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Common fixed point theorems for weakly increasing mappings on ordered orbitally complete metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 85 (2012)
Abstract
In this article, we prove existence results for common fixed points of two or three relatively asymptotically regular mappings satisfying the orbital continuity of one of the involved maps on ordered orbitally complete metric spaces. We furnish suitable examples to demonstrate the validity of the hypotheses of our results.
Mathematics Subject Classification (2010): 47H10; 54H25.
1 Introduction and preliminaries
Browder and Petryshyn introduced the concept of asymptotic regularity of a self-map at a point in a metric space.
Definition 1[1] A self-map on a metric space is said to be asymptotically regular at a point if .
Recall that the set is called the orbit of the self-map at the point .
Definition 2[2] A metric space is said to be -orbitally complete if every Cauchy sequence contained in (for some x in ) converges in .
Here, it can be pointed out that every complete metric space is -orbitally complete for any , but a -orbitally complete metric space need not be complete.
Definition 3[1] A self-map defined on a metric space is said to be orbitally continuous at a point z in if for any sequence (for some ), x n → z as n → ∞ implies as n → ∞.
Clearly, every continuous self-mapping of a metric space is orbitally continuous, but not conversely.
Sastry et al. [3] extended the above concepts to two and three mappings and employed them to prove common fixed point results for commuting mappings. In what follows, we collect such definitions for three maps.
Definition 4 Let be three self-mappings defined on a metric space .
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1.
If for a point , there exits a sequence {x n } in such that , then the set is called the orbit of at x 0.
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2.
The space is said to be -orbitally complete at x 0 if every Cauchy sequence in converges in .
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3.
The map is said to be orbitally continuous at x 0 if it is continuous on .
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4.
The pair is said to be asymptotically regular (in short a.r.) with respect to at x 0 if there exists a sequence {x n } in such that , and as n → ∞.
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5.
If is the identity mapping on , we omit in respective definitions.
On the other hand, fixed point theory has developed rapidly in metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [4] who presented its applications to matrix equations. Subsequently, Nieto and López [5] extended this result for nondecreasing mappings and applied it to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions. Thereafter, several authors obtained many fixed point theorems in ordered metric spaces. For more details, see [6–15] and the references cited therein.
Recently, Nashine and Altun (HK Nashine and I Altun, unpublished work) proved the following ordered version of a result of Zhang [16]:
Theorem 1 Let be a complete partially ordered metric space and let be two weakly increasing mappings such that
holds for each comparable, where F, ψ : [0, +∞) → [0, +∞) are functions such that
(i) F is nondecreasing, continuous, and F(0) = 0 < F(t) for every t > 0;
(ii) ψ is nondecr easing, right continuous, and ψ(t) < t for every t > 0, and
If or is continuous, then and have a unique common fixed point.
In this article, we generalize this theorem of Nashine and Altun (HK Nashine and I Altun, unpublished work) (and, hence, some other related common fixed point results) in two directions. The first is treated in Section 3, where a pair of asymptotically regular mappings in an orbitally complete ordered metric space is considered. The existence and (under additional assumptions) uniqueness of their common fixed point is obtained under assumptions that these mappings are strictly weakly isotone increasing, one is orbitally continuous and they satisfy a generalized weakly contractive condition.
In Section 4, we consider the case of three self-mappings where the pair is -relatively asymptotically regular and relatively weakly increasing, while the contractive condition is given with the help of two control functions.
We furnish suitable examples to demonstrate the validity of the hypotheses of our results.
2 Notation and definitions
First, we introduce some further notation and definitions that will be used later.
If is a partially ordered set then are called comparable if x ≼ y or y ≼ x holds. A subset of is said to be well ordered if every two elements of are comparable. If is such that, for , x ≼ y implies , then the mapping is said to be nondecreasing.
Definition 5 Let be a partially ordered set and .
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1.
The mapping is called dominating if for each [17].
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2.
The pair is called weakly increasing if and for all [18, 19].
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3.
The mapping is said to be -weakly isotone increasing if for all we have [18–20].
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4.
The mapping is said to be -strictly weakly isotone increasing if, for all such that , we have (HK Nashine, B Samet, and C Vetro, unpublished work).
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5.
Let be such that and , and denote , for . We say that and are weakly increasing with respect to if and only if for all , we have [10]:
and
Example 1[17] Let be endowed with the usual ordering. Let be defined by . Since for all is a dominating map.
Remark 1(1) None of two weakly increasing mappings need be nondecreasing. There exist some examples to illustrate this fact in [21].
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(2)
If are weakly increasing, then is -weakly isotone increasing.
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(3)
can be -strictly weakly isotone increasing, while some of these two mappings can be not strictly increasing (see the following example).
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(4)
If is the identity mapping ( for all ), then and are weakly increasing with respect to if and only if they are weakly increasing mappings.
Example 2 Let be endowed with the usual ordering and define as
Clearly, we have for all , and so, is -strictly weakly isotone increasing; is not strictly increasing.
Definition 6[22, 23]. Let be a metric space and .
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1.
If w = fx = gx, for some , then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. If w = x, then x is a common fixed point of f and g.
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2.
The mappings f and g are said to be compatible if limn→∞ d(fgx n , gfx n ) = 0, whenever {x n } is a sequence in such that limn→∞ fx n = limn→∞ gx n = t for some .
Definition 7 Let be a nonempty set. Then is called an ordered metric space if
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(i)
is a metric space,
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(ii)
is a partially ordered set.
The space is called regular if the following hypothesis holds: if {z n } is a nondecreasing sequence in with respect to ≼ such that as n → ∞, then z n ≼ z.
3 Common fixed points for -strictly weakly isotone increasing mappings
In this section, we improve the results of Nashine and Altun (HK Nashine and I Altun, unpublished work) by considering the following:
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1.
a pair of asymptotically regular mappings;
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2.
orbital continuity of one of the involved maps;
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3.
strictly weakly isotone increasing condition;
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generalized weakly contractive condition, and
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5.
an ordered orbitally complete metric space.
We will denote by and Ψ the set of functions F, ψ : [0, +∞) → [0, +∞), respectively, such that:
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(i)
F is nondecreasing, continuous, and F(0) = 0 < F(t) for every t > 0;
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(ii)
ψ is nondecreasing, right continuous, and ψ(0) = 0.
The first main result of this section is as follows:
Theorem 2 Let be an ordered metric space. Let be two mappings satisfying
for all(for some x0) such that x and y are comparable, where, ψ ∈ Ψ and
We assume the following hypotheses:
(i)is a.r. at x0;
(ii)is -orbitally complete at x0;
(iii)oris-orbitally continuous at x0;
(iv)is-strictly weakly isotone increasing;
(v) there exists ansuch that.
Then and have a common fixed point. Moreover, the set of common fixed points of in is well ordered if and only if it is a singleton.
Proof First of all we show that, if or has a fixed point, then it is a common fixed point of and . Indeed, let z be a fixed point of . Now assume . If we use the inequality (3.1), for x = y = z, we have
wherefrom , which is a contradiction. Thus and so z is a common fixed point of and . Analogously, one can observe that if z is a fixed point of , then it is a common fixed point of and .
Since is a.r. at x0 in , there exists a sequence {x n } in such that
and
If or for some n0, then the proof is finished. So assume x n ≠ xn+1for all n.
Since is -strictly weakly isotone increasing, we have
and continuing this process we get
Next, we claim that {x n } is a Cauchy sequence in the metric space . We proceed by negation and suppose that {x n } is not a Cauchy sequence. That is, there exists ε > 0 such that d(x n ,x m ) ≥ ε for infinitely many values of m and n with m < n. This assures that there exist two sequences {m(k)}, {n(k)} of natural numbers, with m(k) < n(k), such that for each k ∈ ℕ
It is not restrictive to suppose that n(k) is the least positive integer exceeding m(k) and satisfying (3.6). We have
and letting k → ∞, we have d(x2m(k), x2n(k)+1) → ε. We note that
and thus as k → ∞. We have
and so letting k → ∞, we have . Therefore, we have
and letting k → ∞ in the above equation, F being continuous and ψ right continuous, we get
a contradiction. Therefore, {x n } is a Cauchy sequence in . Since is -orbitally complete at x0, there exists with limn→∞x n = z.
If or is orbitally continuous, then clearly
Theorem 3 Letandsatisfy all the conditions of Theorem 2, except that condition (iii) is substituted by
(iii') is regular.
Then the same conclusions as in Theorem 2 hold.
Proof Following the proof of Theorem 2, we have that {x n } is a Cauchy sequence in which is -orbitally complete at x0. Then, there exists such that
Now suppose that . From regularity of , we have for all n ∈ ℕ. Hence, we can apply the considered contractive condition. Then, setting and y = z in (3.1), we obtain:
where
Letting n → ∞ in the above inequality and using the continuity of F and right continuity of ψ, we have
a contradiction. Therefore, and thus . Hence, z is a common fixed point of and .
Corollary 1 Let be an ordered metric space. Let be a mapping satisfying
for all(for some x0) such that x and y are comparable, where, ψ ∈ Ψ and
We assume the following hypotheses:
(i)is a.r. at some point x0;
(ii)is-orbitally complete at x0;
(iii) is orbitally continuous at x 0 or is regular.
Also suppose that for all such that and there exists an such that . Then has a fixed point. Moreover, the set of fixed points of in is well ordered if and only if it is a singleton.
We also state a corollary of Theorem 2 involving a contraction of integral type.
Corollary 2 Letandsatisfy the conditions of Theorem 2, except that condition (3.1) is replaced by the following: there exists a positive Lebesgue integrable function u on ℝ+such thatfor each ε > 0 and that
Then, and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.
We present an example showing how our results can be used.
Example 3 Let , where and B = (1, +∞), be equipped with Euclidean metric d and the order ≼ given by
Consider the mappings given by
It is easy to check that and satisfy conditions (i)-(v) of Theorem 2 with . Take defined by
and ψ ∈ Ψ, given as . In order to check the contractive condition (3.1), take with, say x ≺ y, i.e., x > y (the case x = y is trivial). Then and for some m, n ∈ ℕ, m > n. We get that and
Hence, (3.1) is fulfilled. Applying Theorem 2, we conclude that and have a (unique) common fixed point (z = 0).
Note that and do not satisfy the contractive condition for arbitrary .
4 Common fixed points for relatively weakly increasing mappings
In this section, we improve and generalize the results of Nashine and Altun (HK Nashine and I Altun, unpublished work) by taking into account the following for three maps :
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1.
is a pair of asymptotically regular mappings with respect to ;
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2.
orbital continuity of one of the involved maps;
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3.
is a pair of weakly increasing mappings with respect to ;
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4.
is a pair of dominating maps;
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5.
is a pair of compatible maps, and
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6.
the basic space is an ordered orbitally complete metric space.
We will denote by Φ the set of functions φ : [0 + ∞) → [0, +∞), such that φ is right continuous, φ(0) = 0 and φ(t) < t for every t > 0.
The first result of this section is the following.
Theorem 4 Let be a regular ordered metric space and let and be self-maps on satisfying
for all(for some x0) such thatandare comparable, where, φ ∈ Φ and
We assume the following hypotheses:
(i)is a.r. with respect toat;
(ii)is ()-orbitally complete at x0;
(iii)andare weakly increasing with respect to;
(iv) and are dominating maps;
(v)is monotone and orbitally continuous at x0.
Assume either
(a) and are compatible; or
(b)andare compatible.
Then and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.
Proof Since is a.r. with respect to at x0 in , there exists a sequence {x n } in such that
and
holds. We claim that
To this aim, we will use the increasing property with respect to satisfied by the mappings and . From (4.3), we have
Since , then , and we get
Again,
Since , we get
Hence, by induction, (4.5) holds. Therefore, we can apply (4.1) for x = x p and y = x q for all p and q.
Now, we assert that is a Cauchy sequence in the metric space . We proceed by negation and suppose that is not Cauchy. Then, there exists ε > 0 for which we can find two sequences of positive integers {m(k)} and {n(k)} such that for all positive integers k,
From (4.6) and using the triangular inequality, we get
Letting k → ∞ in the above inequality and using (4.4), we obtain
Again, the triangular inequality gives us
Letting k → ∞ in the above inequality and using (4.4) and (4.7), we get:
On the other hand, we have
Letting k → ∞ in the above inequality and using (4.4), (4.7) and properties of , we have
Applying (4.1), we get:
One can check easily that for k large enough, we have:
where d k ≥ 0 and d k → 0 as k → ∞. From (4.10), for k large enough, we have
Letting k → ∞ in (4.11) and using properties of F and φ, we have
Combining (4.9) and (4.12), we get F(ε) < F(ε), a contradiction.
Hence, we deduce that is a Cauchy sequence in . Since is -orbitally complete at x0, there exists some such that
We will prove that z is a common fixed point of the three mappings and .
We have
and
Suppose that (a) holds, i.e., and are compatible. Then, using condition (v),
From (4.13) and the orbitally continuity of , we have also
Now, using (iv), and since is monotone, and are comparable. Thus, we can apply (4.1) to obtain
where
Letting n → ∞ in (4.18), using (4.13)-(4.17), we obtain
unless
Now, and as n → ∞, so by the assumption we have x2n+1≼ z and and are comparable. Hence (4.1) gives
Passing to the limit as n → ∞ in the above inequality and using (4.19), it follows that
which holds unless
Similarly, and as n → ∞, implies that , hence and are comparable. From (4.1) we get
Passing to the limit as n → ∞, we have
which gives that
Therefore, , hence z is a common fixed point of and .
Similarly, the result follows when condition (b) holds.
Now, suppose that the set of common fixed points of and in is well ordered. We claim that there is a unique common fixed point of and in . Assume to the contrary that and but u ≠ v. By supposition, we can replace x by u and y by v in (4.1) to obtain
a contradiction. Hence, u = v. The converse is trivial.
We obtain the following corollaries from Theorem 4.
Corollary 3 Let be a regular ordered metric space and let and be self-maps on satisfying
for all(for some x0) such that x and y are comparable, where, φ ∈ Φ and
We assume the following hypotheses:
(i)is a.r. at some point;
(ii)is-orbitally complete at x0;
(iii) and are weakly increasing;
(iv) and are dominating maps.
Then and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.
Corollary 4 Let be a regular ordered metric space and let and be self-maps on satisfying
for all(for some x0) such thatandare comparable, where, φ ∈ Φ and
We assume the following hypotheses:
(i)is a.r. with respect toat;
(ii)is-orbitally complete at x0;
(iii)is weakly increasing with respect to;
(iv) is a dominating map;
(v)is monotone and orbitally continuous at x0.
Then and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.
Corollary 5 Let be a regular ordered metric space and let be a self-map on satisfying for all such that x and y are comparable,
where, φ ∈ Φ and
We assume the following hypotheses:
(i)is a.r. at some point x0of;
(ii)is-orbitally complete at x0;
(iii)for all;
(iv) is a dominating map.
Then has a fixed point. Moreover, the set of fixed points of in is well ordered if and only if it is a singleton.
We also state a corollary of Theorem 4 involving a contraction of integral type.
Corollary 6 Letandsatisfy the conditions of Theorem 4, except that condition (4.1) is replaced by the following: there exists a positive Lebesgue integrable function u on ℝ+such thatfor each ε > 0 and that
Then, and have a common fixed point. Moreover, the set of common fixed points of and in is well ordered if and only if it is a singleton.
Example 4 Let the set be equipped with the usual metric d and the order defined by
Consider the following self-mappings on :
Take . Then it is easy to show that
and , and all the conditions (i)-(v) and (a)-(b) of Theorem 4 are fulfilled (condition (iii) on . Take and of the form F(t) = kt, k > 0. Then contractive condition (4.1) takes the form
for . Using substitution y = tx, t ≥ 0, the last inequality reduces to
and can be checked by discussion on possible values for t ≥ 0. Hence, all the conditions of Theorem 4 are satisfied and have a unique common fixed point in (which is 0).
Remark 2 It was shown by examples in [24] that (in similar situations):
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(1)
if the contractive condition is satisfied just on , there might not exist a (common) fixed point;
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(2)
under the given hypotheses (common) fixed point might not be unique in the whole space .
References
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J Math Anal Appl 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Ćirić LjB: A generalization of Banach's contraction principle. Proc Am Math Soc 1974, 45: 267–273.
Sastry KPR, Naidu SVR, Rao IHN, Rao KPR: Common fixed points for asymptotically regular mappings. Indian J Pure Appl Math 1984, 15: 849–854.
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc Am Math Soc 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4
Nieto JJ, López RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5
Abbas M, Sintunavarat W, Kumam P: Coupled fixed point in partially ordered G -metric spaces. Fixed Point Theory Appl 2012, 2012: 31. 10.1186/1687-1812-2012-31
Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Appl Anal 2008, 87: 1–8. 10.1080/00036810701714164
Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240
Nashine HK, Altun I: Fixed point theorems for generalized weakly contractive condition in ordered metric spaces. Fixed Point Theory Appl 2011, 2011: 20. Article ID 132367 10.1186/1687-1812-2011-20
Nashine HK, Samet B: Fixed point results for mappings satisfying ( ψ , φ )-weakly contractive condition in partially ordered metric spaces. Nonlinear Anal 2011, 74: 2201–2209. 10.1016/j.na.2010.11.024
Nashine HK, Samet B, Vetro C: Monotone generalized nonlinear contractions and fixed point theorems in ordered metric spaces. Math Comput Model 2011, 54: 712–720. 10.1016/j.mcm.2011.03.014
Nashine HK, Shatanawi W: Coupled common fixed point theorems for pair of commuting mappings in partially ordered complete metric spaces. Comput Math Appl 2011, 62: 1984–1993. 10.1016/j.camwa.2011.06.042
O'Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J Math Anal Appl 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026
Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems c -distance in ordered cone metric spaces. Comput Math Appl 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
Sintunavarat W, Kumam P: Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Acta Math Scientia
Zhang X: Common fixed point theorems for some new generalized contractive type mappings. J Math Anal Appl 2007, 333: 780–786. 10.1016/j.jmaa.2006.11.028
Abbas M, Nazir T, Radenović S: Common fixed point of four maps in partially ordered metric spaces. Appl Math Lett 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038
Dhage BC: Condensing mappings and applications to existence theorems for common solution of differential equations. Bull Korean Math Soc 1999, 36: 565–578.
Dhage BC, ORegan D, Agarwal RP: Common fixed point theorems for a pair of countably condensing mappings in ordered Banach spaces. J Appl Math Stoch Anal 2003, 16: 243–248. 10.1155/S1048953303000182
Vetro C: Common fixed points in ordered Banach spaces. Le Matematiche 2008, 63: 93–100.
Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl 2010, 2010: 17. Article ID 621492
Jungck G: Commuting mappings and fixed points. Am Math Monthly 1976, 83: 261–263. 10.2307/2318216
Jungck G: Compatible mappings and common fixed points. Int J Math Math Sci 1986, 9: 771–779. 10.1155/S0161171286000935
Babu GVR, Sailaja PD: A fixed point theorem of generalized weakly contractive maps in orbitally complete metric spaecs. Thai J Math 2011, 9: 1–10.
Acknowledgements
The authors are highly indebted to the referees for their careful reading of the manuscript and valuable suggestions. H-S Ding acknowledges the support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province (20114BAB211002), the Jiangxi Provincial Education Department (GJJ12173), and the Program for Cultivating Youths of Outstanding Ability in Jiangxi Normal University. Z. Kadelburg is thankful to the Ministry of Science and Technological Development of Serbia.
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Ding, HS., Kadelburg, Z. & Nashine, H.K. Common fixed point theorems for weakly increasing mappings on ordered orbitally complete metric spaces. Fixed Point Theory Appl 2012, 85 (2012). https://doi.org/10.1186/1687-1812-2012-85
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DOI: https://doi.org/10.1186/1687-1812-2012-85