A fixed point theorem for generalized contractions involving w-distances on complete quasi-metric spaces

Abstract

We obtain a fixed point theorem for generalized contractions on complete quasi-metric spaces, which involves w-distances and functions of Meir-Keeler and Jachymski type. Our result generalizes in various directions the celebrated fixed point theorems of Boyd and Wong, and Matkowski. Some illustrative examples are also given. MSC: 47H10; 54H25; 54E50

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

“…Recently, Alegre, Marín and Romaguera [2] have obtained a fixed point theorem for generalized contractions with respect to w-distances on complete quasi-metric spaces from which they deduce w-distance versions of Boyd and Wong's fixed point theorem [4] and of Matkowski's fixed point theorem [16]. Its approach uses a kind of functions considered by Jachymski in [11, Corollary of Theorem 2] and that generalizes the notion of function of Meir-Keeler type [1].…”

“…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]). …”

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].…”

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

“…We would like to mention the result of Alegre [1]. In [1], Alegre et al obtained a fixed point theorem for generalized contractions on complete quasi-metric spaces, which involves w-distances and functions of Meir-Keeler and Jachymski type. They established the following result.…”

“…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.…”