正则余弦算子函数的内插和外插定理

该文研究正则余弦算子函数的内插和外插．证明了线性算子A在Banach空间X中生成一个指数有界的C-正则余弦函数当且仅当存在Banach空间Y和线性算子B使得：［R这里是C在Y中的有界扩张，B在Y中生成一个强连续余弦算子函数且A＝B|x．     

抽象Cauchy问题的适定性与算子半群

在算子A非稠定、问题解非指数有界的情况下，研究抽象Cauchy问题的适定性及其与A生成的算子族之间的关系．首先，引进（ACP1）的C适定性概念和C半群生成元的全新定义，证明：（ACP1）是C适定的充要条件是A生成C半群．并给出A生成非指数有界C半群的充分条件．另外，引进（ACP2）的（n，k）适定性定义，并讨论（n，k）适定性与积分余弦函数的关系．     

退化正则半群

引入了退化正则半群的定义，给出退化正则半群的一些基本性质，并证明了用多值线性算子刻划的指数有界退化正则半群的生成定理．     

C—正则光滑分布群与谱映射定理

我们在不假设生成元稠定、Ｃ具有稠值域的条件下，引进了Ｃ－正则光滑分布群的概念，建立了Ｃ－正则光滑分布群、无界算子演算、多项式有界积分Ｃ－群等概念之间的关系，得到一个推广的Ｓｔｏｎｅ定理。我们还给出与光滑分布群相联系的无界算子演算的谱映射定理。     

双连续半群的概率饱和定理

利用Riemann-stieltjes随机过程、矩生成函数及算子值数学期望讨论了双连续C_0半群的概率逼近问题给出了指数有界的双连续C_0半群的概率饱和定理.     

∑e^1型Banach空间中某些C0半群和余弦族的生成元的连续性

给出在∑e^1型Banach空间中一致有界函半群的生成元是有界线性算子的若干充分条件．证明了在∑e^1型Banach空间中由Hermitian算子或由等距算子组成的函半群的生成元都是有界线性算子．证明了在∑e^1型Banach空间中每个强连续非拟解析余弦族的生成元必是有界线性算子．     

C(t)-I是紧算子的余弦算子函数C(t)

C(t)，t∈R，是Banach空间X中的余弦算子函数，生成元是A，证明了：C(t)-I，A↓t∈R，紧的充要条件是生成元A紧．     

局部积分余弦算子函数的生成定理

本文讨论局部k次积分Cosine算子函数，在不假定其生成元稠定每件下，建立一个Hille—yosida型定理．     

分布余弦函数

本给出分布余弦函数的定义，其中包括生成元为多值算子情形，并讨论退化性型分布余弦函数与退化型二阶Cauchy问题、退化型积分余弦函数的关系，最后说明了非退化分布余弦函数的生成元亦生成正则余弦函数。     

C半群的概率逼近问题

借助Pettis积分、随机过程、矩生成函数及算子值数学期望,给出了一般形式的C半群概率逼近指数公式、生成定理及其收敛速度的估计式,也从另一个角度得出C半群概率表示的Vonorovskaya型渐近公式.     

双连续n次积分C余弦函数的逼近定理

基于双连续半群概念,引入一致双连续半群序列概念,借助Laplace变换和Trotter-Kato定理,考察双连续n次积分C余弦函数与C-预解式之间的关系,得到逼近定理的稳定性条件,进而得出双连续n次积分C余弦函数逼近定理.从而对Banach空间强连续半群逼近定理和双连续半群逼近定理进行了推广,为相应抽象的Cauchy问题提供了解决方案.     

On Integrated Solutions to Abstract Cauchy Problem

51.IntroductionSinceArendt[2j,Kellermann[3j,KellermannandHieder[4j,andNeubrarder[5]es-tablishedthetheoryof"integratedsemigroups",theexponentiallyboundedintegratedsemi-groupshavebeenstudiedintensivly-Butthereexistintegratedsemigroupswicharenotexpo-nentiallybounded.ItiswellknownthatmanyresultsforexponentiallyboundedintegratedsemigroupsarefoundbyusingLapIacetransformtechniques.ltshouldbenotedthatLaplacetransformtechniguesarenotavailableforintegratedsemigroupswhicharenotexponentiallybounded.It…     

STRONGLY CONTINUOUS INTEGRATED COSINE OPERATOR FUNCTIONS WITH GROWTH ω

For a continuous increasing function ω: [0, ∞) → (0, ∞) of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O(ω). It includes the well-known polynomially bounded and exponentially bounded cases.     

Liouville theorem and coupling on negatively curved Riemannian manifolds

By using probabilistic approaches, Liouville theorems are proved for a class of Riemannian manifolds with Ricci curvatures bounded below by a negative function. Indeed, for these manifolds we prove that all harmonic functions (maps) with certain growth are constant. In particular, the well-known Liouville theorem due to Cheng for sublinear harmonic functions (maps) is generalized. Moreover, our results imply the Brownian coupling property for a class of negatively curved Riemannian manifolds. This leads to a negative answer to a question of Kendall concerning the Brownian coupling property.     

LOCAL C-SEMIGROUPS AND LOCAL C-COSINE FUNCTIONS

0IntroductionLetB(X)betheBanachspaceofallboundedlinearoperatorsonaBanachspaceX.CEB(X)isinjective.Alloperatorsarelinear.D(A),Im(A),andP(A)denotethedomain,theimage,andtheresolventsetofoperatorA,respectivelyNandRdenotethesetofallnaturalnumbersandthesetofallrealnumbers.TheconceptoflocalC-sernigroupsalldlocalC-cosinefunctionwereintroducedbyTanakaandOkazawain[1]tF.HuangandT.Huangin[21,respectively.Aspointedoutin[3]and[2]thatnotalllocalC-sendgroups(C--cosinefunctions)canbeextendedtobeC-se…     

STRONGLY CONTINUOUS INTEGRATED COSINE OPERATOR FUNCTIONS WITH GROWTH ω

For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includes the well-known polynomially bounded and exponentially bounded cases.     

一类修正的和与积分混合型算子的$L_p$逼近

The generalized summation integral type operators with Beta basis functions are widely studied. At present, the investigations for the properties of these operators are only limited to the functions of bounded variation. Some authors studied the rate of point-wise rate of convergence, asymptotic formula of Voronovskaja type, and some direct results about these type of operators. The present paper considers the direct, inverse and equivalence theorems of modified summation integral type operators in the Lp spaces.     

Representation Theorem for Mild C-existence Families

As we know,the representation theorem is one of the most important theorems foroperator semigroup theory.The representation theorem for exponentially bounded C-semigroup with dense range R(C) was given in [1 ] ,which was generalized to the casethat C does notnecessarily have dense range(see[2 ] ) .In this paper,more general caseis considered.We get a representation theorem for exponentially bounded mild C-exis-tence families.According to this,a simple proof of a construction theorem for C-s…     

C-cosine Operator Functions and the Second Order Abstract Cauchy Problem

Ｃ－ｃｏｓｉｎｅＯｐｅｒａｔｏｒＦｕｎｃｔｉｏｎｓａｎｄｔｈｅＳｅｃｏｎｄＯｒｄｅｒ　ＡｂｓｔｒａｃｔＣａｕｃｈｙＰｒｏｂｌｅｍＷａｎｇＨａｉｙａｎ（王海燕）（ＤｅｐａｒｔｍｅｔｎｏｆＭａｔｈｅｍａｔｉｃｓａｎｄＭｅｃｈａｎｉｃｓ，Ｓｏｕｔｈｅａｓｔ...