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

Hassen Aydi,¹ Sana Hadj Amor,² and Erdal Karapinar³

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 [1–29].

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 [30–32] 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 [33–37]. 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

Letting in (77), we get

On letting in (76), and using the properties of and and (78), 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 (48), we have

where

Similarly, we can prove that

where

If

By (80) and (81), we get

a contradiction. Then

By (82) and (83), we get

a contradiction. Thus, , and hence the fixed point of is unique.

Finally, we provide the following example.

Example 14. Let and be defined by

Then is a -complete -metric space. Let be defined by

Take and for all . We examine the following cases.

(i) Let . Then

where .

(ii) Let . Then

where .

(iii) Let . Then

where .

(iv) Let . Then

where . Then

for all . Then the conditions of Theorem 13, hold and has a unique fixed point. Notice that is the desired fixed point of .

References

Z. Mustafa and B. Sims, “A new approach to generalized metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 7, no. 2, pp. 289–297, 2006.

Z. Mustafa, A new structure for generalized metric spaces with applications to fixed point theory [Ph.D. thesis], The University of Newcastle, Callaghan, Australia, 2005.

M. Abbas, W. Sintunavarat, and P. Kumam, “Coupled fixed point of generalized contractive mappings on partially ordered -metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 31, 14 pages, 2012.

M. Abbas, T. Nazir, and P. Vetro, “Common fixed point results for three maps in -metric spaces,” Filomat, vol. 25, no. 4, pp. 1–17, 2011.

R. P. Agarwal and E. Karapınar, “Remarks on some coupled fixed point theorems in -metric spaces,” Fixed Point Theory and Applications, vol. 2013, article 2, 2013.

H. Aydi, W. Shatanawi, and C. Vetro, “On generalized weak -contraction mapping in -metric spaces,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4223–4229, 2011.

H. Aydi, E. Karapınar, and W. Shatanawi, “Tripled fixed point results in generalized metric spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 314279, 10 pages, 2012.

Z. Mustafa, H. Aydi, and E. Karapınar, “On common fixed points in -metric spaces using (E.A) property,” Computers & Mathematics with Applications. An International Journal, vol. 64, no. 6, pp. 1944–1956, 2012.

H. Aydi, M. Postolache, and W. Shatanawi, “Coupled fixed point results for -weakly contractive mappings in ordered -metric spaces,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 298–309, 2012.

H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered -metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2443–2450, 2011.

Y. J. Cho, B. E. Rhoades, R. Saadati, B. Samet, and W. Shatanawi, “Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type,” Fixed Point Theory and Applications, vol. 2012, article 8, 14 pages, 2012.

B. S. Choudhury and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 8, pp. 2524–2531, 2010.

B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 73–79, 2011.

H. S. Ding and E. Karapınar, “A note on some coupled fixed point theorems on G-metric space,” Journal of Inequalities and Applications, vol. 2012, article 170, 2012.

Z. Mustafa, H. Aydi, and E. Karapınar, “On common fixed points in -metric spaces using (E.A) property,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1944–1956, 2012.

Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on complete -metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 189870, 12 pages, 2008.

Z. Mustafa, M. Khandagji, and W. Shatanawi, “Fixed point results on complete -metric spaces,” Studia Scientiarum Mathematicarum Hungarica, vol. 48, no. 3, pp. 304–319, 2011.

Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete -metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 917175, 10 pages, 2009.

Z. Mustafa, W. Shatanawi, and M. Bataineh, “Existence of fixed point results in -metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2009, Article ID 283028, 10 pages, 2009.

Z. Mustafa and H. Obiedat, “A fixed point theorem of Reich in -metric spaces,” Cubo, vol. 12, no. 1, pp. 83–93, 2010.

H. K. Nashine, “Coupled common fixed point results in ordered -metric spaces,” Journal of Nonlinear Science and its Applications, vol. 5, no. 1, pp. 1–13, 2012.

W. Shatanawi, “Coupled fixed point theorems in generalized metric spaces,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 3, pp. 441–447, 2011.

W. Shatanawi, “Fixed point theory for contractive mappings satisfying -maps in -metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 181650, 9 pages, 2010.

W. Shatanawi, “Some fixed point theorems in ordered -metric spaces and applications,” Abstract and Applied Analysis, vol. 2011, Article ID 126205, 11 pages, 2011.

W. Shatanawi, “Coupled fixed point theorems in generalized metric spaces,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 3, pp. 441–447, 2011.

W. Shatanawi, M. Abbas, and T. Nazir, “Common coupled coincidence and coupled fixed point results in two generalized metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 80, p. 13, 2011.

N. Tahat, H. Aydi, E. Karapınar, and W. Shatanawi, “Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in -metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 48, 9 pages, 2012.

B. Samet, C. Vetro, and F. Vetro, “Remarks on G-metric spaces,” International Journal of Analysis, vol. 2013, Article ID 917158, 6 pages, 2013.

M. Jleli and B. Samet, “Remarks on -metric spaces and fixed point theorems,” Fixed Point Theory and Applications, vol. 2012, article 210, 7 pages, 2012.

V. Berinde, “Approximating fixed points of weak contractions using the Picard iteration,” Nonlinear Analysis Forum, vol. 9, no. 1, pp. 43–53, 2004.

V. Berinde, “General constructive fixed point theorems for Ćirić-type almost contractions in metric spaces,” Carpathian Journal of Mathematics, vol. 24, no. 2, pp. 10–19, 2008.

V. Berinde, “Some remarks on a fixed point theorem for Ćirić-type almost contractions,” Carpathian Journal of Mathematics, vol. 25, no. 2, pp. 157–162, 2009.

A. Aghajani, S. Radenović, and J. R. Roshan, “Common fixed point results for four mappings satisfying almost generalized -contractive condition in partially ordered metric spaces,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5665–5670, 2012.

G. V. R. Babu, M. L. Sandhya, and M. V. R. Kameswari, “A note on a fixed point theorem of Berinde on weak contractions,” Carpathian Journal of Mathematics, vol. 24, no. 1, pp. 8–12, 2008.

L. Ćirić, M. Abbas, R. Saadati, and N. Hussain, “Common fixed points of almost generalized contractive mappings in ordered metric spaces,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5784–5789, 2011.

M. Păcurar, “Sequences of almost contractions and fixed points in -metric spaces,” Analele Universitatii de Vest, Timisoara Seria Matematica Informatica, vol. 48, no. 3, pp. 125–137, 2010.

W. Shatanawi and A. Al-Rawashdeh, “Common fixed points of almost generalized )-contractive mappings in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 80, 14 pages, 2012.