Abstract and Applied Analysis

Volume 2012 (2012), Article ID 630457, 13 pages

http://dx.doi.org/10.1155/2012/630457

## Common Fixed Point Theorems of Altman Integral Type Mappings in -Metric Spaces

Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Zhejiang, Hangzhou 310036, China

Received 5 August 2012; Accepted 18 October 2012

Academic Editor: Yongfu Su

Copyright © 2012 Feng Gu and Hongqing Ye. 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

We introduce the concept of -weakly commuting self-mapping pairs in -metric space. Using this concept, we establish a new common fixed point theorem of Altman integral type for six self-mappings in the framework of complete -metric space. An example is provided to support our result. The results obtained in this paper differ from the recent relative results in the literature.

#### 1. Introduction and Preliminaries

Metric fixed point theory is an important mathematical discipline because of its applications in areas as variational and linear inequalities, optimization theory. Many results have been obtained by many authors considering different contractive conditions for self-mappings in metric space. In 1975, Altman [1] proved a fixed point theorem for a mapping which satisfies the condition , where is an increasing function satisfying the following conditions:(i);(ii) is a decreasing function;(iii) for some positive number , there holds .

*Remark 1.1. *By condition (i) and that is increasing, we know that and .

Gu and Deng [2], Liu [3], Zhang [4], and Li and Gu [5] discussed common fixed point theorems for Altman type mappings in metric space. In 2006, a new structure of generalized metric space was introduced by Mustafa and Sims [6] as an appropriate notion of generalized metric space called -metric space. Abbas and Rhoades [7] initiated the study of common fixed point in generalized metric space. Recently, many fixed point and common fixed point theorems for certain contractive conditions have been established in -metric spaces, and for more details, one can refer to [8–42]. Coupled fixed point problems have also been considered in partially ordered -metric spaces (see [43–56]). However, no one has discussed the common fixed point theorems for two or three pairs combining Altman type mappings recently.

Inspired by that, the purpose of this paper is to study common fixed point problem of Altman integral type for six self-mappings in -metric space. We introduce a new concept of -weakly commuting self-mapping pairs in -metric space, and a new common fixed point theorem for six self-mappings has been established through this concept. The results obtained in this paper differ from the recent relative results in the literature.

Throughout the paper, we mean by the set of all natural numbers.

*Definition 1.2 (see [6]). *Let be a nonempty set, and let be a function satisfying the following axioms: if ,, for all with ,, for all with , (symmetry in all three variables), for all (rectangle inequality),then the function is called a generalized metric, or, more specifically a -metric on and the pair is called a -metric space.

*Definition 1.3 (see [6]). *Let be a -metric space and let be a sequence of points in , a point in is said to be the limit of the sequence if , and one says that sequence is -convergent to .

Thus, if in a -metric space , then for any , there exists such that , for all .

Proposition 1.4 (see [6]). *Let be a -metric space, then the followings are equivalent:*(1) is -convergent to , (2) as , (3) as , (4) as .

*Definition 1.5 (see [6]). *Let be a -metric space. A sequence is called -Cauchy sequence if, for each there exists a positive integer such that for all ; that is, as .

*Definition 1.6 (see [6]). *A -metric space is said to be -complete, if every -Cauchy sequence in is -convergent in .

Proposition 1.7 (see [6]). * Let be a -metric space. Then the followings are equivalent.*(1) The sequence is -Cauchy;(2) For every , there exists such that , for all .

Proposition 1.8 (see [6]). *Let be a -metric space. Then the function is jointly continuous in all three of its variables.*

*Definition 1.9 (see [6]). *Let and be -metric space, and be a function. Then is said to be -continuous at a point if and only if for every , there is such that , and imply . A function is -continuous at if and only if it is -continuous at all .

Proposition 1.10 (see [6]). *Let and be -metric space. Then is -continuous at if and only if it is -sequentially continuous at , that is, whenever is -convergent to , is -convergent to .*

Proposition 1.11 (see [6]). * Let be a -metric space. Then, for any in it follows that:*(i) if , then ,(ii),
(iii),
(iv),
(v), (vi).

*Definition 1.12 (see [8]). *Self-mappings and of a -metric space are said to be compatible if and , whenever is a sequence in such that , for some .

In 2010, Manro et al. [9] introduced the concept of weakly commuting mappings, -weakly commuting mappings into -metric space as follows.

*Definition 1.13 (see [9]). *A pair of self-mappings of a -metric space is said to be weakly commuting if

*Definition 1.14 (see [9]). *A pair of self-mappings of a -metric space is said to be -weakly commuting, if there exists some positive real number such that

*Remark 1.15. *If , then -weakly commuting mappings are weakly commuting.

Now we introduce the new concept of -weakly commuting mappings as follow.

*Definition 1.16. *A pair of self-mappings of a -metric space is said to be -weakly commuting, if there exists a continuous function , such that

*Remark 1.17. *Commuting mappings are weakly commuting mappings, but the reverse is not true. For example: let , , for all , define , , through a straightforward calculation, we have: , , , but , hence, , but .

*Remark 1.18. *Weakly commuting mappings are -weakly commuting mappings, but the reverse is not true. For example: let , define , for all , , , then , , , , , when , we get that and are -weakly commuting mappings, but not weakly commuting mappings.

*Remark 1.19. *-weakly commuting mappings are -weakly commuting mappings but the reverse is not true. For example: let , , for all , , thus, we have , , , . Let , then
For any given , since , there exists such that , so we get . Therefore, and are -weakly commuting mappings, but not -weakly commuting mappings.

Lemma 1.20. *Let be Lebesgue integrable, and , for all , let , then is an increasing function in .*

*Definition 1.21. *Let and be self-mappings of a set . If for some in , then is called a coincidence point of and , and is called s point of coincidence of and .

#### 2. Main Results

In this paper, we denote the function satisfying , for all .

Theorem 2.1. *Let be a complete -metric space and let , , , , , and be six mappings of into itself. If there exists an increasing function satisfying the conditions (i)~(iii) and the following conditions:*(iv)*,
*(v)*, for all ,**where is a Lebesgue integrable function which is summable nonnegative such that
**
Then,*(a)* one of the pairs , , and has a coincidence point in ,*(b)* if , , and are three pairs of continuous -weakly commuting mappings, then the mappings , , , , , and have a unique common fixed point in .*

* Proof. *Let be an arbitrary point in , from the condition (iv), there exist such that
By induction, there exist two sequences , in , such that

If for some , with , then is a coincidence point of the pair ; if for some , with , then is a coincidence point of the pair ; if for some , with , then is a coincidence point of the pair .

On the other hand, if there exists such that , then for any . This implies that is a -Cauchy sequence.

In fact, if there exists such that , then applying the contractive condition (v) with , , and , and the property of , we get

From Lemma 1.20 and the property of , we have
Which implies that . So we find for any . This implies that is a -Cauchy sequence. The same conclusion holds if , or for some .

Without loss of generality, we can assume that for all and .

Now we prove that is a -Cauchy sequence in .

Let , then we have
for all . Actually, from the condition (v), (2.3) and the property of , we have
By Lemma 1.20 and the property of , we have

Again, using condition (v), (2.3) and the property of , we get
From Lemma 1.20 and the property of we have

Similarly, we can get
From Lemma 1.20 and property of , we have
Combining (2.8), (2.10), and (2.12), we know that the (2.6) holds. This implies that is a nonnegative sequence which is strictly decreasing, hence, is convergent and , for all .

For any , , by combining , , and (2.6), we have
From the convergence of the sequence and the condition (iii) we assure that
Thus, is a -Cauchy sequence in , since is a complete -metric space, there exists such that , hence
Since are -weakly commuting mappings, thus we have
On taking at both sides, noting that and are continuous mappings, we have
Which gives that . Similarly, we can get .

By using condition (v) and the property of , we get
Thus, by Lemma 1.20, and the property of , noting that , , , we have
Which implies that
By Remark 1.1, we have , therefore, . So, immediately, we can have . Setting
Since are -weakly commuting mappings, we have
Which gives that . By the same argument, we can get , , So we have , , . Again, by condition (v), we have
By the Lemma 1.20 and the property of , we have
Which implies that . Thus, by the property of , we have , hence, . Similarly, we can prove that , , so we get , which means that is a common fixed point of , , , , , and .

Now, we will show the common fixed point of , , , , and is unique. Actually, assume is another common fixed point of , , , , , and , then by condition (v), we have
By Lemma 1.20 and the property of , we have
It is a contradiction, unless , that is, , , , , , and have a unique common fixed point in . This completes the proof of Theorem 2.1.

*Remark 2.2. *If we take: (1) ; (2) ; (3) ( is identity mapping); (4) and ; (5) , ; (6) in Theorem 2.1, then several new results can be obtained.

Corollary 2.3. *Let be a complete -metric space and let , , , , , and be six mappings of into itself. If there exists an increasing function satisfying the conditions (i)~(iii) and the following conditions:*(iv)*, , ,*(v)*, .**Then,*(a)* one of the pairs , , and has a coincidence point in .*(b)* if , , and are three pairs of continuous -weakly commuting mappings, then the mappings , , , , , and have a unique common fixed point in .*

* Proof. *Taking in Theorem 2.1, the conclusion of Corollary 2.3 can be obtained from Theorem 2.1 immediately. This completes the proof of Corollary 2.3.

Now we give an example to support Corollary 2.3.

*Example 2.4. *Let , , for all . Let be defined by , , , , , . Clearly, we can get , , . Through calculation, we have
Thus, we have
Now we choose , and , then we have and satisfies (i)~(iii). Thus, we have

On the other hand, let for all , we have for all . Which means that , , and are three pairs of continuous -weakly commuting mappings in . So that all the conditions of Corollary 2.3 are satisfied. Moreover, is the unique common fixed point for all of the mappings , , , , , and .

Corollary 2.5. *Let be a complete -metric space, are three self-mappings in , and function satisfies conditions (i)~(iii) and the following condition:
**
Then , , and have a unique common fixed point in .*

* Proof. *Taking in Corollary 2.3, where is an identity mapping. Then the conclusion of Corollary 2.5 can be obtained from Corollary 2.3 immediately. This completes the proof of Corollary 2.5.

Corollary 2.6. *Let be a complete -metric space, is a self-mapping in , and function satisfies conditions (i)~(iii) and the following condition:
**
Then has a unique fixed point in .*

* Proof. *Taking in Corollary 2.5, the conclusion of Corollary 2.6 can be obtained from Corollary 2.5 immediately. This completes the proof of Corollary 2.6.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030), and the Physical Experiment Center in Hangzhou Normal University.

#### References

- M. Altman, “A fixed point theorem in compact metric spaces,”
*The American Mathematical Monthly*, vol. 82, no. 2, pp. 827–829, 1975. View at Publisher · View at Google Scholar - F. Gu and B. Deng, “Common fixed point for Altman type mappings,”
*Journal of Harbin Normal University*, vol. 17, no. 5, pp. 44–46, 2001. View at Google Scholar - Z. Liu, “On common fixed points of Altman type mappings,”
*Journal of Liaoning Normal University (Natural Science Edition)*, vol. 16, no. 1, pp. 1–4, 1993. View at Google Scholar - S. Zhang, “Common fixed point theorem of Altman type mappings,”
*Journal of Yantai Normal University*, vol. 16, no. 2, pp. 95–97, 2000. View at Google Scholar - Y. Li and F. Gu, “Common fixed point theorem of Altman integral type mappings,”
*Journal of Nonlinear Science and Applications*, vol. 2, no. 4, pp. 214–218, 2009. View at Google Scholar · View at Zentralblatt MATH - 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. View at Google Scholar · View at Zentralblatt MATH - M. Abbas and B. E. Rhoades, “Common fixed point results for noncommuting mappings without continuity in generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 215, no. 1, pp. 262–269, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. K. Vats, S. Kumar, and V. Sihag, “Some common fixed point theorems for compatible mappings of type(
*A*) in complete*G*-metric space,”*Advances in Fuzzy Mathematics*, vol. 6, no. 1, pp. 27–38, 2011. View at Google Scholar - S. Manro, S. S. Bhatia, and S. Kumar, “Expansion mappings theorems in
*G*-metric spaces,”*International Journal of Contemporary Mathematical Sciences*, vol. 5, no. 51, pp. 2529–2535, 2010. View at Google Scholar - Z. Mustafa, H. Obiedat, and F. Awawdeh, “Some fixed point theorem for mapping on complete
*G*-metric spaces,”*Fixed Point Theory and Applications*, vol. 2008, Article ID 189870, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Mustafa, W. Shatanawi, and M. Bataineh, “Existence of fixed point results in
*G*-metric spaces,”*International Journal of Mathematics and Mathematical Sciences*, vol. 2009, Article ID 283028, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Mustafa and B. Sims, “Fixed point theorems for contractive mappings in complete
*G*-metric spaces,”*Fixed Point Theory and Applications*, vol. 2009, Article ID 917175, 10 pages, 2009. View at Google Scholar · View at Zentralblatt MATH - M. Abbas, T. Nazir, and R. Saadati, “Common fixed point results for three maps in generalized metric space,”
*Advances in Difference Equations*, vol. 49, no. 1, pp. 1–20, 2011. View at Google Scholar - W. Shatanawi, “Fixed point theory for contractive mappings satisfying Φ-maps in
*G*-metric spaces,”*Fixed Point Theory and Applications*, vol. 2010, Article ID 181650, 9 pages, 2010. View at Google Scholar - R. Chugh, T. Kadian, A. Rani, and B. E. Rhoades, “Property
*P*in*G*-metric spaces,”*Fixed Point Theory and Applications*, vol. 2010, Article ID 401684, 12 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - M. Abbas, T. Nazir, and S. Radenović, “Some periodic point results in generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 217, no. 8, pp. 4094–4099, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Mustafa, F. Awawdeh, and W. Shatanawi, “Fixed point theorem for expansive mappings in
*G*-metric spaces,”*International Journal of Contemporary Mathematical Sciences*, vol. 5, no. 50, pp. 2463–2472, 2010. View at Google Scholar - M. Abbas, S. H. Khan, and T. Nazir, “Common fixed points of
*R*-weakly commuting maps in generalized metric spaces,”*Fixed Point Theory and Applications*, vol. 2011, Article ID 784595, 11 pages, 2011. View at Google Scholar - H. Aydi, W. Shatanawi, and C. Vetro, “On generalized weak
*G*-contraction mapping in*G*-metric spaces,”*Computers & Mathematics with Applications*, vol. 62, no. 11, pp. 4223–4229, 2011. View at Publisher · View at Google Scholar - M. Abbas, A. R. Khan, and T. Nazir, “Coupled common fixed point results in two generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 217, no. 13, pp. 6328–6336, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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, 2011. View at Publisher · View at Google Scholar - H. Aydi, “A fixed point result involving a generalized weakly contractive condition in
*G*-metric spaces,”*Bulletin Mathematical Analysis and Applications*, vol. 3, no. 4, pp. 180–188, 2011. View at Google Scholar · View at Zentralblatt MATH - Z. Mustafa, M. Khandagji, and W. Shatanawi, “Fixed point results on complete
*G*-metric spaces,”*Studia Scientiarum Mathematicarum Hungarica*, vol. 48, no. 3, pp. 304–319, 2011. View at Publisher · View at Google Scholar - W. Shatanawi, “Coupled fixed point theorems in generalized metric spaces,”
*Hacettepe Journal of Mathematics and Statistics*, vol. 40, no. 3, pp. 441–447, 2011. View at Google Scholar · View at Zentralblatt MATH - N. Hussain, M. H. Shah, and S. Radenović, “Fixed points of weakly contractions through occasionally weak compatibility,”
*Journal of Computational Analysis and Applications*, vol. 13, no. 3, pp. 532–543, 2011. View at Google Scholar - M. Abbas, T. Nazir, and D. Dorić, “Common fixed point of mappings satisfying (E.A) property in generalized metric spaces,”
*Applied Mathematics and Computation*, vol. 218, no. 14, pp. 7665–7670, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Mustafa, H. Aydi, and E. Karapınar, “On common fixed points in
*G*-metric spaces using (E.A) property,”*Computers & Mathematics with Applications*, vol. 64, no. 6, pp. 1944–1956, 2012. View at Publisher · View at Google Scholar - Lj. Gajić and M. Stojaković, “On Ćirié generalization of mappings with a contractive iterate at a point in
*G*-metric spaces,”*Applied Mathematics and Computation*, vol. 219, no. 1, pp. 435–441, 2012. View at Publisher · View at Google Scholar - H. Ye and F. Gu, “Common fixed point theorems for a class of twice power type contraction maps in
*G*-metric spaces,”*Abstract and Applied Analysis*, vol. 2012, Article ID 736214, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Gu, “Common fixed point theorems for six mappings in generalized metric spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 379212, 2012. View at Publisher · View at Google Scholar - W. Long, M. Abbas, T. Nazir, and S. Radenović, “Common fixed point for two pairs of mappings satisfying (E.A) property in generalized metric spaces,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 394830, 15 pages, 2012. View at Publisher · View at Google Scholar - N. Tahat, H. Aydi, E. Karapinar, and W. Shatanawi, “Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in
*G*-metric spaces,”*Fixed Point Theory and Applications*, vol. 2012, article 48, 2012. View at Publisher · View at Google Scholar - Z. Mustafa, “Some new common fixed point theorems under strict contractive conditions in
*G*-metric spaces,”*Journal of Applied Mathematics*, vol. 2012, Article ID 248937, 21 pages, 2012. View at Publisher · View at Google Scholar - V. Popa and A. M. Patriciu, “A general fixed point theorem for pairs of weakly compatible mappings in
*G*-metric spaces,”*Journal of Nonlinear Science and its Applications*, vol. 5, pp. 151–160, 2012. View at Google Scholar - M. Khandaqji, S. Al-Sharif, and M. Al-Khaleel, “Property
*P*and some fixed point results on ($\psi $,$\phi $)-weakly contractive*G*-metric spaces,”*International Journal of Mathematics and Mathematical Sciences*, vol. 2012, Article ID 675094, 11 pages, 2012. View at Publisher · View at Google Scholar - Z. Mustafa, “Common fixed points of weakly compatible mappings in
*G*-metric spaces,”*Applied Mathematical Sciences*, vol. 6, no. 92, pp. 4589–4600, 2012. View at Google Scholar - M. Aggarwal, R. Chugh, and R. Kamal, “Suzuki-type fixed point results in
*G*-metric spaces and applications,”*International Journal of Computer Applications*, vol. 47, no. 12, pp. 14–17, 2012. View at Google Scholar - M. Gugnani, M. Aggarwal, and R. Chugh, “Common fixed point results in
*G*-metric spaces and applications,”*International Journal of Computer Applications*, vol. 43, no. 11, pp. 38–42, 2012. View at Google Scholar - E. Karapinar, A. Yildiz-Ulus, and İ. M. Erhan, “Cyclic contractions on
*G*-metric spaces,”*Abstract and Applied Analysis*, vol. 2012, Article ID 182947, 15 pages, 2012. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Aydi, “A common fixed point of integral type contraction in generalized metric spaces,”
*Journal of Advanced Mathematical Studies*, vol. 5, no. 1, pp. 111–117, 2012. View at Google Scholar - K. P. R. Rao, K. B. Lakshmi, Z. Mustafa, and V. C. C. Raju, “Fixed and related fixed point theorems for three maps in
*G*-metric spaces,”*Journal of Advanced Studies in Topology*, vol. 3, no. 4, pp. 12–19, 2012. View at Google Scholar - R. Saadati, S. M. Vaezpour, P. Vetro, and B. E. Rhoades, “Fixed point theorems in generalized partially ordered
*G*-metric spaces,”*Mathematical and Computer Modelling*, vol. 52, no. 5-6, pp. 797–801, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized
*G*-metric spaces,”*Mathematical and Computer Modelling*, vol. 54, no. 1-2, pp. 73–79, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered
*G*-metric spaces,”*Mathematical and Computer Modelling*, vol. 54, no. 9-10, pp. 2443–2450, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - W. Shatanawi, “Some fixed point theorems in ordered
*G*-metric spaces and applications,”*Abstract and Applied Analysis*, vol. 2011, Article ID 126205, 11 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. V. Luong and N. X. Thuan, “Coupled fixed point theorems in partially ordered
*G*-metric spaces,”*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1601–1609, 2012. View at Publisher · View at Google Scholar - H. Aydi, M. Postolache, and W. Shatanawi, “Coupled fixed point results for ($\psi $,$\phi $)-weakly contractive mappings in ordered
*G*-metric spaces,”*Computers & Mathematics with Applications*, vol. 63, no. 1, pp. 298–309, 2012. View at Publisher · View at Google Scholar - M. Abbas, T. Nazir, and S. Radenović, “Common fixed point of generalized weakly contractive maps in partially ordered
*G*-metric spaces,”*Applied Mathematics and Computation*, vol. 218, no. 18, pp. 9383–9395, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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, 2012. View at Publisher · View at Google Scholar - H. Aydi, E. Karapinar, and W. Shatanawi, “Tripled common fixed point results for generalized contractions in ordered generalized metric spaces,”
*Fixed Point Theory and Applications*, vol. 2012, article 101, 2012. View at Publisher · View at Google Scholar - M. Abbas, W. Sintunavarat, and P. Kumam, “Coupled fixed point of generalized contractive mappings on partially ordered
*G*-metric spaces,”*Fixed Point Theory and Applications*, vol. 2012, article 31, 2012. View at Publisher · View at Google Scholar - H. K. Nashine, “Coupled common fixed point results in ordered
*G*-metric spaces,”*Journal of Nonlinear Science and its Applications*, vol. 5, no. 1, pp. 1–13, 2012. View at Google Scholar - Z. Mustafa, H. Aydi, and E. Karapmar, “Mixed
*g*-monotone property and quadruple fixed point theorems in partially ordered metric space,”*Fixed Point Theory and Applications*, vol. 2012, article 71, 2012. View at Publisher · View at Google Scholar - W. Shatanawi, M. Abbas, H. Aydi, and N. Tahat, “Common coupled coincidence and coupled fixed points in
*G*-metric spaces,”*Nonlinear Analysis and Application*, vol. 2012, Article ID jnaa-00162, 16 pages, 2012. View at Publisher · View at Google Scholar - A. Razani and V. Parvaneh, “On generalized weakly
*G*-contractive mappings in partially ordered Gmetric spaces,”*Abstract and Applied Analysis*, vol. 2012, Article ID 701910, 18 pages, 2012. View at Publisher · View at Google Scholar