On Common Fixed Point for Single and Set-Valued Maps Satisfying OWC Property in IFMS using Implicit Relation

International Journal of Fuzzy Logic and Intelligent Systems.
2015.
Jun,
15(2):
132-136

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 : January 27, 2015
- Accepted : May 27, 2015
- Published : June 30, 2015

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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.
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, ∞),
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.
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
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,
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.
Conflict of Interest
No potential conflict of interest relevant to this article was reported.
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

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
- (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,

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3. Main Results

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- δM(Su,Tv,t) = 1 andδN(Su,Tv,t) = 0.

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- δM(SAu,Tv,t) = 1 andδN(SAu,Tv,t) = 0,

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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.
Acknowledgements

This author is supported by Chinju National University of Education Research Fund in 2014.

BIO

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Citing 'On Common Fixed Point for Single and Set-Valued Maps Satisfying OWC Property in IFMS using Implicit Relation
'

@article{ E1FLA5_2015_v15n2_132}
,title={On Common Fixed Point for Single and Set-Valued Maps Satisfying OWC Property in IFMS using Implicit Relation}
,volume={2}
, url={http://dx.doi.org/10.5391/IJFIS.2015.15.2.132}, DOI={10.5391/IJFIS.2015.15.2.132}
, number= {2}
, journal={International Journal of Fuzzy Logic and Intelligent Systems}
, publisher={Korean Institute of Intelligent Systems}
, author={Park, Jong Seo}
, year={2015}
, month={Jun}