Abstract
In this paper, we use weakly commuting and weakly compatible conditions of self-mapping pairs, prove some new common fixed point theorems for three pairs of self-mappings in G-metric spaces. An example is provided to support our result. The results presented in this paper extend and improve several well-known comparable results.
MSC:47H10, 54H25, 54E50.
Similar content being viewed by others
1 Introduction and preliminaries
The metric fixed point theory is very important and useful in mathematics. It can be applied in various areas, for instance, approximation theory, optimization and variational inequalities. Many authors have introduced the generalizations of metric spaces, for example, Gähler [1, 2] (called 2-metric spaces) and Dhage [3, 4] (called D-metric spaces). In 2003, Mustafa and Sims [5] found that most of the claims concerning the fundamental topological properties of D-metric spaces are incorrect. Therefore, they [6] introduced a new structure of generalized metric spaces, which are called G-metric spaces, as a generalization of metric spaces, to develop and introduce a new fixed point theory for various mappings in this new structure. Later, several fixed point and common fixed point theorems in G-metric spaces were obtained by [6–51].
The purpose of this paper is to use the concept of weakly commuting mappings and weakly compatible mappings to discuss some new common fixed point problem for six self-mappings in G-metric spaces. The results presented in this paper extend and improve the corresponding results of Abbas et al. [7], Mustafa and Sims [8], Abbas and Rhoades [9], Mustafa et al. [10], Mustafa et al. [11], Abbas et al. [12], Chugh and Kadian [13], Manro et al. [14], Vats et al. [15].
We now recall some definitions and properties in G-metric spaces.
Definition 1.1 [6]
Let X be a nonempty set and let be a function satisfying the following properties:
-
() if ;
-
() for all with ;
-
() for all with ;
-
() , symmetry in all three variables;
-
() for all .
Then the function G is called a generalized metric, or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Definition 1.2 [6]
Let be a G-metric space and let be a sequence of points of X. A point is said to be the limit of the sequence if , and we say that the sequence is G-convergent to x or G-convergent to x.
Thus, in a G-metric space if, for any , there exists such that for all .
Proposition 1.1 [6]
Let be a G-metric space, then the following are equivalent:
-
1.
is G-convergent to x.
-
2.
as .
-
3.
as .
-
4.
as .
Definition 1.3 [6]
Let be a G-metric space. A sequence is called G-Cauchy if, for every , there is such that for all ; that is, as .
Proposition 1.2 [6]
Let be a G-metric space. Then the following are equivalent:
-
1.
The sequence is G-Cauchy.
-
2.
For every , there is such that for all .
Definition 1.4 [6]
Let and be G-metric spaces and let be a function. Then f is said to be G-continuous at a point if and only if, for every , there is such that and imply . A function f is G-continuous at X if only if it is G-continuous at .
Proposition 1.3 [6]
Let be a G-metric space. Then the function is jointly continuous in all three of its variables.
Definition 1.5 [6]
A G-metric space is G-complete if every G-Cauchy sequence in is G-convergent in X.
Definition 1.6 [16]
Two self-mappings f and g of a G-metric space are said to be weakly commuting if for all x in X.
Definition 1.7 [16]
Let f and g be two self-mappings from a G-metric space into itself. Then the mappings f and g are said to be weakly compatible if whenever .
Proposition 1.4 [6]
Let be a G-metric space. Then, for all x, y, z, a in X, it follows that:
-
(i)
If , then ;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
;
-
(vi)
.
2 Common fixed point theorems
Theorem 2.1 Let be a complete G-metric space, and let f, g, h, A, B and C be six mappings of X into itself satisfying the following conditions:
-
(i)
, , ;
-
(ii)
,
(2.1)or
(2.2)where . If one of the following conditions is satisfied:
-
(a)
Either f or A is G-continuous, the pair is weakly commuting, the pairs and are weakly compatible;
-
(b)
Either g or B is G-continuous, the pair is weakly commuting, the pairs and are weakly compatible;
-
(c)
Either h or C is G-continuous, the pair is weakly commuting, the pairs and are weakly compatible.
-
(a)
Then
-
(I)
one of the pairs , and has a coincidence point in X;
-
(II)
the mappings f, g, h, A, B and C have a unique common fixed point in X.
Proof Suppose that mappings f, g, h, A, B and C satisfy condition (2.1).
Let in X be an arbitrary point since , , . There exist the sequences and in X such that
for all .
If there exists such that , then the conclusion (I) of Theorem 2.1 holds. In fact, if there exists such that , then , where . Hence the pair has a coincidence point . If , then , where . Therefore, the pair has a coincidence point . If , then , where . And so the pair has a coincidence point .
On the other hand, if there exists such that , then for any . This implies that is a G-Cauchy sequence.
Actually, if there exists such that , then applying the contractive condition (2.1) with , and , we get
If , then from condition () and Proposition 1.4(iii), we get
which implies that , that is a contradiction, since . So, we find for any . This implies that is a G-Cauchy sequence. The same conclusion holds if , or for some .
Assume for the rest of the paper that for any . Applying again (2.1) with , and and using conditions () and (), we get that
From we obtain
where . Similarly it can be shown that
and
It follows from (2.3), (2.4) and (2.5) that, for all ,
Therefore, for all , , by () and (), we have
Hence is a G-Cauchy sequence in X. Since X is a complete G-metric space, there exists a point such that ().
Since the sequences , and are all subsequences of , then they all converge to u
Now we prove that u is a common fixed point of f, g, h, A, B and C under condition (a).
First, we suppose that A is continuous, the pair is weakly commuting, the pairs and are weakly compatible.
Step 1. We prove that .
By (2.6) and a weakly commuting of mapping pair , we have
Since A is continuous, then (), (). By (2.7) we know that ().
From condition (2.1) we know
Letting and using Proposition 1.4(iii), we have
which implies that , and so since .
Again, by use of condition (2.1), we have
Letting , using (2.6), and Proposition 1.4(iii), we obtain
This implies that and so . Thus we have .
Step 2. We prove that .
Since and , there is a point such that . Again, by use of condition (2.1), we have
Letting , using and Proposition 1.4(iii), we have
which implies that , and so .
Since the pair is weakly compatible, we have
Again, by use of condition (2.1), we have
Letting , using and and Proposition 1.4(iii), we have
This implies that , and so .
Step 3. We prove that .
Since and , there is a point such that . Again, by use of condition (2.1), we have
Using , and Proposition 1.4(iii), we obtain
Hence , and so .
Since the pair is weakly compatible, we have
Again, by use of condition (2.1), we have
Using , , and Proposition 1.4(iii), we have
Thus , and so .
Therefore u is the common fixed point of f, g, h, A, B and C when A is continuous and the pair is weakly commuting, the pairs and are weakly compatible.
Next, we suppose that f is continuous, the pair is weakly commuting, the pairs and are weakly compatible.
Step 1. We prove that .
By (2.6) and a weakly commuting mapping pair , we have
Since f is continuous, then (), (). By (2.6) we know ().
From condition (2.1) we know
Letting and noting Proposition 1.4(iii), we have
which implies that , and so .
Step 2. We prove that .
Since and , there is a point such that . Again, by use of condition (2.1), we have
Letting , using and Proposition 1.4(iii), we have
This implies that , and so .
Since the pair is weakly compatible, we have
Again, by use of condition (2.1), we have
Letting , using , and Proposition 1.4(iii), we have
Therefore, , and so .
Step 3. We prove that .
Since and , there is a point such that . Again, by use of condition (2.1), we have
Letting , using and Proposition 1.4(iii), we obtain
Thus , and so .
Since the pair is weakly compatible, we have
Again, by use of condition (2.1), we have
Letting , using , and Proposition 1.4(iii), we have
which implies that , and so .
Step 4. We prove that .
Since and , there is a point such that . Again, by use of condition (2.1), we have
Using , and Proposition 1.4(iii), we obtain
Hence , and so .
Since the pair is weakly compatible, we have
Therefore u is the common fixed point of f, g, h, A, B and C when S is continuous and the pair is weakly commuting, the pairs and are weakly compatible.
Similarly we can prove the result that u is a common fixed point of f, g, h, A, B and C under the condition of (b) or (c).
Finally, we prove the uniqueness of a common fixed point u.
Let u and q be two common fixed points of f, g, h, A, B and C. By use of condition (2.1), we have
This implies that , and so . Thus the common fixed point is unique.
The proof using (2.2) is similar. This completes the proof. □
Now we introduce an example to support Theorem 2.1.
Example 2.2 Let and let be a G-metric space defined by for all in X. Let f, g, h, A, B and C be self-mappings defined by
Note that A is G-continuous in X, and f, g, h, B and C are not G-continuous in X.
-
(i)
Clearly we can get , , .
Actually, because , , , , , , so we know , and .
-
(ii)
By the definition of the mappings of f and A, for all , , so we can get the pair is weakly commuting.
By the definition of the mappings of g and B, only for , , at this time , so , so we can obtain the pair is weakly compatible. Similarly we can prove the pair is also weakly compatible.
-
(iii)
Now we prove the mappings f, g, h, A, B and C satisfy condition (2.1) of Theorem 2.1 with
Case 1. If , then
Thus we have
Case 2. If , , then
Hence we get
Case 3. If , , then
Therefore we obtain
Case 4. If , , then
Thus we have
Case 5. If , , then
Hence we obtain
Case 6. If , , then
So we have
Case 7. If , , then
Thus we get
Case 8. If , then
Then in all the above cases, the mappings f, g, h, A, B and C satisfy condition (2.1) of Theorem 2.1 with . So that all the conditions of Theorem 2.1 are satisfied. Moreover, is the unique common fixed point for all of the mappings f, g, h, A, B and C.
In Theorem 2.1, if we take (I is identity mapping, the same below), then we have the following corollary.
Corollary 2.3 Let be a complete G-metric space and let f, g and h be three mappings of X into itself satisfying the following conditions:
or
, where . Then f, g and h have a unique common fixed point in X.
Remark 2.4 Corollary 2.3 generalizes and extends the corresponding results in Abbas et al. [[7], Theorem 2.1].
Also, if we take and in Theorem 2.1, then we get the following.
Corollary 2.5 Let be a complete G-metric space and let f be a mapping of X into itself satisfying the following conditions:
or
, where . Then f has a unique fixed point in X.
Remark 2.6 Corollary 2.5 generalizes and extends the corresponding results in Mustafa and Sims [[8], Theorem 2.1].
Remark 2.7 Theorem 2.1, Corollaries 2.3 and 2.5 in this paper also improve and generalize the corresponding results of Abbas and Rhoades [[9], Theorems 2.4 and 2.5], Mustafa et al. [[10], Theorems 2.3, 2.5, 2.8 and Corollary 2.6], Mustafa et al. [[11], Theorem 2.5], Abbas et al. [[12], Theorem 2.1, Corollaries 2.3-2.6] and Chugh and Kadian [[13], Theorem 2.2].
Remark 2.8 In Theorem 2.1, we have taken: (1) ; (2) ; (3) and ; (4) , , several new results can be obtained.
Theorem 2.9 Let be a complete G-metric space and let f, g, h, A, B and C be six mappings of X into itself satisfying the following conditions:
-
(i)
, , ;
-
(ii)
The pairs , and are commuting mappings;
-
(iii)
,
(2.13)
or
where , , then f, g, h, A, B and C have a unique common fixed point in X.
Proof Suppose that mappings f, g, h, A, B and C satisfy condition (2.13). Since , , so that . Similar, we can show that and . From Theorem 2.1, we see that , , , A, B and C have a unique common fixed point u.
It follows from (2.13) that
By condition (ii) we have , note that , and Proposition 1.4(iii), we obtain
which implies that , and so .
By the same argument, we can prove and . Thus we have , so that f, g, h, A, B and C have a common fixed point u in X. Let v be any other common fixed point of f, g, h, A, B and C, then by use of condition (2.13) and Proposition 1.4(iii), we have
This implies that , and so . Thus common fixed point is unique.
The proof using (2.14) is similar. This completes the proof. □
In Theorem 2.9, if we take , then we have the following corollary.
Corollary 2.10 Let be a complete G-metric space and let f, g and h be three mappings of X into itself satisfying the following conditions:
or
, where , , then f, g and h have a unique common fixed point in X.
Remark 2.11 Corollary 2.10 generalizes and extends the corresponding results in Abbas et al. [[7], Corollary 2.3].
Also, if we take and in Theorem 2.9, then we get the following.
Corollary 2.12 Let be a complete G-metric space and let f be a mapping of X into itself satisfying the following conditions:
or
, where , , then f has a unique fixed point in X.
Remark 2.13 Corollary 2.12 generalizes and extends the corresponding results in Mustafa and Sims [[8], Corollary 2.3].
Remark 2.14 Theorem 2.9, Corollaries 2.10 and 2.12 generalize and extend the corresponding results in Mustafa et al. [[10], Corollaries 2.4 and 2.7].
Remark 2.15 In Theorem 2.9, we have taken: (1) ; (2) ; (3) and ; (4) , , several new results can be obtained.
Theorem 2.16 Let be a complete G-metric space and let f, g, h, A, B and C be six mappings of X into itself satisfying the following conditions:
-
(i)
, , ;
-
(ii)
,
(2.19)or
(2.20)where () and . If one of the following conditions are satisfied:
-
(a)
Either f or A is G-continuous, the pair is weakly commuting, the pairs and are weakly compatible;
-
(b)
Either g or B is G-continuous, the pair is weakly commuting, the pairs and are weakly compatible;
-
(c)
Either h or C is G-continuous, the pair is weakly commuting, the pairs and are weakly compatible.
-
(a)
Then
-
(I)
one of the pairs , and has a coincidence point in X;
-
(II)
the mappings f, g, h, A, B and C have a unique common fixed point in X.
Proof Suppose that mappings f, g, h, A, B and C satisfy condition (2.19). For , let
Then
Therefore, it follows from (2.19) that
Taking in Theorem 2.1, the conclusion of Theorem 2.16 can be obtained from Theorem 2.1 immediately.
The proof using (2.20) is similar. This completes the proof. □
Remark 2.17 Theorem 2.16 generalizes and extends the corresponding results in Mustafa et al. [[10], Theorem 2.1], Mustafa et al. [[12], Theorem 2.5].
Remark 2.18 In Theorem 2.16, we have taken: (1) ; (2) ; (3) ; (4) and ; (5) , , several new results can be obtained.
Corollary 2.19 Let be a complete G-metric space and let f, g, h, A, B and C be six mappings of X into itself satisfying the following conditions:
-
(i)
, , ;
-
(ii)
The pairs , and are commuting mappings;
-
(iii)
,
(2.21)or
(2.22)where , () and . Then f, g, h, A, B and C have a unique common fixed point in X.
Proof The proof follows from Theorem 2.9, and from an argument similar to that used in Theorem 2.16 □
Remark 2.20 In Theorem 2.1, we have taken: (1) ; (2) ; (3) ; (4) and ; (5) , , several new results can be obtained.
Remark 2.21 Theorems 2.1, 2.9 and 2.16 in this paper also improve and generalize the corresponding results of Manro et al. [14], Vats et al. [15].
References
Gähler S: 2-Metrische räume und ihre topologische struktur. Math. Nachr. 1963, 26: 115–148. 10.1002/mana.19630260109
Gähler S: Zur geometric 2-metrische räume. Rev. Roum. Math. Pures Appl. 1966, 11: 665–667.
Dhage BC: Generalized metric space and mapping with fixed point. Bull. Calcutta Math. Soc. 1992, 84: 329–336.
Dhage BC: Generalized metric space and topological structure I . An. Stiint. Univ. Al. I. Cuza Iasi. Mat. 2000, 46(1):3–24.
Mustafa Z, Sims B: Some remarks concerning D -metric spaces. Proceedings of the International Conference on Fixed Point Theory and Applications 2003, 189–198., Valencia (Spain), July
Mustafa Z, Sims B: A new approach to a generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.
Abbas M, Nazir T, Saadati R: Common fixed point results for three maps in generalized metric space. Adv. Differ. Equ. 2011., 2011: Article ID 49
Mustafa Z, Sims B: Fixed point theorems for contractive mappings in complete G -metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 917175
Abbas M, Rhoades BE: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 2009, 215: 262–269. 10.1016/j.amc.2009.04.085
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorems for mappings on complete G -metric space. Fixed Point Theory Appl. 2008., 2008: Article ID 189870
Mustafa Z, Shatanawi W, Bataineh M: Existence of fixed point results in G -metric spaces. Int. J. Math. Math. Sci. 2009., 2009: Article ID 283028
Abbas M, Khan SH, Nazir T: Common fixed points of R -weakly commuting maps in generalized metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 784595
Chugh R, Kadian T, Rani A, Rhoades BE: Property P in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 401684
Manro S, Kumar S, Bhatia SS: R -Weakly commuting maps in G -metric spaces. Fasc. Math. 2011, 47: 11–18.
Vats RK, Kumar S, Sihag V: Fixed point theorems in complete G -metric spaces. Fasc. Math. 2011, 47: 127–139.
Abbas M, Nazir T, Vetro P: Common fixed point results for three maps in G -metric spaces. Filomat 2011, 25(4):1–17. 10.2298/FIL1104001A
Mustafa Z, Awawdeh F, Shatanawi W: Fixed point theorem for expansive mappings in G -metric spaces. Int. J. Contemp. Math. Sci. 2010, 5(50):2463–2472.
Rao KPR, Sombabu A, Prasad JR: A common fixed point theorem for six expansive mappings in G -metric spaces. J. Sci. Eng. Technol. 2011, 7(1):113–120.
Shatanawi W: Fixed point theory for contractive mappings satisfying Φ-maps in G -metric spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 181650
Gajić L, Lozanov-Crvenković Z: A fixed point results for mappings with contractive iterate at a point in G -metric spaces. Filomat 2011, 25(2):53–58. 10.2298/FIL1102053G
Vats RK, Kumar S, Sihag V: Some common fixed point theorem for compatible mappings of type in complete G -metric space. Adv. Fuzzy Math. 2011, 6(1):27–38.
Aydi H: A fixed point result involving a generalized weakly contractive condition in G -metric spaces. Bull. Math. Anal. Appl. 2011, 3(4):180–188.
Gajić L, Stojaković M: On Ćirić generalization of mappings with a contractive iterate at a point in G -metric spaces. Appl. Math. Comput. 2012, 219(1):435–441. 10.1016/j.amc.2012.06.041
Kaewcharoen A: Some common fixed point theorems for contractive mappings satisfying Φ-maps in G -metric spaces. Banach J. Math. Anal. 2012, 6(1):101–111.
Mustafa Z, Khandagji M, Shatanawi W: Fixed point results on complete G -metric spaces. Studia Sci. Math. Hung. 2011, 48(3):304–319.
Aydi H, Shatanawi W, Vetro C: On generalized weakly G -contraction mapping in G -metric spaces. Comput. Math. Appl. 2011, 62: 4222–4229. 10.1016/j.camwa.2011.10.007
Abbas M, Nazir T, Radenović S: Some periodic point results in generalized metric spaces. Appl. Math. Comput. 2010, 217: 4094–4099. 10.1016/j.amc.2010.10.026
Obiedat H, Mustafa Z: Fixed point results on a nonsymmetric G -metric spaces. Jordan J. Math. Stat. 2010, 3(2):65–79.
Shatanawi W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat. 2011, 40(3):441–447.
Mustafa Z, Obiedat H: A fixed point theorem of Reich in complete G -metric spaces. CUBO 2010, 12(1):83–93. 10.4067/S0719-06462010000100008
Gu F: Common fixed point theorems for six mappings in generalized metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 379212
Gu F, Ye H: Common fixed point theorems of Altman integral type mappings in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 630457
Ye H, Gu F: Common fixed point theorems for a class of twice power type contraction maps in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 736214
Yin Y, Gu F: Common fixed point theorem about four mappings in G -metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. 2012, 11(6):511–515.
Gu F, Yin Y:Common fixed point for three pairs of self-maps satisfying common property in generalized metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 808092
Gu F, Zhou S: Coupled common fixed point theorems for a pair of commuting mappings in partially ordered G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 64 10.1186/1687-1812-2013-64
Ye H, Lu J, Gu F: A new common fixed point theorem for noncompatible mappings of type in G -metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. 2013, 12(1):50–56.
Zhou S, Gu F: Some new fixed point theorems in G -metric spaces. J. Hangzhou Norm. Univ., Nat. Sci. 2012, 11(1):47–50.
Mustafa Z: Common fixed points of weakly compatible mappings in G -metric spaces. Appl. Math. Sci. 2012, 6(92):4589–4600.
Mustafa Z: Some new common fixed point theorems under strict contractive conditions in G -metric spaces. J. Appl. Math. 2012., 2012: Article ID 248937
Tahat N, Aydi H, Karapinar E, Shatanawi W: Common fixed points for single-valued and multi-valued maps satisfying a generalized contraction in G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 48
Aydi H: A common fixed point of integral type contraction in generalized metric spaces. J. Adv. Math. Stud. 2012, 5(1):111–117.
Abbas M, Nazir T, Doric D:Common fixed point of mappings satisfying property in generalized metric spaces. Appl. Math. Comput. 2012, 218(14):7665–7670. 10.1016/j.amc.2011.11.113
Mustafa Z, Aydi H, Karapinar E: On common fixed points in G -metric spaces using property. Comput. Math. Appl. 2012, 64(6):1944–1956. 10.1016/j.camwa.2012.03.051
Long W, Abbas M, Nazir T, Radenovic S:Common fixed point for two pairs of mappings satisfying property in generalized metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 394830
Gugnani M, Aggarwal M, Chugh R: Common fixed point results in G -metric spaces and applications. Int. J. Comput. Appl. 2012, 43(11):38–42.
Popa V, Patriciu A-M: A general fixed point theorem for pairs of weakly compatible mappings in G -metric spaces. J. Nonlinear Sci. Appl. 2012, 5: 151–160.
Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870 10.1155/2012/815870
Aydi H, Karapinar E, Shatanawi W: Tripled common fixed point results for generalized contractions in ordered generalized metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 101
Aydi H, Karapinar E, Shatanawi W: Tripled fixed point results in generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 314279
Gholizadeh L, Saadati R, Shatanawi W, Vaezpour SM: Contractive mapping in generalized, ordered metric spaces with application in integral equations. Math. Probl. Eng. 2011., 2011: Article ID 380784 10.1155/2011/380784
Acknowledgements
The authors are grateful to the editor and the reviewer for suggestions which improved the contents of the article. This work is supported by the National Natural Science Foundation of China (11271105) and the Natural Science Foundation of Zhejiang Province (Y6110287, LY12A01030).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to this work. Both authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Gu, F., Yang, Z. Some new common fixed point results for three pairs of mappings in generalized metric spaces. Fixed Point Theory Appl 2013, 174 (2013). https://doi.org/10.1186/1687-1812-2013-174
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-174