© W. Shatanawi. 2010
Received: 23 March 2010
Accepted: 1 June 2010
Published: 14 June 2010
The fixed point theorems in metric spaces are playing a major role to construct methods in mathematics to solve problems in applied mathematics and sciences. So the attraction of metric spaces to a large numbers of mathematicians is understandable. Some generalizations of the notion of a metric space have been proposed by some authors. In 2006, Mustafa in collaboration with Sims introduced a new notion of generalized metric space called -metric space . In fact, Mustafa et al. studied many fixed point results for a self-mapping in -metric space under certain conditions; see[1–5]. In the present work, we study some fixed point results for self-mapping in a complete -metric space under some contractive conditions related to a nondecreasing map with for all .
2. Basic Concepts
Definition 2.1 (see ).
Definition 2.2 (see ).
Proposition 2.3 (see ).
Definition 2.4 (see ).
Proposition 2.5 (see ).
Definition 2.6 (see ).
Let and be -metric spaces, and let be a function. Then is said to be -continuous at a point if and only if for every , there is such that and implies . A function is -continuous at if and only if it is -continuous at all .
Proposition 2.7 (see ).
Proposition 2.8 (see ).
Example 2.9 (see ).
Example 2.10 (see ).
Definition 2.11 (see ).
3. Main Results
Following to Matkowski , let be the set of all functions such that be a nondecreasing function with for all . If , then is called -map. If is -map, then it is an easy matter to show that
As an application of Theorem 3.1, we have the following results.
the result follows from Theorem 3.1.
The above corollary has been stated in [7, Theorem 5.1.7], and proved by a different way.
the result follows from Theorem 3.1.
As an application to Theorem 3.6, we have the following results.
The author would like to thank the editor of the paper and the referees for their precise remarks to improve the presentation of the paper. This paper is financially supported by the Deanship of the Academic Research at the Hashemite University, Zarqa, Jordan.
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