Coupled Fixed Point Theorems with Rational Type Contractive Condition in a Partially Ordered -Metric Space

Coupled fixed point theorems for a map satisfying mixed monotone property and a nonlinear, rational type contractive condition are established in a partially ordered -metric space. The conditions for uniqueness of the coupled fixed point are discussed. We also present results for the existence of coupled coincidence points of two maps. 1. Introduction The idea of weakening the contractive condition in a metric space by introducing partial order in the space and considering monotone functions satisfying contractive conditions was first developed by Ran and Reurings [1]. Later, this was extended by Bhaskar and Lakshmikantham [2] to prove a coupled fixed point theorem for functions satisfying mixed monotone property. Since then, there has been considerable interest in the development of coupled fixed point theorems in partially ordered metric spaces with a variety of contractive conditions [3–18]. Nonlinear contractive conditions were considered in [4, 6, 19]. In particular, a rational type contractive condition was considered by Jaggi [19] in a complete metric space and this was extended to a partially ordered complete metric space by Harjani et al. [6] to prove some fixed point theorems. Some coupled fixed point theorems in partially ordered, complete metric spaces were developed by Choudhury and Maity [8] and Saadati et al. [9]. The contractive conditions used in [8] were extensions of that used by Bhaskar and Lakshmikantham [2] into a metric space. A new concept of an distance was introduced in [9]. In this paper we develop a coupled fixed point theorem using a rational type, nonlinear contractive condition in a partially ordered complete metric space. The condition is similar to the rational type contractive condition of iri et al. [3] and may be considered as a generalization of the condition given in [3]. We also find conditions for the uniqueness of the coupled fixed point. Finally we consider the conditions for existence of coupled coincidence points. We begin by introducing the basic definitions and notions used in the paper. 2. General Preliminaries Throughout this work will denote a partial order relation on some given set. For any two elements in some partially ordered set endowed with the partial order relation , and are equivalent. Also by we mean and . Definition 1 (see [20]). Let be a nonempty set and let ？？: be a function satisfying the following properties:(1) if ,(2) for all with ,(3) for all with ,(4) (symmetry in all three variables),(5) for all (rectangle inequality). Then is called a generalized metric or more specifically a metric