Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a Q -Functio
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Research Article Coupled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Quasi-Metric Spaces with a Q-Function N. Hussain,1 M. H. Shah,2 and M. A. Kutbi1 1 2
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Department of Mathematical Sciences, LUMS, DHA Lahore, Lahore 54792, Pakistan
Correspondence should be addressed to N. Hussain, [email protected] Received 20 August 2010; Accepted 16 September 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 N. Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Using the concept of a mixed g-monotone mapping, we prove some coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete quasi-metric spaces with a Q-function q. The presented theorems are generalizations of the recent coupled fixed point theorems due to Bhaskar and Lakshmikantham 2006, ´ c 2009 and many others. Lakshmikantham and Ciri´
1. Introduction The Banach contraction principle is the most celebrated fixed point theorem and has been generalized in various directions cf. 1–31. Recently, Bhaskar and Lakshmikantham 8, Nieto and Rodr´ıguez-Lopez 28, 29, Ran and Reurings 30, and Agarwal et al. 1 ´ presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham 8 noted that their theorem can be used to investigate a large class of problems and discussed the existence and uniqueness of solution for a periodic boundary value problem. For more on metric fixed point theory, the reader may consult the book 22. Recently, Al-Homidan et al. 2 introduced the concept of a Q-function defined on a quasi-metric space which generalizes the notions of a τ-function and a ω-distance and establishes the existence of the solution of equilibrium problem see also 3–7. The aim of ´ c 24 for a mixed monotone this paper is to extend the results of Lakshmikantham and Ciri´ nonlinear contractive mapping in the setting of partially ordered quasi-metric spaces with a Q-function q. We prove some coupled coincidence and coupled common fixed point theorems for a pair of mappings. Our results extend the recent coupled fixed point theorems due to ´ c 24 and many others. Lakshmikantham and Ciri´
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Fixed Point Theory and Applications
Recall that if X, ≤ is a partially ordered set and F : X → X such that for x, y ∈ X, x ≤ y implies Fx ≤ Fy, then a mapping F is said to be nondecreasing. Similarly, a nonincreasing mapping is defined. Bhaskar and Lakshmikantham 8 introduced the following notions of a mixed monotone mapping and a coupled fixed point. Definition 1.1 Bhaskar and Lakshmikantham 8. Let X, ≤ be a partially ordered set and F : X × X → X. The mapping F is said to have the mixed monotone property if F is nondecreasing monotone in its
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