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Hu et al. Fixed Point Theory and Applications 2013, 2013:220 http://www.fixedpointtheoryandapplications.com/content/2013/1/220

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Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces Xin-Qi Hu1* , Ming-Xue Zheng1 , Boško Damjanovi´c2 and Xiao-Feng Shao3 *

Correspondence: [email protected] 1 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China Full list of author information is available at the end of the article

Abstract In this paper, we prove a common fixed point theorem for weakly compatible mappings under φ -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 Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716). We also indicate a minor mistake in Hu (Fixed Point Theory Appl. 2011:363716, 2011, doi:10.1155/2011/363716).

1 Introduction In , Zadeh [] introduced the concept of fuzzy sets. Then many authors gave the important contribution to development of the theory of fuzzy sets and applications. George and Veeramani [, ] gave the concept of a 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 E-infinity theory. Bhaskar and Lakshmikantham [], Lakshmikantham and Ćirić [] 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. [] gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Jin-xuan Fang [] gave some common fixed point theorems for compatible and weakly compatible φ-contractions mappings in Menger probabilistic metric spaces. Xin-Qi Hu [] proved a common fixed point theorem for mappings under ϕ-contractive conditions in fuzzy metric spaces. Many authors [–] proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces. In this paper, we give a new coupled fixed point theorem under weaker conditions than in [] and give an example, which shows that the result is a genuine generalization of the corresponding result in []. 2 Preliminaries First, we give some definitions. Definition . (see []) A binary operation ∗ : [, ] × [, ] → [, ] is a continuous t-norm if ∗ satisfies the following conditions: () ∗ is commutative and associative, © 2013 Hu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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() ∗ is continuous, () a ∗  = a for all a ∈ [, ], () a ∗ b ≤ c ∗ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [, ]. Definition . (see []) Let sup  – δ. Definition . (see []) A -tuple (X, M, ∗) is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous t-norm and M is a fuzzy set on X  × (, +∞) satisfying the following conditions for each x, y, z ∈ X and t, s > , (FM-) M(x, y, t) > , (FM-) M(x, y, t) =  if and only if x = y, (FM-) M(x, y, t) = M(y, x, t), (FM-) M(x, y, t) ∗ M(y, z, s) ≤ M(x, z, t + s), (FM-) M(x, y, ·) : (, ∞) → [, ] is continuous. We shall consider a fuzzy metric space (X, M, ∗), which satisfies the following condition: lim M(x, y, t) = ,

t→+∞

∀x, y ∈ X.

(.)

Let (X, M, ∗) be a fuzzy metric space. For t > , the open ball B(x, r, t) with a center x ∈ X and a radius  < r <  is defined by   B(x, r, t) = y ∈ X : M(x, y, t) >  – r .

(.)

A subset A ⊂ X is called open if for each x ∈ A, there exist t >  and  < r <  such that B(x, r, t) ⊂ A. Let τ denote the family of all open subsets of X. Then τ is called the topology on X, induced by the fuzzy metric M. This topology is Hausdorff and first countable. Example . Let (X, d) be a metric space. Define t-norm a ∗ b = ab or a ∗ b = min{a, b} t and for all x, y ∈ X and t > , M(x, y, t) = t+d(x,y) . Then (X, M, ∗) is a fuzzy metric space. Definition . (see []) Let (X, M, ∗) be a fuzzy metric space. Then () a sequence {xn } in X is said to be convergent to x (denoted by limn→∞ xn = x) if lim M(xn , x, t) = 

n→∞

for all t > .

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() A sequence {xn } in X is said to be a Cauchy sequence if for any ε > , there exists n ∈ N, such that M(xn , xm , t) >  – ε for all t >  and n, m ≥ n . () A fuzzy metric space (X, M, ∗) is said to be complete if and only if every Cauchy sequence in X is convergent. Remark . (see []) Let (X, M, ∗) be a fuzzy metric space. Then () for all x, y ∈ X, M(x, y, ·) is non-decreasing; () if xn → x, yn → y, tn → t, then lim M(xn , yn , tn ) = M(x, y, t);

n→∞

() if M(x, y, t) >  – r for x, y in X, t > ,  < r < , then we can find a t ,  < t < t such that M(x, y, t ) >  – r; () for any r > r , we can find a r such that r ∗ r ≥ r , and for any r , we can find a r such that r ∗ r ≥ r (r , r , r , r , r ∈ (, )). Define = {φ : R+ → R+ }, where R+ = [, +∞) and each φ ∈ satisfies the following conditions: (φ-) φ is non-decreasing, (φ-) φ is upper semi-continuous from the right, ∞ n n+ (φ-) (t) = φ(φ n (t)), n ∈ N. n= φ (t) < +∞ for all t > , where φ It is easy to prove that if φ ∈ , then φ(t) < t for all t > . Lemma . (see []) Let (X, M, ∗) be a fuzzy metric space, where ∗ is a continuous t-norm of H-type. If there exists φ ∈ such that   M x, y, φ(t) ≥ M(x, y, t)

(.)

for all t > , then x = y. Definition . (see []) An element (x, y) ∈ X × X is called a coupled fixed point of the mapping F : X × X → X if F(x, y) = x,

F(y, x) = y.

(.)

Definition . (see []) An element (x, y) ∈ X × X is called a coupled coincidence point of the mappings F : X × X → X and g : X → X if F(x, y) = g(x),

F(y, x) = g(y).

(.)

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Definition . (see []) An element (x, y) ∈ X × X is called a common coupled fixed point of the mappings F : X × X → X and g : X → X if x = F(x, y) = g(x),

y = F(y, x) = g(y).

(.)

Definition . (see []) An element x ∈ X is called a common fixed point of the mappings F : X × X → X and g : X → X if x = g(x) = F(x, x).

(.)

Definition . (see []) The mappings F : X × X → X and g : X → X are said to be compatible if     lim M gF(xn , yn ), F g(xn ), g(yn ) , t = 

(.)

    lim M gF(yn , xn ), F g(yn ), g(xn ) , t = 

(.)

n→∞

and

n→∞

for all t >  whenever {xn } and {yn } are sequences in X, such that lim F(xn , yn ) = lim g(xn ) = x,

n→∞

lim F(yn , xn ) = lim g(yn ) = y,

n→∞

n→∞

n→∞

(.)

for all x, y ∈ X are satisfied. Definition . (see []) The mappings F : X × X → X and g : X → X are called weakly compatible mappings if F(x, y) = g(x), F(y, x) = g(y) implies that gF(x, y) = F(gx, gy), gF(y, x) = F(gy, gx) for all x, y ∈ X. Remark . It is easy to prove that if F and g are compatible, then they are weakly compatible, but the converse need not be true. See the example in the next section.

3 Main results For simplicity, denote 

M(x, y, t)

n

= M(x, y, t) ∗ M(x, y, t) ∗ · · · ∗ M(x, y, t)

 n

for all n ∈ N. Xin-Qi Hu [] proved the following result. Theorem . (see []) Let (X, M, ∗) be a complete FM-space, where ∗ is a continuous tnorm of H-type satisfying (.). Let F : X × X → X and g : X → X be two mappings, and there exists φ ∈ such that       M F(x, y), F(u, v), φ(t) ≥ M g(x), g(u), t ∗ M g(y), g(v), t for all x, y, u, v ∈ X, t > .

(.)

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Suppose that F(X × X) ⊆ g(X), g is continuous, F and g are compatible. Then there exist x, y ∈ X such that x = g(x) = F(x, x); that is, F and g have a unique common fixed point in X. Now we give our main result. Theorem . Let (X, M, ∗) be a FM-space, where ∗ is a continuous t-norm of H-type satisfying (.). Let F : X × X → X and g : X → X be two weakly compatible mappings, and there exists φ ∈ satisfying (.). Suppose that F(X × X) ⊆ g(X) and F(X × X) or g(X) is complete. Then F and g have a unique common fixed point in X. Proof Let x , y ∈ X be two arbitrary points in X. Since F(X × X) ⊆ g(X), we can choose x , y ∈ X such that g(x ) = F(x , y ) and g(y ) = F(y , x ). Continuing this process, we can construct two sequences {xn } and {yn } in X such that g(xn+ ) = F(xn , yn ),

g(yn+ ) = F(yn , xn ),

for all n ≥ .

(.)

The proof is divided into  steps. Step : We shall prove that {gxn } and {gyn } are Cauchy sequences. Since ∗ is a t-norm of H-type, for any λ > , there exists an μ >  such that ( – μ) ∗ ( – μ) ∗ · · · ∗ ( – μ) ≥  – λ

 k

for all k ∈ N. Since M(x, y, ·) is continuous and limt→+∞ M(x, y, t) =  for all x, y ∈ X, there exists t >  such that M(gx , gx , t ) ≥  – μ,

M(gy , gy , t ) ≥  – μ.

On the other hand, since φ ∈ , by condition (φ-), we have any t > , there exists n ∈ N such that

t>

∞

φ k (t ).

(.) ∞

n= φ

n

(t ) < ∞. Then for

(.)

k=n

From condition (.), we have     M gx , gx , φ(t ) = M F(x , y ), F(x , y ), φ(t ) ≥ M(gx , gx , t ) ∗ M(gy , gy , t ),     M gy , gy , φ(t ) = M F(y , x ), F(y , x ), φ(t ) ≥ M(gy , gy , t ) ∗ M(gx , gx , t ).

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Similarly, we have     M gx , gx , φ  (t ) = M F(x , y ), F(x , y ), φ  (t )     ≥ M gx , gx , φ(t ) ∗ M gy , gy , φ(t )     ≥ M(gx , gx , t ) ∗ M(gy , gy , t ) ,     M gy , gy , φ  (t ) = M F(y , x ), F(y , x ), φ  (t )     ≥ M(gy , gy , t ) ∗ M(gx , gx , t ) . From the inequalities above and by induction, it is easy to prove that    n–  n– M gxn , gxn+ , φ n (t ) ≥ M(gx , gx , t ) ∗ M(gy , gy , t ) ,    n–  n– M gyn , gyn+ , φ n (t ) ≥ M(gy , gy , t ) ∗ M(gx , gx , t ) . So, from (.) and (.), for m > n ≥ n , we have  M(gxn , gxm , t) ≥ M gxn , gxm ,  ≥ M gxn , gxm ,

∞

 k

φ (t )

k=n m–

 k

φ (t )

k=n

    ≥ M gxn , gxn+ , φ n (t ) ∗ M gxn+ , gxn+ , φ n+ (t ) ∗ · · ·   ∗ M gxm– , gxm , φ m– (t )  n–  n–  n ≥ M(gy , gy , t ) ∗ M(gx , gx , t ) ∗ M(gy , gy , t )  n  m– ∗ M(gx , gx , t ) ∗ · · · ∗ M(gy , gy , t )  m– ∗ M(gx , gx , t )  m– –n–  m– –n– = M(gy , gy , t ) ∗ M(gx , gx , t ) ≥ ( – μ) ∗ ( – μ) ∗ · · · ∗ ( – μ) ≥  – λ,

 m –n

which implies that M(gxn , gxm , t) >  – λ

(.)

for all m, n ∈ N with m > n ≥ n and t > . So {g(xn )} is a Cauchy sequence. Similarly, we can prove that {g(yn )} is also a Cauchy sequence. Step : Now, we prove that g and F have a coupled coincidence point. Without loss of generality, we can assume that g(X) is complete, then there exist x, y ∈ g(X), and exist a, b ∈ X such that lim g(xn ) = lim F(xn , yn ) = g(a) = x,

n→∞

n→∞

lim g(yn ) = lim F(yn , xn ) = g(b) = y.

n→∞

n→∞

(.)

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From (.), we get       M F(xn , yn ), F(a, b), φ(t) ≥ M gxn , g(a), t ∗ M gyn , g(b), t . Since M is continuous, taking limit as n → ∞, we have   M g(a), F(a, b), φ(t) = , which implies that F(a, b) = g(a) = x. Similarly, we can show that F(b, a) = g(b) = y. Since F and g are weakly compatible, we get that gF(a, b) = F(g(a), g(b)) and gF(b, a) = F(g(b), g(a)), which implies that g(x) = F(x, y) and g(y) = F(y, x). Step : We prove that g(x) = y and g(y) = x. Since ∗ is a t-norm of H-type, for any λ > , there exists an μ >  such that ( – μ) ∗ ( – μ) ∗ · · · ∗ ( – μ) ≥  – λ

 k

for all k ∈ N. Since M(x, y, ·) is continuous and limt→+∞ M(x, y, t) =  for all x, y ∈ X, there exists t >  such that M(gx, y, t ) ≥  – μ and M(gy, x, t ) ≥  – μ.  n On the other hand, since φ ∈ , by condition (φ-), we have ∞ n= φ (t ) < ∞. Thus, for ∞ any t > , there exists n ∈ N such that t > k=n φ k (t ). Since     M gx, gyn+ , φ(t ) = M F(x, y), F(yn , xn ), φ(t ) ≥ M(gx, gyn , t ) ∗ M(gy, gxn , t ), letting n → ∞, we get   M gx, y, φ(t ) ≥ M(gx, y, t ) ∗ M(gy, x, t ).

(.)

Similarly, we can get   M gy, x, φ(t ) ≥ M(gx, y, t ) ∗ M(gy, x, t ). From (.) and (.), we have         M gx, y, φ(t ) ∗ M gy, x, φ(t ) ≥ M(gx, y, t ) ∗ M(gy, x, t ) . From this inequality, we can get           M gx, y, φ n (t ) ∗ M gy, x, φ n (t ) ≥ M gx, y, φ n– (t ) ∗ M gy, x, φ n– (t )  n  n ≥ M(gx, y, t ) ∗ M(gy, x, t )

(.)

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for all n ∈ N. Since t >

∞ k=n

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φ k (t ), then, we have 

M(gx, y, t) ∗ M(gy, x, t) ≥ M gx, y,

∞





φ k (t ) ∗ M gy, x,

k=n

∞

 φ k (t )

k=n

    ≥ M gx, y, φ n (t ) ∗ M gy, x, φ n (t )  n  n ≥ M(gx, y, t ) ∗ M(gy, x, t )

≥ ( – μ) ∗ ( – μ) ∗ · · · ∗ ( – μ) ≥  – λ.

 n

Therefore, for any λ > , we have M(gx, y, t) ∗ M(gy, x, t) ≥  – λ

(.)

for all t > . Hence conclude that gx = y and gy = x. Step : Now, we prove that x = y. Since ∗ is a t-norm of H-type, for any λ > , there exists an μ >  such that ( – μ) ∗ ( – μ) ∗ · · · ∗ ( – μ) ≥  – λ

 k

for all k ∈ N. Since M(x, y, ·) is continuous and limt→+∞ M(x, y, t) = , there exists t >  such that M(x, y, t ) ≥  – μ.  n On the other hand, since φ ∈ , by condition (φ-), we have ∞ n= φ (t ) < ∞. Then, for ∞ any t > , there exists n ∈ N such that t > k=n φ k (t ). From (.), we have     M gxn+ , gyn+ , φ(t ) = M F(xn , yn ), F(yn , xn ), φ(t ) ≥ M(gxn , gyn , t ) ∗ M(gyn , gxn , t ). Letting n → ∞ yields   M x, y, φ(t ) ≥ M(x, y, t ) ∗ M(y, x, t ). Thus, we have  M(x, y, t) ≥ M x, y,

∞

 φ k (t )

k=n

  ≥ M x, y, φ n (t ) n –  n –  ∗ M(y, x, t ) ≥ M(x, y, t ) ≥ ( – μ) ∗ ( – μ) ∗ · · · ∗ ( – μ) ≥  – λ,

 n –

which implies that x = y.

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Thus, we proved that F and g have a common fixed point in X. The uniqueness of the fixed point can be easily proved in the same way as above. This completes the proof of Theorem ..  Taking g = I (the identity mapping) in Theorem ., we get the following consequence. Corollary . Let (X, M, ∗) be a FM-space, where ∗ is a continuous t-norm of H-type satisfying (.). Let F : X × X → X, and there exists φ ∈ such that   M F(x, y), F(u, v), φ(t) ≥ M(x, u, t) ∗ M(y, v, t)

(.)

for all x, y, u, v ∈ X, t > . F(X) is complete. Then there exist x ∈ X such that x = F(x, x); that is, F admits a unique fixed point in X. Remark . Comparing Theorem . with Theorem . in [], we can see that Theorem . is a genuine generalization of Theorem .. () We only need the completeness of g(X) or F(X × X). () The continuity of g is relaxed. () The concept of compatible has been replaced by weakly compatible. Remark . The Example  in [] is wrong, since the t-norm a ∗ b = ab is not the t-norm of H-type. Next, we give an example to support Theorem .. t , for all x, y ∈ X, t > . Example . Let X = {, ,  ,  , . . . , n , . . .}, ∗ = min, M(x, y, t) = |x–y|+t Then (X, M, ∗) is a fuzzy metric space. Let φ(t) = t . Let g : X → X and F : X × X → X be defined as

⎧ ⎪ ⎪ ⎨, g(x) = , ⎪ ⎪ ⎩ 

x=

, x= n+

Let xn = yn =

 . n

⎧ ⎨

x = ,  , n+  , n

We have gxn =

F(x, y) =

 n+

 , (n+)

⎩,

  (x, y) = ( n , n ),

others.

→ , F(xn , yn ) =

 (n+)

→ , but

  M F(gxn , gyn ), gF(xn , yn ), t = M(, , t)  , so g and F are not compatible. From F(x, y) = g(x), F(y, x) = g(y), we can get (x, y) = (, ), and we have gF(, ) = F(g, g), which implies that F and g are weakly compatible. The following result is easy to verify   t t t ≥ min , X +t Y +t Z+t

⇔

X ≤ max{Y , Z},

∀X, Y , Z ≥ , t > .

By the definition of M and φ and the result above, we can get that inequality (.)       M F(x, y), F(u, v), φ(t) ≥ M g(x), g(u), t ∗ M g(y), g(v), t

Hu et al. Fixed Point Theory and Applications 2013, 2013:220 http://www.fixedpointtheoryandapplications.com/content/2013/1/220

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is equivalent to the following      F(x, y) – F(u, v) ≤ max g(x) – g(u), g(y) – g(v) .

(.)

 Now, we verify inequality (.). Let A = { n , n ∈ N}, B = X – A. By the symmetry and without loss of generality, (x, y), (u, v) have  possibilities. Case : (x, y) ∈ B × B, (u, v) ∈ B × B. It is obvious that (.) holds. Case : (x, y) ∈ B × B, (u, v) ∈ B × A. It is obvious that (.) holds.  , then Case : (x, y) ∈ B × B, (u, v) ∈ A × A. If u = v, (.) holds. If u = v, let u = v = n

  F(x, y) – F(u, v) =

   max g(x) – g(u), g(y) – g(v) =

 , (n + )

n , n + 

which implies that (.) holds. Case : (x, y) ∈ B × A, (u, v) ∈ B × A. It is obvious that (.) holds. Case : (x, y) ∈ B × A, (u, v) ∈ A × A. If u = v, (.) holds. If u = v, let x ∈ B, y =  , then u = v = n

 , j

  F(x, y) – F(u, v) =

 , (n + )     max g(x) – g(u), g(y) – g(v) = max

        , – , n +   j +  n +  

or         n   , , – max g(x) – g(u), g(y) – g(v) = max n +  j +  n +   (.) holds. Case : (x, y) ∈ A × A, (u, v) ∈ A × A. If x = y, u = v, (.) holds. If x = y, u = v, let x = i , y = j , i = j, u = v =

 . n

Then

  F(x, y) – F(u, v) =

 , (n + )               – – , max g(x) – g(u), g(y) – g(v) = max  , i +  n +    j +  n +  

(.) holds. If x = y, u = v, let x = y =

 , i

u=v=

 . n

Then

    , – (i + ) (n + )           , – max g(x) – g(u), g(y) – g(v) =  i +  n +  

    F(x, y) – F(u, v) = 

(.) holds. Then all the conditions in Theorem . are satisfied, and  is the unique common fixed point of g and F.

Hu et al. Fixed Point Theory and Applications 2013, 2013:220 http://www.fixedpointtheoryandapplications.com/content/2013/1/220

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors read and approved the final manuscript. Author details 1 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, P.R. China. 2 Faculty of Agriculture, University of Belgrade, Nemanjina 6, Belgrade, Serbia. 3 The General Course Department of Huanggang Polytechnic College, Huanggang, Hubei 438002, P.R. China. Acknowledgements This work of Xin-qi Hu was supported by the National Natural Science Foundation of China (under grant No. 71171150). The research of B. Damjanovi´c was supported by Grant No. 174025 of the Ministry of Education, Science and Technological Development of the Republic of Serbia. Received: 1 March 2013 Accepted: 1 August 2013 Published: 19 August 2013 References 1. Zadeh, LA: Fuzzy sets. Inf. Control 8, 338-353 (1965) 2. George, A, Veeramani, P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395-399 (1994) 3. George, A, Veeramani, P: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90, 365-368 (1997) 4. Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. TMA 65, 1379-1393 (2006) ´ c, LjB: Coupled fixed point theorems for nonlinear contractions in partially ordered metric 5. Lakshmikantham, V, Ciri´ space. Nonlinear Anal. TMA 70, 4341-4349 (2009) 6. Sedghi, S, Altun, I, Shobe, N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Anal. TMA 72, 1298-1304 (2010) 7. Fang, J-X: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Anal. TMA 71, 1833-1843 (2009) 8. Hu, X-Q: Common coupled fixed point theorems for contractive mappings in fuzzy metric spaces. Fixed Point Theory Appl. 2011, Article ID 363716 (2011). doi:10.1155/2011/363716 9. Grabiec, M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 27, 385-398 (1988) ´ c, LjB, Mihe¸t, D, Saadati, R: Monotone generalized contractions in partially ordered probabilistic metric spaces. 10. Ciri´ Topol. Appl. 156(17), 2838-2844 (2009) 11. O’Regan, D, Saadati, R: Nonlinear contraction theorems in probabilistic spaces. Appl. Math. Comput. 195(1), 86-93 (2008) 12. Jain, S, Jain, S, Bahadur Jain, L: Compatibility of type (P) in modified intuitionistic fuzzy metric space. J. Nonlinear Sci. Appl. 3(2), 96-109 (2010) ´ c, Lj, Lakshmikantham, V: Coupled random fixed point theorems for nonlinear contractions in partially ordered 13. Ciri´ metric spaces. Stoch. Anal. Appl. 27(6), 1246-1259 (2009) ´ c, Lj, Caki´c, N, Rajovi´c, M, Ume, JS: Monotone generalized nonlinear contractions in partially ordered metric 14. Ciri´ spaces. Fixed Point Theory Appl. 2008, Article ID 131294 (2008) 15. Aliouche, A, Merghadi, F, Djoudi, A: A related fixed point theorem in two fuzzy metric spaces. J. Nonlinear Sci. Appl. 2(1), 19-24 (2009) ´ c, Lj: Common fixed point theorems for a family of non-self mappings in convex metric spaces. Nonlinear Anal. 16. Ciri´ 71(5-6), 1662-1669 (2009) 17. Rao, KPR, Aliouche, A, Babu, GR: Related fixed point theorems in fuzzy metric spaces. J. Nonlinear Sci. Appl. 1(3), 194-202 (2008) ´ c, L, Caki´c, N: On Common fixed point theorems for non-self hybrid mappings in convex metric spaces. Appl. 18. Ciri´ Math. Comput. 208(1), 90-97 (2009) ´ c, L: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos 19. Ciri´ Solitons Fractals 42, 146-154 (2009) 20. Abbas, M, Ali Khan, M, Radenovic, S: Common coupled fixed point theorems in cone metric spaces for w-compatible mappings. Appl. Math. Comput. 217, 195-202 (2010) ´ c, Lj, Saadati, R: Common fixed point theorem in partially ordered L-fuzzy metric spaces. Fixed Point 21. Shakeri, S, Ciri´ Theory Appl. 2010, Article ID 125082 (2010). doi:10.1155/2010/125082 ´ c, Lj, Samet, B, Vetro, C: Common fixed point theorems for families of occasionally weakly compatible mappings. 22. Ciri´ Math. Comput. Model. 53(5-6), 631-636 (2011) ´ c, Lj, Abbas, M, Saadati, R, Hussain, N: Common fixed points of almost generalized contractive mappings in 23. Ciri´ ordered metric spaces. Appl. Math. Comput. 53(9-10), 1737-1741 (2011) ´ c, L, Abbas, M, Damjanovi´c, B, Saadati, R: Common fuzzy fixed point theorems in ordered metric spaces. Math. 24. Ciri´ Comput. Model. (2010). doi:10.1016/j.mcm.2010.12.050 25. Djoric, D: Nonlinear coupled coincidence and coupled fixed point theorems for not necessary commutative contractive mappings in partially ordered probabilistic metric spaces. Appl. Math. Comput. 219, 5926-5935 (2013) 26. Kamran, T, Caki´c, NP: Hybrid tangential property and coincidence point theorems. Fixed Point Theory 9, 487-496 (2008) 27. Hadži´c, O, Pap, E: Fixed Point Theory in Probabilistic Metric Spaces. Kluwer Academic, Dordrecht (2001)

doi:10.1186/1687-1812-2013-220 Cite this article as: Hu et al.: Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces. Fixed Point Theory and Applications 2013 2013:220.

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