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New conditions on fuzzy coupled coincidence fixed point theorem

Abstract

Recently, Choudhury et al. proved a coupled coincidence point theorem in a partial order fuzzy metric space. In this paper, we give a new version of the result of Choudhury et al. by removing some restrictions. In our result, the mappings are not required to be compatible, continuous or commutable, and the t-norm is not required to be of Hadžić-type. Finally, two examples are presented to illustrate the main result of this paper.

MSC:54E70, 47H25.

1 Introduction

The concept of fuzzy metric spaces was defined in different ways [13]. Grabiec [4] presented a fuzzy version of Banach contraction principle in a fuzzy metric space of Kramosi and Michalek’s sense. Fang [5] proved some fixed point theorems in fuzzy metric spaces, which improve, generalize, unify, and extend some main results of Edelstein [6], Istratescu [7], Sehgal and Bharucha-Reid [8].

In order to obtain a Hausdorff topology, George and Veeramani [9, 10] modified the concept of fuzzy metric space due to Kramosil and Michalek [11]. Many fixed point theorems in complete fuzzy metric spaces in the sense of George and Veeramani [9, 10] have been obtained. For example, Singh and Chauhan [12] proved some common fixed point theorems for four mappings in GV fuzzy metric spaces. Gregori and Sapena [13] proved that each fuzzy contractive mapping has a unique fixed point in a complete GV fuzzy metric space in which fuzzy contractive sequences are Cauchy.

The coupled fixed point theorem and its applications in metric spaces are firstly obtained by Bhaskar and Lakshmikantham [14]. Recently, some authors considered coupled fixed point theorems in fuzzy metric spaces; see [1518].

In [15], the authors gave the following results.

Theorem 1.1 [[15], Theorem 2.5]

Let ab>ab for all a,b[0,1] and (X,M,) be a complete fuzzy metric space such that M has n-property. Let F:X×XX and g:XX be two functions such that

M ( F ( x , y ) , F ( u , v ) , k t ) M(gx,gu,t)M(gy,gv,t)

for all x,y,u,vX, where 0<k<1, F(X×X)g(X) and g is continuous and commutes with F. Then there exists a unique xX such that x=gx=F(x,x).

Let 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 semicontinuous from the right;

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

In [16], Hu proved the following result.

Theorem 1.2 [[16], Theorem 1]

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

M ( F ( x , y ) , F ( u , v ) , ϕ ( t ) ) M(gx,gu,t)M(gy,gv,t)

for all x,y,u,vX, t>0. Suppose that F(X×X)g(X), and g is continuous; F and g are compatible. Then there exists xX such that x=gx=F(x,x), that is, F and g have a unique common fixed point in X.

Choudhury et al. [17] gave the following coupled coincidence fixed point result in a partial order fuzzy metric space.

Theorem 1.3 [[17], Theorem 3.1]

Let (X,M,) be a complete fuzzy metric space with a Hadžić type t-norm M(x,y,t)1 as t for all x,yX. Let be a partial order defined on X. Let F:X×XX and g:XX be two mappings such that F has mixed g-monotone property and satisfies the following conditions:

  1. (i)

    F(X×X)g(X),

  2. (ii)

    g is continuous and monotonic increasing,

  3. (iii)

    (g,F) is a compatible pair,

  4. (iv)

    M(F(x,y),F(u,v),kt)γ(M(g(x),g(u),t)M(g(y),g(v),t)) for all x,y,u,vX, t>0 with g(x)g(u) and g(y)g(v), where k(0,1), γ:[0,1][0,1] is a continuous function such that γ(a)γ(a)a for each 0a1.

Also suppose that X has the following properties:

  1. (a)

    if we have a non-decreasing sequence { x n }x, then x n x for all nN{0},

  2. (b)

    if we have a non-increasing sequence { y n }y, then y n y for all nN{0}.

If there exist x 0 , y 0 X such that g( x 0 )F( x 0 , y 0 ), g( y 0 )F( y 0 , x 0 ), and M(g( x 0 ),F( x 0 , y 0 ),t)M(g( y 0 ),F( y 0 , x 0 ),t)>0 for all t>0, then there exist x,yX such that g(x)=F(x,y) and g(y)=F(y,x), that is, g and F have a coupled coincidence point in X.

Wang et al. [18] proved the following coupled fixed point result in a fuzzy metric space.

Theorem 1.4 [[18], Theorem 3.1]

Let (X,M,) be a fuzzy metric space under a continuous t-norm of H-type. Let ϕ:(0,)(0,) be a function satisfying lim n ϕ n (t)=0 for any t>0. Let F:X×XX and g:XX be two mappings with F(X×X)g(X) and assume that for any t>0,

M ( F ( x , y ) , F ( u , v ) , ϕ ( t ) ) M(gx,gu,t)M(gy,gv,t)

for all x,y,u,vX. Suppose that F(X×X) is complete and g and F are w-compatible, then g and F have a unique common fixed point x X, that is, x =g( x )=F( x , x ).

In this paper, by modifying the conditions on the result of Choudhury et al. [17], we give a new coupled coincidence fixed point theorem in partial order fuzzy metric spaces. In our result, we do not require that the t-norm is of Hadžić-type [19], the mappings are compatible [16], commutable, continuous or monotonic increasing. Our proof method is different from the one of Choudhury et al. Finally, some examples are presented to illustrate our result.

2 Preliminaries

Definition 2.1 [9]

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

  1. (1)

    is associative and commutative,

  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].

Typical examples of the continuous t-norm are a 1 b=ab and a 2 b=min{a,b} for all a,b[0,1].

A t-norm is said to be positive if ab>0 for all a,b(0,1]. Obviously, 1 and 2 are positive t-norms.

Definition 2.2 [9]

The 3-tuple (X,M,) is called a fuzzy metric space if X is an arbitrary non-empty 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:

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

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

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

(KM-4) M(x,y,):(0,)[0,1] is continuous,

(KM-5) M(x,y,t+s)M(x,z,t)M(y,z,s).

Lemma 2.1 [4]

Let (X,M,) be a fuzzy metric space. Then M(x,y,) is non-decreasing for all x,yX.

Lemma 2.2 [20]

Let (X,M,) be a fuzzy metric space. Then M is a continuous function on X 2 ×(0,).

Definition 2.3 [9]

Let (X,M,) be a fuzzy metric space. A sequence { x n } in X is called an M-Cauchy sequence, if for each ϵ(0,1) and t>0 there is n 0 N such that M( x n , x m ,t)>1ϵ for all m,n n 0 . The fuzzy metric space (X,M,) is called M-complete if every M-Cauchy sequence is convergent.

Let (X,) be a partially ordered set and F be a mapping from X to itself. A sequence { x n } in X is said to be non-decreasing if for each nN, x n x n + 1 . A mapping g:XX is called monotonic increasing if for all x,yX with xy, g(x)g(y).

Definition 2.4 [21]

Let (X,) be a partially ordered set and F:X×XX and g:XX be two mappings. The mapping F is said to have the mixed g-monotone property if for all x 1 , x 2 X, g( x 1 )g( x 2 ) implies F( x 1 ,y)F( x 2 ,y) for all yX, and for all y 1 , y 2 X, g( y 1 )g( y 2 ) implies F(x, y 1 )F(x, y 2 ) for all xX.

Definition 2.5 [14]

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).

Here (gx,gy) is called a coupled point of coincidence.

3 Main results

Lemma 3.1 Let γ:[0,1][0,1] be a left continuous function and be a continuous t-norm. Assume that γ(a)γ(a)>a for all a(0,1). Then γ(1)=1.

Proof Let { a n }(0,1) be a non-decreasing sequence with lim n a n =1. By hypothesis we have

γ( a n )γ( a n )> a n ,nN.

Since γ is left continuous and is continuous, we get

γ(1)γ(1)1,

which implies that γ(1)γ(1)=1. Since γ(1)γ(1)γ(1), one has γ(1)=1. This completes the proof. □

Theorem 3.1 Let (X,M,) be a fuzzy metric space with a continuous and positive t-norm. Let be a partial order defined on X. Let ϕ:(0,)(0,) be a function satisfying ϕ(t)t for all t>0 and let γ:[0,1][0,1] be a left continuous and increasing function satisfying γ(a)γ(a)>a for all a(0,1). Let F:X×XX and g:XX be two mappings such that F has the mixed g-monotone property and assume that g(X) is complete. Suppose that the following conditions hold:

  1. (i)

    F(X×X)g(X),

  2. (ii)

    we have

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

for all x,y,u,vX, t>0 with g(x)g(u) and g(y)g(v),

  1. (iii)

    if a non-decreasing sequence { x n }x, then x n x for all nN{0},

  2. (iv)

    if a non-increasing sequence { y n }y, then y n y for all nN{0}.

If there exist x 0 , y 0 X such that g( x 0 )F( x 0 , y 0 ), g( y 0 )F( y 0 , x 0 ) and M(g( x 0 ),F( x 0 , y 0 ),t)M(g( y 0 ),F( y 0 , x 0 ),t)>0 for all t>0, then there exist x , y X such that g( x )=F( x , y ) and g( y )=F( y , x ).

Proof Let x 0 , y 0 X such that g( x 0 )F( x 0 , y 0 ) and F( y 0 , x 0 )g( y 0 ). Define the sequences { x n } and { y n } in X by

g( x n + 1 )=F( x n , y n )andg( y n + 1 )=F( y n , x n ),for all nN{0}.

Along the lines of the proof of [17], we see that

g( x n )g( x n + 1 )andg( y n )g( y n + 1 ),for all nN{0}.
(3.2)

By (3.1) and (3.2) we have

M ( g ( x 1 ) , g ( x 2 ) , t ) M ( g ( x 1 ) , g ( x 2 ) , ϕ ( t ) ) = M ( F ( x 0 , y 0 ) , F ( x 1 , y 1 ) , ϕ ( t ) ) γ ( M ( g ( x 0 ) , g ( x 1 ) , t ) M ( g ( y 0 ) , g ( y 1 ) , t ) ) > M ( g ( x 0 ) , g ( x 1 ) , t ) M ( g ( y 0 ) , g ( y 1 ) , t ) > 0 , t > 0 ,
(3.3)

and

M ( g ( y 1 ) , g ( y 2 ) , t ) M ( g ( y 1 ) , g ( y 2 ) , ϕ ( t ) ) = M ( F ( y 0 , x 0 ) , F ( y 1 , x 1 ) , ϕ ( t ) ) γ ( M ( g ( y 0 ) , g ( y 1 ) , t ) M ( g ( x 0 ) , g ( x 1 ) , t ) ) > M ( g ( y 0 ) , g ( y 1 ) , t ) M ( g ( x 0 ) , g ( x 1 ) , t ) > 0 , t > 0 .
(3.4)

Since is positive, we have

M ( g ( x 1 ) , g ( x 2 ) , t ) M ( g ( y 1 ) , g ( y 2 ) , t ) >0,t>0.

Repeating the process (3.3) and (3.4), we get

M ( g ( x 2 ) , g ( x 3 ) , t ) >0andM ( g ( y 2 ) , g ( y 3 ) , t ) >0,t>0,

and further we have

M ( g ( x 2 ) , g ( x 3 ) , t ) M ( g ( y 2 ) , g ( y 3 ) , t ) >0,t>0.

Continuing the above process, we get, for each nN,

M ( g ( x n ) , g ( x n + 1 ) , t ) >0,t>0,

and

M ( g ( y n ) , g ( y n + 1 ) , t ) >0,t>0.

Since is positive, one has

M ( g ( x n ) , g ( x n + 1 ) , t ) M ( g ( y n ) , g ( y n + 1 ) , t ) >0,nN,t>0.

Now we prove by induction that, for each nN and kN with kn, one has

M ( g ( x n ) , g ( x k ) , t ) M ( g ( y n ) , g ( y k ) , t ) >0,t>0.
(3.5)

Obviously (3.5) holds for k=n. Assume that (3.5) holds for some kN with k>n. Then we have

M ( g ( x n ) , g ( x k + 1 ) , t ) M ( g ( x n ) , g ( x k ) , t / 2 ) M ( g ( x k ) , g ( x k + 1 ) , t / 2 ) .

Since M(g( x n ),g( x k ),t/2)>0, M(g( x k ),g( x k + 1 ),t/2)>0, and is positive, we have

M ( g ( x n ) , g ( x k + 1 ) , t ) >0,t>0.

Similarly, we have

M ( g ( y n ) , g ( y k + 1 ) , t ) >0,t>0.

Therefore, (3.5) holds for all kN with kn.

Now we use the method of Wang [22] to show that both {g( x n )} and {g( y n )} are Cauchy sequences. Fix t>0. Let

a n = inf k n M ( g ( x n ) , g ( x k ) , t ) M ( g ( y n ) , g ( y k ) , t ) .

For kn+1, by (3.1) and (3.2) we have

M ( g ( x n + 1 ) , g ( x k ) , t ) M ( g ( x n + 1 ) , g ( x k ) , ϕ ( t ) ) γ ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) .

Similarly,

M ( g ( y n + 1 ) , g ( y k ) , t ) γ ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) .

So, by (3.5) and the hypothesis we have

M ( g ( x n + 1 ) , g ( x k ) , t ) M ( g ( y n + 1 ) , g ( y k ) , t ) 2 ( γ ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) ) M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) > 0 ,
(3.6)

which implies that

a n + 1 a n >0.

Since { a n } is bounded, there exists a(0,1] such that lim n a n =a. Assume that a<1. Since γ is increasing, we have

2 ( γ ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) ) 2 ( γ ( inf k n + 1 ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) ) )

and further

inf k n + 1 2 ( γ ( ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) ) ) 2 ( γ ( inf k n + 1 ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) ) ) .
(3.7)

From (3.6) and (3.7) it follows that

inf k n + 1 ( M ( g ( x n + 1 ) , g ( x k ) , t ) M ( g ( y n + 1 ) , g ( y k ) , t ) ) 2 ( γ ( inf k n + 1 ( M ( g ( x n ) , g ( x k 1 ) , t ) M ( g ( y n ) , g ( y k 1 ) , t ) ) ) ) ,

i.e.,

a n + 1 γ( a n )γ( a n ),nN.

Since γ is left continuous, by hypothesis we get

aγ(a)γ(a)>a.

This is a contradiction. So a=1.

For any given ϵ>0, there exists n 0 N such that

1 a n <ϵfor all n n 0 .

Thus for each kn n 0 ,

M ( g ( x n ) , g ( x k ) , t ) M ( g ( y n ) , g ( y k ) , t ) >1ϵ,

which implies that

min { M ( g ( x n ) , g ( x k ) , t ) , M ( g ( y n ) , g ( y k ) , t ) } >1ϵ.

It follows that both {g( x n )} and {g( y n )} are Cauchy sequences. Since g(X) is complete, there exist x , y X such that g( x n )g( x ) and g( y n )g( y ) as n.

By hypothesis, we have

g( x n )g ( x ) andg( y n )g ( y ) ,nN.
(3.8)

Now, for all t>0, by (3.1) and (3.8) we have

M ( F ( x , y ) , g ( x ) , t ) M ( F ( x , y ) , F ( x n , y n ) , t / 2 ) M ( F ( x n , y n ) , g ( x ) , t / 2 ) M ( F ( x , y ) , F ( x n , y n ) , ϕ ( t / 2 ) ) M ( F ( x n , y n ) , g ( x ) , ϕ ( t / 2 ) ) γ ( M ( g ( x ) , g ( x n ) , t / 2 ) M ( g ( y ) , g ( y n ) , t / 2 ) ) M ( F ( x n , y n ) , g ( x ) , ϕ ( t / 2 ) ) .
(3.9)

Since γ is left continuous and is continuous, letting n in (3.9), we get

M ( F ( x , y ) , g ( x ) , t ) lim n [ γ ( M ( g ( x ) , g ( x n ) , t / 2 ) M ( g ( y ) , g ( y n ) , t / 2 ) ) M ( F ( x n , y n ) , g ( x ) , ϕ ( t / 2 ) ) ] = γ ( 1 1 ) 1 = 1 , t > 0 .

It follows that F( x , y )=g( x ). Similarly, we can prove that F( y , x )=g( y ). This completes the proof. □

If ϕ(t)=t for all t>0 in Theorem 3.1, we get the following corollary.

Corollary 3.1 Let (X,M,) be a fuzzy metric space with a positive t-norm. Let be a partial order defined on X. Let γ:[0,1][0,1] be a left continuous and increasing function satisfying γ(a)γ(a)>a for all a(0,1). Let F:X×XX and g:XX be two mappings such that F has mixed g-monotone property and assume that g(X) is complete. Suppose that the following conditions hold:

  1. (i)

    F(X×X)g(X).

  2. (ii)

    We have

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

for all x,y,u,vX, t>0 with g(x)g(u) and g(y)g(v).

  1. (iii)

    If we have a non-decreasing sequence { x n }x, then x n x for all nN{0}.

  2. (iv)

    If we have a non-increasing sequence { y n }y, then y n y for all nN{0}.

If there exist x 0 , y 0 X such that g( x 0 )F( x 0 , y 0 ), g( y 0 )F( y 0 , x 0 ) and M(g( x 0 ),F( x 0 , y 0 ),t)M(g( y 0 ),F( y 0 , x 0 ),t)>0 for all t>0, then there exist x , y X such that g( x )=F( x , y ) and g( y )=F( y , x ).

Letting g(x)=x for all xX in Theorem 3.1 and Corollary 3.1, we get the following corollaries.

Corollary 3.2 Let (X,M,) be a complete fuzzy metric space with a positive t-norm. Let be a partial order defined on X. Let ϕ:(0,)(0,) be a function satisfying ϕ(t)t for all t>0 and let γ:[0,1][0,1] be a left continuous and increasing function satisfying γ(a)γ(a)>a for all a(0,1). Let F:X×XX and assume F has mixed monotone property. Suppose that the following conditions hold:

  1. (i)

    We have

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

    for all x,y,u,vX, t>0 with xu and yv.

  2. (ii)

    If we have a non-decreasing sequence { x n }x, then x n x for all nN{0}.

  3. (iii)

    If we have a non-increasing sequence { y n }y, then y n y for all nN{0}.

If there exist x 0 , y 0 X such that x 0 F( x 0 , y 0 ), y 0 F( y 0 , x 0 ) and M( x 0 ,F( x 0 , y 0 ),t)M( y 0 ,F( y 0 , x 0 ),t)>0 for all t>0, then there exist x , y X such that x =F( x , y ) and y =F( y , x ).

Corollary 3.3 Let (X,M,) be a complete fuzzy metric space with a positive t-norm. Let be a partial order defined on X. Let γ:[0,1][0,1] be a left continuous and increasing function satisfying γ(a)γ(a)>a for all a(0,1). Let F:X×XX and assume F has mixed monotone property. Suppose that the following conditions hold:

  1. (i)

    We have

    M ( F ( x , y ) , F ( u , v ) , t ) γ ( M ( x , u , t ) M ( y , v , t ) ) ,

    for all x,y,u,vX, t>0 with xu and yv.

  2. (ii)

    If we have a non-decreasing sequence { x n }x, then x n x for all nN{0}.

  3. (iii)

    If we have a non-increasing sequence { y n }y, then y n y for all nN{0}.

If there exist x 0 , y 0 X such that x 0 F( x 0 , y 0 ), y 0 F( y 0 , x 0 ), and M( x 0 ,F( x 0 , y 0 ),t)M( y 0 ,F( y 0 , x 0 ),t)>0 for all t>0, then there exist x , y X such that x =F( x , y ) and y =F( y , x ).

First, we illustrate Theorem 3.1 by modifying [[17], Example 3.4] as follows.

Example 3.1 Let (X,) is the partially ordered set with X=[0,1] and the natural ordering ≤ of the real numbers as the partial ordering . Define M: X 2 ×(0,) by

M(x,y,t)= e | x y | / t ,x,yX,t>0.

Let ab=ab for all a,b[0,1]. Then (X,M,) is a (complete) fuzzy metric space.

Let ψ(t)=t for all t>0 and γ(s)= s 1 3 for all s[0,1]. It is easy to see that γ(s)γ(s)>s for all s(0,1).

Define the mappings g:XX by

g(x)= x 2 ,xX,

and F:X×XX by

F(x,y)= x 2 y 2 3 + 2 3 ,x,yX.

Then F(X×X)g(X), F satisfies the mixed g-monotone property; see [[17], Example 3.4]. Obviously g(X) is complete.

Let x 0 =0 and y 0 =1, then g( x 0 )F( x 0 , y 0 ) and g( y 0 )F( y 0 , x 0 ); see [[17], Example 3.4]. Moreover, M(g( x 0 ),F( x 0 , y 0 ),t)M(g( y 0 ),F( y 0 , x 0 ),t)>0 for all t>0.

Next we show that for all t>0 and all x,y,u,vX with g(x)g(u) and g(y)g(v), i.e., xu and yv, one has

M ( F ( x , y ) , F ( u , v ) , t ) ( M ( g ( x ) , g ( u ) , t ) M ( g ( y ) , g ( v ) , t ) ) 1 3 .
(3.10)

We prove the above inequality by a contradiction. Assume

M ( F ( x , y ) , F ( u , v ) , t ) < ( M ( g ( x ) , g ( u ) , t ) M ( g ( y ) , g ( v ) , t ) ) 1 3 .

Then

e | ( x 2 y 2 ) / 3 ( u 2 v 2 ) / 3 | / t < e ( | x 2 u 2 | + | y 2 v 2 | ) / 3 t ,

i.e.,

| ( x 2 u 2 ) ( y 2 v 2 ) | > | x 2 u 2 | + | y 2 v 2 | .

This is a contradiction. Thus, (3.10) holds. Therefore, all the conditions of Theorem 3.1 are satisfied. Then by Theorem 3.1 we conclude that there exist x , y such that g( x )=F( x , y ) and g( y )=F( y , x ). It is easy to see that ( x , y )=( 2 3 , 2 3 ), as desired.

Example 3.2 Let (X,) is the partially ordered set with X=[0,1){2} and the natural ordering ≤ of the real numbers as the partial ordering . Define a mapping M: X 2 ×(0,) by M(x,x,t)= e | x y | for all x,yX and t>0. Let ab=ab for all a,b[0,1]. Then (X,M,) is a fuzzy metric space but not complete.

Define the mappings g:XX and F:X×XX by

g(x)={ 1 2 ( 1 x ) , if  0 x < 1 , 0 , if  x = 2 ,

and F(x,y)= y x 16 + 1 8 for all x,yX. Then F(X×X)g(X), F satisfies the mixed g-monotone property, and g(X) is complete. Take ( x 0 , y 0 )=( 23 28 , 1 4 ). By a simple calculation we see that g( x 0 )F( x 0 , y 0 ) and g( y 0 )F( y 0 , x 0 ). Moreover, M(g( x 0 ),F( x 0 , y 0 ),t)M(g( y 0 ),F( y 0 , x 0 ),t)>0 for all t>0.

Let ϕ(t)=t for all t>0. Let γ be a function from [0,1] to [0,1] defined by

γ(s)={ s 3 , if  0 s 1 2 , s 4 , if  1 2 < s 1 .

Obviously, γ is left continuous and increasing, and γ(s)γ(s)>s for all s(0,1).

Let t>0 and x,y,u,vX with g(x)g(u) and g(y)g(v), i.e., ux and yv, since

M ( F ( x , y ) , F ( u , v ) , ϕ ( t ) ) = e | F ( x , y ) F ( u , v ) | = e | x y 16 u v 16 | max { e | x y | + | y v | 6 , e | x y | + | y v | 8 } γ ( M ( g ( x ) , g ( u ) , t ) M ( g ( y ) , g ( v ) , t ) ) .

Hence (3.1) is satisfied. Therefore, all the conditions of Theorem 3.1 are satisfied. Then by Theorem 3.1 F and g have a coincidence point. It is easy to check that ( x , y )=( 3 4 , 3 4 ).

The above two examples cannot be applied to [[17], Theorem 3.1], since is not of Hadžić-type, or g is not monotonic increasing or continuous, or M(x,y,t)1 as t for all x,yX.

4 Conclusion

In this paper, we prove a new coupled coincidence fixed point result in a partial order fuzzy metric space in which some restrictions required in [[17], Theorem 3.1] are removed, such that the conditions required in our result are fewer than the ones required in [[17], Theorem 3.1]. The purpose of this paper is to give some new conditions on the coupled coincidence fixed point theorem. Our result is not an improvement of [[17], Theorem 3.1], since we add some other restrictions such as requiring that the function γ is increasing and M(g( x 0 ),F( x 0 , y 0 ),t)M(g( y 0 ),F( y 0 , x 0 ),t)>0 for all t>0. As pointed out in the conclusion part of [17], it still is an interesting open problem to find simpler or fewer conditions on the coupled coincidence fixed point theorem in a fuzzy metric space.

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Acknowledgements

This work is supported by the Fundamental Research Funds for the Central Universities (Grant Numbers: 13MS109, 2014ZD44) and funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors Alsulami and Ćirić, therefore, acknowledge with thanks the DSR financial support.

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Wang, S., Luo, T., Ćirić, L. et al. New conditions on fuzzy coupled coincidence fixed point theorem. Fixed Point Theory Appl 2014, 153 (2014). https://doi.org/10.1186/1687-1812-2014-153

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Keywords

  • fuzzy metric space
  • contraction mapping
  • coincidence fixed point
  • partial order