An asymmetric Arzelà–Ascoli theorem

Abstract

An Arzelà-Ascoli theorem for asymmetric metric spaces (sometimes called quasi-metric spaces) is proved. One genuinely asymmetric condition is introduced, and it is shown that several classic statements fail in the asymmetric context if this assumption is dropped.

Extending asymmetric convergence and Cauchy condition using ideals

“…As mentioned before, here the I-Cauchy condition (by Dems [10], see also [1] and [22]) gives rise to two Cauchy conditions associated with forward and backward I-convergence in an asymmetric space, which naturally extends the notions of forward and backward Cauchy conditions ( [23], see also [3]). …”

“…Now we state below a genuinely asymmetric property of an asymmetric metric space from [3] (see also [8] where this name is given) which will play a very important role throughout our paper.…”

“…However we need the additional assumption on the asymmetric metric space only to prove the implication (1) =⇒ (3). That the assumption is essential can be easily observed by taking the sequence { n−1 n } n∈N in the asymmetric metric space of Example 4.1, which is I-forward Cauchy but it does not satisfy the condition (3).…”

F-contraction on asymmetric metric spaces

“…There are two notions for each of them, namely forward and backward ones, since we have two topologies which are the forward topology and the backward topology in asymmetric metric spaces (see [5]). Collins and Zimmer [2] studied these notions in the asymmetric context. Asymmetric metrics have many applications in pure and applied mathematics; for example, asymmetric metric spaces have recently been studied with questions of existence and uniqueness of Hamilton-Jacobi equations [8] in mind.…”

New results in quasi cone metric spaces

“…A quasi metric is a distance function in which the symmetry axiom is eliminated in the definition of a metric (see [4,12,16,18,28,29]). Many authors have given different definitions of quasi cone metric, see for instance [3,25].…”