In this paper, we introduce the notion of single and set-valued maps satisfying OWC property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and set-valued maps satisfying OWC properties in IFMS using implicit relation.
1. Introduction
Several authors
[1
–
5]
studied and developed the various concepts in different direction and proved some fixed point in fuzzy metric space. Also, Jungck
[6]
introduced the concept of compatible maps, and Vijayaraju and Sajath
[7]
obtained some common fixed point theorems in fuzzy metric space. Recently, Park et.a..
[8]
introduced the intuitionistic fuzzy metric space (IFMS), Park
[12
,
13]
studied the compatible and weakly compatible maps in IFMS, and proved common fixed point theorem in IFMS. Also, Park
[9]
proved some properties for several types compatible maps, and Park
[10]
defined occasionally weakly semi-compatible map and obtained some fixed point using this maps in IFMS.
In this paper, we introduce the notion of single and set-valued maps satisfying occasionally weakly compatible (OWC) property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and set-valued maps satisfying OWC property in IFMS using implicit relation.
2. Preliminaries
In this part, we recall some definitions, properties and known results in the IFMS as follows : Let us recall (
[11]
) that a continuous
t
−norm is an 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]). Also, a continuous
t
−conorm is an operation ⋄ : [0, 1] × [0, 1] → [0, 1] which satisfies the following conditions: (a)⋄ is commutative and associative, (b)⋄ is continuous, (c)
a
⋄ 0 =
a
for all
a
∈ [0, 1], (d)
a
⋄
b
≥
c
⋄
d
whenever
a
≤
c
and
b
≤
d
(
a
,
b
,
c
,
d
∈ [0, 1]).
Definition 2.1
. (
[8]
) 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
in
X
and all
s
,
t
∈ (0, ∞),
-
(a)M(x,y,t) > 0,
-
(b)M(x,y,t) = 1 if and only ifx=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) = 0 if and only ifx=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,
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
Through out this paper,
X
will represent the IFMS and
CB
(
X
), the set of all non-empty closed and bounded subsets of
X
. For
A
,
B
∈
CB
(
X
) and for every
t
> 0, denote
If
A
consists of a single point
a
, we write
δ
M
(
A
,
B
,
t
) =
δ
M
(
a
,
B
,
t
),
δ
N
(
A
,
B
,
t
) =
δ
N
(
a
,
B
,
t
).
Furthermore, if
B
consists of a single point
b
, we write
δ
M
(
A
,
B
,
t
) =
M
(
a
,
b
,
t
),
δ
N
(
A
,
B
,
t
) =
N
(
a
,
b
,
t
).
It follows immediately from definition that
Also,
δ
M
(
A
,
B
,
t
) = 1 and
δ
N
(
A
,
B
,
t
) = 0 if and only if
A
=
B
= {
a
} for al
A
,
B
∈
CB
(
X
).
Definition 2.2
. Let
X
be an IFMS,
A
:
X
→
X
and
B
:
X
→
CB
(
X
).
(a) A point
x
∈
X
is called a coincidence point of hybrid maps
A
and
B
if
x
=
Ax
∈
Bx
.
(b) Hybrid maps
A
and
B
are said to be compatible if
ABx
∈
CB
(
X
) for all
x
∈
X
and
whenever {
x
n
} is a sequence in
X
such that
Bx
n
→
D
∈
CB
(
X
) and
Ax
n
→
x
∈
D
.
(c) Hybrid maps
A
and
B
are said to be weakly compatible if
ABx
=
BAx
whenever
Ax
∈
Bx
.
(d) Hybrid maps
A
and
B
are said to be occasionally weakly compatible (OWC) if there exists some points
x
∈
X
such that
Ax
∈
Bx
and
ABx
⊆
BAx
.
Example 2.3
. Let
X
= [0, ∞) with
a
∗
b
= min{
a
,
b
},
a
⋄
b
= max{
a
,
b
} for all
a
,
b
∈ [0, 1] and for all
t
> 0,
Define the maps
A
:
X
→
X
and
B
:
X
→
CB
(
X
) by
Here 1 is a coincidence point of
A
and
B
, but
A
and
B
are not weakly compatible as
BA
(1) = [1, 5] ≠
AB
(1) = [2, 5]. Also,
A
and
B
are OWC hybrid maps as
A
and
B
are weakly compatible at
x
= 0 as
A
(0) ∈
B
(0) and 0 =
AB
(0) ⊆
BA
(0) = {0}. Hence weakly compatible hybrid maps are OWC, but the converse is not true in general.
3. Main Results
Theorem 3.1
. Let
X
be an IFMS with
t
∗
t
=
t
and
t
⋄
t
=
t
for all
t
∈ [0, 1]. Also, let
A
,
B
:
X
→
X
and
S
,
T
:
X
→
CB
(
X
) be single and set-valued mappings such that the hybrid pairs (
A
,
S
) and (
B
,
T
) are OWC satisfying
for every
x
,
y
∈
X
,
t
> 0.
Also, let implicit relation Φ = {
ϕ
,
ψ
} such that
ϕ
: [0, 1]
5
→ [0, 1] and
ψ
: [0, 1]
5
→ [0, 1] continuous functions satisfying
(a)
ϕ
(
t
1
,
t
2
,
t
3
,
t
4
,
t
5
) is non-increasing in
t
2
and
t
5
for all
t
> 0.
ψ
(
t
1
,
t
2
,
t
3
,
t
4
,
t
5
) is non-decreasing in
t
2
and
t
5
for all
t
> 0.
(b)
ϕ
(
t
,
t
, 1, 1,
t
) ≥ 0 implies that
t
= 1, and
ψ
(
t
,
t
, 0, 0,
t
) ≤ 1 implies that
t
= 0 for all
t
> 0.
Then
A
,
B
,
S
and
T
have a unique common fixed point in
X
.
Proof
Since the hybrid pairs (
A
,
S
) and (
B
,
T
) are OWC maps, there exist two elements
u
,
v
∈
X
such that
Au
∈
Su
,
ASu
⊆
SAu
and
Bv
∈
Tv
,
BTv
⊆
TBv
.
First, we prove that
Au
=
Bv
.
As
Au
∈
Su
and
Bv
∈
Tv
, so,
If
Au
≠
Bv
, then
δ
M
(
Su
,
Tv
,
t
) < 1 and
δ
N
(
Su
,
Tv
,
t
) > 0. Using (1) for
x
=
u
and
y
=
v
, we have
That is,
Also,
ϕ
,
ψ
satisfies (b), so
-
δM(Su,Tv,t) = 1 andδN(Su,Tv,t) = 0.
This is a contradiction which gives
Au
=
Bv
Now, we prove that
A
2
u
=
Au
. Suppose that
A
2
u
≠
Au
, then
δ
M
(
SAu
,
Tv
,
t
) < 1 and
δ
N
(
SAu
,
Tvt
) > 0. Also, using (1) for
x
=
Au
and
y
=
v
, we get
Also,
Au
∈
Su
and
ASu
∈
SAu
, so
AAu
∈
ASu
⊆
SAu
,
Bv
∈
Tv
and
BTv
⊆
TBv
, hence
Therefore
But
ϕ
,
ψ
satisfies (b), so,
-
δM(SAu,Tv,t) = 1 andδN(SAu,Tv,t) = 0,
a contradiction and hence
A
2
u
=
Au
=
Bv
. Similarly, we can show that
B
2
v
=
Bv
.
Let
Au
=
Bv
=
z
, then
Az
=
z
=
Bz
,
z
∈
Sz
and
z
∈
Tz
. Therefore
z
is a fixed point of
A
,
B
,
S
and
T
.
Finally, we prove the uniqueness of the fixed point. Let
z
≠
z
0
be another fixed point of
A
,
B
,
S
and
T
, then by (1), we have,
From (b), we get
This is a contradiction. Hence
z
=
z
0
. Therefore
z
is unique common fixed point of
A
,
B
,
S
and
T
.
Example 3.2
. Let
X
be an IFMS in which
X
=
R
+
,
a
∗
b
= min{
a
,
b
} and
a
⋄
b
= max{
a
,
b
} for all
a
,
b
∈ [0, 1] such that for all
t
> 0,
Define the maps
A
,
B
,
S
and
T
on
X
by
Define
ϕ
: [0, 1]
→
[0, 1], ψ : [0, 1] → [0, 1] as
Here the pairs (
A
,
S
) and (
B
,
T
) are OWC and the contractive condition is satisfied. Hence 1 is a unique common fixed point of
A
,
B
,
S
and
T
.
Corollary 3.3
. Let
X
be an IFMS,
t
∗
t
=
t
and
t
⋄
t
=
t
for all
t
∈ [0, 1] and let
A
:
X
→
X
and
S
,
T
:
X
→
CB
(
X
) be single and set-valued mappings such that the hybrid pair (
A
,
S
) and (
A
,
T
) are OWC satisfying
for every
x
,
y
∈
X
,
t
> 0 and
ϕ
,
ψ
are satisfies (a) and (b), respectively in Theorem 3.1. Then
A
,
S
and
T
have a unique common fixed point in
X
.
Proof
Suppose that
A
=
B
in Eq. (1) of Theorem 3.1, then we get this corollary.
Corollary 3.4
. Let
X
be an IFMS,
t
∗
t
=
t
and
t
⋄
t
=
t
for all
t
∈ [0, 1] and let
A
:
X
→
X
and
S
:
X
→
CB
(
X
) be single and set-valued mappings such that the hybrid pair (
A
,
S
) is OWC satisfying
for every
x
,
y
∈
X
,
t
> 0 and
ϕ
,
ψ
are functions satisfying (a) and (b), respectively in Theorem 3.1. Then
A
and
S
have a unique common fixed point in
X
.
Proof
Suppose that
A
=
B
and
S
=
T
in Eq. (1) of Theorem 3.1, then we get this corollary.
4. Conclusion
Park et.al.
[8]
introduced the IFMS, and proved common fixed point theorem in IFMS. Also, Park
[9]
proved some properties for several types compatible maps, and Park
[10]
defined occasionally weakly semi-compatible map and obtained some fixed point using this maps in IFMS.
In this paper, we introduce the notion of single and set-valued maps satisfying OWC property in IFMS using implicit relation. Also, we obtain common fixed point theorems for single and set-valued maps satisfying OWC property in IFMS using implicit relation.
Conflict of Interest
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This author is supported by Chinju National University of Education Research Fund in 2014.
BIO
Jong Seo Park received the B.S., M.S. and Ph.D. degrees in mathematics from Dong-A University, Busan, 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.
E-mail: parkjs@cue.ac.kr
Deng Z.
1982
“Fuzzy pseudo-metric spaces,”
Journal of Mathe maticalAnalysis and Applications
http://dx.doi.org/10.1016/0022-247X(82)90255-4
86
(1)
74 -
95
DOI : 10.1016/0022-247X(82)90255-4
Grabiec M.
1988
“Fixed points in fuzzy metric spaces,”
Fuzzy Sets and Systems
http://dx.doi.org/10.1016/0165-0114(88)90064-4
27
(3)
385 -
389
DOI : 10.1016/0165-0114(88)90064-4
Kaleva O.
,
Seikkala S.
1984
“On fuzzy metric spaces,”
Fuzzy Sets and Systems
http://dx.doi.org/10.1016/0165-0114(84)90069-1
12
(3)
215 -
229
DOI : 10.1016/0165-0114(84)90069-1
Kubiaczyk I.
,
Sharma S.
2003
“Common coincidence point in fuzzy metric spaces,”
Journal of Fuzzy Mathematics
11
(1)
1 -
5
Singh B.
,
Chauhan M. S.
2000
“Common fixed points of compatible maps in fuzzy metric spaces,”
Fuzzy Sets and Systems
http://dx.doi.org/10.1016/S0165-0114(98)00099-2
115
(3)
471 -
475
DOI : 10.1016/S0165-0114(98)00099-2
Jungck G.
1986
“Compatible mappings and common fixed points,”
International Journal of Mathematics and Mathematical Sciences
http://dx.doi.org/10.1155/S0161171286000935
9
(4)
771 -
779
DOI : 10.1155/S0161171286000935
Vijayaraju P.
,
Sajath Z. M. I.
2011
“Common fixed points of single and multivalued maps in fuzzy metric spaces,”
Applied Mathematics
http://dx.doi.org/10.4236/am.2011.25079
2
(5)
595 -
599
DOI : 10.4236/am.2011.25079
Park J. H.
,
Park J. S.
,
Kwun Y. C.
2006
“A common fixed point theorem in the intuitionistic fuzzy metric space,”
Proceedings of the 2nd International Conference on Advances in Natural Computation (ICNC) and 3rd International Conference on Fuzzy Systems and Knowledge Discovery (FSKD)
Xian, China
293 -
300
Park J. S.
2011
“Some properties for the compatible mappings in intuitionistic fuzzy metric space,”
Far East Journal of Mathematical Sciences
50
(1)
79 -
86
Park J. S.
2009
“On a common fixed point for occasionally weakly semi-compatible hybrid mappings in an intuitionistic fuzzy metric space,”
JP Journal of Fixed Point Theory and Applications
4
(1)
1 -
10
Park J. S.
2012
“Fixed point theorems for weakly compatible functions using (JCLR) property in intuitionistic fuzzy metric space,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2012.12.4.296
12
(4)
296 -
299
DOI : 10.5391/IJFIS.2012.12.4.296
Park J. S.
2011
“Some common fixed point theorems using compatible maps in intuitionistic fuzzy metric space,”
International Journal of Fuzzy Logic and Intelligent Systems
http://dx.doi.org/10.5391/IJFIS.2011.11.2.108
11
(2)
108 -
112
DOI : 10.5391/IJFIS.2011.11.2.108