Palmer线性化定理的一个推广

本文在对二分性函数a(t)有适当的要求的条件下将Palmer线性化定理推广到线性部分只具有广义指数型二分性的系统,并在一定条件下证明了等价函数的强一致连续性.     

线性抛物型积分微分方程的扩展混合体积元方法

1 引言 考虑线性抛物型积分微分方程初边值问题: {pt(x,t)-▽.{A(x,t)▽p(x,t) +∫t0 B(x,t,τ)▽p(x,τ)dτ}=f(x,t),(x,t)∈Ω×(0,T],(1.1) p(x,0):p0(x), x∈Ω, p(x,t)=0, (x,t)∈(a)Ω×(0,T]. 这里x=(x,y),Ω=(a,b)×(c,d),(e)Ω是区域Ω的边界,p为未知函数,A=(aij)2×2为已知的对称正定矩阵,B=(bij)2×2为已知矩阵,而且aij,bij,(aij)t(i,j=1,2)光滑有界,f∈L2(Ω).     

分数阶布朗运动驱动的随机延迟微分方程数值解

<正>1引言1引言本文考虑由分数阶布朗运动驱动的随机延迟微分方程(FSDDE):■t∈(0,T],(1.1)X_t=Φ(t),t∈[-τ,0].(1.2)的数值解.其中,B=(B~1,...,B~q)是q维分数阶布朗运动(fBm),Hurst指数H∈(1/2,1),τ0是延迟项,系数函数F:R~(2d)→R~d,G=(G~1,...,G~q):R~d→R~(d×q),Φ(t)∈     

变分方法对一类p阶超线性脉冲微分方程的应用

本文研究一类p阶超线性脉冲微分方程—(|z'(t)|~(p-2)z'(t))'=g(t,z(t),z'(t)),t≠t_i,a.e.,t∈[0,T],△z'(t_i)=I_i(z(t_i)),i=1,2,...,l,z(0)=z(T)=0,(0.1)其中p≥2.对上述问题(0.1)的研究将转化为对下列辅助问题的研究:—(|z'(t)|~(p-2)z'(t))'=g(t,z(t),ω'(t)),t≠t_i,a.e.,t∈[0,T],△z'(t_i)=I_i(z(t_i)),i=1,2,...,l,z(0)=z(T)=0.(0.2)与已有的利用变分法研究脉冲方程的文献相比,本文的不同之处有两点,其一是本文没有直接对问题(0.1)应用变分法,原因是本文研究的问题类型不同于已有文献;其二是本文没有作通常的PalaisSmale(简记为(PS))型紧性条件假设,而是采用了更为广泛的假设,本文给出两个例子和一个命题以说明本文假设的广泛性.     

巴拿赫空间中线性离散时间系统的多项式二分性

本文给出了巴拿赫空间中线性差分方程的两个多项式二分性概念, 使其在相应空间中的范数的增长速度不快于指数型增长. 并用实例阐释了相关概念之间的关系. 借助于指数二分性的研究方法讨论了多项式二分性的特征, 所得结论推广了指数稳定性及指数二分性中的一些已有结果.     

A CLASS OF NONLOCAL SIINGULARLY PERTURBED NONLINEAR BOUNDARY VALUE PROBLEMS

Considerthesingularlyperturbednonlinearproblemfory~(y,?'...7/.)"oftheformwhereTy=(Tlyll'..5Tny.)'(Tdenotestransposition)whileEisasmallpositiveparameter,f=(fi,...?fn)",g=(gi,...,g.)',A=(Al,''tA.)',B=(BI,'',B.)'anda20,b20,a 5>0.Manyant,horssuchasHowes[1],…     

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.     

伪双曲型积分-微分方程的H~1-Galerkin混合元法误差估计

<正>1引言考虑如下一类具有Lipschitz连续边界(?)Ω的凸有界区域Ω上的伪双曲型积分微分方程其中Ω(?)R~d,(d=1,2,3)J=(0,T],对于固定的T,0T∞,函数0a_0≤a(x,t)≤     

GLOBAL EXISTENCE OF SOLUTIONS FOR A STRONGLY COUPLED REACTION-DIFFUSION SYSTEM

1IntroductionInthispaper,weinvestigatetheglobalexistencealldtllliqllellessofthesmoothsolutiolltotilefollowingIBVPHerefiCR'1isaboulldeddomainwithsllloothboundaryoff,alldal,uZarcunkllowllfullctiolls,QT=(0,T)xfi,ST=(0,T)xoff,Tisanarbitrarypositiveconstallt,R=(--co, co).BI.'Itk=orb(f,x:)% aam(t,x)'(Lk,kistheoutwardnormalderivative,AistheLaplacianoperator,7t(k=(%,....%)(k=1,2)arethegradients.nisanarbitrarypositiveinteger.IBVP(1.1)coillesfi.ommanymathematicalalldphysicalproblelll.Ill11],fo…     

具无限时滞的非线性积分微分方程的周期解

本文考虑具无限时滞非线性积分微分方程和其中t∈R,T≥0是常数,x∈Rn;A(t,x),C(t,s)为n×n连续的函数矩阵;f(t,x),g(t,x),b(t)是n维连续向量.本文利用线性系统的指数型二分性理论和不动点定理研究此系统,建立了保证其周期解存在性.唯一性的充分条件.得到了一些新的结果,推广了相关文献的主要结果.     

广义指数型二分性一个判别法和有界解存在性

该文给出线性系统(dx/dt)=A(t)x具有广义指数型二分性一个充分条件，作为应用研究非线性扰动系统的有界解的存在性. 推广了［1］与［2］结果.     

Bounds on eigenvalues and chromatic numbers

We give new bounds on eigenvalue of graphs which imply some known bounds. In particular, if T(G) is the maximum sum of degrees of vertices adjacent to a vertex in a graph G, the largest eigenvalue ρ(G) of G satisfies with equality if and only if either G is regular or G is bipartite and such that all vertices in the same part have the same degree. Consequently, we prove that the chromatic number of G is at most with equality if and only if G is an odd cycle or a complete graph, which implies Brook's theorem. A generalization of this result is also given.     

Lyapunov Functions for General Nonuniform Dichotomies

For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.     

THE BEHAVIOR OF SOLUTIONS IN THE VICINITY OF A BOUNDED SOLUTION TO AUTONOMOUS DIFFERENTIAL EQUATIONS

By using the exponential dichotomy,this paper investigates the behavior of solutionsin the vicinity of a bounded solution to the autonomous differential systemdx/dt=f(x).(1)Suppose x=u(t)is a nontrivial bounded solution of system(1).By discussing theequivalent equations of system(1)dθ/dt=1 (p,θ)dp/dt=A(θ)p (p,θ)(2)with respect to the moving orthonormal transformationx=u(θ) s(θ)p,the author proves that if linear system corresponding to(2)admits exponential dichotomy,then the given bounded solution x=u(t)should be periodic.The author also discusses thestadility of the obtained periodic solution.Finally,this paper discusses perturbation of thebounded solution of autonomous system(1).     

Affine-Periodic Solutions and Pseudo Affine-Periodic Solutions for Differential Equations with Exponential Dichotomy and Exponential Trichotomy

It is proved that every $(Q,T)$-affine-periodic differential equation has a $(Q,T)$-affine-periodic solution if the corresponding homogeneous linear equation admits exponential dichotomy or exponential trichotomy. This kind of ``periodic'' solutions might be usual periodic or quasi-periodic ones if $Q$ is an identity matrix or orthogonal matrix. Hence solutions also possess certain symmetry in geometry. The result is also extended to the case of pseudo affine-periodic solutions.     

A GENERALIZATION OF A PROPOSITION ON EXPONENTIAL DICHOTOMY

1 Introduction and Statement of TheoremConsider systemx' = A(t)x f(t, x), (l.1)where x E R", A(t) is a continuous matrix function, f: R x R" - R" is acontinuous function.We say that the linear differential equation X' = A(t)x admits an exponential dichotomy, if it has a fundamental matrix X(t) such thatIX(t)PX--'(s)l 5 K' e--a(t--s) for s S t,(1.2)IX(t)(I -- P)X--'(s)I 5 K' e--a(s--t) for s 2 t, (1'2)where P is a projection (P' = P), K and a are positive constants.Remark Without …     

非线性三点共轭边值问题正解存在性的特征值准则

本文提出了三点边值问题-v″(t)=b(t)f(v(t)),满足v′(0)=0及v(1)=αv(η)的共轭问题-u″(t)=b(t)f(u(t)),u′(0)=u(1)=0及u′_+(η)-u′_-(η)=αu′(1),得到了相应的Green函数．将其转化为Hammertein型积分方程,借助于其相应线性问题的第一特征值,利用锥上的不动点指数理论,给出了共轭问题单个正解及多个正解存在的特征值准则．     

On the exponential dichotomy of linear difference equations

We consider a system of linear difference equationsx ⁿ⁺¹ =A (n)xⁿ in anm-dimensional real or complex spaceVsum with detA(n) = 0 for some or alln εZ. We study the exponential dichotomy of this system and prove that if the sequence {A(n)} is Poisson stable or recurrent, then the exponential dichotomy on the semiaxis implies the exponential dichotomy on the entire axis. If the sequence {A (n)} is almost periodic and the system has exponential dichotomy on the finite interval {k, ...,k +T},k εZ, with sufficiently largeT, then the system is exponentially dichotomous onZ.     

Multiple bifurcation from “simple” eigenvalues

Consider an operator equation G(u,λ) = 0 where λ is a real parameter. Suppose 0 is a “simple” eigenvalue of the Fréchet derivative G_u at (u₀, λ₀). We give a hierarchy of conditions which completely determines the solution structure of the operator equation. It will be shown that multiple bifurcation as well as simple bifurcation can occur. This extends the standard bifurcation theory from a simple eigenvalue in which only one branch bifurcates. We also discuss limit point bifurcations. Applications to semilinear elliptic equations and the homotopy method for the matrix eigenvalue problem are also given.     

THE PERTURBATION OF ALMOST PERIODIC SOLUTION OF ALMOST PERIODIC SYSTEM

By using the exponential dichotomy and the averaging method,a perturbation theoryis established for the almost periodic solutions of an almost differential system.Suppose that the almost periodic differential system(dx)/(dt)=f(x,t) ε~2g(x,t,ε)(1)has an almost periodic solution x=x_0(t,M)for ε=0,where M=(m_1,…,m_k)is theparameter vector.The author discusses the conditions under which(1)has an almostperiodic solution x=x(t,ε)such that x(t,ε)=x_0(t,M)holds uniformly.The results obtained are quite complete.