Open Access

New conditions on fuzzy coupled coincidence fixed point theorem

  • Shenghua Wang1Email author,
  • Ting Luo1,
  • Ljubomir Ćirić2 and
  • Saud M Alsulami3
Fixed Point Theory and Applications20142014:153

https://doi.org/10.1186/1687-1812-2014-153

Received: 11 February 2014

Accepted: 13 June 2014

Published: 22 July 2014

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.

Keywords

fuzzy metric spacecontraction mappingcoincidence fixed pointpartial order

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

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

Let Φ = { ϕ : R + R + } , where R + = [ 0 , + ) and each ϕ Φ 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 ) ) , n N .

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 × X X and g : X X be two mappings and let there exist ϕ 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 > 0 . Suppose that F ( X × X ) g ( X ) , and g is continuous; F and g are compatible. Then there exists x X such that x = g x = 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 , y X . Let be a partial order defined on X. Let F : X × X X and g : X X 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 ) , k t ) γ ( M ( g ( x ) , g ( u ) , t ) M ( g ( y ) , g ( v ) , t ) ) for all x , y , u , v X , 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 0 a 1 .

     
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 n N { 0 } ,

     
  2. (b)

    if we have a non-increasing sequence { y n } y , then y n y for all n N { 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 ) , 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 × X X and g : X X 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 ( g x , g u , t ) M ( g y , g v , t )

for all x , y , u , v X . 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)

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

     
  4. (4)

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

     

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

A t-norm is said to be positive if a b > 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 , z X 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 , y X .

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 ) > 1 ϵ 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 n N , x n x n + 1 . A mapping g : X X is called monotonic increasing if for all x , y X with x y , g ( x ) g ( y ) .

Definition 2.4 [21]

Let ( X , ) be a partially ordered set and F : X × X X and g : X X 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 y X , 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 x X .

Definition 2.5 [14]

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 ) .

Here ( g x , g y ) 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 , n N .
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 × X X and g : X X 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 , v X , 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 n N { 0 } ,

     
  2. (iv)

    if a non-increasing sequence { y n } y , then y n y for all n N { 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 ) and g ( y n + 1 ) = F ( y n , x n ) , for all  n N { 0 } .
Along the lines of the proof of [17], we see that
g ( x n ) g ( x n + 1 ) and g ( y n ) g ( y n + 1 ) , for all  n N { 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 ) > 0 and M ( 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 n N ,
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 , n N , t > 0 .
Now we prove by induction that, for each n N and k N with k n , 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 k N 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 k N with k n .

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 k n + 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 ) , n N .
Since γ is left continuous, by hypothesis we get
a γ ( 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 < ϵ for all  n n 0 .
Thus for each k n n 0 ,
M ( g ( x n ) , g ( x k ) , t ) M ( g ( y n ) , g ( y k ) , t ) > 1 ϵ ,
which implies that
min { M ( g ( x n ) , g ( x k ) , t ) , M ( g ( y n ) , g ( y k ) , t ) } > 1 ϵ .

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 ) and g ( y n ) g ( y ) , n N .
(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 × X X and g : X X 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 , v X , 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 n N { 0 } .

     
  2. (iv)

    If we have a non-increasing sequence { y n } y , then y n y for all n N { 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 x X 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 × X X 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 , v X , t > 0 with x u and y v .

     
  2. (ii)

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

     
  3. (iii)

    If we have a non-increasing sequence { y n } y , then y n y for all n N { 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 × X X 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 , v X , t > 0 with x u and y v .

     
  2. (ii)

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

     
  3. (iii)

    If we have a non-increasing sequence { y n } y , then y n y for all n N { 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 , y X , t > 0 .

Let a b = a b 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 : X X by
g ( x ) = x 2 , x X ,
and F : X × X X by
F ( x , y ) = x 2 y 2 3 + 2 3 , x , y X .

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 , v X with g ( x ) g ( u ) and g ( y ) g ( v ) , i.e., x u and y v , 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 , y X and t > 0 . Let a b = a b for all a , b [ 0 , 1 ] . Then ( X , M , ) is a fuzzy metric space but not complete.

Define the mappings g : X X and F : X × X X 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 , y X . 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 , v X with g ( x ) g ( u ) and g ( y ) g ( v ) , i.e., u x and y v , 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 , y X .

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.

Declarations

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.

Authors’ Affiliations

(1)
School of Mathematics and Physics, North China Electric Power University
(2)
Faculty of Mechanical Engineering, University of Belgrade
(3)
Department of Mathematics, King Abdulaziz University

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