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

K Chakrabarti

doi:10.1155/2014/785357

Abstract

Coupled fixed point theorems for a map satisfying mixed monotone property and a nonlinear, rational type contractive condition are established in a partially orderedG-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.

Highlights

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]
This was extended by Bhaskar and Lakshmikantham [2] to prove a coupled fixed point theorem for functions satisfying mixed monotone property
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

Summary

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. 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. In this paper we develop a coupled fixed point theorem using a rational type, nonlinear contractive condition in a partially ordered complete G-metric space. We begin by introducing the basic definitions and notions used in the paper

General Preliminaries

Main Results

Conclusion