© Z. Mustafa and B. Sims. 2009
Received: 31 December 2008
Accepted: 7 April 2009
Published: 16 April 2009
Metric spaces are playing an increasing role in mathematics and the applied sciences.
Over the past two decades the development of fixed point theory in metric spaces has attracted considerable attention due to numerous applications in areas such as variational and linear inequalities, optimization, and approximation theory.
Different generalizations of the notion of a metric space have been proposed by Gahler [1, 2] and by Dhage [3, 4]. However, HA et al.  have pointed out that the results obtained by Gahler for his metrics are independent, rather than generalizations, of the corresponding results in metric spaces, while in the current authors have pointed out that Dhage's notion of a -metric space is fundamentally flawed and most of the results claimed by Dhage and others are invalid.
In 2003 we introduced a more appropriate and robust notion of a generalized metric space as follows.
Definition 1.1 ().
Example 1.2 ().
We now recall some of the basic concepts and results for -metric spaces that were introduced in ().
Let be a -metric space, let be a sequence of points of , we say that is -convergent to if ; that is, for any there exists such that , for all (throughout this paper we mean by the set of all natural numbers). We refer to as the limit of the sequence and write .
Let and be -metric spaces and let be a function, then is said to be -continuous at a point if given , there exists such that ; implies . A function is -continuous on if and only if it is -continuous at all .
2. The Main Results
We begin with the following theorem.
and (2.13) leads to the following cases,
The proof follows from the previous theorem and the same argument used in Corollary 2.3.
therefore, (2.38) implies two cases.
The proof follows from the previous theorem and the same argument used in Corollary 2.3. The following theorem has been stated in  without proof, but this can be proved by using Theorem (2.6) as follows.
Theorem 2.8 ().
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