A coupled coincidence point theorem in partially ordered metric spaces with an implicit relation

Selma Gülyaz,Erdal Karapınar,İlker Savas Yüce

doi:10.1186/1687-1812-2013-38

Abstract

Abstract In this manuscript, we discuss the existence of a coupled coincidence point for mappings F : X × X → X and g : X → X , where F has the mixed g-monotone property, in the context of partially ordered metric spaces with an implicit relation. Our main theorem improves and extends various results in the literature. We also state some examples to illustrate our work. MSC:47H10, 54H25, 46J10, 46J15.

Highlights

1 Introduction and preliminaries It is well known that fixed point theory is one of the crucial and very efficient tools in nonlinear functional analysis
The effects of fixed point theory are most apparent in fields like economy, computer sciences and engineering including many branches of mathematics
In, Brouwer published a result in this field, which was equivalent to Poincaré’s theorem, which in the simplest terms states that a continuous function from a disk D to itself has a fixed point

Summary

Introduction

Introduction and preliminariesIt is well known that fixed point theory is one of the crucial and very efficient tools in nonlinear functional analysis. This principle can be stated as follows: any contraction in a complete metric space has a unique fixed point. Gnana-Bhaskar and Lakshmikantham [ ] improved the idea of a coupled fixed point in the category of partially ordered metric spaces by introducing the notion of a mixed monotone mapping and presented certain applications on the solution of periodic boundary value problems.

Results

Conclusion