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Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces

Fixed Point Theory and Applications20112011:81

https://doi.org/10.1186/1687-1812-2011-81

  • Received: 4 August 2011
  • Accepted: 18 November 2011
  • Published:

Abstract

Recently, Gordji et al. [Math. Comput. Model. 54, 1897-1906 (2011)] prove the coupled coincidence point theorems for nonlinear contraction mappings satisfying commutative condition in intuitionistic fuzzy normed spaces. The aim of this article is to extend and improve some coupled coincidence point theorems of Gordji et al. Also, we give an example of a nonlinear contraction mapping which is not applied by the results of Gordji et al., but can be applied to our results.

2000 MSC: primary 47H10; secondary 54H25; 34B15.

Keywords

  • intuitionistic fuzzy normed space
  • coupled fixed point
  • coupled coincidence point
  • partially ordered set
  • commutative condition

1. Introduction

The classical Banach's contraction mapping principle first appear in [1]. Since this principle is a powerful tool in nonlinear analysis, many mathematicians have much contributed to the improvement and generalization of this principle in many ways (see [210] and others).

One of the most interesting is study to other spaces such as probabilistic metric spaces (see [1115]). The fuzzy theory was introduced simultaneously by Zadeh [16]. The idea of intuitionistic fuzzy set was first published by Atanassov [17]. Since then, Saadati and Park [18] introduced the concept of intuitionistic fuzzy normed spaces (IFNSs). In [19], Saadati et al. have modified the notion of IFNSs of Saadati and Park [18].

Several researchers have applied fuzzy theory to the well-known results in many fields, for example, quantum physics [20], nonlinear dynamical systems [21], population dynamics [22], computer programming [23], fixed point theorem [2427], fuzzy stability problems [2830], statistical convergence [3134], functional equation [35], approximation theory [36], nonlinear equation [37, 38] and many others.

In the other hand, coupled fixed points and their applications for binary mappings in partially ordered metric spaces were introduced by Bhaskar and Lakshmikantham [39]. They applied coupled fixed point theorems to show the existence and uniqueness of a solution for a periodic boundary value problem. After that, Lakshmikantham and Ćirić [40] proved some more generalizations of coupled fixed point theorems in partially ordered sets.

Recently, Gordji et al. [41] proved some coupled coincidence point theorems for contractive mappings satisfying commutative condition in partially complete IFNSs as follows:

Theorem 1.1 (Gordji et al. [41]). Let (X, ) be a partially ordered set and (X, μ, ν, *, ) a complete IFNS such that (μ, ν) has n-property and
a b a b a * b , a , b [ 0 , 1 ] .
(1.1)
Let F: X × XX and g : XX be two mappings such that F has the mixed g-monotone property and
μ ( F ( x , y ) - F ( u , v ) , k t ) μ ( g x - g u , t ) * μ ( g y - g v , t ) , x , y , u , v X , ν ( F ( x , y ) - F ( u , v ) , k t ) ν ( g x - g u , t ) ν ( g y - g v , t ) , x , y , u , v X ,
(1.2)
for which g(x) g(u) and g(y) z g(v), where 0 <k < 1, F(X × X) g(X), g is continuous and g commuting with F. Suppose that either
  1. (1)

    F is continuous or

     
  2. (2)

    X has the following properties:

     
  3. (a)

    if {x n } is a non-decreasing sequence with {x n } → x, then gx n gx for all n ,

     
  4. (b)

    if {y n } is a non-increasing sequence with {y n } → y, then gy gy n for all n .

     
If there exist x0, y0 X such that
g ( x 0 ) F ( x 0 , y 0 ) , g ( y 0 ) F ( y 0 , x 0 ) ,

then F and g have a coupled coincidence point in X × X.

In this article, we improve the result given by Gordji et al. [41] without using the commutative condition and also give an example to validate the main results in this article. Our results improve and extend some couple fixed point theorems due to Gordji et al. [41] and other couple fixed point theorems.

2. Preliminaries

Now, we give some definitions, examples and lemmas for our main results in this article.

Definition 2.1 ([42]). A binary operation *: [0,1]2 → [0,1] is called a continuous t-norm if ([0,1], *) is an abelian topological monoid, i.e.,
  1. (1)

    * is associative and commutative;

     
  2. (2)

    * is continuous;

     
  3. (3)

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

     
  4. (4)

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

     
Definition 2.2 ([42]). A binary operation : [0,1]2 → [0,1] is called a continuous t-conorm if ([0,1],) is an abelian topological monoid, i.e.,
  1. (1)

    is associative and commutative;

     
  2. (2)

    is continuous;

     
  3. (3)

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

     
  4. (4)

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

     

Using the continuous t-norm and t-conorm, Saadati and Park [18] introduced the concept of IFNSs.

Definition 2.3 ([18]). The 5-tuple (X, μ, ν, *,) is called an IFNS if X is a vector space, * is a continuous t-norm, is a continuous t-conorm and μ, ν are fuzzy sets on X × (0, ∞) satisfying the following conditions: for all x, y X and s, t > 0,

(IF1) μ(x, t) + ν(x, t) ≤ 1;

(IF2) μ(x, t) > 0;

(IF3) μ(x, t) = 1 if and only if x = 0;

(IF4) μ ( α x , t ) = μ x , t | α | for all α ≠ 0;

(IF5) μ(x, t) * μ(y, s) ≤ μ(x + y, t + s);

(IF6) μ(x,.): (0, ∞) → [0,1] is continuous;

(IF7) μ is a non-decreasing function on +,
lim t μ ( x , t ) = 1 , lim t 0 μ ( x , t ) = 0 ;

(IF8) ν(x, t) < 1;

(IF9) ν(x, t) = 0 if and only if x = 0;

(IF10) ν ( α x , t ) = ν x , t | α | for all α ≠ 0;

(IF11) ν(x, t) ν(y, s) ≥ ν(x + y, t + s);

(IF12) ν(x,·): (0, ∞) → [0,1] is continuous;

(IF13) ν is a non-increasing function on +,
lim t ν ( x , t ) = 0 , lim t 0 ν ( x , t ) = 1 .

In this case, (μ, ν) is called an intuitionistic fuzzy norm.

Definition 2.4 ([18]). Let (X, μ, ν, *,) be an IFNS.
  1. (1)
    A sequence {x n } in X is said to be convergent to a point x X with respect to the intuitionistic fuzzy norm (μ, ν) if, for any ε > 0 and t > 0, there exists k such that
    μ ( x n - x , t ) > 1 - ε , ν ( x n - x , t ) < ε , n k .
     
In this case, we write lim n →∞ x n = x. In fact that lim n →∞ x n = x if μ(x n - x, t) → 1 and ν(x n - x, t) → 0 as n → ∞ for every t > 0.
  1. (2)
    A sequence {x n } in X is called a Cauchy sequence with respect to the intuitionistic fuzzy norm (μ, ν) if, for any ε > 0 and t > 0, there exists k such that
    μ ( x n - x m , t ) > 1 - ε , ν ( x n - x m , t ) < ε , n , m k .
     
This implies {x n } is Cauchy if μ(x n - x m , t) → 1 and ν(x n - x m , t) → 0 as n, m → ∞ for every t > 0.
  1. (3)

    An IFNS (X, μ, ν, *, ) is said to be complete if every Cauchy sequence in (X, μ, ν, *, ) is convergent.

     

Definition 2.5 ([43, 44]). Let X and Y be two IFNS. A function g : XY is said to be continuous at a point x0 X if, for any sequence {x n } in X converging to a point x0 X, the sequence {g(x n )} in Y converges to a point g(x0) Y. If g : XY is continuous at each x X, then g : XY is said to be continuous on X.

Example 2.6 ([41]). Let (X, || · ||) be an ordinary normed space and θ an increasing and continuous function from + into (0,1) such that limt→∞θ(t) = 1. Four typical examples of these functions are as follows:
θ ( t ) = t t + 1 , θ ( t ) = sin π t 2 t + 1 , θ ( t ) = 1 - e - t , θ ( t ) = e - 1 t .
Let a * b = ab and a bab for all a, b [0,1]. If, for any t (0, ∞), we define
μ ( x , t ) = [ θ ( t ) ] | | x | | , ν ( x , t ) = 1 - [ θ ( t ) ] | | x | | , x X ,

then (X, μ, ν, *, ) is an IFNS.

The other basic properties and examples of IFNSs are given in [18].

Definition 2.7 ([41]). Let (X, μ, ν, *,) be an IFNS. (μ, ν) is said to satisfy the n-property on X × (0, ∞) if
lim n [ μ ( x , k n t ) ] n p = 1 , lim n [ ν ( x , k n t ) ] n p = 0

whenever x X, k > 1 and p > 0.

For examples for n-property see in [41]. Next, we give some notion in coupled fixed point theory.

Definition 2.8 ([39]). Let X be a non-empty set. An element (x, y) X × X is call a coupled fixed point of the mapping F : X × XX if
x = F ( x , y ) , y = F ( y , x ) .
Definition 2.9 ([40]). Let X be a non-empty set. An element (x, y) X × X is call a coupled coincidence point of the mappings F : X × XX and g : XX if
g ( x ) = F ( x , y ) , g ( y ) = F ( y , x ) .
Definition 2.10 ([39]). Let (X, ) be a partially ordered set and F : X × XX be a mapping. The mapping F is said to has the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any x, y X
x 1 , x 2 X , x 1 x 2 F ( x 1 , y ) F ( x 2 , y )
(2.1)
and
y 1 , y 2 X , y 1 y 2 F ( x , y 1 ) F ( x , y 2 ) .
(2.2)
Definition 2.11 ([40]). Let (X, ) be a partially ordered set and F : X × XX, g : XX be mappings. The mapping F is said to has 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 x, y X,
x 1 , x 2 X , g ( x 1 ) g ( x 2 ) F ( x 1 , y ) F ( x 2 , y )
(2.3)
and
y 1 , y 2 X , g ( y 1 ) g ( y 2 ) F ( x , y 1 ) F ( x , y 2 ) .
(2.4)
Definition 2.12 ([40]). Let X be a non-empty set and F : X × XX, g : XX be mappings. The mappings F and g are said to be commutative if
g ( F ( x , y ) ) = F ( g ( x ) , g ( y ) ) , x , y X .

The following lemma proved by Haghi et al. [45] is useful for our main results:

Lemma 2.13 ([45]). Let X be a nonempty set and g : XX be a mapping. Then, there exists a subset E X such that g(E) = g(X) and g : EX is one-to-one.

3. Main Results

First, we prove a coupled fixed point theorem for a mapping F : X × XX which is an essential tool in the partial order IFNSs to show the existence of coupled fixed point. Although the proof in Theorem 3.1 is not difficult to modify, it is an important theorem which is helpful in proving some coupled coincidence point theorems without commutative condition.

Theorem 3.1. Let (X, ) be a partially ordered set and (X, μ, ν, *, ) a complete IFNS such that (μ, ν) has n-property and
a b a b a * b , a , b [ 0 , 1 ] .
(3.1)
Let F : X × XX be mapping such that F has the mixed monotone property and
μ ( F ( x , y ) - F ( u , v ) , k t ) μ ( x - u , t ) * μ ( y - v , t ) , x , y , u , v X , ν ( F ( x , y ) - F ( u , v ) , k t ) ν ( x - u , t ) ν ( y - v , t ) , x , y , u , v X ,
(3.2)
for which x u and y v, where 0 <k < 1. Suppose that either
  1. (1)

    F is continuous or

     
  2. (2)

    X has the following properties:

     
  3. (a)

    if {x n } is a non-decreasing sequence with {x n } → x, then x n x for all n ,

     
  4. (b)

    if {y n } is a non-increasing sequence with {y n } → y, then y y n for all n .

     
If there exist x0, y0 X such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) ,

then F has a coupled fixed point in X × X.

Proof. Let x0, y0 X be such that
x 0 F ( x 0 , y 0 ) , y 0 F ( y 0 , x 0 ) .
Since F(X × X) X, we can construct the sequences {x n } and {y n } in X such that
x n + 1 = F ( x n , y n ) , y n + 1 = F ( y n , x n ) , n 0 .
(3.3)
Now, we show that
x n x n + 1 , y n y n + 1 , n 0 .
(3.4)
In fact, by induction, we prove this. For n = 0, since x0 F(x0, y0) = x1 and y0 = F(y0, x0) y1, we show that (3.4) holds for n = 0. Suppose that (3.4) holds for any n ≥ 0. Then, we have
x n x n + 1 , y n y n + 1 .
(3.5)
Since F has the mixed monotone property, it follows from (3.5) and (2.1) that
F ( x n , y ) F ( x n + 1 , y ) , F ( y n + 1 , x ) F ( y n , x ) , x , y X ,
(3.6)
and also it follows from (3.5) and (2.2) that
F ( y , x n ) F ( y , x n + 1 ) , F ( x , y n + 1 ) F ( x , y n ) , x , y X .
(3.7)
If we take y = y n and x = x n in (3.6), then we get
x n + 1 = F ( x n , y n ) F ( x n + 1 , y n ) , F ( y n + 1 , x n ) F ( y n , x n ) = y n + 1 .
(3.8)
If we take y = yn+1and x = xn+1in (3.7), then we get
F ( y n + 1 , x n ) F ( y n + 1 , x n + 1 ) = y n + 2 , x n + 2 = F ( x n + 1 , y n + 1 ) F ( x n + 1 , y n ) .
(3.9)
Hence, it follows from (3.8) and (3.9) that
x n + 1 x n + 2 , y n + 1 y n + 2 .
(3.10)
Therefore, by induction, we conclude that (3.4) holds for all n ≥ 0, that is,
x 0 x 1 x 2 x n x n + 1
(3.11)
and
y 0 y 1 y 2 y n y n + 1 .
(3.12)
Define α n (t): = μ(x n - xn+1, t) * μ(y n - yn+1, t). Then, using (3.2) and (3.3), we have
μ ( x n - x n + 1 , k t ) = μ ( F ( x n - 1 , y n - 1 ) - F ( x n , y n ) , k t ) μ ( x n - 1 - x n , t ) * μ ( y n - 1 - y n , t ) = α n - 1 ( t )
(3.13)
and
μ ( y n - y n + 1 , k t ) = μ ( y n + 1 - y n , k t ) = μ ( F ( y n , x n ) - F ( y n - 1 , x n - 1 ) , k t ) μ ( y n - y n - 1 , t ) * μ ( x n - x n - 1 , t ) = μ ( y n - 1 - y n , t ) * μ ( x n - 1 - x n , t ) = α n - 1 ( t ) .
(3.14)
From the t-norm property, (3.13) and (3.14), it follows that
α n ( k t ) α n - 1 ( t ) * α n - 1 ( t ) .
(3.15)
From (3.1), we have
α n - 1 ( t ) * α n - 1 ( t ) [ α n - 1 ( t ) ] 2 .
(3.16)
By (3.15) and (3.16), we get α n (kt) ≥ [αn-1(t)]2 for all n ≥ 1. Repeating this process, we have
α n ( t ) α n - 1 t k 2 α 0 t k n 2 n ,
(3.17)
which implies that
μ ( x n - x n + 1 , k t ) * μ ( y n - y n + 1 , k t ) μ x 0 - x 1 , t k n 2 n * μ y 0 - y 1 , t k n 2 n .
(3.18)
On the other hand, we have
t ( 1 - k ) ( 1 + k + + k m - n - 1 ) < t , m > n , 0 < k < t .
By property of t-norm, we get
μ ( x n - x m , t ) * μ ( y n - y m , t ) μ ( x n - x m , t ( 1 - k ) ( 1 + k + + k m - n - 1 ) ) * μ ( y n - y m , t ( 1 - k ) ( 1 + k + + k m - n - 1 ) ) μ ( x n - x n + 1 , t ( 1 - k ) ) * μ ( y n - y n + 1 , t ( 1 - k ) ) * μ ( x n + 1 - x n + 2 , t ( t - k ) k ) * μ ( y n + 1 - y n + 2 , t ( 1 - k ) k ) * * μ ( x m - 1 - x m , t ( 1 - k ) k m - n - 1 ) * μ ( y m - 1 - y m , t ( t - k ) k m - n - 1 ) μ x 0 - x 1 , ( 1 - k ) t k n * μ y 0 - y 1 , ( 1 - k ) t k n * * μ x 0 - x 1 , ( 1 - k ) t k n * μ y 0 - y 1 , ( 1 - k ) t k n μ x 0 - x 1 , ( 1 - k ) t k n m - n * μ y 0 - y 1 , ( 1 - k ) t k n m - n μ x 0 - x 1 , ( 1 - k ) t k n m * μ y 0 - y 1 , ( 1 - k ) t k n m μ x 0 - x 1 , ( 1 - k ) t k n n p * μ y 0 - y 1 , ( 1 - k ) t k n n p ,
(3.19)
where p > 0 such that m <n p . Sine (μ, ν) has the n-property, we have
lim n μ x 0 - x 1 , ( 1 - k ) t k n n p = 1
and so
lim n μ ( x n - x m ) * μ ( y n - y m ) = 1 .
(3.20)
Next, we claim that
lim n ν ( x n - x m ) ν ( y n - y m ) = 0 .
Define β n (t) := ν(x n - xn+1, t) ν(y n - yn+1, t). It follows from (3.2) and (3.3) that
ν ( x n - x n + 1 , k t ) = ν ( F ( x n - 1 , y n - 1 ) - F ( x n , y n ) , k t ) ν ( x n - 1 - x n , t ) ν ( y n - 1 - y n , t ) = β n - 1 ( t )
(3.21)
and
ν ( y n - y n + 1 , k t ) = ν ( y n + 1 - y n , k t ) = ν ( F ( y n , x n ) - F ( y n - 1 , x n - 1 ) , k t ) ν ( y n - y n - 1 , t ) ν ( x n - x n - 1 , t ) = ν ( y n - 1 - y n , t ) ν ( x n - 1 - x n , t ) = β n - 1 ( t ) .
(3.22)
Thus, it follows from the notion of t-conorm, (3.21) and (3.22) that
β n ( k t ) β n - 1 ( t ) β n - 1 ( t ) .
(3.23)
From (3.1), we have
β n - 1 ( t ) β n - 1 ( t ) [ β n - 1 ( t ) ] 2 .
(3.24)
Thus, by (3.23) and (3.24), we get β n (kt) ≤ [βn-1(t)]2 for all n ≥ 1. Repeating this process again, we have
β n ( t ) β n - 1 t k 2 β 0 t k n 2 n ,
(3.25)
that is,
ν ( x n - x n + 1 , k t ) ν ( y n - y n + 1 , k t ) ν x 0 - x 1 , t k n ν y 0 - y 1 , t k n 2 n .
(3.26)
Since we have
t ( 1 - k ) ( 1 + k + + k m - n - 1 ) < t , m > n , 0 < k < 1 ,
by the t-conorm property, we get
ν ( x n x m , t ) ν ( y n y m , t ) ν ( x n x m , t ( 1 k ) ( 1 + k + + k m n 1 ) ) ν ( y n y m , t ( 1 k ) ( 1 + k + + k m n 1 ) ) ν ( x n x n + 1 , t ( 1 k ) ) ν ( y n y n + 1 , t ( 1 k ) ν ( x n + 1 x n + 2 , t ( 1 k ) k ) ν ( y n + 1 y n + 2 , t ( 1 k ) k ) ν ( x m 1 x m , t ( 1 k ) k m n 1 ) ν ( y m 1 y m , t ( 1 k ) k m n 1 ) ν ( x 0 x 1 , ( 1 k ) t k n ) ( y 0 y 1 , ( 1 k ) t k n ) ν ( x 0 x 1 , ( 1 k ) t k n ) ν ( y 0 y 1 ( 1 k ) t k n ) [ ν ( x 0 x 1 , ( 1 k ) t k n ) ] m n [ ν ( y 0 y 1 , ( 1 k ) t k n ) ] m n [ ν ( x 0 x 1 , ( 1 k ) t k n ) ] m [ ν ( y 0 y 1 , ( 1 k ) t k n ) ] m [ ν ( x 0 x 1 , ( 1 k ) t k n ) ] n p [ ν ( y 0 y 1 , ( 1 k ) t k n ) ] n p ,
(3.27)
where p > 0 such that m <n p . Sine (μ, ν) has the n-property, we have
lim n ν x 0 - x 1 , ( 1 - k ) t k n n p = 0
and so
lim n ν ( x n - x m ) ν ( y n - y m ) = 0 .
(3.28)
From (3.20) and (3.28), we know that the sequences {x n } and {y n } are Cauchy sequences in X. Since X complete, there exist x, y X such that
lim n x n = x , lim n y n = y .
(3.29)
Next, we show that x = F(x, y) and y = F(y, x). If the assumption (1) holds, then we have
x = lim n x n + 1 = lim n F ( x n , y n ) = F ( lim n x n , lim n y n ) = F ( x , y )
(3.30)
and
y = lim n y n + 1 = lim n F ( y n , x n ) = F ( lim n y n , lim n x n ) = F ( y , x ) .
(3.31)

Therefore, x = F(x, y) and y = F(y, x), that is, F has a coupled fixed point.

Suppose that the assumption (2) holds. Since {x n } is non-decreasing and x n x, it follows from (a) that x n x for all n . Similarly, we can conclude that y n y for all n . Then, by (3.2), we get
μ ( x n + 1 - F ( x , y ) , k t ) = μ ( F ( x n , y n ) - F ( x , y ) , k t ) μ ( x n - x , t ) * μ ( y n - y , t ) .
(3.32)
Taking the limit as n → ∞, we have μ(x - F(x, y), kt) = 1 and so x = F(x, y). Using (3.2) again, we have
ν ( y n + 1 - F ( y , x ) , k t ) = ν ( F ( y , x ) - y n + 1 , k t ) = ν ( F ( y , x ) - F ( y n , x n ) , k t ) ν ( y - y n , t ) ν ( x - x n , t ) = ν ( y n - y , t ) ν ( x n - x , t ) .
(3.33)

Taking the limit as n → ∞ in both sides of (3.33), we have ν(y - F(y, x), kt) = 0 and then y = F(y, x). Therefore, F has a coupled fixed point at (x, y). This completes the proof. □

Next, we prove the existence of coupled coincidence point theorem, where we do not require that F and g are commuting.

Theorem 3.2. Let (X, ) be a partially ordered set and (X, μ, ν, *,) a IFNS such that (μ, ν) has n-property and
a b a b a * b , a , b [ 0 , 1 ] .
(3.34)
Let F : X × XX and g : XX be two mappings such that F has the mixed g-monotone property and
μ ( F ( x , y ) - F ( u , v ) , k t ) μ ( g x - g u , t ) * μ ( g y - g v , t ) , x , y , u , v X , ν ( F ( x , y ) - F ( u , v ) , k t ) ν ( g x - g u , t ) ν ( g y - g v , t ) , x , y , u , v X ,
(3.35)
for which gx gu and gy gv, where 0 <k < 1, F(X × X) g(X) and g is continuous, g(X) is complete. Suppose that either
  1. (1)

    F is continuous or

     
  2. (2)

    X has the following property:

     
  3. (a)

    if {x n } is a non-decreasing sequence with {x n } → x, then x n x for all n ,

     
  4. (b)

    if {y n } is a non-increasing sequence with {y n } → y, then y y n for all n .

     
If there exist x0, y0 X such that
g ( x 0 ) F ( x 0 , y 0 ) , g ( y 0 ) F ( y 0 , x 0 ) ,

then F and g have a coupled coincidence point in X × X.

Proof. Using Lemma 2.13, there exists E X such that g(E) = g(X) and g : EX is one-to-one. We define a mapping A : g ( E ) × g ( E ) X by
A ( g x , g y ) = F ( x , y ) , g x , g y g ( E ) .
(3.36)
As g is one to one on g(E), so A is well-defined. Thus, it follows from (3.35) and (3.36) that
μ ( A ( g x , g y ) - A ( g u , g v ) , k t ) μ ( g x - g u , t ) * ( g y - g v , t )
(3.37)
and
ν ( A ( g x , g y ) - A ( g x , g y ) , k t ) ν ( g x - g u , t ) ν ( g y - g v , t )
(3.38)
for all gx, gy, gu, gv g(E) with gx gy and gy gv. Since F has the mixed g-monotone property, for all x, y X, we have
x 1 , x 2 X , g x 1 g x 2 F ( x 1 , y ) F ( x 2 , y )
(3.39)
and
y 1 , y 2 X , g y 1 g y 2 F ( x , y 1 ) F ( x , y 2 ) .
(3.40)
Thus, it follows from (3.36), (3.39) and (3.40) that, for all gx, gy g(E),
g x 1 , g x 2 g ( E ) , g x 1 g x 2 A ( g x 1 , g y ) A ( g x 2 , g y )
(3.41)
and
g y 1 , g y 2 g ( E ) , g y 1 g y 2 A ( g x , g y 1 ) A ( g x , g y 2 ) ,
(3.42)

which implies that A has the mixed monotone property.

Suppose that the assumption (1) holds. Since F is continuous, A is also continuous. Using Theorem 3.1 with the mapping A , it follows that A has a coupled fixed point (u, v) g(X) × g(X).

Suppose that the assumption (2) holds. We can conclude similarly in the proof of Theorem 3.1 that the mapping A has a coupled fixed point (u, v) g(X) × g(X).

Finally, we prove that F and g have a coupled coincidence point in X. Since (u, v) is a coupled fixed point of A , we get
u = A ( u , v ) , v = A ( v , u ) .
(3.43)
Since (u, v) g(X) × g(X), there exists a point ( u ^ , v ^ ) X × X such that
u = g u ^ , v = g v ^ .
(3.44)
Thus, it follows from (3.43) and (3.44) that
g u ^ = A ( g u ^ , g v ^ ) , g v ^ = A ( g v ^ , g u ^ ) .
(3.45)
Also, from (3.36) and (3.45), we get
g u ^ = F ( u ^ , v ^ ) , g v ^ = F ( v ^ , u ^ ) .
(3.46)

Therefore, ( u ^ , v ^ ) is a coupled coincidence point of F and g. This completes the proof. □

Next, we give example to validate Theorem 3.2.

Example 3.3. Let X = , a * b = aba b for all a, b [0,1] and θ ( t ) = e - 1 t . Then, (X, μ, ν, *,) is a complete fuzzy normed space, where
μ ( x , t ) = [ θ ( t ) ] | x | , ν ( x , t ) = 1 - [ θ ( t ) ] | x | , x X ,
that (μ, ν) satisfies the n-property on X × (0, ∞). If X is endowed with the usual order as x y y - x [0, ∞), then (X, ) is a partially ordered set. Define mappings F : X × XX and g : XX by
F ( x , y ) = 1 , ( x , y ) X × X
and
g ( x ) = x - 1 , x X .
Since
g ( F ( x , y ) ) = g ( 1 ) = 0 1 = F ( g x , g y )
for all x, y X, the mappings F and g do not satisfy the commutative condition. Hence, Theorem 2.5 of Gordji et al. [41] cannot be applied to this example. But, by simple calculation, we see that F(X × X) g(X), g and F are continuous and F has the mixed g-monotone property. Moreover, there exist x0 = 1 and y0 = 3 with
g ( 1 ) = 0 1 = F ( 1 , 3 )
and
g ( 3 ) = 2 1 = F ( 3 , 1 ) .
Now, for any x, y, u, v X with gx gu and gy gv, we get
μ ( F ( x , y ) - F ( u , v ) , k t ) = μ ( 0 , k t ) = 1 μ ( g x - g u , t ) * μ ( g y - g v , t )
(3.47)
and
ν ( F ( x , y ) - F ( u , v ) , k t ) = ν ( 0 , k t ) = 0 ν ( g x - g u , t ) ν ( g y - g v , t ) ,
(3.48)

where 0 <k < 1. Therefore, all the conditions of Theorem 3.2 hold and so F and g have a coupled coincidence point in X × X. In fact, a point (2,2) is a coupled coincidence point of F and g.

Remark 3.4. Although Theorem 2.5 of Gordji et al. [41] is essential tool in the partially ordered fuzzy normed spaces to claim the existence of coupled coincidence points of two mappings. However, some mappings do not have the commutative property as in the above example. Therefore, it is very interesting to use Theorem 3.2 as another auxiliary tool to claim the existence of a coupled coincidence point.

Declarations

Acknowledgements

The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the third author would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under the Project NRU-CSEC No. 54000267) for financial support during the preparation of this manuscript. This project was partially completed while the first and third authors visit Department of Mathematics Education, Gyeongsang National University, Korea. Also, the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).

Authors’ Affiliations

(1)
Department of Mathematics, King Mongkut's University of Technology Thonburi, Bang-Mod, Bangkok, 10140, Thailand
(2)
Department of Mathematics Education and the Rins, Gyeongsang National University, Chinju, 660-701, Korea

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© Sintunavarat et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. 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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