Partial b_{v}(s) and b_{v}({\theta}) metric spaces and related fixed point theorems

In this paper, we introduced two new generalized metric spaces called partial b_{v}(s) and b_{v}({\theta}) metric spaces which extend b_{v}(s) metric space, b-metric space, rectangular metric space, v-generalized metric space, partial metric space, partial b-metric space, partial rectangular b-metric space and so on. We proved some famous theorems such as Banach, Kannan and Reich fixed point theorems in these spaces. Also, we give definition of partial v-generalized metric space and show that these fixed point theorems are valid in this space. We also give numerical examples to support our definitions. Our results generalize several corresponding results in literature.Comment: 15 page

Topology from enrichment: the curious case of partial metrics

For any small quantaloid \Q, there is a new quantaloid \D(\Q) of diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa), and then it is particularly interesting to compare categories enriched in \Q with categories enriched in \D(\Q). Taking Lawvere's quantale of extended positive real numbers as base quantale, \Q-categories are generalised metric spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-enriched category has a canonical closure operator on its set of objects: this makes for a functor from quantaloid-enriched categories to closure spaces. Under mild necessary-and-sufficient conditions on the base quantaloid, this functor lands in the category of topological spaces; and an involutive quantaloid is Cauchy-bilateral (a property discovered earlier in the context of distributive laws) if and only if the closure on any enriched category is identical to the closure on its symmetrisation. As this now applies to metric spaces and partial metric spaces alike, we demonstrate how these general categorical constructions produce the "correct" definitions of convergence and Cauchyness of sequences in generalised partial metric spaces. Finally we describe the Cauchy-completion, the Hausdorff contruction and exponentiability of a partial metric space, again by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a revised section on Cauchy sequences in a partial metric spac

Partial metric spaces with negative distances and fixed point theorems

In this paper we consider partial metric spaces in the sense of O'Neill. We introduce the notions of strong partial metric spaces and Cauchy functions. We prove a fixed point theorem for such spaces and functions that improves Matthews' contraction mapping theorem in two ways. First, the existence of fixed points now holds for a wider class of functions and spaces. Second, our theorem also allows for fixed points with nonzero self-distances. We also prove fixed point theorems for orbitally $r$-contractive and orbitally $\phi_r$-contractive maps. We then apply our results to give alternative proofs of some of the other known fixed point theorems in the context of partial metric spaces.Comment: 19 page

From metric spaces to partial metric spaces

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