非自反Banach空间中的度量投影

该文给出非自反Banach空间中一类超平面上度量投影的表达式．在近严格凸Banach空间中,研究了它们的连续性．对于对偶Banach空间X*,给出弱*闭子集上度量投影的一些连续性结果．     

凸性与度量投影的连续性

本文研究近强凸、近非常凸Banach空间中度量投影的连续性。获得如下结果：若A是近强凸（近非常凸）空间中的逼近凸集,则度量投影PA是范-范上半连续的（范-弱上半连续的）。此外,我们还利用度量投影的连续性给出Banach空间为近强凸、近非常凸的一些充分必要条件。     

近可凹性和Banach空间的逼近紧性及度量投影算子的连续性

本文定义了近可凹的Banach 空间. 利用Banach 空间几何技巧证得: X 是逼近紧的当且仅当(1) X 是近可凹的; (2) X 是近严格凸的. 还证明了如果Banach 空间X 是近可凹的, 则对任意闭凸集C, 度量投影算子P_C 是上半连续的. 最后作者给出了近可凹性在广义逆理论中的应用.     

Banach空间中线性算子的(集值)度量广义逆的表示及应用

在Banach空间Y无自反和从Banach空间X到Y的线性算子T无闭值域和稠定的假定下,利用Banach空间几何方法证明了Banach空间中线性算子的度量广义逆是具有闭凸值的集值映射,建立了该度量广义逆的存在性、唯一性和等价表达式,并给出了此表达式的一个应用示例．所得的部分结果本质地拓广王玉文和潘少荣在Banach空间Y自反,从X到Y的线性算子T为闭值域和稠定的假定下的近期相应结果．     

本质上是Banach空间闭凸子集的凸度量空间

利用凸度量空间中凸结构的性质,在保持凸组合关系与等距嵌入意义下,获得了一个凸度量空间本质上是某个Banach空间中闭凸子集的充分必要条件,并举例说明了其应用.     

Banach空间中拟线性投影算子

证明了 :在自反 Banach空间＄X＄中 ,每个闭子空间 L,都存在 X到 L上的拟线性投影算子 SL.一般说来 ,SL 既非度量投影算子 ,又非线性算子 .     

Banach空间中线性流形上的度量投影的表达式

借助于正规对偶映射,建立了一般Banach空间中线性流形上的(集值)度量投影存在的 充要条件,同时给出了度量投影的表达式和点到线性流形上的距离公式.这些本质地推广和改进了 王玉文和于金凤在空间自反、严格凸和光滑强假定下的相应结果.     

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.     

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.     

Banach空间中集值度量广义逆齐性单值选择的判据

本文研究了Banach空间(X,‖·‖),(Y,‖·‖)上具有闭值域的稠定闭算子T:X→Y的(集值)度量广义逆.在限定X为自反的、Y为一般的Banach空间且算子值域R(T)为空间Y中Chebyshev子空间时,证明了算子T具有非空闭凸集值的度量广义逆的存在性,运用Banach空间中广义正交分解定理,得出算子T的集值度量广义逆具有唯一齐性单值选择,并且该单值选择恰为赋等价严格凸范数的空间Xr=(X,‖·‖_r)上算子T的Moore-Penrose度量广义逆.特别地,将抽象的Banach空间X与Y具体化为有限维Banach空间l₁ⁿ=(Rn,‖·‖₁)(即n维空间Rn赋l₁范数)与有限维Hilbert空间(即m维欧式空间l₂^m=(Rm,‖·‖₂),亦即m维空间赋l₂范数),线性算子T可具体表示为m×n阶矩阵A,得到了从n维空间l₁ⁿ到m维空间l     

Approximating fixed points of asymptotically nonexpansive mappings in Banach spaces by metric projections

In this paper, a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex and smooth Banach space is proved by using metric projections. This theorem extends and improves the recent strong convergence theorem due to Matsushita and Takahashi [S. Matsushita, W. Takahashi, Approximating fixed points of nonexpansive mappings in a Banach space by metric projections, Appl. Math. Comput. 196 (2008) 422–425] which was established for nonexpansive mappings.     

Banach空间中一类度量投影的判据及表达式

X为自反、严格凸Ｂａｎａｃｈ 空间,L为X中闭子空间,P：X→L为单值算子,该文给出P成为L上度量投影P_L的判据及P_L为线性算子的充分必要条件．在自反Ｂａｎａｃｈ空间中,利用对偶映射,给出超平面上（值）度量投影的表达式．对于自反、严格凸、光滑的Ｂａｎａｃｈ 空间中线性流形上的（单值）度量投影,利用广义右逆的表示,求出其表达式．在后继文章中将给出此表达式的应用．     

On the uniform continuity of metric projections in Banach spaces

Let M be a convex Chebyshev subset of a uniformly convex and uniformly smooth Banach space. It is proved that the metric projection P_M of X onto M is uniformly continuous on every bounded subset of X. Moreover, a global and explicit estimate on the modulus of continuity of the metric projection is obtained.     

On support points and continuous extensions

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by ${\mathcal{BCC}(X)}A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by BCC(X){\mathcal{BCC}(X)} the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S : BCC(X) ? X{S : \mathcal{BCC}(X) \rightarrow X} such that S(K) is a support point of K for each K ? BCC(X){K \in \mathcal{BCC}(X)}. Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X){\mathcal{BCC}(X)}.     

Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces

Recently, two retractions (projections), which are different from the metric projection and the sunny nonexpansive retraction in a Banach space, were found. In this article, using nonlinear analytic methods and new retractions, we prove a nonlinear ergodic theorem for positively homogeneous and nonexpansive mappings in a uniformly convex Banach space. The limit points are characterized by using new retractions.     

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.     

On the quantitative asymptotic behavior of strongly nonexpansive mappings in banach and geodesic spaces

We give explicit rates of asymptotic regularity for iterations of strongly nonexpansive mappings T in general Banach spaces as well as rates of metastability (in the sense of Tao) in the context of uniformly convex Banach spaces when T is odd. This, in particular, applies to linear norm-one projections as well as to sunny nonexpansive retractions. The asymptotic regularity results even hold for strongly quasi-nonexpansive mappings (in the sense of Bruck), the addition of error terms and very general metric settings. In particular, we get the first quantitative results on iterations (with errors) of compositions of metric projections in CAT(?)-spaces (? > 0). Under an additional compactness assumption we obtain, moreover, a rate of metastability for the strong convergence of such iterations.