like |Xv|^p.     

线性常微分方程组解的特征数的估计

复数域上线性系统ｘ＝Ａ（ｔ）ｘ，当Ａ（ｔ）＝（ａｉｊ（ｔ））ｎ×ｎ具有（ｎ，Ｎ，ｒ） 差异性质且ｒｎ时，解的特征数ｊ有估计λｊ－ｌｉｍｔ→∞１ｔ∫ｔｔ０Ｒｅａｊ（τ）ｄτｎ－１ｒ＋１－ｎｌｉｍｔ→∞１ｔ∫ｔｔ０Ａ（τ）ｄτ，ｊ＝１，２，…，ｎ，其中Ａ（ｔ）＝ｍａｘ｛｜ａｉｊ（ｔ）｜：ｉ，ｊ＝１，２，…，ｎ，ｉ≠ｊ．｝     

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.     

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     

一个关于马尔可夫链的极限定理

给出一个关于可列非齐次马尔可夫链M元状态序组出现频率的新形式的强极限定理,所得结论对任意可列非齐次马尔可夫链普遍成立.     

由H\"{o}rmander向量场构成的抛物方程的 $W_{\ast}^{1,p}$ 正则性

设 $X_{1},\cdots ,X_{q}\ (q相似文献   

On the Behaviour of Solutions of a Class of Third Order Difference Equations

In this paper, the structure of the solution space of y n +3 + ry n +2 + qy n +1 + py n =0, n S 0, is studied, keeping oscillatory/nonoscillatory behaviour of solutions of the equation in view, where p , q and r are constants. Some of these results are generalized partially to hold for y n +3 + r n y n +2 + q n y n +1 + p n y n =0, n S 0, where { p n }, { q n } and { r n } are sequences of real numbers.     

有限个非扩张非自映像隐迭代格式的逼近

假设E为一致凸Banach空间,K为E的非空闭凸子集且为E的非扩张收缩,P为非扩张收缩映像.{Ti:i=1,2,…,N}:K→E为非扩张映像且F(T)=∩ from i=1 to N F(Ti)≠■.定义{xn}如下:x0∈K,xn=P(αnxn-1+(1-αn)TnP[βnxn-1+(1-βn)Tnxn]),n≥1,这里{αn},{βn}为[δ,1-δ]中的实序列,其中δ∈(0,1).若{Ti:i=1,2,…,N}满足条件(B),则{xn}强收敛于x*∈F(T).     

一类4-正则平面图的邻点可区别关联色数

所谓图R_n是指具有如下结构的平面图:R_n=(V,E),其中顶点集合V={u_1,u_2,…,u_n}U{v_1,v_2,…,v_n},边集合E={u_iu_(i+1),v_iv_(i+1),u_iv_i,u_iv_(i+1)|i=1,2,…,n},其中u_(n+1)=u_1,v_(n+1)=v_1.通过研究R_n的邻点可区别关联着色,给出了当n=4,n是3或者5的正整数倍时,R_n的邻点可区别关联色数.     

Asymptotically Constant Solutions of Functional Difference Systems

We consider the functional difference system ( A ) j x i ( n )= f i ( n ; X ), 1 h i h k , where X =( x 1 ,…, x k ) and f 1 (·; X ),…, f k (·; X ) are real-valued functionals of X , which may depend quite arbitrarily on values of X ( l ) for multiple values of l ] Z . We give sufficient conditions for ( A ) to have solutions that approach specified constant vectors as n M X . Some of the results guarantee only that the solutions are defined for n sufficiently large, while others are global. The proof of the main theorem is based on the Schauder-Tychonoff theorem. Applications to specific quasi-linear systems are included.     

方差分量模型中回归系数的线性Minimax估计

考虑方差分量(混合线性)模型y=Xβ+U1ξ1+U2ξ2+…+Ukξk,这里Xn×p,Ui,n×ti为已知设计矩阵,βp×1是固定效应,iξ是ti×1随机效应向量,满足E(iξ)=0,cov(iξ)=σ2iIti,iξ都不相关.往往Uk=In,ξk=ek,即最后一项为随机误差,热β∈RP和i2σ>0(i=1,2,…,k)为未知参数.我们考虑β的可估函数Sβ,选取二次损失函数L(d,Sβ)=(d-Sβ)′(d-Sβ)∑ki=1ciσi2+β′X′Vk-1Xβ,然后在线性估计类中给出Sβ的惟一的mini max估计.     

Quadratic derivative nonlinear Schrödinger equations in two space dimensions

We study the global in time existence of small classical solutions to the nonlinear Schrödinger equation with quadratic interactions of derivative type in two space dimensions $\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&;t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&;x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)$ where the quadratic nonlinearity has the form ${\mathcal{N}( \nabla u,\nabla v) =\sum_{k,l=1,2}\lambda _{kl} (\partial _{k}u) ( \partial _{l}v) }We study the global in time existence of small classical solutions to the nonlinear Schr?dinger equation with quadratic interactions of derivative type in two space dimensions

$\left\{{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \right.\quad\quad\quad\quad\quad\quad (0.1)$\left\{\begin{array}{l@{\quad}l}i \partial _{t} u+\frac{1}{2}\Delta u=\mathcal{N}\left( \nabla u,\nabla u\right),&t >0 ,\;x\in {\bf R}^{2},\\ u\left( 0,x\right) =u_{0} \left( x\right),&x\in {\bf R}^{2}, \end{array}\right.\quad\quad\quad\quad\quad\quad (0.1)      18. Inverse scattering for the nonlinear Schrödinger equation II. Reconstruction of the potential and the nonlinearity in the multidimensional case      Ricardo Weder 《Proceedings of the American Mathematical Society》2001,129(12):3637-3645 We solve the inverse scattering problem for the nonlinear Schrödinger equation on : We prove that the small-amplitude limit of the scattering operator uniquely determines . Our proof gives a method for the reconstruction of the potentials . The results of this paper extend our previous results for the problem on the line.       19. 最大最小次序统计量的联合分布      陈光曙 《大学数学》2006,22(5):134-137 X1,X2,…,Xn是来自总体X的简单随机样本,Nk=min1≤i≤k{Xi},Mk=max1≤i≤k{Xi}(k=1,2,…,n),本文给出了最小次序统计量与最大次序统计量的联合分布函数.      20. 广义Mycielski图的邻强边色数和邻点可区别全色数的两个上界      李沐春  强会英  张忠辅 《大学数学》2009,25(2) 对简单图G,|V(G)|=p,n是自然数,Mn(G)被称为图G的广义Mycielski图,如果V(Mn(G))={v01,v02,…,v0p;v11,v12,…,v1p;…;vn1,vn2,…,vnp},E(Mn(G))=E(G)∪{vijv(i+1)k|v0jv0k∈E(G),1≤j,k≤p,i=0,1,…,n-1}.文中针对简单图G与它的广义Mycielski图之间的关系,给出了G的广义Mycielski图的邻强边色数和邻点可区别全色数的两个上界.

Inverse scattering for the nonlinear Schrödinger equation II. Reconstruction of the potential and the nonlinearity in the multidimensional case

We solve the inverse scattering problem for the nonlinear Schrödinger equation on :

We prove that the small-amplitude limit of the scattering operator uniquely determines . Our proof gives a method for the reconstruction of the potentials . The results of this paper extend our previous results for the problem on the line.

     

最大最小次序统计量的联合分布

X1,X2,…,Xn是来自总体X的简单随机样本,Nk=min1≤i≤k{Xi},Mk=max1≤i≤k{Xi}(k=1,2,…,n),本文给出了最小次序统计量与最大次序统计量的联合分布函数.     

广义Mycielski图的邻强边色数和邻点可区别全色数的两个上界

对简单图G,|V(G)|=p,n是自然数,Mn(G)被称为图G的广义Mycielski图,如果V(Mn(G))={v01,v02,…,v0p;v11,v12,…,v1p;…;vn1,vn2,…,vnp},E(Mn(G))=E(G)∪{vijv(i+1)k|v0jv0k∈E(G),1≤j,k≤p,i=0,1,…,n-1}.文中针对简单图G与它的广义Mycielski图之间的关系,给出了G的广义Mycielski图的邻强边色数和邻点可区别全色数的两个上界.