  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.
