Remarks on -Metric Spaces

In 2005, Mustafa and Sims (2006) introduced and studied a new class of generalized metric spaces, which are called -metric spaces, as a generalization of metric spaces. We establish some useful propositions to show that many fixed point theorems on (nonsymmetric) -metric spaces given recently by many authors follow directly from well-known theorems on metric spaces. Our technique can be easily extended to other results as shown in application. 1. Introduction The literature of the last decades is rich of papers that focus on all matters related to the generalized metric spaces (i.e., ; , a cone in an ordered Banach space; -metric spaces; probabilistic metric spaces, etc, see, e.g., [1–4]). In 2005, Mustafa and Sims introduced a new class of generalized metric spaces (see [5, 6]), which are called -metric spaces as a generalization of metric spaces. Subsequently, many fixed point results on such spaces appeared in [6–11]. Here, we present the necessary definitions and results in -metric spaces, which will be useful for the rest of the paper. However, for more details, we refer to [5, 6]. Definition 1. Let be a nonempty set. Suppose that is a function satisfying the following conditions: (a) if and only if ; (b) for all with ; (c) for all with ; (d) (symmetry in all three variables); (e) for all . Then is called a -metric on and is called a -metric space. Definition 2. A -metric space is said to be symmetric if for all . Definition 3. Let be a -metric space. We say that is (i)a -Cauchy sequence if, for any , there is an (the set of all positive integers) such that for all , ; (ii)a -convergent sequence to if, for any , there is an such that for all , . A -metric space is said to be complete if every -Cauchy sequence in is -convergent in . Proposition 4. Let be a -metric space. The following are equivalent: (1) is -convergent to ; (2) as ; (3) as . Proposition 5. Let be a -metric space. Then the following are equivalent: (i)the sequence is -Cauchy; (ii) as . An interesting observation is that any -metric space induces a metric on given by Moreover, is -complete if and only if is complete. It was observed that in the symmetric case ( is symmetric), many fixed point theorems on -metric spaces are particular cases of existing fixed point theorems in metric spaces. In this paper, we shall show that also in the nonsymmetric case, many results given recently on such spaces follow directly from existing results on metric spaces. This is done by using as key results some propositions. Our technique can be easily extended to other results as shown in application.