%0 Journal Article
%T Set-Valued Hardy-Rogers Type Contraction in 0-Complete Partial Metric Spaces
%A Satish Shukla
%A Stojan Radenovi£¿
%A Calogero Vetro
%J International Journal of Mathematics and Mathematical Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/652925
%X 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
%U http://www.hindawi.com/journals/ijmms/2014/652925/