Partial Metric Topology

Abstract

Abstract. In this paper we introduce the notion of canonical partial metric associated to a norm to study geometric properties of normed spaces. In particular, we characterize strict convexity and uniform convexity of normed spaces in terms of the canonical partial metric defined by its norm. We prove that these geometric properties can be considered, in this sense, as topological properties that appear when we compare the natural metric topology of the space with the non translation invariant topology induced by the canonical partial metric in the normed space.2000 AMS Classification: 54E35, 46B04.

Common Fixed Point of Maps in Complete Partial Metric Spaces

“…Partial metric space is a generalized metric space in which each object does not necessarily have to have a zero distance from itself [10]. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, give a modified version of the Banach contraction principle, more suitable in this context [10]. Subsequently, Valero [14], Oltra and Valero [12] and Altun et al [2] gave some generalizations of the result of Matthews.…”

“…The reader interested in fixed point theory in partial metric spaces is referred to the work of [1,8,10,12,13,14] and references therein.…”

“…A basic example of a partial metric space is the pair (R + , p), where p(x, y) = max{x, y} for all x, y ∈ R + . Other examples of the partial metric spaces which are interesting from a computational point of view may be found in [5] and [10].…”

On Fixed Points of Strictly Causal Functions

“…And since c ∈ [0, 1), by (39) and (55) Therefore, there is a strictly contracting function on X, namely F , that is not a contraction mapping on X, d .…”

“…In Banach's fixed-point theorem, and any reasonable generalization of it (e.g., see [39], [10]), it is of course not the convergence of the orbit of s under 1m2 F , but ultimately, the convergence of the orbit of s under F that is exploited. It is therefore interesting to see what the orbit of s under 1m2 F does when the orbit of s under F converges.…”