Some fixed point theorems for hybrid mappings in partial metric spaces

This research project forms part of the study of possible generalizations of some metric fixed point results for hybrid mappings to the setup of partial metric spaces. Partial metric spaces are one of generalizations of the notion of metric spaces such that the distance of a point from itself is not necessarily zero. The notion of non-zero self-distance is desirable, in particular, in modeling of partially computed information in computer sciences and in the study of non-Hausdorff topological spaces, although it may looks unnatural. We adapt some existing approaches of metric fixed point theory to establish our main results. The notion of compatible mappings and its generalizations proved to be very useful in the study of fixed points as the existing literature of Fixed Point Theory contains numerous metric fixed point results for the mappings. We proved some metric fixed point theorems for hybrid pair of compatible mappings as well as of weak compatible mappings in complete partial metric spaces. In many situations, the mappings arising need not be self-mappings. It is well known that metrically convex metric spaces are desirable for the study of fixed points for non-self-mappings. We proved a common fixed point theorems for two multi-valued and one single-valued weakly compatible mappings in complete metrically convex partial metric spaces. Our main results, in particular, generalize some existing metric fixed point results for compatible, weak compatible, and weakly compatible mappings due to Kaneko and Sessa 1989, Pathak 1995 and Ciric and Ume 2006, respectively to partial metric spaces.