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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 165420, 11 pages
http://dx.doi.org/10.1155/2013/165420
Research Article

Some Almost Generalized -Contractions in -Metric Spaces

1Department of Mathematics, Jubail College of Education, Dammam University, P.O. Box 12020, Industrial Jubail 31961, Saudi Arabia
2Laboratoire Physique Mathématique, Fonctions Spéciales et Applications (MAPSFA) LR11ES35, Ecole Supérieure des Sciences et de la Technologie de Hammam Sousse, Université de Sousse, Rue Lamine el Abbassi, 4011 Hammam Sousse, Tunisia
3Department of Mathematics, Atilim University, İncek, 06836 Ankara, Turkey

Received 20 July 2013; Accepted 27 August 2013

Academic Editor: Chi-Ming Chen

Copyright © 2013 Hassen Aydi 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 this paper, we introduce some almost generalized -contractions in the setting of -metric spaces. We prove some fixed points results for such contractions. The presented theorems improve and extend some known results in the literature. An example is also presented.

1. Introduction and Preliminaries

In 2006, a new structure of a generalized metric space was introduced by Mustafa and Sims [1] as an appropriate notion of a generalized metric space called a -metric space. Fixed point theory in this space was initiated in [2]. Particularly, Banach contraction mapping principle was established in this work. Since then the fixed point theory in -metric spaces has been studied and developed by many authors, see [129].

The following definitions and results will be needed in the sequel.

Definition 1 (see [1]). Let be a nonempty set, be a function satisfying the following properties:(G1) if ,(G2) for all with ,(G3) for all with ,(G4) (symmetry in all three variables),(G5) for all (rectangle inequality).Then the function is called a generalized metric, or, more specially, a -metric on , and the pair is called a -metric space.

Every -metric on defines a metric on by

Example 2. Let be a metric space. The function , defined by or for all , is a -metric on .

Definition 3 (see [1]). Let be a -metric space, and let be a sequence of points of ; therefore, we say that is -convergent to if ; that is, for any , there exists such that , for all . We call the limit of the sequence and write or .

Proposition 4 (see [1]). Let be a -metric space. The following are equivalent:(1) is -convergent to ,(2) as ,(3) as .

Definition 5 (see [1]). Let be a -metric space. A sequence is called a -Cauchy sequence if, for any , there is such that for all ; that is, as .

Proposition 6 (see [1]). Let be a -metric space. Then the following are equivalent:(1)the sequence is -Cauchy,(2)for any , there exists such that , for all .

Definition 7 (see [1]). A -metric space is called -complete if every -Cauchy sequence is -convergent in .

Definition 8. Let be a -metric space. A mapping is said to be -continuous if for any -convergent sequence to , then is -convergent to .

Now, let denote the set of functions such that if and only if . We denote by and the subsets of such that There are a lot of fixed point theorems for different type contractions in the literature. In particular, Berinde [3032] introduced the concept of an almost contraction in metric spaces and studied many interesting fixed point theorems for a Ćirić strong almost contraction. For other fixed point results on generalized almost contractions, see [3337]. In this paper, we introduce some almost generalized -contractions in the setting of -metric spaces, and we establish some fixed points results for such contractions.

2. Main Results

Let be a -metric space. First, we consider the following expressions: for all .

Our first result for almost generalized -contractions is the following.

Theorem 9. Let be a complete -metric space. Let be a self-mapping. Suppose there exist , , and such that for all , Then has a unique fixed point; say .

Proof. Let , and define a sequence in such that If for some , then , and the proof is completed. Thus, we may assume that for all . By (6) we have where From (8) and (9), we get If then by (10) we have Thus , and hence . Therefore, which is a contradiction. So, Therefore, we get and (10) becomes Thus, by (14), the sequence is monotone nonincreasing. It follows that as for some . Next we claim that . On taking limit as in (15), we obtain Hence and we get . Hence Next, we show that is a -Cauchy sequence. On contrary, assume that is not a -Cauchy sequence. Then, there is an for which we can find subsequences of with such that Using (18) and (G5), we have Taking limit as and using (17), we have By (G5), we get On taking limit as in the above inequality and using (17) and (20), we obtain Again, by (G5), we have On taking limit as in the above inequality and using (17) and (22), we obtain We have also, by (G5), On taking limit as in the above inequality and using (17) and (G3), we obtain We have also, by (G3) and (G5), Now, on taking limit as in (27), and using (17), (20), and (26), we obtain Furthermore, by (G3) and (G5), we get On taking limit as in the above inequality, and using (17), (20), (22), and (28), we obtain Now, we have Letting , and using (17), (20), (22), (24), (28), and (30), and the properties of and , we have Thus and hence , a contradiction. Thus is a -Cauchy sequence in .
Now, since is -complete, there are such that is -convergent to ; that is By (6), we get where Letting in (35), we get On letting in (34), and using the properties of and and (36), we obtain Therefore and hence . Thus is a fixed point of .
Now our purpose is to check that such point is unique. Suppose that there are two fixed points of ; say such that . By (6), we have where Similarly, we can prove that where If By (38) and (39), we get a contradiction. Then By (40) and (41), we get a contradiction. Thus, , and hence the fixed point of is unique.

As consequence of Theorem 9, we present the following corollaries.

Corollary 10. Let be a complete -metric space. Let be a self mapping. Suppose there exist and such that for all Then has a unique fixed point; say .

Proof. It suffices to take and in Theorem 9.

Corollary 11. Let be a complete -metric space. Let be a self mapping. Suppose there exist and such that for all Then has a unique fixed point; say .

Proof. Note that in Corollary 10.

Remark 12. Corollary 11 is a generalization of Mustafa’s result [2].

Our second main result is given as follows.

Theorem 13. Let be a complete -metric space. Let be a self mapping. Suppose there exist ,   and such that for all : where for all . Then has a unique fixed point; say .

Proof. Let be an arbitrary point in and define a sequence in such that If for some , then and the proof is completed. Thus, we may assume that for all . By (48), we have where From (51) and (52), we get If then by (53) we have Thus and hence . Therefore, , which is a contradiction. So, Therefore, we get and (53) becomes Thus, by (57), the sequence is monotone nonincreasing. It follows that as for some . Next we claim that . On taking limit as in (58), we obtain Hence and we get . Then Next, we show that is a -Cauchy sequence. On contrary, assume that is not a -Cauchy sequence. Then, there is an for which we can find subsequences of with such that Using (61) and (G5), we have Taking limit as and using (60), we have By (G5), and using (60) and (63), we obtain, as stated in the proof of Theorem 9, Again, by (G5), and using (60) and (64), similarly as in the proof of Theorem 9, we obtain Furthermore, by (G3) and (G5), we get By the proof of Theorem 9, we have stated that So, on taking limit as in (66), and using (60), (64), and (67), we obtain Furthermore, by (G3) and (G5), we get On taking limit as in above inequalities, and using (60) and (63), we have Also, by (G3) and (G5), we get On taking limit as in above inequalities, and using (60), (63), and (70), we obtain Now, we have Letting , and using (60), (65), (68), (70), and (72), and the properties of and , we have Thus and hence , a contradiction. Thus is a -Cauchy sequence in .
Now, since is -complete, there is such that is -convergent to ; that is By (48), we get where