Characterization of completeness for $$m$$ m -metric spaces and a related fixed point theorem
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ORIGINAL RESEARCH PAPER
Characterization of completeness for m-metric spaces and a related fixed point theorem Sushanta Kumar Mohanta1 • Deep Biswas1 Received: 14 June 2020 / Accepted: 19 September 2020 Ó Forum D’Analystes, Chennai 2020
Abstract We provide a Kirk type characterization and a Weston type characterization of 0completeness for m-metric spaces in terms of fixed point theory and l-point respectively. As a consequence of this study, we obtain a fixed point result in this new framework. Keywords m-Metric 0-Completeness rl -Caristi Fixed point
Mathematics Subject Classification 54H25 47H10
1 Introduction The study of fixed points of mappings in various spaces is an interesting study because the study finds applications in different fields of mathematics. The most celebrated work in study of fixed point of mappings goes back in 1922 due to S. Banach by whose name we are familiar with Banach contraction theorem [4]. Several authors successfully extended Banach contraction theorem in many directions (see [2, 3, 6, 12–16, 18, 19]). In [11], Matthews introduced the concept of partial metric spaces as a part of the study of denotational semantics of dataflow networks. It is interesting to note that in partial metric spaces, self distance of an arbitrary point need not be equal to zero. Thereafter, a lot of articles have been dedicated to the improvement of fixed point theory in partial metric spaces (see [8–10] and references therein). Very recently, Asadi et al. [1] extended the notion of partial metric spaces to m-metric spaces and obtained some important fixed point results in this new framework. & Sushanta Kumar Mohanta [email protected] 1
Department of Mathematics, West Bengal State University, Barasat, 24 Parganas (North), Kolkata, West Bengal 700126, India
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S. K. Mohanta, D. Biswas
In [7], Kirk proved that a metric space ðX; dÞ is complete if and only if every Caristi mapping has a fixed point. In this paper, we introduce a new kind of Caristi mapping and provide a Kirk type characterization of 0-completeness for m-metric spaces. In [25], Weston had shown that completeness criterion of metric spaces has got some relation with the family of real valued semicontinuous functions carried over the space. In fact, he had proved a necessary and sufficient condition for the metric space ðX; dÞ to be complete in terms of the notion of d-point for lower semicontinuous functions. Some successful characterization of metric completeness in terms of fixed point theory can be found in [13, 17–21, 23, 24]. Finally, we introduce the concept of l-point and present a Weston type characterization of 0completeness for m-metric spaces in terms of l-point. As a consequence of this study, we obtain a fixed point result in this setting.
2 Some basic concepts We begin with some basic notations, definitions, and necessary results in m-metric spaces. Definition 2.1 [11] A partial metric on a nonempty set X is a function p : X X ! Rþ such that for all x; y; z 2
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