SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEM FOR NONLINEAR HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS INVOLVING TWO SMALL PARAMETERS

The singularly perturbed boundary value problem for nonlinear higher order ordinary differential equation involving two small parameters has been considered. Under appropriate assumptions, for the three cases:ε/μ2→0(μ→0),μ2/ε→0 (ε→0) andε=μ2, the uniformly valid asymptotic solution is obtained by using the expansion method of two small parameters and the theory of differential inequality.     

SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM FOR NONLINEAR SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS INVOLVING TWO SMALL PARAMETERS

In this paper, a class of singular perturbation of boundary value problem for nonlinear second order ordinary differential equation involving two parameters is considered. For the three cases ε/μ→0(μ→0), μ2/ε→0(ε→0) and ε=μ2, uniformly valid asymptotic solutions are obtained using the method of two steps expansions and a fixed-point theorem.     

非线性弹性杆中孤波的演变

1.基本方程非线性弹性杆内的纵向应变波的广义Korteweg-de Vries-Burgers方程为(1·1)为了明确起见,本文研究n=2的硬非线性材料(软非线性材料μ<0仿此处理)情形,此时μ>0.作变换x→(6μ)~(1/2)x,t→6(6μ)~(1/2)t,u→-u,(1·1)变为(1·2)其中ε=(6/μ)~(1/2)δ,方程(1·1)与(1·2)右端项为粘弹性阻尼的贡献,右端第三项来自横向惯性所引起的应力.当忽略耗散时,ε=0,(1·2)就是通常的KdV方程,此时杆内有纵向传播的稳定孤波.本文研究当耗散较小(ε<<1),但不可忽略时,它对应变孤波运动的影响,此时(1·2)看作是含微扰的KdV方程.     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

非线性边值问题的奇摄动

本文研究边值问题:εy"=f(x,y,y',ε,μ)(μ0(ε,μ)y(x,ε,μ)|(x=1-μ)=φ₁(ε,μ)其中ε,μ是两个正的小参数 在f_y’≤-k<0和其他适当的限制下,存在一个解且满足其中y₀,0(x)是退化问题 f(x,y,y',0,0)=0(01(0,0)的解,而y_i-j,j(x)(j=0,1,…,i;i=1,2,…m)能够从某些线性方程逐次求得.     

一类向量四阶非线性微分方程边值问题的奇摄动

我们研究伴有边界摄动的向量边值问题:
ε2y(4)=f(x,y,y″,ε,μ)(μy(x,ε,μ)|_x=μ=A₁(ε,μ),y(x,ε,μ)|_x=1-μ=B₁(ε,μ)
y″(x,ε,μ)|_x=μ=A₂(ε,μ),y″(x,ε,μ)|_x=1-μ=B₂(ε,μ)
其中y,f,A_j和B_j(j=1,2)是n维向量函数和ε,μ是两个正的小参数.虽然纯量边值问题曾有人研究过,但这样的向量边值问题尚未被研究.在适当的假设下,利用微分不等式方法,我们找到向量边值问题的一个解和获得一致有效的渐近展开式.     

次分数Brown运动一个积分泛函的中心极限定理及其应用

假设S~H是Hurst参数为0H1的次分数Brown运动.本文研究积分过程1/(η(ε))∫_0~T((S_(s+ε)~H-S_s~H)~2-ε~(2H))ds,ε0的渐近分布,其中T0,η(ε)表示一个当ε→0时的无穷小量.当0H3/4和η(ε)=ε~(2H+1/2时,本文证明了上述积分弱收敛于一个标准Brown运动B的常数倍;当H=3/4和η(ε)=ε(-logε)~1/2时,证明了存在另一标准Brown运动W,使得上述积分弱收敛于3/4W.为应用,本文利用广义二次变差建立了Ornstein-Uhlenbeck(O-U)过程X_t~H=X_0~H+ σS_t~H-β∫_0~tX_s~Hds,中参数σ0的估计量,并给出其渐近正态性.     

A Bishop–Phelps–Bollobás Theorem for Asplund Operators

By characterizing Asplund operators through Fréchet differentiability property of convex functions, we show the following Bishop–Phelps–Bollobás theorem: Suppose that X is a Banach space,T : X → C(K) is an Asplund operator with ║T║= 1, and that x_0 ∈ S_X, 0 ε satisfy ║T(x_0)║ 1-ε~2/2.Then there exist x_ε∈ S_X and an Asplund operator S : X → C(K) of norm one so that ║S(x_ε)║ = 1, x_0-x_ε ε and ║T-S║ ε.Making use of this theorem, we further show a dual version of Bishop–Phelps–Bollobás property for a strong Radon–Nikodym operator T : ?_1 → Y of norm one: Suppose that y_0~*∈ S_(Y~*), ε≥ 0 satisfy T~*(y_0~*) 1-ε~2/2. Then there exist y_ε~*∈ S_(Y~*), x_ε∈(±e_n), y_ε∈ S_Y, and a strong Radon–Nikodym operator S : ?_1 → Y of norm one so that (ⅰ)║S(x_ε)║= 1;(ⅱ) S(x_ε) = y_ε;(ⅲ)║T-S║ ε;(ⅳ)║S~*(y_ε~*)║=y_ε~*, y_ε= 1;(ⅴ)║y_0~*-y_ε~*║ ε and (ⅵ)║T~*-S~*║ ε,where(e_n) denotes the standard unit vector basis of ?_1.     

拟对角扩张的近似等距提升

设 设 设0→I→A—A/I→0为C~*-代数的短正合列且A含有单位元.如果扩张0→I→A→A/I→0为拟对角扩张,则证明对任给的A/I中正元(投影,部分等距,酉元)均有同形式的提升且提升与I中一列由投影组成的拟中心近似单位元相交换.进而证明对任给正数ε,任意A/I中的两正元(投影,部分等距,酉元)a~-和b~-,以及a~-的正元(投影,部分等距,酉元)的提升a,存在b~-的正元(投影,部分等距,酉元)的提升b,使得‖a-b‖<‖a~--b~-‖ ε.作为上述结论的应用证明了对任意的正数ε和u~-∈U(A/I)_0,存在u~-的提升u∈U(A)_0,使得cel(u)相似文献   

一类具强非线性源的半线性热方程解的性质

讨论了具强非线性源的半线性热方程ut=△u m in{-ε1,up}边值问题解的性质,证明了TK(uε)在L2(0,T;W10,2(Ω))中关于ε是一致有界的,且(TK(uε))t在L2(Q)中关于ε是一致有界的,从而存在一子列和一可测函数u(x,t)使得当ε→0时,uε→u a.e.于Q.其中TK(r)=m in{K,m ax(r,-K)},Q=Ω×(0,T).     

奇摄动半线性系统的边界层和角层性质

一些作者已对纯量边值问题εy"=h(t,y),a相似文献   

奇摄动向量问题的边界层和内层现象

本文考虑非线性向量边值问题:εy″=f(x,y,z,y',ε), y(0)=A₁ y(1)=B₁ εz″=f(x,y,z,z',ε), z(0)=A₂ z(1)=B₂其中ε是正的小参数,0≤x≤1,f,g是R4中的连续函数。在适当的假设下,利用微分不等式理论,我们证明了上述问题的解的存在性,并得到包括边界层和内层在内的解的估计.     

Homogenization of a composite medium with a thermal barrier

In this work, we consider a heat transfer problem between two periodic connected media exchanging a heat flux throughout their common interface. The interfacial exchange coefficient λ is assumed to tend to zero or to infinity following a rate λ=λ(ε) when the size εof the basic cell tends to zero. Three homogenized problems are determined according to the value of δ=lim_ε→0λ/ε. Copyright © 2004 John Wiley & Sons, Ltd.     

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρ_i(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε-domain perturbation considered in this paper.     

Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1

We consider a boundary-value problem for the Poisson equation in a thick junction Ω_ε, which is the union of a domain Ω₀ and a large number of ε-periodically situated thin curvilinear cylinders. The following nonlinear Robin boundary condition ∂_νu_ε + εκ(u_ε)=0 is given on the lateral surfaces of the thin cylinders. The asymptotic analysis of this problem is performed as ε → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove the convergence theorem and show that the nonlinear Robin boundary condition is transformed (as ε → 0) in the blow-up term of the corresponding ordinary differential equation in the region that is filled up by the thin cylinders in the limit passage. The convergence of the energy integral is proved as well. Using the method of matched asymptotic expansions, the approximation for the solution is constructed and the corresponding asymptotic error estimate in the Sobolev space H¹(Ω_ε) is proved. Copyright © 2007 John Wiley & Sons, Ltd.     

Multidimensional Oscillations for Non-linear Hyperbolic Mixed Problem: The Justified Non-linear Geometric Optics Method

The mixed-Neumann problem for the non-linear wave equation □u−a(u)(∣∂_tu)∣²−∣∇u∣² = f_ε(z) is studied. The function f_ε(z) = ∑_k∈Kf_k(z,ε⁻¹ϕ_k(z),ε), ε∈[0,1], K is finite, f_k(z,θ_k,ε) are 2π-periodic with respect to θ_k. The existence of solution u_ε on a domain z = (t,x,y)∈[0,T]×ℝ₊×ℝ^d, d = 1 or 2, is proved when ε is sufficiently small; T does not depend on ε. By the non-linear geometric optics method the asymptotic (with respect to ε→0) solution ũ _ε is constructed. The estimation for the rest ε²r_ε = u_ε−ũ_ε is derived and the limit r_ε, ε→0, is studied.     

Homogenization of eigenvalue problem for Laplace–Beltrami operator on Riemannian manifold with complicated ‘bubble‐like’ microstructure

We study the asymptotic behavior of the eigenvalues and the eigenfunctions of the Laplace–Beltrami operator on a Riemannian manifold M^ε depending on a small parameter ε>0 and whose structure becomes complicated as ε→0. Under a few assumptions on scales of M^ε we obtain the homogenized eigenvalue problem. In addition we study the behavior of the heat equation on M^ε and investigate the large time behavior of the homogenized equation. Copyright © 2009 John Wiley & Sons, Ltd.     

Γ-CONVERGENCE OF INTEGRAL FUNCTIONALS DEPENDING ON VECTOR-VALUED FUNCTIONS OVER PARABOLIC DOMAINS

51.IntroductionWebeginwiththecharacterizationofr--convergencein[1,2].Definition1.1.Let(X,T)beafirstcountabletOPologicalspaceand{F'}7=,bease-quenceOjfunctionalshemXtoR=RU{--co,co},u6X,AER.Wecallifandonlyifforeverysequence{u'}concealingtouin(X,T)andthereedestsasequence{u'}conveneingtouin(X,T)suchthatWecallA~r(T)timF"(u)ifandonlyifjoreveryah-- a(h-co)Throughoutthispaper,weassumethatfiisaboundedopensetinR".Letp>1,T>0,andmbeapositiveinteger.Denoteforavectorvaluedfunctionu.ConsiderthefUc…