Common Fixed Point and Example for Type(β) Compatible Mappings with Implicit Relation in an Intuitionistic Fuzzy Metric Space

International Journal of Fuzzy Logic and Intelligent Systems.
2014.
Mar,
14(1):
66-72

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

- Received : December 03, 2013
- Accepted : February 13, 2014
- Published : March 01, 2014

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In this paper, we establish common fixed point theorem for type(
β
) compatible four mappings with implicit relations defined on an intuitionistic fuzzy metric space. Also, we present the example of common fixed point satisfying the conditions of main theorem in an intuitionistic fuzzy metric space.
X
as following:
Let us recall (see
[13]
) that a continuous
t
–norm is a binary operation ⁎ : [0, 1] × [0, 1] → [0, 1] which satisfies the following conditions:(a)⁎ is commutative and associative; (b)⁎ is continuous; (c)
a
⁎ 1 =
a
for all
a
∈ [0, 1]; (d)
a
⁎
b
≤
c
⁎
d
whenever
a
≤
c
and
b
≤
d
(
a, b, c, d
∈ [0, 1]).
Similarly, a continuous
t
–conorm is a binary operation
: [0, 1] × [0, 1] → [0, 1] which satisfies the following conditions:
Definition 2.1.
(
[14]
) The 5..tuple (
X,M,N
, ⁎,
) is said to be an intuitionistic fuzzy metric space (IFMS) if
X
is an arbitrary set, ⁎ is a continuous
t
–norm,
is a continuous
t
–conorm and
M,N
are fuzzy sets on
X
^{2}
× (0,∞) satisfying the following conditions; for all
x, y, z
∈
X
, such that
Note that (
M, N
) is called an IFM on
X
. The functions M(
x, y, t
) and N(
x, y, t
) denote the degree of nearness and the degree of non-nearness between
x
and
y
with respect to
t
, respectively.
Definition 2.2.
(
[6]
) Let
X
be an IFMS.
(a) {
x_{n}
} is said to be convergent to a point
x
∈
X
if, for any 0 < 𝜖 < 1 and
t
> 0, there exists
n
_{0}
∈
N
such that
M (x_{n}, x, t ) > 1 – 𝜖, N (x_{n}, x, t ) < 𝜖
for all
n
≥
n
_{0}
.
(b) {
x_{n}
} is called a Cauchy sequence if for any 0 < 𝜖 < 1 and
t
> 0, there exists
n
_{0}
∈
N
such that
M (x_{n}, x_{m}, t ) > 1 – 𝜖, N (x_{n}, x_{m}, t ) < 𝜖
for all
m, n
≥
n
_{0}
.
(c)
X
is complete if every Cauchy sequence converges in
X
.
Lemma 2.3.
(
[8]
) Let
X
be an IFMS. If there exists a number
k
∈ (0, 1) such that for all
x, y
∈
X
and
t
> 0,
M (x, y, kt ) ≥ M (x, y, t ), N (x, y, kt ) ≤ N (x, y, t ),
then
x
=
y
.
Definition 2.4.
(
[7]
) Let
A,B
be mappings from IFMS
X
into itself. The mappings are said to be type(𝛽) compatible if for all
t
> 0,
whenever {
x_{n}
} ⊂
X
such that
for some
x
∈
X
.
Proposition 2.5.
(
[15]
) Let
X
be an IFMS with
t
⁎
t
≥
t
and
t
t
≤
t
for all
t
∈ [0, 1].
A,B
be type(𝛽) compatible maps from
X
into itself and let {
x_{n}
} be a sequence in
X
such that
Ax_{n},Bx_{n}
→
x
for some
x
∈
X
. Then we have the following
Implicit relations on fuzzy metric spaces have been used in many articles (
[3
,
16]
). Let
= {
ϕ
_{M}
, 𝜓
_{N}
},
I
= [0, 1],
ϕ
_{M}
, 𝜓
_{N}
:
I
^{6}
→
R
be continuous functions and ⁎,
be a continuous t-norm, t-conorm. Now, we consider the following conditions (
[6]
):
(I)
ϕ
_{M}
is decreasing and 𝜓
_{N}
is increasing in sixth variables.
(II) If, for some
k
∈ (0, 1), we have
for any fixed
t
> 0, any nondecreasing functions
u, v
: (0,∞) →
I
with 0 <
u
(
t
),
v
(
t
) ≤ 1, and any nonincreasing functions
x, y
: (0,∞) →
I
with 0 <
x
(
t
),
y
(
t
) ≤ 1, then there exists
h
∈ (0, 1) with
u (ht ) ≥ v (t ) ⁎ u (t ), x (ht ) ≤ y (t ) x (t ).
(III) If, for some
k
∈ (0, 1), we have
ϕ _{M}(u (kt ), u (t ), 1, 1, u (t ), u (t )) ≥ 1
for any fixed
t
> 0 and any nondecreasing function
u
: (0,∞) →
I
, then
u
(
kt
) ≥
u
(
t
). Also, if, for some
k
∈ (0, 1), we have
𝜓_{N}(x (kt ), x (t ), 0, 0, x (t ), x (t )) ≤ 1
for any fixed
t
> 0 and any nonincreasing function
x
: (0,∞) →
I
, then
x
(
kt
) ≤
x
(
t
).
Example 2.6.
(
[6]
) Let
a
⁎
b
= min{
a, b
} and
a
b
= max{
a, b
},
Also, let
t
> 0, 0 <
u
(
t
),
v
(
t
),
x
(
t
),
y
(
t
) ≤ 1,
k
∈ (0, ½ ) where
u, v
: [0,∞) →
I
are nondecreasing functions and
x, y
: [0,∞) →
I
are nonincreasing functions. Now, suppose
then from Park
[6]
,
ϕ
_{M}
, 𝜓
_{N}
∈
.
Theorem 3.1.
Let
X
be a complete intuitionistic fuzzy metric space with
a
⁎
b
= min{
a, b
},
a
b
= max{
a, b
} for all
a, b
∈
I
and
A, B, S
and
T
be mappings from
X
into itself satisfying the conditions:
for all x, y ∈ X and t > 0.
Then
A, B, S
and
T
have a unique common fixed point in
X
.
Proof.
Let
x
_{0}
be an arbitrary point of
X
. Then from Theorem 3.1 of (
[6]
), we can construct a Cauchy sequence {
y_{n}
} ⊂
X
. Since
X
is complete, {
y_{n}
} converges to a point
x
∈
X
. Since {
Ax
_{2n+2}
}, {
Bx
_{2n+1}
}, {
Sx
_{2n}
} and {
Tx
_{2n+1}
} ⊂ {
y_{n}
}, we have
Now, let
A
is continuous. Then
By Proposition 2.5,
Using condition (d), we have, for any
t
> 0,
and by letting
n
→ ∞,
ϕ
_{M}
, 𝜓
_{N}
are continuous, we have
Therefore, by (III), we have
M
(
Ax, x, kt
) ≥
M
(
Ax, x, t
),
N
(
Ax, x, kt
) ≤
N
(
Ax, x, t
).
Hence
Ax
=
x
from Lemma 2.3. Also, we have, by condition (d),
and, let
n
→ ∞, we have
On the other hand, since
ϕ
_{M}
is nonincreasing and 𝜓
_{N}
is nondecreasing in the fifth variable, we have, for any
t
> 0,
which implies that
Sx
=
x
. Since
S
(
X
) ⊆
B
(
X
), there exists a point
y
∈
X
such that
By
=
x
. Using condition (d), we have
which implies that
x
=
Ty
. Since
By
=
Ty
=
x
and
B, T
are type(𝛽) compatible, we have
TTy
=
BBy
. Hence
Tx
=
TTy
=
BBy
=
Bx
. Therefore, from (d), we have, for any
t
> 0,
From (III), we have
M
(
x, Tx, kt
) ≥
M
(
x, Tx, t
),
M
(
x, Tx, kt
) ≤
M
(
x, Tx, t
).
Therefore, we have
x
=
Tx
=
Bx
from Lemma 2.3. Hence
x
is a common fixed point of
A,B, S
and
T
. The same result holds if we assume that
B
is continuous instead of
A
.
Now, suppose that
S
is continuous. Then
By Proposition 2.5,
Using (d), we have for any
t
> 0,
and by
n
→ ∞, since
ϕ
_{M}
, 𝜓
_{N}
∈
are continuous, we have
Thus, we have, from (III),
M
(
Sx, x, kt
) ≥
M
(
Sx, x, t
),
N
(
Sx, x, kt
) ≤
N
(
Sx, x, t
).
Hence
Sx
=
x
by Lemma 2.3. Since
S
(
X
) ⊆
B
(
X
), there exists a point
z
∈
X
such that
Bz
=
x
. Using (d), we have
letting
n
→ ∞, we get
which implies that
x
=
Tz
. Since
Bz
=
Tz
=
x
and
B, T
are type(𝛽) compatible, we have
TBz
=
BBz
and so
Tx
=
TBz
=
BBz
=
Bx
. Thus, we have
letting
n
→ ∞,
Thus,
x
=
Tx
=
Bx
. Since
T
(
X
) ⊆
A
(
X
), there exists
w
∈
X
such that
Aw
=
x
. Thus, from (d),
Hence we have
x
=
Sw
=
Aw
. Also, since
A, S
are type(𝛽) compatible,
x = Sx = SSw = AAw = Ax .
Hence
x
is a common fixed point of
A,B, S
and
T
. The same result holds if we assume that
T
is continuous instead of
S
.
Finally, suppose that
A, B, S
and
T
have another common fixed point
u
. Then we have, for any
t
> 0,
Therefore, from (III),
x
=
u
. This completes the proof.
Example 3.2.
Let
X
be a intuitionistic fuzzy metric space with
X
= [0, 1], ⁎,
be t-norm and t-conorm defined by
a ⁎ b = min{a, b }, a b = max{a, b }
for all
a, b
∈
X
. Also, let
M, N
be fuzzy sets on
X
^{2}
× (0,∞) defined by
Let
ϕ
_{M}
, 𝜓
_{N}
:
X
^{6}
→
R
be defined as in Example 2.6 and define the maps
A, B, S, T
:
X
→
X
by
Ax
=
x
,
and
. Then, for some
, we have
≤ max{N (Ax, By, t ), N (Sx, Ax, t ), N (Ty, By, t ), N (Sx, By, t ), N (Ty, Ax, t )}.
Thus the condition (d) of Theorem 3.1 is satisfied. Also, it is obvious that the other conditions of the theorem are satisfied. Therefore 0 is the unique common fixed point of
A, B, S
and
T
.
Jong Seo Park received the B.S., M.S. and Ph.D. degrees in mathematics from Dong-A University, Pusan, Korea, in 1983, 1985 and 1995, respectively. He is currently Professor in Chinju National University of Education, Jinju, Korea. His research interests include fuzzy mathematics, fuzzy fixed point theory and fuzzy differential equation, etc.
Tel: +82-55-740-1238
Fax: +82-55-740-1230
E-mail: parkjs@cue.ac.kr

1. Introduction

Zadeh
[1]
introduced the concept of fuzzy sets in 1965 and in the next decade, Grabiec
[2]
obtained the Banach contraction principle in setting of fuzzy metric spaces, Also, Altun and Turkoglu
[3]
proved some fixed theorems using implicit relations in fuzzy metric spaces. Furthermore, Park et al.
[4]
defined the intuitionistic fuzzy metric space, and Park et al.
[5]
proved a fixed point theorem of Banach for the contractive mapping of a complete intuitionistic fuzzy metric space. Recently, Park
[6
,
7]
, Park et al.
[8]
obtained a unique common fixed point theorem for type(𝛼) and type(𝛽) compatible mappings defined on intuitionistic fuzzy metric space. Also, authors proved the fixed point theorem using compatible properties in many articles
[9
–
12]
.
In this paper, we will obtain a unique common fixed point theorem and example for this theorem under the type(𝛽) compatible four mappings with implicit relations defined on intuitionistic fuzzy metric space.
2. Preliminaries

We will give some definitions, properties of the intuitionistic fuzzy metric space
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- (a)is commutative and associative;
- (b)is continuous;
- (c)a0 =afor alla∈ [0, 1];
- (d)ab≥cdwhenevera≤candb≤d(a, b, c, d∈ [0, 1]).

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- (a)M(x,y,t) > 0,
- (b)M(x,y,t) = 1x=y,
- (c)M(x,y,t) =M(y, x, t),
- (d)M(x,y,t) ⁎M(y, z, s) ≤M(x, z, t+s),
- (e)M(x,y,⋅) : (0,∞) → (0, 1] is continuous,
- (f)N(x,y,t) > 0,
- (g)N(x,y,t) = 0x=y,
- (h)N(x,y,t) =N(y, x, t),
- (i)N(x,y,t)N(y, z, s) ≥N(x, z, t+s),
- (j)N(x,y,⋅) : (0,∞) → (0, 1] is continuous.

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- (a)BBxn→AxifAis continuous atx,
- (b)AAxn→BxifBis continuous atx,
- (c)ABx=BAxandAx=BxifAandBare continuous atx.

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3. Main Results and Example

Now, we will prove some common fixed point theorem for four mappings on complete IFMS as follows:
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- (a)S(X) ⊆B(X) andT(X) ⊆A(X),
- (b) One of the mappingsA, B, S, Tis continuous,
- (c)AandSas well asBandTare type(𝛽) compatible
- (d) There existk∈ (0, 1) andϕM, 𝜓N∈such that

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4. Conclusion

Park et al.
[4
,
5]
defined an IFMS and proved uniquely existence fixed point for the mappings satisfying some properties in an IFMS. Also, Park et al.
[8]
studied the type(𝛼) compatible mapping, and Park
[7]
proved some properties of type(𝛽) compatibility in an IFMS.
In this paper, we obtain a unique common fixed point and example for type(𝛽) compatible mappings under implicit relations in an IFMS. This paper attempted to develop a proof method according to some conditions based on the fundamental properties and results in this space. I think that this results will be extended and applied to the other spaces, and further research should be conducted to determine how to combine the collaborative learning algorithm with our proof method in the future.
Conflict of Interest

No potential conflict of interest relevant to this article was reported.
BIO

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8
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,
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,
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Citing 'Common Fixed Point and Example for Type(β) Compatible Mappings with Implicit Relation in an Intuitionistic Fuzzy Metric Space
'

@article{ E1FLA5_2014_v14n1_66}
,title={Common Fixed Point and Example for Type(β) Compatible Mappings with Implicit Relation in an Intuitionistic Fuzzy Metric Space}
,volume={1}
, url={http://dx.doi.org/10.5391/IJFIS.2014.14.1.66}, DOI={10.5391/IJFIS.2014.14.1.66}
, number= {1}
, journal={International Journal of Fuzzy Logic and Intelligent Systems}
, publisher={Korean Institute of Intelligent Systems}
, author={Park, Jong Seo}
, year={2014}
, month={Mar}