Remark on a Paper of Sababheh and Khalil

We show that any Banach space X that has a sequence of unit vectors weakly converging to 0 has a closed and bounded convex set that is not remotal. This extends the main result of Sababheh and Khalil, (Numer. Funct. Anal. Optim. 2008; 29:1166–1170).     

Approximation by matrices with restricted spectra

We investigate the properties of the approximation of a matrix by matrices whose spectra are in a closed convex set of the complex plane. We explain why the Khalil and Maher characterization of an approximant, which spectrum is in a strip, is not quite correct. We prove that their characterization is valid but for another kind of approximation. We formulate a conjecture which leads to some algorithm for computing approximants. The conjecture is motivated by numerical experiments and some theoretical considerations. Separately we consider the approximation of normal matrices.     

On the illumination of unbounded closed convex sets

In this note we prove that the illumination of an almost bounded closed convex set by minimum number of affine subspaces of given dimension can be reduced to the illumination of a bounded closed convex set of lower dimension. The work was supported by Hung. Nat. Found. for Sci. Research No. 326-0213     

On extreme points in separable conjugate spaces

It is proved that every bounded closed and convex subset of an arbitrary conjugate separable Banach space is the closed convex hull of its extreme points.     

On strongly exposed points and frechet differentiability

Using properties of convex functionals, it is shown that closed and bounded convex sets in a class of Banach spaces which includes separable conjugate spaces are the closed convex hulls of their strongly exposed points. This research was supported by the National Research Council of Canada, Grant A-3999.     

Approximate fixed points of nonexpansive mappings in unbounded sets

It follows from Banach’s fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, as we show in the present paper, there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points.     

A counterexample to the Bishop-Phelps Theorem in complex spaces

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX ^*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.     

一种反向集值鞅型序列的收敛性

本文研究了在非空有界闭凸集的上鞅收敛性和有界闭凸集的适应可积集值序列的收敛性.引入了反向集值mil的基础上,得到了收敛性定理及定理证明.     

Almost Chebyshev set with respect to bounded subsets

The uniqueness and existence of restricted Chebyshev center with respect to arbitrary subset are investigated. The concept of almost Chebyshev sets with respect to bounded subsets is introduced. It is proved that each closed subset in a reflexive locally uniformly convex (uniformly convex, respectively) Banach space is an almost Chebyshev subset with respect to compact convex subsets (bounded convex subsets and bounded subsets, respectively). Project supported by the National Natural Science Foundation of China, Natural Science Foundation of Zhejiang Province, and the State Major Key Project for Basic Researchers of China.     

Dentability and extreme points in Banach spaces

It is shown that a Banach space E has the Radon-Nikodym property (equivalently, every bounded subset of E is dentable) if and only if every bounded closed convex subset of E is the closed convex hull of its strongly exposed points. Using recent work of Namioka, some analogous results are obtained concerning weak¹ strongly exposed points of weak¹ compact convex subsets of certain dual Banach spaces.     

关于真有效点集的连通性

在局部有界的Ｈａｕｓｄｏｒｆｆ局部凸空间中讨论了集合的真有效点集的连通性问题。证明了当序锥具有基底时,任何非空紧凸集的真有效点集是连通的。     

Scalarization of Henig Proper Efficient Points in a Normed Space

In a general normed space equipped with the order induced by a closed convex cone with a base, using a family of continuous monotone Minkowski functionals and a family of continuous norms, we obtain scalar characterizations of Henig proper efficient points of a general set and a bounded set, respectively. Moreover, we give a scalar characterization of a superefficient point of a set in a normed space equipped with the order induced by a closed convex cone with a bounded base.     

Convex subrings of partially ordered rings

The relationship between a poring and a convex subring has been studied most successfully for porings with bounded inversion. The best conceivable results are known for real closed rings. The present paper focuses on the connections between the prime spectra and between the real spectra of a poring and a convex subring. Examples show that for arbitrary porings one may not expect a very close relationship. But with the assumption of bounded inversion or spectral compatibility the results are similar to those for real closed rings (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Banach空间中关于有界集的同时远达问题的适定性

本文研究Banach空间中关于有界集的同时远达问题的适定性,在集合的Hausdorff距离下,证明了：对自反局部一致凸Banach空间中的闭有界集K,使所有关于K的同时远达问题是适定的紧凸子集A全体在紧凸子集全体中是Gδ型集．     

Preservation of closed sets under non-archimedean operators

In recent years some characterizations of the operators which preserve closed sets, closed bounded sets and closed bounded convex sets had been given in the literature. The kinds of operators involved are semi-Fredholm and tauberian operators. In this paper we introduce non-archimedean tauberian operators and prove that they coincide with semi-Fredholm operators when the ground field is spherically complete. Also, we give some theorems of preservation of closed sets under these classes of operators.     

A fixed-point theorem for asymptotically contractive mappings

We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets.

     

Uniformly bounded set-valued Nemytskij operators acting between generalized Hölder function spaces

We show that the generator of any uniformly bounded set-valued Nemytskij composition operator acting between generalized Hölder function metric spaces, with nonempty, bounded, closed, and convex values, is an affine function.     

Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups

We give three equivalent conditions for weak convergence of almost orbits of an asymptotically nonexpansive commutative semigroup acting on a nonempty bounded closed convex subset of a uniformly convex Banach space whose dual has the Kadec property.     

A Note on the First Eigenvalue of Hypersurfaces

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔＬｅｔＭｂｅａｎｏｒｉｅｎｔｅｄｃｌｏｓｅｄｈｙｐｅｒｓｕｒｆａｃｅｏｆＭ,ａｎ（ｎ＋１）－ｄｉｍｅｎｓｉｏｎａｌｏｒｉｅｎｔｅｄｃｌｏｓｅｄＲｉｅｍａｎｎｍａｉｎｆｏｌｄ（ｃｌｏｓｅｄｍｅａｎｓｃｏｍｐａｃｔａｎｄｗｉｔｈｏｕｔｂｏｕｎｄａｒｙ）．ＤｅｎｏｔｅｔｈｅｆｉｒｓｔｅｉｇｅｎｖａｌｕｅｏｆｔｈｅＬａｐｌａｃｉａｎｆｏｒＭｂｙλ１．ＳｕｐｐｏｓｅｔｈａｔｔｈｅＲｉｃｃｉｃｕｒｖａｔｕｒｅｏｆＭ,Ｒｉｃ（Ｍ）,ｉｓｂｏｕｎｄ…     

有限区间上的函数K<1集压缩映象的性质

有界闭集上的函数Ｋ＜１集压缩映象是紧压缩映象，且是闭映象．有界闭凸集到自身的函数Ｋ＜１集压缩映象一定有不动点．