%0 Journal Article
%T Coupled Fixed Point Theorems for ()-Contractive Mixed Monotone Mappings in Partially Ordered Metric Spaces and Applications
%A Manish Jain
%A Neetu Gupta
%A Sanjay Kumar
%J International Journal of Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/586096
%X The object of this paper is to establish the existence and uniqueness of coupled fixed points under a ( , )-contractive condition for mixed monotone operators in the setup of partially ordered metric spaces. Presented work generalizes the recent results of Berinde (2011, 2012) and weakens the contractive conditions involved in the well-known results of Bhaskar and Lakshmikantham (2006), and Luong and Thuan (2011). The effectiveness of our work is validated with the help of a suitable example. As an application, we give a result of existence and uniqueness for the solutions of a class of nonlinear integral equations. 1. Introduction and Preliminaries Fixed point theory is an important tool for studying the phenomenon of nonlinear analysis and is a bridge bond between pure and applied mathematics. The theory has its wide applications in engineering, computer science, physical and life sciences, economics, and other fields. Banach [1] introduced the well-known classical and valuable theorem in nonlinear analysis, which is named after him, known as the Banach contraction principle. This celebrated principle has been extended and improved by various authors in many ways over the years (see for instance [2¨C17]). Nowadays, fixed point theory has been receiving much attention in partially ordered metric spaces, that is, metric spaces endowed with a partial ordering. Ran and Reurings [17] were the first to establish the results in this direction. The results were then extended by Nieto and Rodr¨ªguez-L¨®pez [10] for nondecreasing mappings. Works noted in [18¨C24] are some examples in this direction. The work of Bhaskar and Lakshmikantham [25] is worth mentioning, as they introduced the new notion of fixed points for the mappings having domain the product space , which they called coupled fixed points, and thereby proved some coupled fixed point theorems for mappings satisfying the mixed monotone property in partially ordered metric spaces. As an application, they discussed the existence and uniqueness of a solution for a periodic boundary value problem. Definition 1 (see [25]). Let be a partially ordered set. The mapping is said to have the mixed monotone property if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any , , Definition 2 (see [25]). An element is called a coupled fixed point of the mapping if and . Bhaskar and Lakshmikantham [25] gave the following result. Theorem 3 (see [25]). Let be a partially ordered set and suppose there exists a metric on such that is a complete metric space. Let be a continuous mapping having the mixed
%U http://www.hindawi.com/journals/ijanal/2014/586096/