Coupled Fixed Point Theorems with New Implicit Relations and an Application

We introduce two new classes of implicit relations and where is a proper subset of , and these classes are more general than the class of implicit relations defined by Altun and Simsek (2010). We prove the existence of coupled fixed points for the maps satisfying an implicit relation in . These coupled fixed points need not be unique. In order to establish the uniqueness of coupled fixed points we use an implicit relation , where . Our results extend the fixed point theorems on ordered metric spaces of Altun and Simsek (2010) to coupled fixed point theorems and generalize the results of Gnana Bhaskar and Lakshimantham (2006). As an application of our results, we discuss the existence and uniqueness of solution of Fredholm integral equation. 1. Introduction Existence of fixed point theorems in partially ordered metric spaces with a contractive condition has been considered by several authors (see [1–6]). Guo and Lakshmikantham [7] introduced mixed monotone operators. Gnana Bhaskar and Lakshmikantham [8] established the existence of coupled fixed points of mappings satisfying mixed monotone property in partially ordered metric spaces. Later, Lakshmikantham and Ciric [9] extended this property to two maps by introducing mixed -monotone property and established the existence of coupled coincidence point and coupled common fixed points for a pair of commuting maps. Choudhury and Kundu [10] extended the existence of coupled coincidence and coupled common fixed points for a pair of noncommuting maps, particularly for a pair of compatible maps. Definition 1 (see [7]). Let be a nonempty set. An element in is called a coupled fixed point of the mapping if and . A point is called a fixed point of if . Definition 2 (see [7]). Let be a partially ordered set and be a mapping. We say that satisfies mixed monotone property if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any : Theorem 3 (see [8]). Let be a partially ordered set and suppose that is a metric on such that is a complete metric space. Let be a mapping satisfying mixed monotone property. Assume that there exists a with Suppose that either is continuous or the following conditions hold in : (i)if a nondecreasing sequence with , then for all and(ii)if a nonincreasing sequence with , then for all . If there exist such that and then has a coupled fixed point. In 2011, Luong and Thuan [11] proved the following coupled fixed point theorem. Theorem 4 (see [11]). Let be a partially ordered set and suppose that is a metric on such that is a complete metric space. Let be a mapping