动力系统在两种尺度意义下的Lipschitz半稳定性

本文将半稳定性（即在某种限制条件下的稳定性）和Ｌｉｐｓｃｈｉｔｚ稳定性,两尺度稳定性概念结合,提出动力系统的（ｈ０,ｈ）－半稳定和（ｈ０,ｈ）－Ｌｉｐｓｃｈｉｔｚ（局部）半稳定性概念,并且Ｌｉａｐｕｎｏ类似函数给出相应的充要条件。     

Lipschitz常数缩减的散乱数据插值

在计算机辅助设计几何中，变差缩减是一个非常重要的概念，本文分析了函数的变差和Ｌｉｐｓｃｈｉｔｚ常数的关系，指出可以用Ｌｉｐｓｃｈｉｔｚ常数来控制变差，由于变差的概念只限于一维的情形，而Ｌｉｐｓｃｈｉｔｚ常数适用于任意维，这样在高维时就可用Ｌｉｐｓｃｈｉｔｚ常数缩减的概念来代替变差缩减的概念，文中构造性地证明了Ｌｉｐｓｃｈｉｔｚ常数缩减的散乱数据插值函数的存在性，并且对这类函数的性质及光滑性条件进行了讨论．     

非线性Lipschitz算子的Lipschitz对偶算子及其应用

在文山中我们对非线性Ｌｉｐｓｃｈｉｔｚ算子定义了其Ｌｉｐｓｃｈｉｔｚ对偶算子，并证明了任意非线性Ｌｉｐｓｃｈｉｔｚ算子的Ｌｉｐｓｃｈｉｔｚ对偶算子是一个定义在Ｌｉｐｓｃｈｉｔｚ对偶空间上的有界线性算子．本文还进一步证明：设Ｃ为 Ｂａｎａｃｈ空间 Ｘ的闭子集，Ｃ＊Ｌ为Ｃ的 Ｌｉｐｓｃｈｉｔｚ对偶空间，Ｕ为 Ｃ＊Ｌ上的有界线性算子，则当且仅当 Ｕ为 ｗ＊－ｗ＊连续的同态变换时，存在Ｌｉｐｓｃｈｉｔｚ连续算子Ｔ，使Ｕ为Ｔ的Ｌｉｐｓｃｈｉｔｚ对偶算子．这一结论的理论意义在于：它表明一个非线性Ｌｉｐｓｃｈｉｔｚ算子的可逆性问题可转化为有界线性算子的可逆性问题．作为应用，通过引入一个新概念──ＰＸ－对偶算子，在一般框架下给出了非线性算子半群的生成定理．     

局部凸空间上的泛函的非光滑分析及多目标规划

该文对定义于局部凸线性拓扑空间Ｘ上的泛函引入广义方向导数、广义梯度及满足Ｌｉｐｓ－ｃｈｉｔｚ条件等概念,证明了它们的几个重要性质,并举例说明这里满足Ｌｉｐｓｃｈｉｔｚ条件的概念是Ｂａ－ｎａｃｈ空间情形的严格推广．最后,作为上述结论及方法的应用,讨论定义于Ｘ的多目标数学规划,得出若干关于弱有效解的最优性条件．     

非线性Lipschitz连续算子的定量性质(Ⅲ)──glb-Lipschitz数

本文引进非线性Ｌｉｐｓｃｈｉｔｚ算子Ｔ的ｇｌｂ－Ｌｉｐｓｃｈｉｔｚ数ｌ（Ｔ）,并证明：ｌ（Ｔ）定量刻画非线性Ｌｉｐｓｃｈｉｔｚ连续算子全体所构成的赋半范算子空间中可逆算子Ｔ保持可逆的最大扰动半径,因而具有特别重要意义。所获结果被应用来建立“非线性扰动引理”、非线性算子条件数、推广线性算子逼近理论和建立与矩阵理论中Ｇｅｒｓｃｈｇｏｒｉｎ圆盘定理对应的非线性Ｌｉｐｓｃｈｉｔｚ连续算子谱集的包含域。     

非可微分二层Lipschitz规划的广义微分和广义方向导数

本文对构成函数为Ｌｉｐｓｃｈｉｔｚ函数的二层规划问题,利用非光滑分析工具,讨论了下层极值函数和上层复合目标函数的Ｌｉｐｓｃｈｉｔｚ连续性,给出了这些函数的广义微分和广义方向导数的估计式。本文得到的结果为进一步研究非可微二层Ｌｉｐｓｃｈｉｔｚ规划的最优性条件和有效算法等理论和方法问题奠定了基础。     

非线性Lipschitz-α算子的若干性质

本文引入并研究Ｌｉｐｓｃｈｉｔｚ－α算子（简称Ｌｉｐ－α算子）我们首先给出这类算子的定义及其基本性质;然后,讨论Ｌｉｐ－α算子的可逆性并引入它的α－阶条件数,并给出其在研究非线性算子方程扰动问题中的一个应用;其次,还研究了Ｌｉｐ－α算子列的收敛性,引入并研究了Ｌｉｐ－α极限与Ｌｉｐ－α　Ｃａｕｃｈｙ列,证明过零Ｌｉｐ－α算子空间是一个Ｂａｎａｃｈ空间．     

关于广义梯度＊弱上半连续的注记

在本文中，我们建立了集值映射的拟闭性和序列拟闭性的概念。在讨论巴拿赫空间上局部Ｌｉｐｓｃｈｉｔｚ函数的广义梯度的性质时，得到了集值映射ｘ→δｆ（ｘ）的＊弱上半连续与＊弱拟闭性等价的结果。并通过例子说明了＊弱拟闭性弱于ｘ弱序列拟闭性，从而改正了文（１）中的某些错误。     

Banach空间中渐近正则的Lipschitz半群的不动点定理

本文首先定义了渐近正则的Ｌｉｐｓｃｈｉｔｚ半群的概念．其次，证明了ｐ一致凸Ｂａｎａｃｈ空间中渐近正则的Ｌｉｐｓｃｈｉｔｚ半群的不动点定理．同时也证明了具有正规结构系数的一致凸Ｅａｎａｃｈ空间中的渐近正则的Ｌｉｐｓｃｈｉｔｚ半群的一个新的不动点定理．     

吸引子的强稳定性与Ω-极限集的性质

本文讨论紧Ｈａｕｓｄｏｒｆｆ拓扑空间上流的吸引子在小扰动下的坚持性和强稳定性,以及Ω－极限集的一些性质．     

On attractors and stability for semigroup actions and control systems

The present paper is dedicated to the study of various aspects of attraction and stability for semigroup actions on topological spaces. The main purpose is to present the connections among the distinct notions of attractors and stable sets. The concept of Conley attractor is also investigated and related to the other notions of attractors. All the results are applied to the theory of control systems.     

Topological properties of asymptotically stable sets

Topological properties of the domain of attraction for dynamical systems are investigated. The main purpose of this paper is to prove that a compact, asymptotically stable attractor of a dynamical system defined on a locally compact metric space is a deformation retract of its domain of attraction, in a weak sense that is made precise. Under additional local assumptions, the attractor can be shown to be a retract, a deformation retract, or a strong deformation retract. The well known result that the domain of attraction of an asymptotically stable equilibrium is contractible follows as a corollary.     

Pointed shape and global attractors for metrizable spaces

In this paper we consider two notions of attractors for semidynamical systems defined in metric spaces. We show that Borsuk's weak and strong shape theories are a convenient framework to study the global properties which the attractor inherits from the phase space.Moreover we obtain pointed equivalences (even in the absence of equilibria) which allow to consider also pointed invariants, like shape groups.     

On global attractors of the 3D Navier-Stokes equations

In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier-Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.     

非线性Sobolev-Galpern方程的有限维整体吸引子

本文研究非线性Sobolev-Galpern方程解的渐近性态．首先证明了该方程在H^2(Ω)∩H0^1(Ω)中整体弱吸引子的存在性，然后利用一个能量方程证明了整体弱吸引子实际上是整体强吸引子，建立了整体吸引子的有限维性．     

GLOBAL BEHAVIOR IN A RATIONAL RECURSIVE SEQUENCE

In this paper, we study the global stability, and the periodic character of the rational recursive sequence. We show that the positive equilibrium of the sequence is a global attractor with a basin which depends on certain conditions posed on the coefficients.     

Can Soliton Attractors Exist in Realistic 3+1-D Conservative Systems?

High-energy physicists already know that stable attractors (solitons) can exist in 3+1-dimensional conservative Lagrangian systems, so long as the definition of an attractor is based on weak notions of stability and the fields admit topological charge. This paper explores the possibility of attractors in Lagrangian field theories without topological charge, using a new, stronger concept of stability—Convective quantized Asymptotic Orbital Stability (ChAOS) . Under certain conditions, ChAOS is related to additive Liapunov stability or energetic stability. Russian physicists have argued that such stability tends to require topological charge; however, this paper describes systems which avoid those arguments, and suggests how numerical examples might be constructed. Solitons have been proposed to explain the existence and nature of elementary particles within the Feynman version of quantum theory; Section 6cites this literature, as well as new possibilities for alternative versions with testable nuclear implications.     

二阶非线性差分方程的全局吸引性

[摘　要] 研究了差分方程xn+1=a-bxn-1A-xn(a≥0,A≥b≥0)的全局稳定性和正解的周期性.证明了方程的一个正平衡点是一个全局吸引子,并给出了相应的吸引域.     

LIPSCHITZ STABILITY OF GENERAL CONTROL SYSTEMS Ⅱ.IN TERMS OF TWO MEASURES

1IntroductionStabilityintermsoftwomeasures,thisconceptenableustounifyava-rietyofstabilityconceptsandtoofferageneralandunifiedframeworkforinvestigationofstabilitytheory.AndtheoriginalLiapunovtheoremshavebeenextended,generalizedandweakenedinvariousaspects[3…     

高阶非线性差分方程的全局吸引性

研究了差分方程 xn 1 =a - bxn- k A - xn( a≥ 0 ,A≥ b≥ 0 )的全局稳定性和正解的周期性质 .证明了方程的一个正平衡点是一个全局吸引子 ,并给出了相应的吸引域