METRIC GENERALIZED INVERSE OF LINEAR OPERATOR IN BANACH SPACE＊＊＊

The Moore-Penrose metric generalized inverse T of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moore-Penrose metric generalized inverse T is usually homogeneous and nonlinear in general. By means of the methods of geometry of Banach Space, the necessary and sufficient conditions for existence, continuitv, linearity and minimum property of the Moore-Penrose metric generalized inverse T will be given, and some properties of T will be investigated in this paper.     

线性算子双半群的扰动定理

本文研究有界线性算子强连续双半群的扰动问题。文中首先研究与强连续双半群母元有关的算子方程的可解性与算子的相似性。在此基础上证明了在一定条件下可化为指数衰减的强连续双半群经适当扰动后仍是一个可化为指数衰减的强连续双半群。     

THE MINIMAL PROPERTY OF THE CONDITION NUMBER OF INVERTIBLE LINEAR BOUNDED OPERATORS IN BANACH SPACES

In this paper we show that in error estimates, the condition number k(T) of any in-vertible linear bounded operator T in Banach spaces is minimal. We also extend the Hahn-Ba-nach theorem and other related results.     

Banach空间中算子的秩定理

设E和F是Banach空间,B(E,F)表示映E到F的有界线性算子全体.记T0+∈B(F,E)为To∈B(E,F)的一个广义逆.本文证明,每一个具有||T0+(T-T0)|| J<1的算子T∈B(E,F),B≡(I+T0+(T-T0))-1T0+是T的广义逆当且仅当(I-T0+T0)N(T)=N(T0),其中N(·)表示括弧中算子的零空间.这一结果改进了Nashed和Cheng的一个有用的定理,并进一步证明Nashed和Cheng的一个引理对半-Fredholm算子有效但一般未必成立。     

Banach空间中算子的秩定理

设E和F是Banach空间,B(E,F)表示映E到F的有界线性算子全体.记T+0 ∈ B(F,E)为T0 ∈ B(E,F)的一个广义逆.本文证明,每一个具有‖T+0(T-T0)‖＜1的算子T ∈ B(E,F),B≡(I+T+0(T-T0))-1T+0是T的广义逆当且仅当(I-T+0T0)N(T)=N(T0),其中N(·)表示括弧中算子的零空间.这一结果改进了Nashed和Cheng的一个有用的定理,并进一步证明Nashed和Cheng的一个引理对半-Fredholm算子有效但一般未必成立.     

有界线性算子空间中最佳逼近的强惟一性

研究了有界线性算子空间中最佳逼近的强惟一性问题，给出了线性和非线性最佳逼近的强惟一性定理。     

Banach空间中线性算子的Tseng度量广义逆

在 Ｂａｎａｃｈ空间中,利用 Ｂａｎａｃｈ几何方法及度量投影算子,将 Ｅ．Ｈ．Ｍｏｏｒｓ的学生,曾远荣（Ｙ．Ｙ． Ｔｓｅｎｇ）在 Ｈｉｌｂｅｒｔ空间中为线性算子引入的 Ｔｓｅｎｇ广义道,推广到 Ｂａｎａｃｈ空间,引入 Ｔｓｅｎｇ度量广义逆（此时的 Ｔｓｅｎｇ度量广义逆一般为齐性算子,而非线性算子）,利用 Ｂａｎａｃｈ空间对偶映射与广义正交分解定理给出 Ｔｓｅｎｇ度量广义道存在的充分必要条件．讨论了最大Ｔｓｅｎｇ度量广义逆在最优化,控制论及微分方程不适定问题有着直接应用的一些基础性质．     

ON THE GENERALIZED RESOLVENT OF LINEAR PENCILS IN BANACH SPACES

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil λ→ T λ S are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.     

Banach空间中一类广义Lipschitz非线性算子迭代序列的收敛定理

1　引言与预备知识设 X为一实 Banach空间 ,X*是 X的对偶空间 ,正规对偶映射 J:X→ 2 X*定义为 :J( x) ={ f∈ X*;〈x,f〉 =‖ f‖ .‖ x‖ ,‖ f‖ =‖ x‖ }其中〈· ,·〉表示 X和 X*的广义对偶组 .用 j(· )表示单值的正规对偶映射 .设 K是 X的一非空子集 ,算子 T:K→ X称为φ-强增生的[1 ,2 ] ,如果存在一个严格增加函数φ:[0 ,+∞ )→ [0 ,+∞ ) ,φ( 0 ) =0满足 x,y∈ K, j( x-y)∈ J( x-y)使得〈Tx -Ty,j( x -y)〉≥φ(‖ x -y‖ ) .‖ x -y‖ ( 1 )( 1 )中若 φ( t) =kt(其中 k>0 ) ,相应地称 T为强增生算子 ,k称为 T的…     

（1.2） INVERSES OF OPERATORS BETWEEN BANACH SPACES AND LOCAL CONJUGACY THEOREM

1.(1.2)InverseandLocalFinePropertyofaFamilyofOperatorsTxLetEandFbebothBanachspaces,andB(E,F)thesetofallboundedlinearoperatorsfromEintoFAnoperatorT B(F,E)issaidtobea(1.2)inverseofTifTT T=TandT TT =T .IfT satisfiesonlythefirstcondition,thenT issaidtobe...     

A NEW CLASS OF BILEVEL GENERALIZED MIXED EQUILIBRIUM PROBLEMS IN BANACH SPACES

A new class of bilevel generalized mixed equilibrium problems involving setvalued mappings is introduced and studied in a real Banach space. By using the auxiliary principle technique, new iterative algorithms for solving the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems involving set-valued mappings are suggested and analyzed. Existence of solutions and strong convergence of the iterative sequences generated by the algorithms are proved under quite mild conditions. The behavior of the solution set of the generalized mixed equilibrium problems and bilevel generalized mixed equilibrium problems is also discussed. These results are new and generalize some recent results in this field.     

Banach空间有界线性算子强连续双半群

本文在Ｂａｎａｃｈ空间上研究单参数有界线性算子族－强连续双半群。     

A STRONG CONVERGENCE THEOREM FOR QUASI-EQUILIBRIUM PROBLEMS IN BANACH SPACES

In this paper,we study an extragradient algorithm for approximating solutions of quasi-equilibrium problems in Banach spaces.We prove strong convergence of the sequence generated by the extragradient method to a solution of the quasi-equilibrium problem.     

Banach空间中带扰动的m-增生算子的零点与映象定理

设X为实Banach空间,TX D(T)→2x为m-增生算子,CD(T)→X为有界算子(未必连续),而C(T+I)-1为紧算子.假设     

A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES

This article is committed to deal with measure of non-compactness of operators in Banach spaces. Firstly, the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication) is a normed semigroup, and the mapping J from C(X) onto F(?) is a fully order-preserving positively linear surjective isometry, where ? is the closed unit ball of X*and F(?) the collection of all continuous and w*-lower semicontinuous sublinear functions on X*but restricted to ?. Furthermore, both ■ and ■ are Banach lattices and EK is a lattice ideal of EC. The quotient space EC/EK is an abstract M space, hence,order isometric to a sublattice of C(K) for some compact Haudorspace K, and(FQJ)C which is a closed cone is contained in the positive cone of C(K), where Q : E_C → E_C/E_K is the quotient mapping and F : E_C/E_K → C(K) is a corresponding order isometry. Finally, the representation of the measure of non-compactness of operators is given: Let B_X be the closed unit ball of a Banach space X, then■     

Banach空间中一类广义变分不等式的强收敛定理

应用广义f-投影算子引入了一类新的迭代方法,并用此方法在Banach空间的非紧子集上证明了一个关于一类广义变分不等式的强收敛定理;该定理中所涉及到的映射既不具备紧性也不具备单调性。所得结果推广了最近一些相关定理。     

EXPONENTIAL STABILITY OF LINEAR SYSTEMS IN BANACH SPACES

In this paper the author proves a new fundamental lemma of Hardy-Lebesgne class $\[{H^2}(\sigma )\]$ and by this lemma obtains some fundamental results of exponential stability of $\[{C_0}\]$-semigroup of bounded linear operators in Banach spaces. Specially, if $\[{\omega _s} = \sup \{ {\mathop{\rm Re}\nolimits} \lambda ;\lambda \in \sigma (A) < 0\} \]$ and $\[\sup \{ \left\| {{{(\lambda - A)}^{ - 1}}} \right\|;{\mathop{\rm Re}\nolimits} \lambda \ge \sigma \} < \infty \]$ , where \[\sigma \in ({\omega _s},0)\]) and A is the infinitesimal generator of a $\[{C_0}\]$-semigroup in a Banach space $X$, then $\[(a)\int_0^\infty {{e^{ - \sigma t}}\left| {f({e^{tA}}x)} \right|} dt < \infty \]$, $\[\forall f \in {X^*},x \in X\]$; (b) there exists $\[M > 0\]$ such that $\[\left\| {{e^{tA}}x} \right\| \le N{e^{\sigma t}}\left\| {Ax} \right\|\]$, $\[\forall x \in D(A)\]$; (c) there exists a Banach space $\[\hat X \supset X\]$ such that $\[\left\| {{e^{tA}}x} \right\|\hat x \le {e^{\sigma t}}\left\| x \right\|\hat x,\forall x \in X.\]$.     

Banach 空间中极大单调算子扰动的值域

设X为实Banach空间, T:D(T)(?)X→2X*为极大单调算子, C: D(T)(?)X→X*为有界算子(未必连续),而C(T+J)-1为紧算子．本文在上述假设条件下,通过附加一定的边界条件应用Leray-Schauder度理论研究了下述包含关系:0∈(T+C)(D(T)∩ BQ(0)),0∈(T+C)(D(T)∩ BQ(0));以及S(?)R(T+C), intS(?)intR(T+C)(其中S(?) X*);B+D(?)R(T+C),int(B+D)(?)intR(T+C)(其中 B(?)X*,D(?)X*)的可解性,得出了一些新的结论．     

A CHARACTERIZATION OF LINEAR OPERATORS IN CERTAIN PARANORMED SEQUENCE SPACES

Abstract

We define a new class B of sets of strongly summable and bounded sequences, determine their first and second α- and β-duals, and characterize linear operators on the spaces in B.     

环上矩阵的广义Moore-Penrose逆

本文研究环上矩阵的广义Moore-Penros逆,利用矩阵行空间与列空间的包含关系,给出其存在的充要条件及表达式．推广了以往文献的相应结果。