Abstract
Abstract Recently, some (common) coupled fixed theorems in various abstract spaces have appeared as a generalization of existing (usual) fixed point results. Unexpectedly, we noticed that most of such (common) coupled fixed theorems are either weaker or equivalent to existing fixed point results in the literature. In particular, we prove that the very recent paper of Turkoglu and Sangurlu ‘Coupled fixed point theorems for mixed g-monotone mappings in partially ordered metric spaces [Fixed Point Theory and Applications 2013, 2013:348]’ can be considered as a consequence of the existing fixed point theorems on the topic in the literature. Furthermore, we give an example to illustrate that the main results of Turkoglu and Sangurlu (Fixed Point Theory Appl. 2013:348, 2013) has limited applicability compared to the mentioned existing fixed point result. MSC:47H10, 54H25.
Highlights
1 Introduction and preliminaries In many recent publications in fixed point theory auxiliary functions are used to generalize the contractive conditions on the maps defined on various spaces
We first recall the definition of the following auxiliary functions and some of their basic properties
Theorem . [ ] Let (X, ≤) be a partially ordered set and suppose there exists a metric d on X such that (X, d) is a complete metric space
Summary
Introduction and preliminariesIn many recent publications in fixed point theory auxiliary functions are used to generalize the contractive conditions on the maps defined on various spaces. Let (X, ≤) be a partially ordered set and F : X × X → X be a given mapping.
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