一般化凸空间上的一个连续选择定理(英文)

本文研究一般化凸空间上的连续选择定理.利用在D■X的条件下,一般化凸空间(X,D;Γ)上Γ-凸子集的概念,得到了两类一般化凸空间之间,以及φ映射和Γ-凸映射之间的关系,并且得到了一个连续选择定理.本文推广了一般化凸空间上凸子集的概念.     

Continuous selection from the sets of best and near-best approximation

The paper studies approximation and structural geometric-topological properties of sets in normed and more general (asymmetric) spaces for which there exists a continuous selection for the best and near-best approximation operators. Sufficient conditions on the metric projection of sets which ensure the existence of a continuous selection for this projection are obtained, and the structural properties of such sets are determined. The existence of a continuous selection for the near-best approximation operator on a finite-dimensional space more general than a normed space is investigated. It is shown that the lower semicontinuity of the metric projection is sufficient for the existence of a continuous selection for the near-best approximation operator in the general case.     

度量射影的连续选择

本文讨论了集值映射的弱连续选择并应用于度量射影．设Y是Banach空间X的子空间且Y是可分的,在相差一个第一纲集的情况下,于弱拓扑下,支撑于Y上的度量射影是下半连续的并有连续选择．     

Characterizations of intervals via continuous selections

We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.     

A selection theorem

It is shown that under certain assumptions the existence of a continuous selection on every compact subset of a metric space implies the existence of a continuous selection on the whole space. An application of this result to a problem concerning the extension of compact operators is given. Research supported in part by the National Science Foundation, U.S.A. (NSF-GP-378).     

Continuous Selections and Fixed Points for $ϕ$-Maps with Their Applications to Section Problems

The concept of finitely continuous topological space is introduced and the basic properties of the space are given. Several continuous selection theorems and fixed point theorems for $ϕ$-maps are established, and as applications of the above fixed point theorems, some section problems are discussed. The results generalize and improve many corresponding conclusions.     

度量射影的Lipschitz连续选择

It has shown that, for Lipschitz continuous metric projections, it has a Lips-chitz continuous selection for l-Chebyshev subspace and, for each subspace M which is ismorphism to a Hilbert space ,P_M admits a selection s which is locally Lipschitz continuous with index 1/2.     

On Maps Of Bounded p-Variation With p>1

This paper addresses properties of maps of bounded p-variation (p>1) in the sense of N. Wiener, which are defined on a subset of the real line and take values in metric or normed spaces. We prove the structural theorem for these maps and study their continuity properties. We obtain the existence of a Hölder continuous path of minimal p-variation between two points and establish the compactness theorem relative to the p-variation, which is an analog of the well-known Helly selection principle in the theory of functions of bounded variation. We prove that the space of maps of bounded p-variation with values in a Banach space is also a Banach space. We give an example of a Hölder continuous of exponent 0<<1 set-valued map with no continuous selection. In the case p=1 we show that a compact absolutely continuous set-valued map from the compact interval into subsets of a Banach space admits an absolutely continuous selection.     

乘积拓扑空间内的重合点组定理及应用(Ⅰ)

首先引入了无凸性结构的有限连续拓扑空间(简称FC-空间)新概念．其次在FC-空间内建立了一个新的连续选择定理．应用此定理,在很弱的假设下,对定义在非紧FC-空间的乘积空间上的两个集值映射簇证明了某些新的重合点定理．这些结果推广了最近文献中的许多已知结果．某些应用将在后继文章中给出．     

Selezioni continue o misurabili

We give a characterization of set-valued mappings from a topological (measure) space into the class of real non-empty intervals admitting a continuous (measurable) selection. As an application we obtain a characterization of set-valued functions defined on IR ⁿ admitting an approximately continuous or an approximately continuous and almost everywhere continuous selection.     

Topological convexities, selections and fixed points

A convexity on a set X is a family of subsets of X which contains the whole space and the empty set as well as the singletons and which is closed under arbitrary intersections and updirected unions. A uniform convex space is a uniform topological space endowed with a convexity for which the convex hull operator is uniformly continuous. Uniform convex spaces with homotopically trivial polytopes (convex hulls of finite sets) are absolute extensors for the class of metric spaces; if they are completely metrizable then a continuous selection theorem à la Michael holds. Upper semicontinuous maps have approximate selections and fixed points, under the usual assumptions.     

Continuous selections as parametrically defined integrals

An analog of the classical Michael theorem on continuous single-valued selections of lower semicontinuous maps whose values are closed and convex in a Fréchet space is proved for maps into metrizable (non-locally-convex) vector spaces. It turns out that, instead of the local convexity of the whole space containing these values, it is sufficient to require that the family of values of the map be uniformly locally convex. In contrast to the standard selection theorems, the proof bypasses the process of successively improving the approximations, and the desired selection is constructed as the result of pointwise integration with respect to a suitable probability distribution.     

Weak selections and weak orderability of function spaces

It is proved that for a zero-dimensional space X, the function space C _p (X, 2) has a Vietoris continuous selection for its hyperspace of at most 2-point sets if and only if X is separable. This provides the complete affirmative solution to a question posed by Tamariz-Mascarúa. It is also obtained that for a strongly zero-dimensional metrizable space E, the function space C _p (X, E) is weakly orderable if and only if its hyperspace of at most 2-point sets has a Vietoris continuous selection. This provides a partial positive answer to a question posed by van Mill and Wattel.     

Minimum selections and fixed points of set-valued operators in Banach spaces with some uniform convexity

A new continuity theorem of minimum selection is presented for a continuous set-valued operator from a topological space into a Banach space with some uniform convexity. As applications, some problems concerning minimum right inverses for linear operators and minimum fixed points for condensing set-valued nonlinear operators are discussed. Also, the existence of minimum solutions for an integral inclusion is proved.     

Selections and hyperspace topologies via special metrics

If (Y, d) is a complete metric space with a non-Archimedean metric d, then there exists a selection on the set of all nonempty closed subsets of Y which is continuous with respect to both the Vietoris and Hausdorff topologies on .     

A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces

We prove a continuous selection theorem for quasi-lower semicontinuous mappings with values that are closed sub-admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces.     

Evolutionary game theory on measure spaces: Well-posedness

An attempt is made to find a comprehensive mathematical framework in which to investigate the problems of well-posedness and asymptotic analysis for fully nonlinear evolutionary game theoretic models. The model should be rich enough to include all classical nonlinearities, e.g., Beverton–Holt or Ricker type. For several such models formulated on the space of integrable functions, it is known that as the variance of the payoff kernel becomes small the solution converges in the long term to a Dirac measure centered at the fittest strategy; thus the limit of the solution is not in the state space of integrable functions. Starting with the replicator–mutator equation and a generalized logistic equation as bases, a general model is formulated as a dynamical system on the state space of finite signed measures. Well-posedness is established, and then it is shown that by choosing appropriate payoff kernels this model includes all classical density models, both selection and mutation, and discrete and continuous strategy (trait) spaces.     

Continuous selection of the solution map for one-sided Lipschitz differential inclusions

In the paper we deal with differential inclusions with one-sided Lipschitz (OSL) continuous right-hand sides, and prove the existence of a continuous selection of the solution map which assigns to any point the set of solutions to the multivalued Cauchy problem. As an application we study the problem of the existence of viable trajectories in a prescribed closed subset of a Banach space via the generalized Wa?ewski retract method.     

Monotone inequalities,dynamical systems,and paths in the positive orthant of Euclidean <Emphasis Type="Italic">n</Emphasis>-space

Given monotone operators on the positive orthant in n-dimensional Euclidean space, we explore the relation between inequalities involving those operators, and induced monotone dynamical systems. Attractivity of the origin implies stability for these systems, as well as a certain inequality, the no-joint-increase condition. Under the right perspective the converse is also true. In addition we construct an unbounded path in the set where trajectories of the dynamical system decay monotonically, i.e., we solve a positive continuous selection problem.     

A continuous selection of the metric projection in matrix spaces,addendum

Summary In an earlier paper [1] the authors proved that the metric projection of the space ofm byn matrices onto the subset of matrices of rank less than or equal tok has a continuous selection. In this note we sharpen the result to show that the selection is in fact globally Lipschitz-continuous.Dedicated to R. S. Varga on the occasion to his sixtieth birthday