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International Journal of Analysis
Volume 2013 (2013), Article ID 917158, 6 pages
http://dx.doi.org/10.1155/2013/917158
Research Article

Remarks on -Metric Spaces

1Department of Mathematics, King Saud University, Saudi Arabia
2Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
3DEIM, Università degli Studi di Palermo, 90128 Palermo, Italy

Received 13 September 2012; Revised 5 November 2012; Accepted 5 November 2012

Academic Editor: Serge Nicaise

Copyright © 2013 Bessem Samet et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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., [14]).

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 [611].

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.

2. Preliminaries: Fixed Point Results on Metric Spaces

In this section, we recall some well-known fixed point theorems on metric spaces.

The following Ćirić’s theorem for quasicontractive mappings [12] plays a key role in our paper.

Theorem 6 (see Ćirić [12]). Let be a complete metric space and let be a mapping with the property: for all , where is a constant such that . Then has a unique fixed point.

The next result is a consequence of Theorem 2.1 of [13]. For details about this refer to Das and Naik [14].

Theorem 7. Let be a subset of a metric space , and let be weakly compatible self-mappings of . Assume that the range of contains the range of  , is a complete subspace of , and and satisfy the condition: for all , where is a constant such that . Then and have a unique common fixed point.

The following result is a consequence of Theorem 2.3 of [15].

Theorem 8. Let be self-mappings of a metric space such that . Assume that there exist such that (i) is nondecreasing, continuous, and for every ; (ii) is nondecreasing, right continuous, and for every ; (iii), for all , where If one of and is a complete subspace of , then and have a coincidence point. Further, if T and are weakly compatible, then and have a unique common fixed point.

In , Edelstein [16] proved the following version of the Banach Contraction Principle.

Theorem 9. Let be a compact metric space and let be a given mapping. Assume that for all with . Then has a unique fixed point.

3. Main Results

For our purpose we need the following propositions which are key results.

Proposition 10. Let be a -metric space. Let be defined by for all . Then is a metric space.

Proof. Let . Clearly, . Now, suppose that . This implies that , thus . Finally, we have to prove the triangular inequality. We have Then we get that

Proposition 11. If is a complete -metric space then is a complete metric space.

Proof . Let be a Cauchy sequence in . For all , we have Letting in the above inequality we get that , which implies that is a -Cauchy sequence. Since is a complete -metric space, there exists some such that is -convergent to . Then we have which implies that .

Definition 12. Let be a -metric space and let be a nonempty subset of . The subset is called sequentially -compact if, for any sequence in , there exists a subsequence of which -converges to some .

Proposition 13. If is a sequentially -compact -metric space, then is a compact metric space.

Proof. Let be a sequence in . Since is a sequentially -compact -metric space, there exists some and a subsequence of such that is -convergent to . Then we have which implies that , and hence is a compact metric space.

3.1. Discussion on Fixed Point Results of Mustafa and Obiedat

Recently, Mustafa and Obiedat [7] established the following result.

Theorem 14 (see Mustafa and Obiedat [7]). Let be a complete -metric space and let be a mapping satisfying the following condition: for all , where is a constant such that . Then has a unique fixed point.

Now, we prove the following.

Theorem 15. Theorem 14 is a particular case of Theorem 6.

Proof. Taking in (12), we get for all . Thus, we have for all , where . Note that . Similarly, we can write that for all . It follows from (14), (15), and the definition of that for all . The existence and uniqueness of the fixed point are followed immediately by Theorem 6, Propositions 10 and 11.

In the same paper, the authors presented the following fixed point theorem.

Theorem 16 (see Mustafa and Obiedat [7]). Let (X, G) be a complete G-metric space and let be a mapping satisfying the following condition: for all , where are nonnegative constants such that . Then has a unique fixed point.

Proceeding similarly as above we can prove the following.

Theorem 17. Theorem 16 is a particular case of Theorem 6.

Proof. Taking in (17), we get for all . Thus, we have for all , where . Note that . Similarly, we can write that for all . It follows from (19), (24), and the definition of that for all . The existence and uniqueness of the fixed point are followed immediately by Theorem 6, Propositions 10 and 11.

Now, we consider another theorem.

Theorem 18 (see Mustafa and Obiedat [7]). Let be a complete -metric space and let be a mapping satisfying the following condition: for all , where are nonnegative constants such that . Then has a unique fixed point.

In view of this result we give the following.

Theorem 19. Theorem 18 is a particular case of Theorem 6.

Proof. Taking in (22), we get for all . Thus, we have for all , where . Note that . Similarly, we can write that for all . It follows from (24), (25), and the definition of that for all . The existence and uniqueness of the fixed point are followed immediately by Theorem 6, Propositions 10 and 11.

3.2. Discussion on Fixed Point Results of Mustafa and Sims

Recently, Mustafa and Sims [9] established the following result.

Theorem 20 (see Mustafa and Sims [9]). Let be a complete -metric space and let be a mapping satisfying the following condition: for all , where is a constant such that . Then has a unique fixed point.

We give the following.

Theorem 21. Theorem 20 is a particular case of Theorem 6.

Proof. Taking in (27), we get for all . Similarly, we can write that for all .
It follows from (28), (29), and the definition of that for all . The existence and uniqueness of the fixed point are followed immediately by Theorem 6, Propositions 10 and 11.

Remark 22. From the proof of Theorem 21, it follows that Theorem 20 holds also if .

4. Additional Results

In this section, using our previous technique, we establish very easily some fixed point theorems on -metric spaces.

4.1. Common Fixed Point Theorem for Quasicontractive Condition

We have the following result.

Theorem 23. Let be a -metric space and let be weakly compatible self-mappings of . Suppose that the mappings and satisfy one of the following conditions: or for all , where is a constant. If the range of contains the range of  and is a -complete subspace of , then and have a unique common fixed point.

Proof. Suppose that and satisfy inequality (31). Then we have It follows from (31), (33), and the definition of that The existence and uniqueness of the common fixed point are followed immediately by Theorem 7, Propositions 10 and 11.

4.2. Edelstein Type Theorem

Here, we give a fixed point result of Edelstein type in the setting of -metric spaces.

Theorem 24. Let be a sequentially -compact -metric space and let be a mapping such that for all with . Then has a unique fixed point.

Proof. From (35), we have It follows from (35), (36), and the definition of that By Proposition 13, is compact and by Theorem 9, the mapping has a unique fixed point.

4.3. -Contractive Condition

To conclude this paper, we study the existence and uniqueness of common fixed points for two self-mappings satisfying a -contractive condition. The following result is a consequence of Theorem 8.

Theorem 25. Let be self-mappings of a -metric space such that . Assume that there exist such that (i) is nondecreasing, continuous, and for every ; (ii) is nondecreasing, right continuous, and for every ; (iii), , , for all .
If one of and is a -complete subspace of , then and have a coincidence point. Further, if and are weakly compatible, then and have a unique common fixed point.

Proof. From condition (iii), we have Combining (ii) with the above inequality, using the monotone property of and , we get for all . Now, the result follows from Theorem 8, Propositions 10 and 11.

5. Conclusions

In this paper, we showed that some fixed point generalizations in fixed point theory are not real generalizations as they could easily be obtained from the corresponding fixed point theorems in metric spaces. Here, we reviewed only some theorems which appeared recently in the literature. It follows that researchers in fixed point theory should take care in obtaining real generalizations.

Acknowledgments

The authors are grateful to editors and anonymous reviewers for their constructive comments and suggestions. The second author is supported by Università degli Studi di Palermo, Local Project R.S. ex 60%.

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