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Common coupled fixed point theorems for weakly compatible mappings in fuzzy metric spaces

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 1965, Zadeh [1] 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 [2, 3] 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 [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 Jin-xuan Fang [7] gave some common fixed point theorems for compatible and weakly compatible ϕ-contractions mappings in Menger probabilistic metric spaces. Xin-Qi Hu [8] proved a common fixed point theorem for mappings under φ-contractive conditions in fuzzy metric spaces. Many authors [926] 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 [8] and give an example, which shows that the result is a genuine generalization of the corresponding result in [8].

2 Preliminaries

First, we give some definitions.

Definition 2.1 (see [2])

A binary operation :[0,1]×[0,1][0,1] is a continuous t-norm if satisfies the following conditions:

  1. (1)

    is commutative and associative,

  2. (2)

    is continuous,

  3. (3)

    a1=a for all a[0,1],

  4. (4)

    abcd whenever ac and bd for all a,b,c,d[0,1].

Definition 2.2 (see [27])

Let sup 0 < t < 1 Δ(t,t)=1. A t-norm Δ is said to be of H-type if the family of functions { Δ m ( t ) } m = 1 is equicontinuous at t=1, where

Δ 1 (t)=tΔt, Δ m + 1 (t)=tΔ ( Δ m ( t ) ) ,m=1,2,,t[0,1].
(2.1)

The t-norm Δ M =min is an example of t-norm of H-type, but there are some other t-norms Δ of H-type [27].

Obviously, Δ is a t-norm of H-type if and only if for any λ(0,1), there exists δ(λ)(0,1) such that Δ m (t)>1λ for all mN, when t>1δ.

Definition 2.3 (see [2])

A 3-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 2 ×(0,+) satisfying the following conditions for each x,y,zX and t,s>0,

(FM-1) M(x,y,t)>0,

(FM-2) M(x,y,t)=1 if and only if x=y,

(FM-3) M(x,y,t)=M(y,x,t),

(FM-4) M(x,y,t)M(y,z,s)M(x,z,t+s),

(FM-5) M(x,y,):(0,)[0,1] is continuous.

We shall consider a fuzzy metric space (X,M,), which satisfies the following condition:

lim t + M(x,y,t)=1,x,yX.
(2.2)

Let (X,M,) be a fuzzy metric space. For t>0, the open ball B(x,r,t) with a center xX and a radius 0<r<1 is defined by

B(x,r,t)= { y X : M ( x , y , t ) > 1 r } .
(2.3)

A subset AX is called open if for each xA, there exist t>0 and 0<r<1 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 2.4 Let (X,d) be a metric space. Define t-norm ab=ab or ab=min{a,b} and for all x,yX and t>0, M(x,y,t)= t t + d ( x , y ) . Then (X,M,) is a fuzzy metric space.

Definition 2.5 (see [2])

Let (X,M,) be a fuzzy metric space. Then

  1. (1)

    a sequence { x n } in X is said to be convergent to x (denoted by lim n x n =x) if

    lim n M( x n ,x,t)=1

    for all t>0.

  2. (2)

    A sequence { x n } in X is said to be a Cauchy sequence if for any ε>0, there exists n 0 N, such that

    M( x n , x m ,t)>1ε

    for all t>0 and n,m n 0 .

  3. (3)

    A fuzzy metric space (X,M,) is said to be complete if and only if every Cauchy sequence in X is convergent.

Remark 2.6 (see [9])

Let (X,M,) be a fuzzy metric space. Then

  1. (1)

    for all x,yX, M(x,y,) is non-decreasing;

  2. (2)

    if x n x, y n y, t n t, then

    lim n M( x n , y n , t n )=M(x,y,t);
  3. (3)

    if M(x,y,t)>1r for x, y in X, t>0, 0<r<1, then we can find a t 0 , 0< t 0 <t such that M(x,y, t 0 )>1r;

  4. (4)

    for any r 1 > r 2 , we can find a r 3 such that r 1 r 3 r 2 , and for any r 4 , we can find a r 5 such that r 5 r 5 r 4 ( r 1 , r 2 , r 3 , r 4 , r 5 (0,1)).

Define Φ={ϕ: R + R + }, where R + =[0,+) and each ϕΦ satisfies the following conditions:

(ϕ-1) ϕ is non-decreasing,

(ϕ-2) ϕ is upper semi-continuous from the right,

(ϕ-3) n = 0 ϕ n (t)<+ for all t>0, where ϕ n + 1 (t)=ϕ( ϕ n (t)), nN.

It is easy to prove that if ϕΦ, then ϕ(t)<t for all t>0.

Lemma 2.7 (see [7])

Let (X,M,) be a fuzzy metric space, where is a continuous t-norm of H-type. If there exists ϕΦ such that

M ( x , y , ϕ ( t ) ) M(x,y,t)
(2.4)

for all t>0, then x=y.

Definition 2.8 (see [4])

An element (x,y)X×X is called a coupled fixed point of the mapping F:X×XX if

F(x,y)=x,F(y,x)=y.
(2.5)

Definition 2.9 (see [5])

An element (x,y)X×X is called a coupled coincidence point of the mappings F:X×XX and g:XX if

F(x,y)=g(x),F(y,x)=g(y).
(2.6)

Definition 2.10 (see [5])

An element (x,y)X×X is called a common coupled fixed point of the mappings F:X×XX and g:XX if

x=F(x,y)=g(x),y=F(y,x)=g(y).
(2.7)

Definition 2.11 (see [5])

An element xX is called a common fixed point of the mappings F:X×XX and g:XX if

x=g(x)=F(x,x).
(2.8)

Definition 2.12 (see [8])

The mappings F:X×XX and g:XX are said to be compatible if

lim n M ( g F ( x n , y n ) , F ( g ( x n ) , g ( y n ) ) , t ) =1
(2.9)

and

lim n M ( g F ( y n , x n ) , F ( g ( y n ) , g ( x n ) ) , t ) =1
(2.10)

for all t>0 whenever { x n } and { y n } are sequences in X, such that

lim n F( x n , y n )= lim n g( x n )=x, lim n F( y n , x n )= lim n g( y n )=y,
(2.11)

for all x,yX are satisfied.

Definition 2.13 (see [20])

The mappings F:X×XX and g:XX 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,yX.

Remark 2.14 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

for all nN.

Xin-Qi Hu [8] proved the following result.

Theorem 3.1 (see [8])

Let (X,M,) be a complete FM-space, where is a continuous t-norm of H-type satisfying (2.2). Let F:X×XX and g:XX be two mappings, and there exists ϕΦ such that

M ( F ( x , y ) , F ( u , v ) , ϕ ( t ) ) M ( g ( x ) , g ( u ) , t ) M ( g ( y ) , g ( v ) , t )
(3.1)

for all x,y,u,vX, t>0.

Suppose that F(X×X)g(X), g is continuous, F and g are compatible. Then there exist x,yX 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 3.2 Let (X,M,) be a FM-space, where is a continuous t-norm of H-type satisfying (2.2). Let F:X×XX and g:XX be two weakly compatible mappings, and there exists ϕΦ satisfying (3.1).

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 0 , y 0 X be two arbitrary points in X. Since F(X×X)g(X), we can choose x 1 , y 1 X such that g( x 1 )=F( x 0 , y 0 ) and g( y 1 )=F( y 0 , x 0 ). Continuing this process, we can construct two sequences { x n } and { y n } in X such that

g( x n + 1 )=F( x n , y n ),g( y n + 1 )=F( y n , x n ),for all n0.
(3.2)

The proof is divided into 4 steps.

Step 1: We shall prove that {g x n } and {g y n } are Cauchy sequences.

Since is a t-norm of H-type, for any λ>0, there exists an μ>0 such that

for all kN.

Since M(x,y,) is continuous and lim t + M(x,y,t)=1 for all x,yX, there exists t 0 >0 such that

M(g x 0 ,g x 1 , t 0 )1μ,M(g y 0 ,g y 1 , t 0 )1μ.
(3.3)

On the other hand, since ϕΦ, by condition (ϕ-3), we have n = 1 ϕ n ( t 0 )<. Then for any t>0, there exists n 0 N such that

t> k = n 0 ϕ k ( t 0 ).
(3.4)

From condition (3.1), we have

M ( g x 1 , g x 2 , ϕ ( t 0 ) ) = M ( F ( x 0 , y 0 ) , F ( x 1 , y 1 ) , ϕ ( t 0 ) ) M ( g x 0 , g x 1 , t 0 ) M ( g y 0 , g y 1 , t 0 ) , M ( g y 1 , g y 2 , ϕ ( t 0 ) ) = M ( F ( y 0 , x 0 ) , F ( y 1 , x 1 ) , ϕ ( t 0 ) ) M ( g y 0 , g y 1 , t 0 ) M ( g x 0 , g x 1 , t 0 ) .

Similarly, we have

M ( g x 2 , g x 3 , ϕ 2 ( t 0 ) ) = M ( F ( x 1 , y 1 ) , F ( x 2 , y 2 ) , ϕ 2 ( t 0 ) ) M ( g x 1 , g x 2 , ϕ ( t 0 ) ) M ( g y 1 , g y 2 , ϕ ( t 0 ) ) [ M ( g x 0 , g x 1 , t 0 ) ] 2 [ M ( g y 0 , g y 1 , t 0 ) ] 2 , M ( g y 2 , g y 3 , ϕ 2 ( t 0 ) ) = M ( F ( y 1 , x 1 ) , F ( y 2 , x 2 ) , ϕ 2 ( t 0 ) ) [ M ( g y 0 , g y 1 , t 0 ) ] 2 [ M ( g x 0 , g x 1 , t 0 ) ] 2 .

From the inequalities above and by induction, it is easy to prove that

M ( g x n , g x n + 1 , ϕ n ( t 0 ) ) [ M ( g x 0 , g x 1 , t 0 ) ] 2 n 1 [ M ( g y 0 , g y 1 , t 0 ) ] 2 n 1 , M ( g y n , g y n + 1 , ϕ n ( t 0 ) ) [ M ( g y 0 , g y 1 , t 0 ) ] 2 n 1 [ M ( g x 0 , g x 1 , t 0 ) ] 2 n 1 .

So, from (3.3) and (3.4), for m>n n 0 , we have

which implies that

M(g x n ,g x m ,t)>1λ
(3.5)

for all m,nN with m>n n 0 and t>0. So {g( x n )} is a Cauchy sequence.

Similarly, we can prove that {g( y n )} is also a Cauchy sequence.

Step 2: 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,yg(X), and exist a,bX such that

lim n g ( x n ) = lim n F ( x n , y n ) = g ( a ) = x , lim n g ( y n ) = lim n F ( y n , x n ) = g ( b ) = y .
(3.6)

From (3.1), we get

M ( F ( x n , y n ) , F ( a , b ) , ϕ ( t ) ) M ( g x n , g ( a ) , t ) M ( g y n , g ( b ) , t ) .

Since M is continuous, taking limit as n, we have

M ( g ( a ) , F ( a , b ) , ϕ ( t ) ) =1,

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 3: We prove that g(x)=y and g(y)=x.

Since is a t-norm of H-type, for any λ>0, there exists an μ>0 such that

for all kN.

Since M(x,y,) is continuous and lim t + M(x,y,t)=1 for all x,yX, there exists t 0 >0 such that M(gx,y, t 0 )1μ and M(gy,x, t 0 )1μ.

On the other hand, since ϕΦ, by condition (ϕ-3), we have n = 1 ϕ n ( t 0 )<. Thus, for any t>0, there exists n 0 N such that t> k = n 0 ϕ k ( t 0 ). Since

M ( g x , g y n + 1 , ϕ ( t 0 ) ) = M ( F ( x , y ) , F ( y n , x n ) , ϕ ( t 0 ) ) M ( g x , g y n , t 0 ) M ( g y , g x n , t 0 ) ,

letting n, we get

M ( g x , y , ϕ ( t 0 ) ) M(gx,y, t 0 )M(gy,x, t 0 ).
(3.7)

Similarly, we can get

M ( g y , x , ϕ ( t 0 ) ) M(gx,y, t 0 )M(gy,x, t 0 ).
(3.8)

From (3.7) and (3.8), we have

M ( g x , y , ϕ ( t 0 ) ) M ( g y , x , ϕ ( t 0 ) ) [ M ( g x , y , t 0 ) ] 2 [ M ( g y , x , t 0 ) ] 2 .

From this inequality, we can get

M ( g x , y , ϕ n ( t 0 ) ) M ( g y , x , ϕ n ( t 0 ) ) [ M ( g x , y , ϕ n 1 ( t 0 ) ) ] 2 [ M ( g y , x , ϕ n 1 ( t 0 ) ) ] 2 [ M ( g x , y , t 0 ) ] 2 n [ M ( g y , x , t 0 ) ] 2 n

for all nN. Since t> k = n 0 ϕ k ( t 0 ), then, we have

Therefore, for any λ>0, we have

M(gx,y,t)M(gy,x,t)1λ
(3.9)

for all t>0. Hence conclude that gx=y and gy=x.

Step 4: Now, we prove that x=y.

Since is a t-norm of H-type, for any λ>0, there exists an μ>0 such that

for all kN.

Since M(x,y,) is continuous and lim t + M(x,y,t)=1, there exists t 0 >0 such that M(x,y, t 0 )1μ.

On the other hand, since ϕΦ, by condition (ϕ-3), we have n = 1 ϕ n ( t 0 )<. Then, for any t>0, there exists n 0 N such that t> k = n 0 ϕ k ( t 0 ).

From (3.1), we have

M ( g x n + 1 , g y n + 1 , ϕ ( t 0 ) ) = M ( F ( x n , y n ) , F ( y n , x n ) , ϕ ( t 0 ) ) M ( g x n , g y n , t 0 ) M ( g y n , g x n , t 0 ) .

Letting n yields

M ( x , y , ϕ ( t 0 ) ) M(x,y, t 0 )M(y,x, t 0 ).

Thus, we have

which implies that x=y.

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 3.2. □

Taking g=I (the identity mapping) in Theorem 3.2, we get the following consequence.

Corollary 3.3 Let (X,M,) be a FM-space, where is a continuous t-norm of H-type satisfying (2.2). Let F:X×XX, and there exists ϕΦ such that

M ( F ( x , y ) , F ( u , v ) , ϕ ( t ) ) M(x,u,t)M(y,v,t)
(3.10)

for all x,y,u,vX, t>0. F(X) is complete.

Then there exist xX such that x=F(x,x); that is, F admits a unique fixed point in X.

Remark 3.4 Comparing Theorem 3.2 with Theorem 3.1 in [8], we can see that Theorem 3.2 is a genuine generalization of Theorem 3.2.

  1. (1)

    We only need the completeness of g(X) or F(X×X).

  2. (2)

    The continuity of g is relaxed.

  3. (3)

    The concept of compatible has been replaced by weakly compatible.

Remark 3.5 The Example 3 in [8] is wrong, since the t-norm ab=ab is not the t-norm of H-type.

Next, we give an example to support Theorem 3.2.

Example 3.6 Let X={0,1, 1 2 , 1 3 ,, 1 n ,}, =min, M(x,y,t)= t | x y | + t , for all x,yX, t>0. Then (X,M,) is a fuzzy metric space.

Let ϕ(t)= t 2 . Let g:XX and F:X×XX be defined as

g(x)={ 0 , x = 0 , 1 , x = 1 2 n + 1 , 1 2 n + 1 , x = 1 2 n , F(x,y)={ 1 ( 2 n + 1 ) 4 , ( x , y ) = ( 1 2 n , 1 2 n ) , 0 , others .

Let x n = y n = 1 2 n . We have g x n = 1 2 n + 1 0, F( x n , y n )= 1 ( 2 n + 1 ) 4 0, but

M ( F ( g x n , g y n ) , g F ( x n , y n ) , t ) =M(0,1,t)0,

so g and F are not compatible. From F(x,y)=g(x), F(y,x)=g(y), we can get (x,y)=(0,0), and we have gF(0,0)=F(g0,g0), which implies that F and g are weakly compatible.

The following result is easy to verify

t X + t min { t Y + t , t Z + t } Xmax{Y,Z},X,Y,Z0,t>0.

By the definition of M and ϕ and the result above, we can get that inequality (3.1)

M ( F ( x , y ) , F ( u , v ) , ϕ ( t ) ) M ( g ( x ) , g ( u ) , t ) M ( g ( y ) , g ( v ) , t )

is equivalent to the following

2 | F ( x , y ) F ( u , v ) | max { | g ( x ) g ( u ) | , | g ( y ) g ( v ) | } .
(3.11)

Now, we verify inequality (3.11). Let A={ 1 2 n ,nN}, B=XA. By the symmetry and without loss of generality, (x,y), (u,v) have 6 possibilities.

Case 1: (x,y)B×B, (u,v)B×B. It is obvious that (3.11) holds.

Case 2: (x,y)B×B, (u,v)B×A. It is obvious that (3.11) holds.

Case 3: (x,y)B×B, (u,v)A×A. If uv, (3.11) holds. If u=v, let u=v= 1 2 n , then

2 | F ( x , y ) F ( u , v ) | = 2 ( 2 n + 1 ) 4 ,max { | g ( x ) g ( u ) | , | g ( y ) g ( v ) | } = 2 n 2 n + 1 ,

which implies that (3.11) holds.

Case 4: (x,y)B×A, (u,v)B×A. It is obvious that (3.11) holds.

Case 5: (x,y)B×A, (u,v)A×A. If uv, (3.11) holds. If u=v, let xB, y= 1 2 j , u=v= 1 2 n , then

2 | F ( x , y ) F ( u , v ) | = 2 ( 2 n + 1 ) 4 , max { | g ( x ) g ( u ) | , | g ( y ) g ( v ) | } = max { 1 2 n + 1 , | 1 2 j + 1 1 2 n + 1 | } ,

or

max { | g ( x ) g ( u ) | , | g ( y ) g ( v ) | } =max { 2 n 2 n + 1 , | 1 2 j + 1 1 2 n + 1 | } ,

(3.11) holds.

Case 6: (x,y)A×A, (u,v)A×A.

If xy, uv, (3.11) holds.

If xy, u=v, let x= 1 2 i , y= 1 2 j , ij, u=v= 1 2 n . Then

2 | F ( x , y ) F ( u , v ) | = 2 ( 2 n + 1 ) 4 , max { | g ( x ) g ( u ) | , | g ( y ) g ( v ) | } = max { | 1 2 i + 1 1 2 n + 1 | , | 1 2 j + 1 1 2 n + 1 | } ,

(3.11) holds.

If x=y, u=v, let x=y= 1 2 i , u=v= 1 2 n . Then

2 | F ( x , y ) F ( u , v ) | = 2 | 1 ( 2 i + 1 ) 4 1 ( 2 n + 1 ) 4 | , max { | g ( x ) g ( u ) | , | g ( y ) g ( v ) | } = | 1 2 i + 1 1 2 n + 1 | ,

(3.11) holds.

Then all the conditions in Theorem 3.2 are satisfied, and 0 is the unique common fixed point of g and F.

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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. Damjanović was supported by Grant No. 174025 of the Ministry of Education, Science and Technological Development of the Republic of Serbia.

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Correspondence to Xin-Qi Hu.

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The authors declare that they have no competing interests.

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Keywords

  • Fixed Point Theorem
  • Cauchy Sequence
  • Common Fixed Point
  • Unique Fixed Point
  • Compatible Mapping