Coincidence point and common fixed point theorems in the product spaces of mixed-monotonically complete quasi-ordered metric spaces and their applications to the systems of integral equations and ordinary differential equations

Hsien-Chung Wu

doi:10.1186/1029-242x-2014-518

Abstract

The main purpose of this paper is to study the coincidence point and common fixed point theorems in the product spaces of mixed-monotonically complete quasi-ordered metric spaces based on some new types of contractive inequalities. In order to investigate the existence and chain-uniqueness of solutions for the systems of integral equations and ordinary differential equations, we shall also study the fixed point theorems for the functions having mixed monotone property or comparable property in the product space of quasi-ordered metric space. MSC:47H10, 54H25.

Highlights

The existence of coincidence point has been studied in [ – ] and the references therein
We shall establish some new coincidence point and common fixed point theorems in the product spaces of mixed-monotonically complete quasi-ordered metric spaces in which the fixed points of functions having mixed monotone property or mixed comparable property that are defined in the product space of quasi-ordered metric space can be subsequently obtained
We shall present the interesting applications to the existence and chain-uniqueness of solutions for the systems of integral equations and ordinary differential equations according to the fixed points of functions having mixed monotone property

Summary

Introduction

The existence of coincidence point has been studied in [ – ] and the references therein. Assume that the function F : (Xm, I) → (Xm, I) has the I-mixed monotone property on Xm. Given the initial element x ∈ Xm, we define the sequence {xn}n∈N by xn = F(xn– ). M}, assume that the function F : (Xm, d, I) → (Xm, d, I) is continuous on Xm and has the I-mixed monotone property, and that there exist a function ρ : Xm × Xm → R+ and a function of contractive factor φ : [ , ∞) → [ , ) such that, for any x, y ∈ Xm with y I x or x I y, the following inequalities are satisfied: for all k = , . We can generate a I -mixed monotone sequence {xn}n∈N, which says that the initial element x is a mixed I -monotone seed element in Xm. the results follow immediately from Theorem .

Then there exists a

In order to obtain d