Partial metric spaces with negative distances and fixed point theorems

TL;DR

This paper extends fixed point theorems in partial metric spaces by introducing strong partial metric spaces and Cauchy functions, allowing for broader applicability and fixed points with nonzero self-distances.

Contribution

It introduces strong partial metric spaces and Cauchy functions, improving fixed point theorems to cover more general spaces and functions, including nonzero self-distances.

Findings

Fixed point existence for a wider class of functions and spaces.

Fixed points with nonzero self-distances are now possible.

Alternative proofs of existing theorems using new fixed point results.

Abstract

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.