Remarks on the fixed point problem of 2-metric spaces
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Remarks on the fixed point problem of 2-metric spaces Nguyen Van Dung, Nguyen Trung Hieu* , Nguyen Thi Thanh Ly and Vo Duc Thinh *
Correspondence: [email protected]; [email protected] Department of Mathematics, Dong Thap University, Dong Thap, Dong Thap, Vietnam
Abstract In this paper, we prove a fixed point theorem on a 2-metric space and show that the main results in Lahiri et al. (Taiwan. J. Math. 15:337-352, 2011) and Singh et al. (J. Adv. Math. Stud. 5:71-76, 2012) may be obtained easily from the axioms of a 2-metric space. Examples are given to validate the results.
1 Introduction and preliminaries There have been some generalizations of a metric space and its fixed point problem such as -metric spaces, D-metric spaces, G-metric spaces, cone metric spaces, complex-valued metric spaces. The notion of a -metric space was introduced by Gähler in []. Notice that a -metric is not a continuous function of its variables, whereas an ordinary metric is. This led Dhage to introduce the notion of a D-metric space in []. After that, in [], Mustafa and Sims showed that most of topological properties of D-metric spaces were not correct. Then, in [], they introduced the notion of a G-metric space and many fixed point theorems on G-metric spaces have been obtained. Unfortunately, in [], Jleli and Samet showed that most of the obtained fixed point theorems on G-metric spaces can be deduced immediately from fixed point theorems on metric spaces or quasi-metric spaces. In [], Huang and Zhang defined the notion of a cone metric space, which generalized a metric and a metric space, and proved some fixed point theorems for contractive maps on this space. After that, many authors extended some fixed point theorems on metric spaces to cone metric spaces. In [], Feng and Mao introduced a metric on a cone metric space and then proved that a complete cone metric space is always a complete metric space. They verified that a contractive map on a cone metric space is a contractive map on a metric space, then fixed point theorems on a cone metric space are, essentially, fixed point theorems on a metric space. In [], Azam, Fisher and Khan introduced the notion of a complex-valued metric space and some fixed point theorems on this space were stated. But in [], Sastry, Naidu and Bekeshie showed that some fixed point theorems recently generalized to complex-valued metric spaces are consequences of their counter parts in the setting of metric spaces and hence are redundant. Notice that in the above generalizations, only a -metric space is not topologically equivalent to an ordinary metric. Then there was no easy relationship between results obtained in -metric spaces and metric spaces. In particular, the fixed point theorems on -metric spaces and metric spaces may be unrelated easily. For the fixed point theorems on -metric spaces, the readers may refer to [–]. © 2013 Dung et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (h
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