Coupled Fixed Point Theorems with New Implicit Relations and an Application

G V R Babu,P D Sailaja

doi:10.1155/2014/389646

Abstract

We introduce two new classes of implicit relations S and S′ where S′ is a proper subset of S, 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 S. These coupled fixed points need not be unique. In order to establish the uniqueness of coupled fixed points we use an implicit relation S′, where S′⊂S. 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.

Highlights

Existence of fixed point theorems in partially ordered metric spaces with a contractive condition has been considered by several authors
We prove the existence of coupled fixed points for the maps satisfying an implicit relation in S
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)

Summary

Introduction

Existence of fixed point theorems in partially ordered metric spaces with a contractive condition has been considered by several authors (see [1,2,3,4,5,6]). Gnana Bhaskar and Lakshmikantham [8] established the existence of coupled fixed points of mappings satisfying mixed monotone property in partially ordered metric spaces. Lakshmikantham and Ciric [9] extended this property to two maps by introducing mixed g-monotone property and established the existence of coupled coincidence point and coupled common fixed points for a pair of commuting maps. Let F : X × X → X be a mapping satisfying mixed monotone property on X and there exist x0, y0 ∈ X such that x0 ⪯ F(x0, y0) and y0 ⪰ F(y0, x0). These coupled fixed points need not be unique (Example 17).

Implicit Relations

Existence of Coupled Fixed Points Using a New Implicit Relation

An Application