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16 pages, 306 KB

Open AccessFeature PaperArticle

On Protected Quasi-Metrics

by Salvador Romaguera

Axioms 2024, 13(3), 158; https://doi.org/10.3390/axioms13030158 - 28 Feb 2024

Cited by 1 | Viewed by 2187

In this paper, we introduce and examine the notion of a protected quasi-metric. In particular, we give some of its properties and present several examples of distinguished topological spaces that admit a compatible protected quasi-metric, such as the Alexandroff spaces, the Sorgenfrey line, the Michael line, and the Khalimsky line, among others. Our motivation is due, in part, to the fact that a successful improvement of the classical Banach fixed-point theorem obtained by Suzuki does not admit a natural and full quasi-metric extension, as we have noted in a recent article. Thus, and with the help of this new structure, we obtained a fixed-point theorem in the framework of Smyth-complete quasi-metric spaces that generalizes Suzuki’s theorem. Combining right completeness with partial ordering properties, we also obtained a variant of Suzuki’s theorem, which was applied to discuss types of difference equations and recurrence equations. Full article

(This article belongs to the Section Geometry and Topology)

13 pages, 312 KB

Open AccessArticle

Basic Contractions of Suzuki-Type on Quasi-Metric Spaces and Fixed Point Results

by Salvador Romaguera

Mathematics 2022, 10(21), 3931; https://doi.org/10.3390/math10213931 - 23 Oct 2022

Cited by 11 | Viewed by 2195

Abstract

This paper deals with the question of achieving a suitable extension of the notion of Suzuki-type contraction to the framework of quasi-metric spaces, which allows us to obtain reasonable fixed point theorems in the quasi-metric context. This question has no an easy answer; in fact, we here present an example of a self map of Smyth complete quasi-metric space (a very strong kind of quasi-metric completeness) that fulfills a simple and natural contraction of Suzuki-type but does not have fixed points. Despite it, we implement an approach to obtain two fixed point results, whose validity is supported with several examples. Finally, we present a general method to construct non-

T_{1}

quasi-metric spaces in such a way that it is possible to systematically generate non-Banach contractions which are of Suzuki-type. Thus, we can apply our results to deduce the existence and uniqueness of solution for some kinds of functional equations which is exemplified with a distinguished case. Full article

(This article belongs to the Special Issue New Progress in General Topology and Its Applications)

13 pages, 2816 KB

Open AccessArticle

Mardivirus Infection and Persistence in Feathers of a Chicken Model Harboring a Local Autoimmune Response

by Gisela F. Erf, Gilles Le Pape, Sylvie Rémy and Caroline Denesvre

Microorganisms 2020, 8(10), 1613; https://doi.org/10.3390/microorganisms8101613 - 20 Oct 2020

Cited by 1 | Viewed by 2775

Abstract

Herpesvirus of turkey (HVT) is commonly used as a vaccine to protect chickens against Marek’s disease. Following vaccination, HVT infects feathers where it can be detected in all chicken lines examined. Unlike the parental Brown line (BL), Smyth line (SL) chickens develop vitiligo, due to autoimmune destruction of melanocytes in feathers. Previous reports showed a strong inflammatory response in Smyth chickens’ feathers at vitiligo onset, that subsided once melanocytes were destroyed, and depigmentation was complete. Here, we questioned whether the local autoimmune response in the Smyth model influences HVT infection and persistence in feathers. For this, one-day-old SL and BL chickens were vaccinated with Newcastle disease (rHVT-ND). Vitiligo was scored and HVT loads in pigmented and non-pigmented growing feathers were quantified regularly over 20 weeks. Chickens of both lines showed moderate HVT loads in feathers. At the onset of active vitiligo, the HVT load was significantly higher in SL compared to BL feathers. However, no difference in HVT loads was noticed between pigmented and non-pigmented feathers from SL chickens. Therefore, surprisingly, the inflammatory response in feathers of SL chickens did not inhibit HVT infection and persistence, but on the contrary, temporarily promoted HVT infection in feathers. Full article

(This article belongs to the Special Issue Marek’s Disease Virus)

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