Set-Valued Hardy-Rogers Type Contraction in 0-Complete Partial Metric Spaces

In this paper we introduce set-valued Hardy-Rogers type contraction in 0-complete partial metric spaces and prove the corresponding theorem of fixed point. Our results generalize, extend, and unify several known results, in particular the recent Nadler’s fixed point theorem in the context of complete partial metric spaces established by Aydi et al. (2012). As an application of our results, a homotopy theorem for such mappings is derived. Also, some examples are included which show that our generalization is proper. 1. Introduction and Preliminaries The well-known Banach contraction mapping principle states that, if is a complete metric space and is a self-mapping such that for all , where , then has a fixed point in . Because of simplicity and several applications, Banach principle was generalized by several authors, in different directions. For instance, Hardy and Rogers [1] used the contractive condition for all , , where are nonnegative constants such that , and proved fixed point result. Note that condition (2) generalizes the contractive conditions of Banach, Kannan, Reich, Chatterjea, and iri (see [2]). It is well known that the theory of set-valued mappings has application in control theory, convex optimization, differential equations, and economics. Nadler Jr. [3] generalized the Banach contraction mapping principle to set-valued mappings and proved the following fixed point theorem. Theorem 1. Let be a complete metric space and let be a mapping from into (here denotes the set of all nonempty closed bounded subset of ) such that, for all , where . Then has a fixed point. In recent years, Matthews [4] introduced the notion of partial metric space as a part of the study of denotational semantics of dataflow networks, with the interesting property of “nonzero self-distance” in space. He showed that the Banach contraction mapping principle can be generalized to the partial metric context for applications in program verification. For a more detailed explanation we refer the reader to Bukatin et al. [5] where the motivation for introducing nonzero distance is explained, which is also leading to interesting research in foundations of topology. Later on, Romaguera [6] introduced the notions of 0-Cauchy sequence and 0-complete partial metric spaces and proved some characterizations of partial metric spaces in terms of completeness and 0-completeness. Very recently, Aydi et al. [7] introduced the notion of partial Hausdorff metric and extended the Nadler's theorem in partial metric spaces. In this paper, we discuss some properties of partial metric spaces