Fixed Point Theorems and Asymptotically Regular Mappings in Partial Metric Spaces

The notion of asymptotically regular mapping in partial metric spaces is introduced, and a fixed point result for the mappings of this class is proved. Examples show that there are cases when new results can be applied, while old ones (in metric space) cannot. Some common fixed point theorems for sequence of mappings in partial metric spaces are also proved which generalize and improve some known results in partial metric spaces. 1. Introduction Matthews [1] introduced partial metric spaces as a part of the study of denotational semantics of data flow networks. In partial metric space, the usual metric was replaced by partial metric, with a property that the self-distance of any point may not be zero. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation. Partial metric has applications in the branches of science where the size of data point is represented by its self-distance. The fixed point of a contraction mapping in partial metric space has zero self-distance; that is, fixed point is a total object. Every metric space is a partial metric space with zero self-distance that is, partial metric spaces are the generalization of metric spaces. O’Neill [2] generalized the concept of partial metric space a bit further by admitting negative distances. The partial metric defined by O’Neill is called dualistic partial metric. Heckmann [3] generalized it by omitting small self-distance axiom. The partial metric defined by Heckmann is called weak partial metric. Banach contraction principle ensures the existence and uniqueness of a fixed point of a contractive self-map of metric space and has many applications in applied sciences. The fixed point result of Matthews is the generalization of the following Banach contraction principle. Let be a complete metric space and let be a self-map on . If there exists such that for all , then has a unique fixed point in . The fixed point result of Matthews is generalized by several authors for single self map in partial metric spaces (see, e.g., [4–6]). Almost all contractive conditions in these papers imply the asymptotic regularity of the mapping under consideration. The purpose of this paper is to prove some common fixed point theorems for a sequence of self maps on partial metric spaces and generalize the result of Matthews. The notion of asymptotically regular mapping in partial metric spaces is introduced and a fixed point result for the mappings of this class is also proved. 2. Definitions and Preliminaries First, we recall some