Coupled common fixed point theorems for mixed weakly monotone mappings in partially ordered metric spaces

Madjid Eshaghi Gordji,Esmat Akbartabar,Yeol Je Cho,Maryam Ramezani

doi:10.1186/1687-1812-2012-95

Abstract

In this paper, we introduce the concept of a mixed weakly monotone pair of mappings and prove some coupled common fixed point theorems for a contractive-type mappings with the mixed weakly monotone property in partially ordered metric spaces. Our results are generalizations of the main results of Bhaskar and Lakshmikantham and Kadelburg et al.Mathematics Subject Classification 2000: 54H25.

Highlights

In 1922, Banach gave a theorem, which is well-known as Banach’s Fixed Point Theorem to establish the existence of solutions for nonlinear operator equations and integral equations
The existence of coupled fixed points for some kinds of contractive-type mappings in partially ordered metric spaces, cone metric spaces, fuzzy metric spaces and other spaces with applications has been investigated by some authors, for example, Bhaskar and Lakshmikantham [5], Cho et al [12,13,14], Dhage et al [15], Gordji et al [16,17], Kadelburg et al [18], Nieto and Lopez [10], Ran and Rarings [11], Sintunavarat et al [19,20], Yang et al [21] and others
In [5], Bhaskar and Lakshmikantham introduced the notions of a mixed monotone mapping and a coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings and discussed the existence and uniqueness of solution for periodic boundary value problems

Summary

Introduction

In 1922, Banach gave a theorem, which is well-known as Banach’s Fixed Point Theorem (or Banach’s Contractive Principle) to establish the existence of solutions for nonlinear operator equations and integral equations. Let f: X × X ® X be a mapping having the mixed monotone property on X.

Objectives

Results

Conclusion