Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric

Abbas M, Ali, Ali Basit

Description

Coincidence point theorems for hybrid pairs of single-valued and multi-valued mappings on an arbitrary non-empty set with values in a partial metric space using a partial Hausdorff metric have been proved. As an application of our main result, the existence and uniqueness of common and bounded solutions of functional equations arising in dynamic programming are discussed. MSC: 47H10, 54H25, 54E50.

Sujets

• Coincidence point

Informations

• Publié par: biomed
• Publié le: 01 janvier 2013
• Nombre de lectures: 12

Extrait

Abbas and Ali Fixed Point Theory and Applications 2013, 2013 :21 http://www.fixedpointtheoryandapplications.com/content/2013/1/21

R E S E A R C H Fixed point of Suzuki-Zamfirescu hybrid contractions in partial metric spaces via partial Hausdorff metric M Abbas 1 and Basit Ali 1,2* * Correspondence: [e-mail] 1 Department of Mathematics, Lahore University of Management Sciences, Lahore, 54792, Pakistan Full list of author information is available at the end of the article

Open Access

Abstract Coincidence point theorems for hybrid pairs of single-valued and multi-valued mappings on an arbitrary non-empty set with values in a partial metric space using a partial Hausdorff metric have been proved. As an application of our main result, the existence and uniqueness of common and bounded solutions of functional equations arising in dynamic programming are discussed. MSC: 47H10; 54H25; 54E50 Keywords: coincidence point; orbitally complete; common fixed point; partial metric space

1 Introduction and preliminaries Fixed point theory plays a fundamental role in solving functional equations [ ] arising in several areas of mathematics and other related disciplines as well. The Banach contraction principle is a key principle that made a remarkable progress towards the development of metric fixed point theory. Markin [ ] and Nadler [] proved a multi-valued version of the Banach contraction principle employing the notion of a Hausdorff metric. Afterwards, a number of generalizations (see [ –]) were obtained using different contractive condi- tions. The study of hybrid type contractive conditions involving single-valued and multi- valued mappings is a valuable addition to the metric fixed point theory and its applications (for details, see [–]). Among several generalizations of the Banach contraction princi- ple, Suzuki’s work [, Theorem .] led to a number of results (for details, see [ , –]). On the other hand, Matthews [ ] introduced the concept of a partial metric space as a part of the study of denotational semantics of dataflow networks. He obtained a mod- ified version of the Banach contraction principle, more suitable in this context (see also [, ]). Since then, results obtained in the framework of partial metric spaces have been used to constitute a suitable framework to model the problems related to the theory of computation (see [, –]). Recently, Aydi et al. [] initiated the concept of a partial Hausdorff metric and obtained an analogue of Nadler’s fixed point theorem [ ] in partial metric spaces. The aim of this paper is to obtain some coincidence point theorems for a hybrid pair of single-valued and multi-valued mappings on an arbitrary non-empty set with values in a partial metric space. Our results extend, unify and generalize several known results in the existing literature (see [ , , , ]). As an application, we obtain the existence and © 2013 Abbas and Ali; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.