- Research Article
- Open access
- Published:
Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces
Fixed Point Theory and Applications volume 2011, Article number: 363716 (2011)
Abstract
We prove a common fixed point theorem for mappings under -contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of S.Sedghi et al. (2010)
1. Introduction
Since Zadeh [1] introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space which have very important applications in quantum particle physics particularly in connection with both string and -infinity theory.
Bhaskar and Lakshmikantham [4], Lakshmikantham and Ćirić [5] discussed the mixed monotone mappings and gave some coupled fixed point theorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al. [6] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang [7] gave some common fixed point theorems under -contractions for compatible and weakly compatible mappings in Menger probabilistic metric spaces. Many authors [8–23] have proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.
In this paper, using similar proof as in [7], we give a new common fixed point theorem under weaker conditions than in [6] and give an example which shows that the result is a genuine generalization of the corresponding result in [6].
2. Preliminaries
First we give some definitions.
Definition 1 (see [2]).
A binary operation is continuous
-norm if
is satisfying the following conditions:
(1)is commutative and associative;
(2) is continuous;
(3) for all
;
(4) whenever
and
for all
.
Definition 2 (see [24]).
Let . A
-norm
is said to be of H-type if the family of functions
is equicontinuous at
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ1_HTML.gif)
The -norm
is an example of
-norm of H-type, but there are some other
-norms
of H-type [24].
Obviously, is a H-type
norm if and only if for any
, there exists
such that
for all
, when
.
Definition 3 (see [2]).
A 3-tuple is said to be a fuzzy metric space if
is an arbitrary nonempty set,
is a continuous
-norm, and
is a fuzzy set on
satisfying the following conditions, for each
and
:
(FM-1);
(FM-2) if and only if
;
(FM-3);
(FM-4);
(FM-5) is continuous.
Let be a fuzzy metric space. For
, the open ball
with a center
and a radius
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ2_HTML.gif)
A subset is called open if, for each
, there exist
and
such that
. Let
denote the family of all open subsets of
. Then
is called the topology on
induced by the fuzzy metric
. This topology is Hausdorff and first countable.
Example 1.
Let be a metric space. Define
-norm
and for all
and
,
. Then
is a fuzzy metric space. We call this fuzzy metric
induced by the metric
the standard fuzzy metric.
Definition 4 (see [2]).
Let be a fuzzy metric space, then
(1)a sequence in
is said to be convergent to
(denoted by
) if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ3_HTML.gif)
for all ;
(2)a sequence in
is said to be a Cauchy sequence if for any
, there exists
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ4_HTML.gif)
for all and
;
(3)a fuzzy metric space is said to be complete if and only if every Cauchy sequence in
is convergent.
Remark 1 (see [25]).
-
(1)
For all
,
is nondecreasing.
-
(2)
It is easy to prove that if
,
,
, then
(2.5)
-
(3)
In a fuzzy metric space
, whenever
for
in
,
,
, we can find a
,
such that
.
-
(4)
For any
, we can find an
such that
and for any
we can find a
such that
  
).
Definition 5 (see [6]).
Let be a fuzzy metric space.
is said to satisfy the
-property on
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ6_HTML.gif)
whenever ,
and
.
Lemma 1.
Let be a fuzzy metric space and
satisfies the
-property; then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ7_HTML.gif)
Proof.
If not, since is nondecreasing and
, there exists
such that
, then for
,
when
as
and we get
, which is a contraction.
Remark 2.
Condition (2.7) cannot guarantee the -property. See the following example.
Example 2.
Let be an ordinary metric space,
for all
, and
be defined as following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ8_HTML.gif)
where . Then
is continuous and increasing in
,
and
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ9_HTML.gif)
then is a fuzzy metric space and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ10_HTML.gif)
But for any ,
,
,
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ11_HTML.gif)
Define , where
and each
satisfies the following conditions:
(-1) is nondecreasing;
(-2) is upper semicontinuous from the right;
(-3) for all
, where
,
.
It is easy to prove that, if , then
for all
.
Lemma 2 (see [7]).
Let be a fuzzy metric space, where
is a continuous
-norm of H-type. If there exists
such that if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ12_HTML.gif)
for all , then
.
Definition 6 (see [5]).
An element is called a coupled fixed point of the mapping
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ13_HTML.gif)
Definition 7 (see [5]).
An element is called a coupled coincidence point of the mappings
and
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ14_HTML.gif)
Definition 8 (see [7]).
An element is called a common coupled fixed point of the mappings
and
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ15_HTML.gif)
Definition 9 (see [7]).
An element is called a common fixed point of the mappings
and
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ16_HTML.gif)
Definition 10 (see [7]).
The mappings and
are said to be compatible if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ17_HTML.gif)
for all whenever
and
are sequences in
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ18_HTML.gif)
for all are satisfied.
Definition 11 (see [7]).
The mappings and
are called commutative if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ19_HTML.gif)
for all .
Remark 3.
It is easy to prove that, if and
are commutative, then they are compatible.
3. Main Results
For convenience, we denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ20_HTML.gif)
for all .
Theorem 1.
Let be a complete FM-space, where
is a continuous
-norm of H-type satisfying (2.7). Let
and
be two mappings and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ21_HTML.gif)
for all ,
.
Suppose that , and
is continuous,
and
are compatible. Then there exist
such that
, that is,
and
have a unique common fixed point in
.
Proof.
Let be two arbitrary points in
. Since
, we can choose
such that
and
. Continuing in this way we can construct two sequences
and
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ22_HTML.gif)
The proof is divided into 4 steps.
Step 1.
Prove that and
are Cauchy sequences.
Since is a
-norm of H-type, for any
, there exists a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ23_HTML.gif)
for all .
Since is continuous and
for all
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ24_HTML.gif)
On the other hand, since , by condition (
) we have
. Then for any
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ25_HTML.gif)
From condition (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ26_HTML.gif)
Similarly, we can also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ27_HTML.gif)
Continuing in the same way we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ28_HTML.gif)
So, from (3.5) and (3.6), for , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ29_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ30_HTML.gif)
for all with
and
. So
is a Cauchy sequence.
Similarly, we can get that is also a Cauchy sequence.
Step 2.
Prove that and
have a coupled coincidence point.
Since complete, there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ31_HTML.gif)
Since and
are compatible, we have by (3.12),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ32_HTML.gif)
for all . Next we prove that
and
.
For all , by condition (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ33_HTML.gif)
for all . Let
, since
and
are compatible, with the continuity of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ34_HTML.gif)
which implies that . Similarly, we can get
.
Step 3.
Prove that and
.
Since is a
-norm of H-type, for any
, there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ35_HTML.gif)
for all .
Since is continuous and
for all
, there exists
such that
and
.
On the other hand, since , by condition
we have
. Then for any
, there exists
such that
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ36_HTML.gif)
letting , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ37_HTML.gif)
Similarly, we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ38_HTML.gif)
From (3.18) and (3.19) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ39_HTML.gif)
By this way, we can get for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ40_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ41_HTML.gif)
So for any we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ42_HTML.gif)
for all . We can get that
and
.
Step 4.
Prove that .
Since is a
-norm of H-type, for any
, there exists an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ43_HTML.gif)
for all .
Since is continuous and
, there exists
such that
.
On the other hand, since , by condition
we have
. Then for any
, there exists
such that
.
Since for ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ44_HTML.gif)
Letting yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ45_HTML.gif)
Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ46_HTML.gif)
which implies that .
Thus we have proved that and
have a unique common fixed point in
.
This completes the proof of the Theorem 1.
Taking (the identity mapping) in Theorem 1, we get the following consequence.
Corollary 1.
Let be a complete FM-space, where
is a continuous
-norm of H-type satisfying (2.7). Let
and there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ47_HTML.gif)
for all ,
.
Then there exist such that
, that is,
admits a unique fixed point in
.
Let , where
, the following by Lemma 1, we get the following.
Corollary 2 (see [6]).
Let for all
and
be a complete fuzzy metric space such that
has
-property. Let
and
be two functions such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ48_HTML.gif)
for all , where
,
and
is continuous and commutes with
. Then there exists a unique
such that
.
Next we give an example to demonstrate Theorem 1.
Example 3.
Let ,
for all
.
is defined as (2.8). Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ49_HTML.gif)
for all . Then
is a complete FM-space.
Let ,
and
be defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ50_HTML.gif)
Then satisfies all the condition of Theorem 1, and there exists a point
which is the unique common fixed point of
and
.
In fact, it is easy to see that ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ51_HTML.gif)
For all and
. (3.28) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ52_HTML.gif)
Since , we can get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ53_HTML.gif)
From (3.33), we only need to verify the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ54_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ55_HTML.gif)
We consider the following cases.
Case 1 ().
Then (3.36) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ56_HTML.gif)
it is easy to verified.
Case 2 ().
Then (3.36) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ57_HTML.gif)
which is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ58_HTML.gif)
since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ59_HTML.gif)
that is
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ60_HTML.gif)
holds for all . So (3.36) holds for
.
Case 3 ().
Then (3.36) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ61_HTML.gif)
Let , we only need to verify
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F363716/MediaObjects/13663_2010_Article_1400_Equ62_HTML.gif)
for all that
. We can verify it holds.
Thus it is verified that the functions ,
,
satisfy all the conditions of Theorem 1;
is the common fixed point of
and
in
.
References
Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.
George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 1994,64(3):395–399. 10.1016/0165-0114(94)90162-7
George A, Veeramani P: On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems 1997,90(3):365–368. 10.1016/S0165-0114(96)00207-2
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis. Theory, Methods & Applications 2006,65(7):1379–1393. 10.1016/j.na.2005.10.017
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,70(12):4341–4349. 10.1016/j.na.2008.09.020
Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2010,72(3–4):1298–1304. 10.1016/j.na.2009.08.018
Fang J-X: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(5–6):1833–1843. 10.1016/j.na.2009.01.018
Ćirić LB, Miheţ D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topology and its Applications 2009,156(17):2838–2844. 10.1016/j.topol.2009.08.029
O'Regan D, Saadati R: Nonlinear contraction theorems in probabilistic spaces. Applied Mathematics and Computation 2008,195(1):86–93. 10.1016/j.amc.2007.04.070
Jain S, Jain S, Bahadur Jain L: Compatibility of type (P) in modified intuitionistic fuzzy metric space. Journal of Nonlinear Science and its Applications 2010,3(2):96–109.
Ćirić LB, Ješić SN, Ume JS: The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 2008,37(3):781–791. 10.1016/j.chaos.2006.09.093
\'Cirić L, Lakshmikantham V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stochastic Analysis and Applications 2009,27(6):1246–1259. 10.1080/07362990903259967
Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.
Aliouche A, Merghadi F, Djoudi A: A related fixed point theorem in two fuzzy metric spaces. Journal of Nonlinear Science and its Applications 2009,2(1):19–24.
Ćirić L: Common fixed point theorems for a family of non-self mappings in convex metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(5–6):1662–1669. 10.1016/j.na.2009.01.002
Rao KPR, Aliouche A, Babu GR: Related fixed point theorems in fuzzy metric spaces. Journal of Nonlinear Science and its Applications 2008,1(3):194–202.
Ćirić L, Cakić N: On common fixed point theorems for non-self hybrid mappings in convex metric spaces. Applied Mathematics and Computation 2009,208(1):90–97. 10.1016/j.amc.2008.11.012
Ćirić L: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos, Solitons and Fractals 2009,42(1):146–154. 10.1016/j.chaos.2008.11.010
Shakeri S, Ćirić LJB, Saadati R: Common fixed point theorem in partially ordered -fuzzy metric spaces. Fixed Point Theory and Applications 2010, 2010:-13.
Ćirić L, Samet B, Vetro C: Common fixed point theorems for families of occasionally weakly compatible mappings. Mathematical and Computer Modelling 2011,53(5–6):631–636. 10.1016/j.mcm.2010.09.015
Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Applied Mathematics and Computation 2011,217(12):5784–5789. 10.1016/j.amc.2010.12.060
Ćirić L, Abbas M, Damjanović B, Saadati R: Common fuzzy fixed point theorems in ordered metric spaces. Mathematical and Computer Modelling 2011,53(9–10):1737–1741. 10.1016/j.mcm.2010.12.050
Kamran T, Cakić N: Hybrid tangential property and coincidence point theorems. Fixed Point Theory 2008,9(2):487–496.
Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and its Applications. Volume 536. Kluwer Academic, Dordrecht, The Netherlands; 2001:x+273.
Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 1988,27(3):385–389. 10.1016/0165-0114(88)90064-4
Acknowledgment
The author is grateful to the referees for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
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
Hu, XQ. Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces. Fixed Point Theory Appl 2011, 363716 (2011). https://doi.org/10.1155/2011/363716
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2011/363716