Generalized Contractive Set-Valued Maps on Complete Preordered Quasi-Metric Spaces

Abstract

By using a suitable modification of the notion of a w-distance we obtain some fixed point results for generalized contractive set-valued maps on complete preordered quasi-metric spaces. We also show that several distinguished examples of non-metrizable quasi-metric spaces and of cones of asymmetric normed spaces admit w-distances of this type. Our results extend and generalize some well-known fixed point theorems.

Modified w-distances on quasi-metric spaces and a fixed point theorem on complete quasi-metric spaces

“…Later Park [17] extended the notion of w−distance and generalized several results from [12] to quasi-metric spaces. Since then, the w-distance has been used in some directions in order to obtain fixed point results on complete metric and quasi-metric spaces ( [1], [2], [3], [14], [15]). …”

Fixed point results for generalized contractive mappings involving altering distance functions on complete quasi-metric spaces and applications

“…Al-Homidan et al [2] introduced the concept of Q-function defined on a quasi-metric space as a generalization of w-distances, and then established a Caristi-Kirk-type fixed point theorem, a Takahashi minimization theorem, and versions of Ekeland's principle in the setting of quasi-metric space with a Q-function, generalizing the main results of [19]. This approach has been continued by Marín et al [17]. We would like to mention the result of Alegre [1].…”

“…One of the generalizations of metric spaces is the so-called quasi-metric spaces in which the commutativity condition does not hold in general [1,2,11,17,19]. Park [19] extended the notion of w-distance to quasi-metric spaces and obtained far-reaching generalized forms of Ekeland's principle and its six equivalents.…”

A new concept of (α,Fd )-contraction on quasi metric space

“…We can find some recent fixed point results for single valued and multivalued mappings on quasi metric spaces in [1,2,11,18,19,20].…”