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.
Highlights
In their interesting and germinal paper [1], Samet, Vetro, and Vetro obtained various fixed point theorems in terms of α–ψ contractions which allowed them to deduce, in an elegant and direct way, several important and well-known fixed point results from [2,3,4,5]
In this note 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 [1] (Theorem 2.2) characterizes the metric completeness
We show that T is an α–ψ-SVV contractive mapping for α given by α(0, n) = α(n, n + 1) = 1 for all n ∈ N, and α(ζ, η ) = 0 otherwise; and ψ ∈ Ψ given by ψ(t) = t/2 for all t ≥ 0
Summary
Introduction and PreliminariesIn their interesting and germinal paper [1], Samet, Vetro, and Vetro obtained various fixed point theorems in terms of α–ψ contractions which allowed them to deduce, in an elegant and direct way, several important and well-known fixed point results from [2,3,4,5]. In this note 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 [1] (Theorem 2.2) characterizes the metric completeness (see Corollary 1 at the end of the paper). As in the metric case [1] (Definition 2.1), given a quasi-metric space (X , ρ) we say that a mapping
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
