Strong Convergence Theorems for a Countable Family of Multi-Valued Bregman Quasi-Nonexpansive Mappings in Reflexive Banach Spaces

This article uses the shrinking projection method introduced by Takahashi, Kubota and Takeuchi to propose an iteration algorithm for a countable family of Bregman multi-valued quasi-nonexpansive mappings in order to have the strong convergence under a limit condition in the framework of reflexive Banach spaces. We apply our results to a zero point problem of maximal monotone mappings and equilibrium problems in reflexive Banach spaces. The results presented in the article improve and extend the corresponding results of that found in the literature.     

广义正交分解定理与Tseng度量广义逆

本文将Banach空间中广义正交分解定理从线性子空间拓广至非线性集—太阳集,分别给出了一算子为度量投影算子和一度量投影算子为有界线性算子的充要条件;得到了判别Banach空间中子空间广义正交可补的充要条件;建立了王玉文和季大琴(2000年)新近引入的Banach空间中的线性算子的Tseng度量广义逆存在的特征刻划条件;这些工作本质地把王玉文等人的新近结果从自反空间拓广至非自反空间的情形.     

On the Borelness of the intersection operation

We study the intersection operation of closed linear subspaces in a separable Banach space. We show that if the ambient space is quasi-reflexive, then the intersection operation is Borel. On the other hand, if the space contains a closed subspace with a Schauder decomposition into infinitely many non-reflexive spaces, then the intersection operation is not Borel. As a corollary, for a closed subspace of a Banach space with an unconditional basis, the intersection operation of the closed linear subspaces is Borel if and only if the space is reflexive. We also consider the intersection operation of additive subgroups in an infinite-dimensional separable Banach space, and show that if this intersection operation is Borel then the space is hereditarily indecomposable.     

Embedding Banach spaces with unconditional bases into spaces with symmetric bases

Every reflexive Banach space with unconditional basis is isomorphic to a complemented subspace of a reflexive Banach space with symmetric basis.     

Ｐ型基和良性有界算子

在自反Ｂａｎａｃｈ空间上一个有界线性算子是（Ｂ）型良性有界的当且仅当它的共扼算子也是（Ｂ）型的。但在非自反Ｂａｎａｃｈ空间上这种性质不成立。本文证明在一大类非自反Ｂａｎａｃｈ空间上总存在一个（Ｂ）型良性有界线性算子，它的共扼算子不是（Ｂ）型的。同时也证明了在具有基的Ｂａｎａｃｈ空间上，任何Ｐ型基序列一定有一个子序弄能够扩张成该空间的一个基。     

Reflexive cones

Reflexive cones in Banach spaces are cones with weakly compact intersection with the unit ball. In this paper we study the structure of this class of cones. We investigate the relations between the notion of reflexive cones and the properties of their bases. This allows us to prove a characterization of reflexive cones in term of the absence of a subcone isomorphic to the positive cone of $\ell _{1}$ . Moreover, the properties of some specific classes of reflexive cones are investigated. Namely, we consider the reflexive cones such that the intersection with the unit ball is norm compact, those generated by a Schauder basis and the reflexive cones regarded as ordering cones in Banach spaces. Finally, it is worth to point out that a characterization of reflexive spaces and also of the spaces with the Schur property by the properties of reflexive cones is given.     

On a proximal point method for convex optimization in banach spaces

We analyze the behavior of a parallel proximal point method for solving convex optimization problems in reflexive Banach spaces. Similar algorithms were known to converge under the implicit assumption that the norm of the space is Hilbertian. We extend the area of applicability of the proximal point method to solving convex optimization problems in Banach spaces on which totally convex functions can be found. This includes the class of all smooth uniformly convex Banach spaces. Also, our convergence results leave more flexibility for the choice of the penalty function involved in the algorithm and, in this way, allow simplification of the computational procedure.     

The dual quasisemigroup and controllability of evolution equations

We study the notion of dual quasisemigroups of bounded linear operators as a generalization of that for strongly continuous semigroup and prove some properties similar to the dual of a semigroup, among other things we prove that for reflexive Banach spaces the dual quasisemigroup is strongly continuous on (0,+∞). This allows us to extend some recent criteria of controllability to a general class of evolution equations in reflexive Banach spaces.     

自反Banach空间中锥拟凸映射多目标最优化问题锥有效解集的弱连通性

本文在自反Banach空间中引进了锥弱连续映射和点集的弱连通概念．在讨论锥弱连续和锥拟凸映射以及锥最小上界的几个性质的基础上,证明了当象集为锥凸集时定义在自反Ba- nach空间中的有界闭凸集上的锥弱连续和锥拟凸映射多目标最优化问题的锥有效解集是弱连通的．     

Strong convergence of a multiple-step iterative process for nonexpansive mappings

In this paper, we introduce a multiple-step iterative process for approximating a fixed point of nonexpansive mappings in the framework of uniformly smooth Banach spaces and reflexive Banach spaces which have a weakly continuous duality map, respectively. The results presented in this paper improve and extend the corresponding results of announced by many others.     

A characterization of conjugate WCG banach spaces

It is proved that a WCG Banach space X is isomorphic to a conjugate Banach space if and only if there exists a closed subspace V of its conjugate space X' with positive characteristic such that X possesses the following summability property with respect to V: For every bounded sequence in X there exists a regular essentially positive summability method A such that the A-means of the sequence are σ(X,V)-convergent in X. This extends a well-known theorem of Nishiura-Waterman [8] and yields an analogous characterization of quasi-reflexive spaces. Conjugate spaces of smooth Banach spaces can also be characterized by the above summability condition.     

On reflexive spaces having the metric approximation property

It is proved that some conjectures concerning non-separable reflexive Banach spaces are true for reflexive spaces having the metric approximation property.     

Weak and strong stationarity in generalized bilevel programming and bilevel optimal control

In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.     

Boundary conditions for linear manifolds. III. Infinite dimensional adjoints via solution spaces

A formula is given for the orthogonal complement of any vector subspace of l₂. Countably infinite adjoint subspaces in a Banach space are characterized via solution spaces. In particular, infinite dimensional self-adjoint subspaces in a reflexive Banach space are characterized via solution spaces, generalizing a result in Dunford and Schwartz [“Linear Operators, II,” Interscience, New York, 1963]. Applications are made to closed linear manifolds in l₂ ⊕ l₂ as well as infinite dimensional, generalized ordinary differential subspaces in a Hilbert space with the boundary conditions imposed on real sequences. The results are also expressed via solution spaces.     

Risk measures on ordered non-reflexive Banach spaces

This work is devoted to the study of coherent and convex risk measure on non-reflexive Banach spaces. An extension of dual representation and continuity results which hold in the case of reflexive spaces is established in this paper for the class of non-reflexive Banach spaces. This study also relies on the fact that the riskless bond is replaced by some numeraire asset which defines a base on the cone of the spot-price functionals.     

Monotone generalized variational inequalities and generalized complementarity problems

Some existence results for generalized variational inequalities and generalized complementarity problems involving quasimonotone and pseudomonotone set-valued mappings in reflexive Banach spaces are proved. In particular, some known results for nonlinear variational inequalities and complementarity problems in finite-dimensional and infinite-dimensional Hilbert spaces are generalized to quasimonotone and pseudomonotone set-valued mappings and reflexive Banach spaces. Application to a class of generalized nonlinear complementarity problems studied as mathematical models for mechanical problems is given.The research of the first author was supported by the National Natural Science Foundation of P. R. China and by the Ethel Raybould Fellowship, University of Queensland, St. Lucia, Brisbane, Australia.     

自反Banach空间上的半强制单调变分问题

仅用变分法的经典定理，研究了自反Ｂａｎａｃｈ空间上的半强制单调变分问题．获得了自反Ｂａｎａｃｈ空间上半线性方程解的存在性     

Bases and quasi‐reflexivity in Fréchet spaces

A Fréchet space E is quasi‐reflexive if, either dim(E″/E) < ∞, or E″[β(E″,E′)]/E is isomorphic to ω. A Fréchet space E is totally quasi‐reflexive if every separated quotient is quasi‐reflexive. In this paper we show, using Schauder bases, that E is totally quasi‐reflexive if and only if it is isomorphic to a closed subspace of a countable product of quasi‐reflexive Banach spaces. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Banach空间上闭线性子空间强正交可补的充分必要条件

证明了闭的极大线性子空间是强正交可补的充分必要条件是,空间X是自反严格凸的.     

Uniformly convex Banach spaces are reflexive—constructively

We propose a natural definition of what it means in a constructive context for a Banach space to be reflexive, and then prove a constructive counterpart of the Milman‐Pettis theorem that uniformly convex Banach spaces are reflexive.