Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations
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RESEARCH
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Coupled fixed point results on quasi-Banach spaces with application to a system of integral equations Nawab Hussain1 , Peyman Salimi2 and Saleh Al-Mezel1* Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday * Correspondence: [email protected] 1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract The aim of this paper is to obtain coupled fixed point theorems for self-mappings defined on an ordered closed and convex subset of a quasi-Banach space. Our method of proof is different and constructive in nature. As an application of our coupled fixed point results, we establish corresponding coupled coincidence point results without any type of commutativity of underlying maps. Moreover, an application to integral equations is given to illustrate the usability of the obtained results. MSC: Primary 47H10; 54H25; 55M20 Keywords: quasi-Banach space; metric-type space; coupled fixed point; mixed monotone property; ordered set; integral equation
1 Introduction It is well known that the Banach contraction principle is one of the most important results in classical functional analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics (see [–]). The study of coupled fixed points in partially ordered metric spaces was first investigated in by Guo and Lakshmikantham [], and then it attracted many researchers; see, for example, [, ] and references therein. Recently, Bhaskar and Lakshmikantham [] presented coupled fixed point theorems for contractions in partially ordered metric spaces. Luong and Thuan [] presented nice generalizations of these results. Alsulami et al. [] further extended the work of Luong and Thuan to coupled coincidences in partial metric spaces. For more related work on coupled fixed points and coincidences, we refer the readers to recent results in [–]. In recent years, several authors have obtained coupled fixed point results for various classes of mappings on the setting of many generalized metric spaces. The concept of metric-type space appeared in some works, such as Czerwik [], Khamsi [] and Khamsi and Hussain []. Metric-type space is a symmetric space with some special properties. A metric-type space can also be regarded as a triplet (X, d, K), where (X, d) is a symmetric space and K ≥ is a real number such that d(x, z) ≤ K d(x, y) + d(y, z) ©2013 Hussain et al.; 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.
Hussain et al. Fixed Point Theory and Applications 2013, 2013:261 http://www.fixedpointtheoryandapplications.com/content/2013/1/261
for any x, y, z
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