Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property

Ravi P Agarwal,Wutiphol Sintunavarat,Poom Kumam

doi:10.1186/1687-1812-2013-22

Abstract

Abstract In this paper, we prove the coupled coincidence point theorems for a w ∗ -compatible mapping in partially ordered cone metric spaces over a solid cone without the mixed g-monotone property. In the case of a totally ordered space, these results are automatically obvious under the assumption given. Therefore, these results can be applied in a much wider class of problems. We also prove the uniqueness of a common coupled fixed point in this setup and give some example which is not applied to the existence of a common coupled fixed point by using the mixed g-monotone property but can be applied to our results. MSC:47H10, 54H25.

Highlights

The famous Banach contraction principle states that if (X, d) is a complete metric space and T : X → X is a contraction mapping (i.e., d(Tx, Ty) ≤ αd(x, y) for all x, y ∈ X, where α is a non-negative number such that α < ), T has a unique fixed point
Nashine et al [ ] established common coupled fixed point theorems for mixed g-monotone and w∗-compatible mappings satisfying more general contractive conditions in ordered cone metric spaces over a cone that is only solid which improve works of Karapınar [ ] and Shatanawi [ ]
We show that the mixed g-monotone property in common coupled fixed point theorems in ordered cone metric spaces can be replaced by another property due to Ðorić et al [ ]

Summary

Introduction

The famous Banach contraction principle states that if (X, d) is a complete metric space and T : X → X is a contraction mapping (i.e., d(Tx, Ty) ≤ αd(x, y) for all x, y ∈ X, where α is a non-negative number such that α < ), T has a unique fixed point. Nashine et al [ ] established common coupled fixed point theorems for mixed g-monotone and w∗-compatible mappings satisfying more general contractive conditions in ordered cone metric spaces over a cone that is only solid (i.e., has a nonempty interior) which improve works of Karapınar [ ] and Shatanawi [ ]. This result is an ordered version extension of the results of Abbas et al [ ].

Coupled coincidence point theorems lacking the mixed g-monotone property

Common coupled fixed point theorems lacking the mixed monotone property

Methods