度量投影的收敛性（英文）

讨论了赋范空间中度量投影的收敛性．得到了在局部紧集控制下，Chebyshev凸集序列的度量投影的收敛性与K－M收敛，Wijsman收敛和Kuratowski收敛都等价．本文的结论完善了M．Tsukada在[1]和[2]结果．     

On Differentiability of Metric Projections onto Moving Convex Sets

We consider properties of the metric projections onto moving convex sets in normed linear spaces. Under certain conditions about the norm, directional differentiability of first and higher order of the metric projections at boundary points is characterized. The conditions are formulated in terms of differentiability of multifunctions and properties of the set-derivatives are shown.     

Simultaneous metric projection onto closed sets

We examine simultaneous metric projection by closed sets in a class of ordered normed spaces. First, we study simultaneous metric projection onto downward and upward sets and separation properties of these sets. The results obtained are used for examination of simultaneous metric projection by arbitrary closed sets, and we examine the minimization of the distance from a bounded set to an arbitrary closed set in a class of ordered normed spaces.     

Best Approximation on Convex Sets in Metric Linear Spaces

In this paper the concepts of strictly convex and uniformly convex normed linear spaces are extended to metric linear spaces. A relationship between strict convexity and uniform convexity is established. Some existence and uniqueness theorems on best approximation in metric linear spaces under different conditions are proved.     

Minimizing Irregular Convex Functions: Ulam Stability for Approximate Minima

The main concern of this article is to study Ulam stability of the set of ε-approximate minima of a proper lower semicontinuous convex function bounded below on a real normed space X, when the objective function is subjected to small perturbations (in the sense of Attouch & Wets). More precisely, we characterize the class all proper lower semicontinuous convex functions bounded below such that the set-valued application which assigns to each function the set of its ε-approximate minima is Hausdorff upper semi-continuous for the Attouch–Wets topology when the set $\mathcal{C}(X)$ of all the closed and nonempty convex subsets of X is equipped with the Hausdorff topology. We prove that a proper lower semicontinuous convex function bounded below has Ulam-stable ε-approximate minima if and only if the boundary of any of its sublevel sets is bounded.     

Characterization of inner product spaces in V- and L-spaces

In [4], Freese and Murphy introduce a new class of spaces, the V-spaces, which include Banach spaces, hyperbolic spaces, and other metric spaces. In this class of spaces they investigate conditions which are equivalent to strict convexity in Banach spaces, and extend some of the Banach space results to this new class of spaces. It is natural to ask if known characterizations of real inner product spaces among Banach spaces can also be extended to this larger class of spaces. In the present paper it will be shown that a metrization of an angle bisector property used in [3] to characterize real inner product spaces among Banach spaces also characterizes real inner product spaces among V-spaces, and among another class of spaces, the L-spaces, which include hyperbolic spaces and strictly convex Banach spaces. In the process it is shown that in a complete, convex, externally convex metric space M, if the foot of a point on a metric line is unique, then M satisfies the monotone property, thus answering a question raised in [4].     

Pythagorean euclidean four point properties

Characterizations of real inner product spaces among a class of metric spaces have been obtained based on homogeneity of metric pythagorean orthogonality, a metrization of the concept of pythagorean orthogonality as defined in normed linear spaces. In the present paper a considerable weakening of this hypothesis is shown to characterize real inner product spaces among complete, convex, externally convex metric spaces, generalizing a result of Kapoor and Prasad [9], and providing a connection with the many characterizations of such spaces using euclidean four point properties.     

线性距离空间的一致凸性与自反性

本文研究了线性距离空间的一致凸性与自反性，同时对吴从Ｘｉｎ等提出的严格缩条件能否去掉的问题给出了肯定的答案。     

Probabilistische Topologien

Referring only to closed L-fuzzy sets we introduce a concept of probabilistic topological spaces including random metric spaces ([17]) statistical metric spaces ([9][15]) and fuzzy uniform spaces studied by Lowen [11]. In particular probabilistic topologies in the sense of Frank [5] satisfying the additional property (R3) are equivalent to systems of closed [0, 1]-fuzzy sets. Moreover random topologies as well as fuzzy topologies ([3],[13]) equipped with the property (03) can be considered as probabilistic topologies.     

On the Structure of the Complements of Chebyshev Sets

A set is called a Chebyshev set if it contains a unique best approximation element. We study the structure of the complements of Chebyshev sets, in particular considering the following question: How many connected components can the complement of a Chebyshev set in a finite-dimensional normed or nonsymmetrically normed linear space have? We extend some results from [A. R. Alimov, East J. Approx, 2, No. 2, 215--232 (1996)]. A. L. Brown's characterization of four-dimensional normed linear spaces in which every Chebyshev set is convex is extended to the nonsymmetric setting. A characterization of finite-dimensional spaces that contain a strict sun whose complement has a given number of connected components is established.     

Monotone generalized contractions in partially ordered probabilistic metric spaces

In this paper, a concept of monotone generalized contraction in partially ordered probabilistic metric spaces is introduced and some fixed and common fixed point theorems are proved. Presented theorems extend the results in partially ordered metric spaces of Nieto and Rodriguez-Lopez [Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], Ran and Reurings [A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443] to a more general class of contractive type mappings in partially ordered probabilistic metric spaces and include several recent developments.     

Linear selections for the metric projection

A characterization is given of those proximinal subspaces of a normed linear space whose (set-valued) metric projections admit linear selections. This characterization is applied in each of the classical Banach spaces C₀(T) and L_p (1 ? p ? ∞), resulting in an intrinsic characterization of those one-dimensional subspaces whose metric projections admit linear selections.     

The Set-Valued Dynamic System and Its Applications to Pareto Optima

In this paper, we first study the existence of endpoints for set-valued dynamic systems which are either upper or lower semicontinuous in metric spaces. Then the existence, uniqueness and algorithms of endpoints for set-valued dynamic systems which are either generalized contractions (defined in metric spaces) or topological contractions (defined in topological spaces which do not necessarily have any metric). These results are then applied to derive the existence of Pareto optima for mappings which take values in ordered Banach spaces. Finally, the stability of (generalized) nucleolus sets is also established.     

Rigid sets and nonexpansive mappings

We introduce a new class of normed spaces (not necessarily finite dimensional), which contains the finite dimensional normed spaces with polyhedral norm. We study the properties of rigid sets of the spaces of this class and we apply the results to limit sets of the sequences of iterates of nonexpansive maps.

     

On an equivalence of topological vector space valued cone metric spaces and metric spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces.     

TWO POROSITY RESULTS IN MONOTONIC ANALYSIS

ABSTRACT

In this work we consider spaces of increasing functions defined on a subset of an ordered normed space. We equip each of these spaces with a natural metric and show that the complement of the subset of all strictly increasing functions is σ-porous. We also discuss some properties of normal sets and strictly normal sets.     

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.     

Some Low Distortion Metric Ramsey Problems

In this note we consider the metric Ramsey problem for the normed spaces $\ell_p$. Namely, given some $1\le p \le \infty$ and $\alpha \ge 1$, and an integer $n$, we ask for the largest $m$ such that every $n$-point metric space contains an $m$-point subspace which embeds into $\ell_p$ with distortion at most $ \alpha$. In [1] it is shown that in the case of $\ell_2$, the dependence of $m$ on $\alpha$ undergoes a phase transition at $\alpha =2$. Here we consider this problem for other $\ell_p$, and specifically the occurrence of a phase transition for $p\neq 2$. It is shown that a phase transition does occur at $\alpha=2$ for every $p\in [1,2]$. For $p > 2$ we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every $1 < p < \infty$ there are arbitrarily large metric spaces, no four points of which embed isometrically in $\ell_p$.     

Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

复赋范空间中的同时太阳集的最佳逼近

研究了复赋范空间中的同时太阳集对无穷序列的最佳逼近问题，得到了特征及唯一性定理．