RESEARC H Open Access
Coupled coincidence point theorems for
contractions without commutative condition in
intuitionistic fuzzy normed spaces
Wutiphol Sintunavarat
1
, Yeol Je Cho
2*
and Poom Kumam
1*
* Correspondence: ;

Full list of author information is
available at the end of the article
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 coinci-
dence 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
[2-10] and others).
One of the most interesting is study to ot her spaces such as probabilistic metric
spaces (see [11-15]). The fuzzy theory was introduced simultaneously by Zadeh [16].

Let F: X × X ® Xandg: X ® X b e two mappings such that F has t he mixed g-
monotone property and
μ(F(x, y) − F(u, v), kt) ≥ μ(gx − gu, t) ∗ μ(gy − gv, t), ∀x, y, u, v ∈ X
,
ν
(
F
(
x, y
)
− F
(
u, v
)
, kt
)
≤ ν
(
gx − gu, t
)
♦ν
(
gy − gv, t
)
, ∀x, y, u, v ∈ X
,
(1:2)
for which g(x) ≼ g(u) and g(y) ≽ zg(v), where 0<k <1,F(X × X) ⊆ g(X), g is continu-
ous and g commuting with F. Suppose that either
(1) F is continuous or

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 binar y operation *: [0,1]
2
® [0,1] is called a continuous t-
norm if ([0,1], *) is an abelian topological monoid, i.e.,
(1) * is associative and commutative;
(2) * is continuous;


x,
t
|α|

for all a ≠ 0;
(IF
5
) μ(x, t)*μ(y, s) ≤ μ(x + y, t + s);
(IF
6
) μ(x,.): (0, ∞) ® [0,1] is continuous;
(IF
7
) μ is a non-decreasing function on ℝ
+
,
lim
t→∞
μ(x, t) = 1, lim
t
→
0
μ(x, t)=0
;
(IF
8
) ν(x, t)<1;
(IF
9

In this case, (μ, ν) is called an intuitionistic fuzzy norm.
Definition 2.4 ([18]). Let (X, μ, ν, *,◊) be an IFNS.
(1) A sequence {x
n
}inX is said to be convergent to a point x Î X with respect to the
intuitionistic fuzzy norm (μ, ν) if, for any ε > 0 and t > 0, there exists k Î N 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
m
, t) ® 1andν(x
n
- x
m
, t) ® 0asn, m ® ∞
for every t >0.
Sintunavarat et al. Fixed Point Theory and Applications 2011, 2011:81
/>Page 3 of 13
(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 : X ® Y is said to
be continuous at a point x
0
Î X if, for any sequence {x
n
}inX converging to a point x
0
Î X, the sequence {g(x
n
)} in Y converges to a point g(x
0
) Î Y. If g : X ® Y is continu-
ous at each x Î X, then g : X ® Y 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 ℝ
+
into (0,1) s uch that l im
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-prop-
erty on X × (0, ∞)if
lim
n
→
∞
[μ(x, k
n
t)]
n
p
= 1, lim
n
→
∞
[ν(x, k
n
t)]
n

= F
(
y, x
).
Definition 2.10 ([39]). Let (X, ≼) be a partially ordered set and F : X × X ® X 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 Î X
x
1
, x
2
∈ X, x
1
 x
2
⇒ F
(
x
1
, y
)
 F
(
x
2
, y
)
(2:1)
and

(
x
1
)
 g
(
x
2
)
⇒ F
(
x
1
, y
)
 F
(
x
2
, y
)
(2:3)
and
y
1
, y
2
∈ X, g
(
y

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 : X ® Xbeamapping.Then,
there exists a subset E ⊆ X such that g(E)=g(X) and g : E ® X is one-to-one.
3. Main Results
First, we prove a coupled fixed point theorem for a mapping F : X × X ® X which is
an essential tool in the partial order IFNSs to show the existence of coupled fixed
point. Altho ugh the pr oof 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 ≤ ab ≤ a ∗ b, ∀a, b ∈
[
0, 1
]
.
(3:1)
Let F : X × X ® X be mapping such that F has the mixed monotone property and
μ(F(x, y) − F(u, v), kt) ≥ μ(x − u, t) ∗ μ(y − v, t), ∀x, y, u, v ∈ X
,
ν
(

≼ x for all n Î N,
(b) if {y
n
} is a non-increasing sequence with {y
n
} ® y, then y ≼ y
n
for all n Î N.
If there exist x
0
, y
0
Î X such that
x
0
 F
(
x
0
, y
0
)
, y
0
 F
(
y
0
, x
0

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


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

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
(

, 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)

2
 ···
y
n

y
n+1
 ···
.
(3:12)
Define a
n
(t): = μ(x
n
- x
n+1
, t)*μ(y
n
- y
n+1
, t). Then, using (3.2) and (3.3), we have
μ(x
n
− x
n+1
, kt)=μ(F(x
n−1
, y
n−1
) − F(x

= μ(F(y
n
, x
n
) − F(y
n−1
, x
n−1
), kt)
≥ 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
)
≥ [α
n−1
(
t
)
]
2
.
(3:16)
By (3.15) and (3.16), we get a
n
(kt) ≥ [a
n-1
(t)]
2
for all n ≥ 1. Repeating this process,
we have
Sintunavarat et al. Fixed Point Theory and Applications 2011, 2011:81
/>Page 6 of 13
α
n
(t ) ≥

α
n−1

t

− x
1
,
t
k
n

2n
∗

μ

y
0
− y
1
,
t
k
n

2n
.
(3:18)
On the other hand, we have
t
(
1 − k
)(
1+k + ···+ k

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

,(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
≥

μ

x
0
− x
1
,(1− k)
t
k
n

np
∗

μ


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

n
, t)♦ν(y
n−1
− y
n
, t
)
= β
n−1
(
t
)
(3:21)
Sintunavarat et al. Fixed Point Theory and Applications 2011, 2011:81
/>Page 7 of 13
and
ν( y
n
− y
n+1
, kt)=ν(y
n+1
− y
n
, kt)
= ν(F(y
n
, x
n
) − F(y

Thus, it follows from the notion of t-conorm, (3.21) and (3.22) that
β
n
(
kt
)
≤ β
n−1
(
t
)
♦β
n−1
(
t
)
.
(3:23)
From (3.1), we have
β
n−1
(
t
)
♦β
n−1
(
t
)
≤ [β

0

t
k
n

2
n
,
(3:25)
that is,
ν( x
n
− x
n+1
, kt)♦ν(y
n
− y
n+1
, kt) ≤

ν

x
0
− x
1
,
t
k

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

,(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


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)

k
n

n
p
,
(3:27)
Sintunavarat et al. Fixed Point Theory and Applications 2011, 2011:81
/>Page 8 of 13
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

∞
y
n
= y
.
(3:29)
Next, we show that x = F(x, y)andy = 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

, 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 Î N. Similarly, we can c onclude that y
n
≽ y for
all n Î N. Then, by (3.2), we get
μ(x
n+1
− F(x, y), kt)=μ(F(x
n
, y
n
) − F(x, y), kt)
≥ μ
(

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 )=0and
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 ≤ ab ≤ a ∗ b, ∀a, b ∈
[
0, 1
].
(3:34)
Let F : X × X ® Xandg: X ® X be two mappings such that F has the mixed g-
monotone property and
μ
(
F
(
x, y

)
≤ ν
(
gx − gu, t
)
♦ν
(
gy − gv, t
)
, ∀x, y, u, v ∈ X
,
(3:35)
Sintunavarat et al. Fixed Point Theory and Applications 2011, 2011:81
/>Page 9 of 13
for which gx ≼ gu and gy ≽ gv, where 0<k <1,F(X × X) ⊆ g(X) , g(X) is complete and
g is continuous. Suppose that either
(1) F is continuous or
(2) X has the following property:
(a) if {x
n
} is a non-decreasing sequence with {x
n
} ® x, then x
n
≼ x for all n Î N,
(b) if {y
n
} is a non-increasing sequence with {y
n
} ® y, then y ≼ y

),
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)andg : E ® X is
one-to-one. We define a mapping
A
: g
(
E
)
× g
(
E
)
→ X
by
A
(
gx, gy
)
= F
(
x, y
)
, ∀gx, gy ∈ 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

gx, gy
)
, kt
)
≤ ν
(
gx − gu, t
)
♦ν
(
gy − gv, t
)
(3:38)
for all gx, gy, gu, gv Î g(E)withgx ≼ gy and gy ≽ gv.SinceF has the mixed g-mono-
tone property, for all x, y Î X, we have
x
1
, x
2
∈ X, gx
1
 gx
2
⇒ F
(
x
1
, y
)
 F

, gx
2
∈ g
(
E
)
, gx
1
 gx
2
⇒ A
(
gx
1
, gy
)
 A
(
gx
2
, gy
)
(3:41)
and
gy
1
, gy
2
∈ g
(

Theorem 3.1 that the mapping
A
has a coupled fixed point (u, v) Î g(X)×g(X).
Finally, we prove that F and g have a coupled coincidence point in X. Since (u, v)isa
coupled fixed point of
A
, we get
u
=
A(
u, v
)
, v =
A(
v, u
).
(3:43)
Sintunavarat et al. Fixed Point Theory and Applications 2011, 2011:81
/>Page 10 of 13
Since (u, v) Î g(X)×g(X), there exists a point
(

u,

v
)
∈ X ×
X
such that
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.
□

]
|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 Î [0, ∞), then (X, ≼) is a partially ordered set. Define mappings
F : X × X ® X and g : X ® X by
F
(
x, y
)
=1, ∀
(
x, y
)
∈ X ×
X
and
g
(
x
)
= x − 1, ∀x ∈ X
.
Since
g(
F
(
x, y
))

)
.
Now, for any x, y, u, v Î X with gx ≼ gu and gy ≽ gv, we get
μ(F(x, y) − F(u, v), kt)=μ(0, kt)
=1
≥ μ
(
gx − gu, t
)
∗ μ
(
gy − gv, t
)
(3:47)
and
ν( F(x, y) − F(u, v), kt)=ν(0, kt)
=0
≤ ν
(
gx − gu, t
)
♦ν
(
gy − gv, t
),
(3:48)
Sintunavarat et al. Fixed Point Theory and Applications 2011, 2011:81
/>Page 11 of 13
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 coinci-

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Cite this article as: Sintunavarat et al.: Coupled coincidence point theorems for contractions without
commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory and Applications 2011 2011:81.