Hausdorff dimension and doubling measures on metric spaces

Volberg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any , the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most .

     

Common fixed points in cone metric spaces

In this paper we consider a notion of g-weak contractive mappings in the setting of cone metric spaces and we give results of common fixed points. This results generalize some common fixed points results in metric spaces and some of the results of Huang and Zhang in cone metric spaces. Supported by Universitá degli Studi di Palermo, R. S. ex 60%.     

Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces

Generalizations of the Edelstein-Suzuki theorem [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal. TMA 71 (2009), 5313-5317], including versions of the Kannan, Chatterjea and Hardy-Rogers-type fixed point results for compact metric spaces, are proved. Also, abstract metric versions of these results are obtained. Examples are presented to distinguish our results from the existing ones.     

A fixed point theorem in metric spaces

We discuss a fixed point theorem for a function f mapping a complete metric space X into itself. For all x ? X{x \in X} the iterates of f(x) are shown to converge to x_* = f(x_*){{x_{\star} = f(x_{\star})}} and an explicit estimate of the convergence rate is given.     

Continuous reducibility and dimension of metric spaces

If (X, d) is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space (X, d) of positive dimension, there are uncountably many Borel subsets of (X, d) that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \((X,\tau )\) is called the Wadge quasi-order for \((X,\tau )\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \((X,\tau )\), is a well-quasiorder if and only if \((X,\tau )\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs.     

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.     

A best approximation theorem in hyperconvex metric spaces

In this paper, we present a best approximation theorem for set-valued mappings in hyperconvex metric spaces, which generalize the well-known result of Kirk, Sims and Yuan [W.A. Kirk, B. Sims, X.Z. Yuan, The Knaster–Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications, Nonlinear Anal. 39 (2000) 611–627].     

Coupled fixed point theorems in cone metric spaces

In this paper, by considering two metrics, we obtain some coupled fixed point theorems in cone metric spaces by assuming that the cone has nonempty interior as well as employing a number of contractive-type conditions. Our results generalize and extend some recently announced results in the literature.     

Some fixed point theorems in convex metric spaces

The concept of a convex metric space was introduced by Takahashi [10]. He observed that it is possible to generalize fixed point theorems in Banach spaces. Subsequently, Machado [8], Itoh [5], Naimpally, Singh and Whitfield [9] and Beg and Azam [2], among others have studied fixed point theorems in convex metric spaces. This paper is a continuation of these investigations.     

Small inductive dimension of completions of metric spaces

We construct a 0-dimensional metric space which under a special set-theoretic assumption, denoted in the paper as S(), does not have a 0-dimensional completion. Shortly after the submission of the paper for publication R. Dougherty has shown the consistency of S(). (S() disagrees with the continuum hypothesis.)

     

Common fixed point theorems in probabilistic metric spaces

We extend the notion of R-weak commutativity and its variants to probabilistic metric spaces and prove common fixed point theorems concerning them. Examples are included to reflect upon the distinctiveness of the types of mappings defined in the paper.      

Some characterizations of upper doubling conditions on metric measure spaces

We provide several equivalent characterizations for the upper doubling condition introduced in the framework of T. Hytönen for non‐homogeneous metric measure spaces. We also introduce the “smooth strong upper doubling” condition and provide equivalent characterizations, which is related to the development of Littlewood–Paley theory on this non‐homogeneous setting.     

Sobolev embeddings in metric measure spaces with variable dimension

In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities with optimal exponents. All of these questions lead naturally to function spaces with variable exponents. Supported the Research Council of Norway, Project 160192/V30.     

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.     

Efficient piecewise-linear function approximation using the uniform metric

We given anO(n logn)-time method for finding a bestk-link piecewise-linear function approximating ann-point planar point set using the well-known uniform metric to measure the error, ε≥0, of the approximation. Our methods is based upon new characterizations of such functions, which we exploit to design an efficient algorithm using a plane sweep in “ε space” followed by several applications of the parametric-searching technique. The previous best running time for this problems wasO(n ²). This research was announced in preliminary form at the 10th ACM Symposium on Computational Geometry. The author was partially supported by the NSF and DARPA under Grant CCR-8908092, and by the NSF under Grants IRI-9116843 and CCR-9300079.     

Finite metric spaces needing high dimension for lipschitz embeddings in banach spaces

We construct a sequence of metric spaces (M _n) with cardM _n=3n satisfying that for everyc<2, there exists a real numbera(c)>0 such that, if the Lipschitz distance fromM _n to a subset of a Banach spaceE is less thanc, then dim(E) ≥a(c)n. We also prove several results about embeddings of metric spaces whose non-zero distance values are in the interval [1,2].     

Best approximation, coincidence and fixed point theorems for quasi-lower semicontinuous set-valued maps in hyperconvex metric spaces

Suppose X is a compact admissible subset of a hyperconvex metric spaces M, and suppose F:XM is a quasi-lower semicontinuous set-valued map whose values are nonempty admissible. Suppose also G:XX is a continuous, onto quasi-convex set-valued map with compact, admissible values. Then there exists an x₀X such that

As applications, we give some coincidence and fixed point results for weakly inward set-valued maps. Our results, generalize some well-known results in literature.