- Research
- Open access
- Published:
Coupled coincidence point theorems for compatible mappings in partially ordered intuitionistic generalized fuzzy metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 265 (2013)
Abstract
In this work, we introduce the notion of intuitionistic generalized fuzzy metric space by using the idea of intuitionistic fuzzy set due to Atanassov. We determine some coupled coincidence point results for compatibility of two mappings, that is, and , in the framework of intuitionistic generalized fuzzy metric spaces endowed with partial order. An interesting example is also displayed here in support of our result.
1 Preliminaries, background and notation
Zadeh [1] suggests the creation of what are called fuzzy sets which base their development on the idea that the membership of an element to a set is indicated by a number between 0 and 1, having non-membership for 0, membership for 1 and different degrees of membership for the numbers between 0 and 1. Such sets have proved very useful in the description of phenomena governed by imprecise parameters as well as for the development of non-bivalent logic models. Using the idea of fuzzy set, many authors have introduced the concept of fuzzy metric in different point of views [2–4]. George and Veeramani [5] modified the concept of fuzzy metric space due to Kramosil and Michalek [4].
Atanassov [6] suggests a generalization of fuzzy sets making the degrees of membership and non-membership intervene to describe the vinculation of an element to a set, so that the sum of these degrees is always less or equal to 1, that is, an intuitionistic fuzzy set. Park [7] introduced and discussed the concept of intuitionistic fuzzy metric space which is based on the idea of intuitionistic fuzzy set and the notion of fuzzy metric space given by George and Veeramani [5]. Afterward, it was followed by the notion of intuitionistic fuzzy normed space and intuitionistic fuzzy bounded linear operators [8–18]. The authors of [19–21] established an interesting relationship between three various disciplines: the theory of fuzzy normed spaces, the theory of stability of functional equations and fixed point theory.
The concept of generalized metric space was introduced and studied by Mustafa and Sims [22] and was later used to determine coupled fixed point theorems and related results by a number of authors [23–33]. Sun and Yang [34] defined the notion of generalized fuzzy metric space with the help of generalized metric space and fuzzy sets, and further studied it in [35, 36] to deal with some fixed point theory.
In this work, we present an interesting generalization of generalized fuzzy metric space with the help of an intuitionistic fuzzy set and call it an intuitionistic generalized fuzzy metric space. We also define the notions of convergence, Cauchy sequences and compatibility of two mappings in this setup. Further, we establish coupled coincidence point and coupled fixed point results for compatible mappings in partially ordered intuitionistic generalized fuzzy metric spaces and construct an example in support of our result.
Now, we recall some definitions and notations which we will used throughout the article. We shall assume throughout this paper that the symbols ℝ and ℕ denote the set of real and natural numbers, respectively. In this section, we recall some definitions and preliminary results which we will use throughout the paper. Mustafa and Sims [22] defined the notion of generalized metric space as follows.
Let X be a nonempty set and a mapping . Then is called a generalized metric (for short, -metric) on X and a generalized metric space or simply a -metric space if the following conditions are satisfied:
-
(i)
if ,
-
(ii)
for all and ,
-
(iii)
for all and ,
-
(iv)
(symmetry in all three variables),
-
(v)
for all (rectangle inequality).
We remark that every -metric on X defines a metric on X by for all .
For example, let be a metric space. The function is defined by
or
for all . Then is a -metric space [22].
Bhaskar and Lakshmikantham [37] presented the definitions of mixed monotone property and coupled fixed point for the contractive mapping and established some coupled fixed point theorems for a mixed monotone operator. As an application of the coupled fixed point theorems, they determined the existence and uniqueness of the solution of a periodic boundary value problem. Afterward, Lakshmikantham and Ćirić [38] presented the notions of mixed g-monotone property and coupled coincidence point and proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces. Many authors obtained important fixed point theorems, for details and background of fixed point theory, we refer to [39–48] and references therein.
Definition 1.1 [37]
Let be a partially ordered set and be a mapping. Then a map F is said to have the mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y; that is, for any ,
and
Definition 1.2 [37]
An element is said to be a coupled fixed point of the mapping if
Definition 1.3 [38]
Let be a partially ordered set, and let and be two mappings. Then a map F is said to have the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any ,
and
Definition 1.4 [38]
An element is said to be a coupled coincidence point of mappings and if
Definition 1.5 [38]
Let X be a nonempty set, and let and be two mappings. Then F and g are commutative if for all , we have
Definition 1.6 [41]
Let be an IGFM-space. The mappings F and g, where and , are said to be compatible if
and
whenever and are sequences in X such that and for some .
Afterward, the concept of compatible mappings was introduced in fuzzy metric spaces by Hu [49]. In the recent past, Hu and Luo [36] defined and studied this notion in the framework of generalized fuzzy metric spaces.
2 Intuitionistic generalized fuzzy metric space
Let us recall [50] that a binary operation is said to be a continuous t-norm if it satisfies the following conditions:
-
(a)
∗ is associative and commutative,
-
(b)
∗ is continuous,
-
(c)
for all ,
-
(d)
whenever and for each .
Similarly, a binary operation is said to be a continuous t-conorm if it satisfies the following conditions:
(a′) ♢ is associative and commutative,
(b′) ♢ is continuous,
(c′) for all ,
(d′) whenever and for each .
Remark 2.1 The concepts of triangular norms (t-norms) and triangular conorms (t-conorms) are known as the axiomatic skeletons that we use for characterizing fuzzy intersections and unions, respectively.
In 2004, Park [7] presented the notion of intuitionistic fuzzy metric space as follows: The 5-tuple X, M, N, ∗, ♢ is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, ∗ is a continuous t-norm, ♢ is a continuous t-conorm and M, N are fuzzy sets on satisfying the following conditions for all , : (i) , (ii) , (iii) if and only if , (iv) , (v) , (vi) is continuous, (vii) , (viii) if and only if , (ix) , (x) , and (xi) is continuous.
Now, we introduce the notion of intuitionistic generalized fuzzy metric space by using the concepts of continuous t-norm and t-conorm.
Definition 2.2 The 5-tuple is said to be an intuitionistic generalized fuzzy metric space (for short, IGFM-space) if X is an arbitrary nonempty set, ∗ is a continuous t-norm, ♢ is a continuous t-conorm, and G, H are fuzzy sets on satisfying the following conditions. For every and ,
-
(i)
,
-
(ii)
for ,
-
(iii)
for ,
-
(iv)
if and only if ,
-
(v)
, where p is a permutation function,
-
(vi)
,
-
(vii)
is continuous,
-
(viii)
G is a non-decreasing function on ,
-
(xi)
for ,
-
(x)
for ,
-
(xi)
if and only if ,
-
(xii)
, where p is a permutation function,
-
(xiii)
,
-
(xiv)
is continuous,
-
(xv)
H is a non-increasing function on ,
In this case, the pair is called an intuitionistic generalized fuzzy metric on X.
Example 2.3 Let be a -metric space. For all and every , consider G, H to be fuzzy sets on defined by
and denote
Then is an intuitionistic generalized fuzzy metric space. Notice that the above example holds even with the t-norm and the t-conorm . We remark that this intuitionistic generalized fuzzy metric is induced by a -metric , the standard intuitionistic generalized fuzzy metric.
Remark 2.4 In an intuitionistic generalized fuzzy metric space, is non-decreasing and is non-increasing for all .
Definition 2.5 Let , where is an IGFM-space. Then, for and , the set
is said to be an open ball with center x and radius r with respect to t. Note that every open ball is an open set.
Remark 2.6 Let be an IGFM-space. Define . Then is a topology on X (induced by the intuitionistic generalized fuzzy metric ).
Recall that a topological space is first countable if each point has a countable (decreasing) local base. Since is a local base at x, then topology is first countable.
Definition 2.7 Let be an IGFM-space. Then a sequence is said to be convergent to with respect to the intuitionistic generalized fuzzy metric if for every and , there exists such that and for all . In this case, we write or .
Theorem 2.8 Let be an IGFM-space and be the topology induced by the fuzzy metric. Then, for a sequence in X, if and only if and as .
Proof Let be convergent to x with respect to an intuitionistic generalized fuzzy metric , i.e., . Then, for every and , there is a number such that for all . It follows that and and hence and . Thus, and as .
Conversely, suppose that and as for each . Then, for any and , there exists such that and for all . It follows that and for all . Therefore for all . Hence is convergent to x with respect to . □
Definition 2.9 Let be an IGFM-space. Then is a Cauchy sequence with respect to the intuitionistic generalized fuzzy metric if, for every and , there exists such that and for all . An IGFM-space is said to be complete if every Cauchy sequence with respect to the intuitionistic generalized fuzzy metric is convergent with respect to .
3 Coupled coincidence results for compatible mappings
In this section we establish coupled coincidence theorems for compatibility of two mappings in partially ordered intuitionistic generalized fuzzy metric spaces. Before proceeding further, first we define the notion of compatible mappings with respect to the intuitionistic generalized fuzzy metric as follows.
Definition 3.1 Let be an IGFM-space. The mappings F and g, where and , are said to be compatible with respect to if for all ,
and
whenever and are sequences in X such that and for some .
Secondly, we prove the following lemmas which we will used to prove our coupled coincidence theorem.
Lemma 3.2 Let be an IGFM-space. Suppose that is defined by
for all , and . Then, for each , there exists such that
for all .
Proof Given choose such that and . Then, for any , write
and, similarly,
This implies
Since was arbitrary, we have
□
Denote by Φ the family of strictly increasing functions such that for all , where is the n th iterate of ϕ and satisfies (i) ψ is upper semi-continuous, (ii) , (iii) for all .
Lemma 3.3 Let be an IGFM-space and be a sequence in X. Suppose that there exists such that
for all . Then is a Cauchy sequence with respect to the intuitionistic generalized fuzzy metric .
Proof Let be given by (3.1). For simplicity in notation, write for each . We have to show that
Since ϕ is upper semi-continuous from right, for given and each , there exists such that . It follows from (3.1) that
From (3.2), we have
and
Again, from (3.1), we have
Since was arbitrary, we have
Then . Otherwise, which is not possible. Proceeding along the same lines as in the proof of Lemma 2.5 in [36], for a given , there exists such that
for all . Hence is a Cauchy sequence with respect to the intuitionistic generalized fuzzy metric . □
Now, we are ready to determine a coupled coincidence theorem for compatible mappings in partially ordered intuitionistic generalized fuzzy metric spaces.
Theorem 3.4 Let be a partially ordered set and be a complete IGFM-space. Suppose that and are mappings such that F has the mixed g-monotone property, and also assume that there exists such that
for all and with and , or and . Suppose that , g is continuous and F and g are compatible with respect to , and also suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is convergent to x with respect to , then for all n,
-
(ii)
if a non-increasing sequence is convergent to y with respect to , then for all n.
If there exist with and , then there exist such that and ; that is, F and g have a coupled coincidence point.
Proof Let be such that and . Since , we can choose such that and . Again, using the assumption , choose such that and . Continuing this process, we can construct two sequences and in X as follows:
for all . We have to show that
for all . Since and and as and , we have and . This proves that (3.7) and (3.8) hold for . Let us suppose that (3.7) and (3.8) hold for some fixed . Since and , so by the mixed g-monotone property of F, we have
Also,
It follows that
Thus, by the mathematical induction, we conclude that (3.7) and (3.8) hold for all .
Substituting , , and in (3.6), we obtain
and
for all . Using (3.7) in the above inequalities, we get
and
for all . Hence, by Lemma 3.3, is a Cauchy sequence with respect to the intuitionistic generalized fuzzy metric . Again, by substituting , , , and in (3.6), we get
and, similarly,
By using (3.7), the above inequalities become
and
for all . This proves that is a Cauchy sequence with respect to . That is, and are Cauchy sequences with respect to . Since is a complete IGFM-space, there exist such that
Therefore
Since is a compatible pair with respect to , we have
for all . Now, suppose that assumption (a) holds. From (3.10) and (3.11), by using the continuity of F and g, we have
for all . Similarly, for all , we get
From (3.12) and (3.13), we obtain and . This proves that F and g have a coupled coincidence point with respect to the intuitionistic generalized fuzzy metric .
Lastly, let us assume that (b) holds. Since is a non-decreasing sequence and is convergent to x with respect to the intuitionistic generalized fuzzy metric , and also is a non-increasing sequence and is convergent to y with respect to , by our assumption we have and for all n. Since is a compatible pair, using the continuity of g, we have
and
Write
and
for all k and . Taking limit as , using the fact that G is continuous and from the definition of IGFM-space, we have
for all . Since and , using equalities (3.14) and (3.15) in (3.16), we get
Using (3.6) in the right-hand side of (3.17) and then from (3.14) and (3.15), we obtain
and, similarly, we have
Letting in the last two inequalities, we get
for all . From the definition of IGFM-space, we conclude that and and hence F and g have a coupled coincidence point in X. □
Corollary 3.5 Let be a partially ordered set and be a complete IGFM-space. Suppose that is a mapping having the mixed monotone property such that there exist with and . Suppose that there exists such that
for all and , for which and , or and . Also, suppose that either
-
(a)
F is continuous, or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence is convergent to x with respect to , then for all n,
-
(ii)
if a non-increasing sequence is convergent to y with respect to , then for all n.
Then there exist such that and , that is, F has a coupled fixed point in X.
Proof The proof follows by putting , the identity mapping, in Theorem 3.4. □
We consider the following example in support of our Theorem 3.4.
Example 3.6 Let be a partially ordered set with . Suppose that and . Consider G, H to be fuzzy sets on defined by
for all and . Then is a complete IGFM-space. Let the mapping be defined by
and let the mapping be defined by
for all . Then F satisfies the mixed g-monotone property F. Let be such that for all . Suppose that and are two sequences in X such that
Then and . For all , we define
and
From the above, we see that
and
This proves that F and g are compatible with respect to . Also, suppose that and are two points in X such that
and
Now it is left to show that (3.6) of Theorem 3.4 is satisfied with as defined above. Let be such that and , that is, , . We have the following possible cases.
Case 1: When and . Then we get
and
Case 2: If , , then we see that this assumption cannot happen since .
Case 3: When and . Then
and
Case 4. If and , then both and . Therefore
Obviously, assumption (3.6) is fulfilled.
Thus all the hypotheses of Theorem 3.4 are fulfilled. So, we conclude that F and g have a coupled coincidence point. In this case, is a coupled coincidence point of F and g in X.
Remark 3.7 Proceeding along the same technique as given by Sintunavarat et al. [16], we can also obtain our coupled coincidence point theorem without using the commutative condition.
References
Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S0019-9958(65)90241-X
Erceg MA: Metric spaces in fuzzy set theory. J. Math. Anal. Appl. 1979, 69: 205–230. 10.1016/0022-247X(79)90189-6
Kaleva O, Seikkala S: On fuzzy metric spaces. Fuzzy Sets Syst. 1984, 12: 215–229. 10.1016/0165-0114(84)90069-1
Kramosil O, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetika 1975, 11: 326–334.
George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/0165-0114(94)90162-7
Atanassov, K: Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Science-Technical Library of Bulg. Academy of Science, 1697/84) (in Bulgarian)
Park JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051
Al-Fhaid AS, Mohiuddine SA: On the Ulam stability of mixed type QA mappings in IFN-spaces. Adv. Differ. Equ. 2013., 2013: Article ID 203
Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos Solitons Fractals 2006, 27: 331–344. 10.1016/j.chaos.2005.03.019
Mohiuddine SA: Stability of Jensen functional equation in intuitionistic fuzzy normed space. Chaos Solitons Fractals 2009, 42: 2989–2996. 10.1016/j.chaos.2009.04.040
Mohiuddine SA, Alotaibi A, Obaid M: Stability of various functional equations in non-Archimedean intuitionistic fuzzy normed spaces. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 234727
Mohiuddine SA, Şevli H: Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space. J. Comput. Appl. Math. 2011, 235: 2137–2146. 10.1016/j.cam.2010.10.010
Mursaleen M, Ansari KJ: Stability results in intuitionistic fuzzy normed spaces for a cubic functional equation. Appl. Math. Inform. Sci. 2013, 7(5):1685–1692. 10.12785/amis/070505
Mursaleen M, Mohiuddine SA: On stability of a cubic functional equation in intuitionistic fuzzy normed spaces. Chaos Solitons Fractals 2009, 42: 2997–3005. 10.1016/j.chaos.2009.04.041
Mursaleen M, Mohiuddine SA: Nonlinear operators between intuitionistic fuzzy normed spaces and Fréhet differentiation. Chaos Solitons Fractals 2009, 42: 1010–1015. 10.1016/j.chaos.2009.02.041
Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81
Abbas M, Ali B, Sintunavarat W, Kumam P: Tripled fixed point and tripled coincidence point theorems in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 187
Sintunavarat W, Kumam P: Fixed point theorems for a generalized intuitionistic fuzzy contraction in intuitionistic fuzzy metric spaces. Thai J. Math. 2012, 10(1):123–135.
Mohiuddine SA, Cancan M, Şevli H: Intuitionistic fuzzy stability of a Jensen functional equation via fixed point technique. Math. Comput. Model. 2011, 54: 2403–2409. 10.1016/j.mcm.2011.05.049
Mohiuddine SA, Alotaibi A: Fuzzy stability of a cubic functional equation via fixed point technique. Adv. Differ. Equ. 2012., 2012: Article ID 48
Mohiuddine SA, Alghamdi MA: Stability of functional equation obtained through a fixed-point alternative in intuitionistic fuzzy normed spaces. Adv. Differ. Equ. 2012., 2012: Article ID 141
Mustafa Z, Sims B: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 2006, 7(2):289–297.
Mustafa Z, Obiedat H, Awawdeh F: Some fixed point theorem for mapping on complete G -metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 189870
Mohiuddine SA, Alotaibi A: On coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 897198
Mohiuddine SA, Alotaibi A: Some results on tripled fixed point for nonlinear contractions in partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 179
Abbas M, Sintunavarat W, Kumam P: Coupled fixed point in partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31
Aydi H, Karapinar E, Shatnawi W: Tripled fixed point results in generalized metric spaces. J. Appl. Math. 2012., 2012: Article ID 314279
Choudhury BS, Maity P: Coupled fixed point results in generalized metric spaces. Math. Comput. Model. 2011, 54: 73–79. 10.1016/j.mcm.2011.01.036
Karapinar E, Kumam P, Erhan IM: Coupled fixed point theorems on partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 174
Karapinar E, Erhan M, Ulus AY: Cyclic contractions on G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 182947
Luong NV, Thuan NX: Coupled fixed point theorems in partially ordered G -metric spaces. Math. Comput. Model. 2012, 55: 1601–1609. 10.1016/j.mcm.2011.10.058
Shatanawi W: Some fixed point theorems in ordered G -metric spaces and applications. Abstr. Appl. Anal. 2011., 2011: Article ID 126205
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
Sun G, Yang K: Generalized fuzzy metric spaces with properties. Res. J. Appl. Sci. 2010, 2: 673–678.
Rao KPR, Altun I, Bindu SH: Common coupled fixed-point theorems in generalized fuzzy metric spaces. Adv. Fuzzy Syst. 2011., 2011: Article ID 986748
Hu X-Qi, Luo Q: Coupled coincidence point theorems for contractions in generalized fuzzy metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 196
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 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 Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151
Aydi H, Damjanović B, Samet B, Shatanawi W: Coupled fixed point theorems for nonlinear contractions in partially ordered G -metric spaces. Math. Comput. Model. 2011, 54: 2443–2450. 10.1016/j.mcm.2011.05.059
Choudhury BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73: 2524–2531. 10.1016/j.na.2010.06.025
Mursaleen M, Mohiuddine SA, Agarwal RP: Coupled fixed point theorems for α - ψ -contractive type mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 228 (Corrigendum to ‘Coupled fixed point theorems for α-ψ-contractive type mappings in partially ordered metric spaces’. Fixed Point Theory Appl. 2013, 127 (2013))
Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5
Nieto JJ, Rodriguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23(12):2205–2212. 10.1007/s10114-005-0769-0
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
Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for weak contraction mapping under F -invariant set. Abstr. Appl. Anal. 2012., 2012: Article ID 324874
Sintunavarat W, Petruşel A, Kumam P:Common coupled fixed point theorems for -compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo 2012, 61: 361–383. 10.1007/s12215-012-0096-0
Sintunavarat W, Kumam P, Cho YJ: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl. 2012., 2012: Article ID 170
Hu XQ: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 363716
Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 314–334.
Acknowledgements
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. 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
Mohiuddine, S.A., Alotaibi, A. Coupled coincidence point theorems for compatible mappings in partially ordered intuitionistic generalized fuzzy metric spaces. Fixed Point Theory Appl 2013, 265 (2013). https://doi.org/10.1186/1687-1812-2013-265
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-265