Ascoli-type theorems in the cone metric space setting

Antonio Boccuto,Xenofon Dimitriou

doi:10.1186/1029-242x-2014-420

Abstract

We give some necessary and sufficient conditions for (global) continuity of the limit of a pointwise convergent net of cone metric space-valued functions, defined on a Hausdorff topological space, in terms of weak filter exhaustiveness. In this framework, we prove some Ascoli-type theorems, considering also possibly asymmetric and extended real-valued distance functions. Furthermore, we pose some open problems.MSC:26E50, 28A12, 28A33, 28B10, 28B15, 40A35, 46G10, 54A20, 54A40, 06F15, 06F20, 06F30, 22A10, 28A05, 40G15, 46G12, 54H11, 54H12, 47H10.

Highlights

1 Introduction In the literature there have been several studies about cone metric spaces, namely abstract structures endowed with a distance function taking values in an ordered vector or a normed space, which includes in particular metric semigroups, whose an example is the set of fuzzy numbers, which is not a group
These structures are closely related with order vector spaces endowed with abstract convergences satisfying suitable axioms, but in which in general convergence of subsequences of convergent sequences is not required, like filter convergence
In this paper we investigate some properties of continuity of the limit of a net of functions, taking values in cone metric spaces, in terms of weak filter exhaustiveness, extending earlier results proved in [ – ], and relate filter exhaustiveness with filter uniform convergence

Summary

Introduction

In the literature there have been several studies about cone metric spaces, namely abstract structures endowed with a distance function taking values in an ordered vector or a normed space, which includes in particular metric semigroups, whose an example is the set of fuzzy numbers, which is not a group (see for instance [ – ]). In [ ] and [ ] these convergences, together with the concept of exhaustiveness, are considered in the filter/ideal context, and in this setting some Ascoli-type theorems for real-valued functions are extended. We consider symmetric or asymmetric distances with values in lattice groups and use the tool of (weak) filter exhaustiveness in connection with (global) continuity of the limit function and uniform convergence on compact sets. The concepts of (weak, forward, backward) filter exhaustiveness can be given analogously for sequences of functions, by taking = N with the usual order. Let (σp)p, (σp∗)p, (τp)p be three (O)-sequences in Y , related with F -exhaustiveness of (fλ)λ, (ROF )-convergence on X and global continuity of f on X respectively, let C ⊂ X be any compact set, and choose arbitrarily p ∈ N and x ∈ C. Since (fλ)λ (ROF )-converges to f on X, in correspondence with p ∈ N and x , . . . , xk there is a set F ∈ F with ρ fλ(xj), f (xj) ≤ σp∗ for each j ∈ [ , k] and λ ∈ F

Pick arbitrarily z

It is not difficult to check that for every free filter