A Characterization of Quasi-Metric Completeness in Terms of α–ψ-Contractive Mappings Having Fixed Points

Abstract

We obtain a characterization of Hausdorff left K-complete quasi-metric spaces by means of α – ψ -contractive mappings, from which we deduce the somewhat surprising fact that one the main fixed point theorems of Samet, Vetro, and Vetro (see “Fixed point theorems for α – ψ -contractive type mappings”, Nonlinear Anal. 2012, 75, 2154–2165), characterizes the metric completeness.

α-ψ-contractive type mappings on quasi-metric spaces

“…However, we shall show (Theorem 1) that, despite this, it is still possible to obtain a quasi-metric extension of Theorem A by using right K-sequential completeness and α −1 − ψ-contractivity. At this point it seems interesting to recall that by replacing "α(x n , x) ≥ 1 for all n ∈ N" with "α(x, x n ) ≥ 1 for all n ∈ N" in condition (iii) of Theorem A, a fixed point theorem for Hausdorff left K-sequentially complete quasi-metric spaces was obtained in [17]. We also have that T is an α − ψ-contractive mapping for ψ ∈ Ψ given by ψ(t) = t/2 for all t ≥ 0.…”

“…Theorem 3 [17,Theorem 1]. Let (X, d) be a Hausdorff left K-sequentially complete quasi-metric space and T : X → X be an α − ψ-contractive mapping satisfying the following conditions:…”

“…[1,3,4,7,10,11,19]). In particular, a slight variant of one of these theorems was recently extended to the setting of quasi-metric spaces in [17], where the authors also proved that this extension actually characterizes left K-sequential completeness for Hausdorff quasi-metric spaces and consequently that variant of Theorem 2.2 of [18] characterizes metric completeness. The fact that the α-admissible functions are not necessarily symmetric provides in the quasi-metric setting various appealing types of α − ψ-contractive mappings that are useful to obtain some fixed point results not only for Hausdorff left K-sequentially complete quasimetric spaces but also for Hausdorff right K-sequentially complete quasi-metric spaces, a relevant type of quasi-metric completeness to the study of completeness of hyperspaces and function spaces, and in characterizing the weak form of Ekeland's Variational Principle (see e.g.…”

Characterizing Complete Fuzzy Metric Spaces Via Fixed Point Results

“…Remarkable is also the contribution of Suzuki and Takahashi who gave in [7] (Theorem 4) a necessary and sufficient condition for a metric space to be complete by means of weakly contractive mappings having a fixed point. In a paper published in 2008, Suzuki [8] obtained a nice and elegant characterization of the metric completeness with the help of a weak form of the Banach contraction principle, and, very recently, it was proved in [9] that an important fixed point theorem by Samet, Vetro and Vetro [10] (Theorem 2.2) is also able of characterizing complete metric spaces.…”

New Results on Start-Points for Multi-Valued Maps

“…For the sake of completeness, we recollect, in the present manuscript, the necessary notations and fundamental concepts from the literature. We first recall the basic notions regarding quasi-metric spaces as well as some additional definitions that are related to multi-valued maps on these spaces [14][15][16]. For a general approach in metric fixed point theory for multi-valued operators, see [17][18][19].…”