Fuzzy Fixed Points of Contractive Fuzzy Mappings
Akbar Azam,1*, Muhammad Arshad2
1
Department of Mathematics, COMSATS Institute of Information Technology,ChakShahzad, 44000, Islamabad, Pakistan,
[email protected] & akbar_azam @comsats.edu.pk,
2
Department of Mathematics, Address International Islamic University,H-10, Islamabad, 44000, Pakistan,
ICM 2012, 11-14 March, Al Ain
ABSTRACT
We prove the existence of fuzzy fixed points of a general class of
fuzzy mappings satisfying a contractive condition on a metric space
with the Hausdorff metric on the family of fuzzy sets and apply it to
obtain fuzzy fixed points of fuzzy locally contractive mappings.
Keywords: Fuzzy fixed point; contractive type mappings; fuzzy set;
fuzzy mapping.
E Ad, B  { : A  N d ( , B), B  N d ( , A)}, where
d ( x, A)  inf{ d ( x, y ) : y  A} .
The Hausdorff metric d H on
as
1
INTRODUCTION
CB( X )  { A : A is nonempty closed and bounded
subset of X },
C ( X )  { A : A is nonempty compact subset of X }
A, B  CB( X )
E Ad, B
and
  0 the sets N d ( , A)
x, y  X
, an
x0 , x1 , x2 ,, xn
d ( x j , x j 1 )  
 -chain from x to
such that
for all
and
x  x0 , x n  y
and
j  0.1,2, , n  1 .
A fuzzy set in X is a function with domain X and values in
. If A is a fuzzy set and x  X , then the function values
[0 , 1]
A(x )
is
called the membership grade of x in A. The  -level set of A, denoted by

A , and is defined by

A  {x : A( x)   } if   ( 0 , 1 ] ,
0
A  {x : A( x)  0} .
B
Here
denotes the closure of the set B.
A fuzzy set A in a metric linear space X is said to be an ap
A is compact and convex in
for each   [0 , 1] and sup A( x)  1 .The family of all
proximate quantity if and only if
X
approximate quantities in a metric linear space X is denoted by
W(X) . We denote the fuzzy set
stated, where
A
 {x}
by {x} unless and until it is
is the characteristic function of the crisp set
A.
are defined as follows:
N d ( , A)  {x  X : d ( x, a)   for some a  A}
Let F(X) be the collection of all fuzzy sets in a metric space
X and

E( X )  A  F ( X ) :
*
y is a finite set of points
xX
.
For
induced by d is defined
d H ( A, B)  inf E Ad, B .
For
Heilpern [16] first introduced the concept of fuzzy mappings and
established a fixed point theorem for fuzzy contraction mappings.
Afterwards many researcher [e.g.,see 1, 2, 3, 4, 9, 10, 21, 22, 23,
24 and reference therein] extended the result of Heilpern and studied fixed point theorems for fuzzy generalized contractive mappings. Recently in [1, 2] the authors obtained Heilpern fixed points
of fuzzy contractive and fuzzy locally contractive mappings on a
compact metric space with the d -metric for fuzzy sets . In [4]
the authors studied fixed point theorems of a wider class of fuzzy
mappings and obtained some d -metric fixed point results of the
literature as corollaries.
In the present paper we prove theorems concerning common fixed
points of the same wider class [4] of fuzzy contractive and fuzzy
locally contractive mappings and obtain some d -metric fixed
point results of [2] as corollaries. Our results also generalize/fuzzify several other known results [e.g., see 7, 13, 16, 18, 25].
Let ( X , d ) be a metric space and
CB ( X )


A  CB( X ),    [0,1] .
Akbar Azam (corresponding Author)
1
Azam A. and Arshad M.
A  F( X ) :

EC ( X )   
.
A  C ( X ),    [0,1]

Lemma 1.1 (Nadler [25]) Let
A, B  CB( X )
For A , B  F ( X ) , A  B means
  [0,1] such that
A, B  CB( X ) then define
P ( A, B)   inf  d ( x, y ),
A, y

B
then define

7, 18, 20, 25) to locally contractive fuzzy mappings and obtained a
fuzzy fixed points for such mappings.

X
is another metric on
P  ( A, B) 
x

inf
A, y
rems for locally contractive mappings. We extend the concept of
locally contractive mappings of Edelstein [12,13] (see also 1, 3, 6,
D( A, B)  sup D ( A, B).
*
be a metric space and
mappings. Section 3 deals with the study of fuzzy fixed point theo-

d
.
In section 2 we extend Edelstein fixed point theorem to fuzzy

A,  B  CB( X ) for each   [0,1]
P( A, B)  sup P ( A, B),
If
(X ,d)
d ( a, b)  
d (a, B)  d H ( A, B) .
D ( A, B)  d H ( A, B).
If
such that
a  A,
A, B  CB( X ) , then for each a  A ,

x
bB
Lemma 1.2 (Nadler [25]) Let
If there exists an

be a metric space and
with d H ( A, B)   , then for each
there exists an element
A( x)  B( x) for each x  X
(X ,d)
then
*

d ( x, y ),
2
B
D * ( A, B)  d * H ( A, B).
FIXED POINTS OF FUZZY CONTRACTIVE
MAPS
One very pretty and significant fixed point theorem, originally due
d  : E ( X )  E ( X )  R,
Hausdorff metric d H ) as
d  ( A, B)  sup D ( A, B).
Now define
(induced by the
(X , d)
d
(CB( X ), d H )
and
( E( X ), d  )
( X , d )  (CB( X ), d H )  ( E( X ), d  ),
are isometrics embeddings by means
A  A
x  {x}
(crisp set) and
respectively.
called fuzzy mapping if T is a mapping from X into F(Y). A fuzzy
mapping T is a fuzzy subset on
T (x ) . A point x  X
point of a fuzzy mapping T if
2
X Y
with membership func-
T ( x )( y ) . The function T ( x )( y )
ship of y in
is a contractive mapping (i.e.
for each
x, y  X
). Then there exists
ther studied/extended by Daffer and Kaneko[11], Hu and Rosen
[18]. Beg [5] proved random analogue of this result and obtained
random fixed points of contractive random mappings. Recently
Grabiec [15] and Mihet [24] extended this result to fuzzy metric
spaces. In the following theorem, we extend the above result to a
general class of fuzzy mappings.
Theorem 2.1 Let ( X , d ) be a compact metric space and
Let X be an arbitrary set, Y be a metric space. A mapping T is
tion
is a compact metric space and
a unique fixed point of T. Edelstein fixed point theorem was fur-
is a metric on E ( X ) and the completeness of
implies that
are complete. Moreover
T:X X
d (Tx, Ty  d ( x, y )

We note that
(X ,d)
to Edelstein [13] is that if
is the grade of member-
is said to be fuzzy fixed
{x}  T ( x) .
T : X  F(X )
x X
be a fuzzy mapping such that for each
there exists  ( x )  (0,1] such that
nonempty, compact and for each
*
X
T ( x)
x, y  X , x  y
d H ( ( x ) T ( x), ( y ) T ( y))  d ( x, y)
Then there exists x
 ( x)
*
such that x
.
  ( z )T ( x* )
*
.
is
Proof. For each
x  X , pick  ( x)  (0,1] such that
nonempty, compact and
 ( x)
T ( x) is
1
2
define a real valued function

T ( x)  t  X : T ( x)t  

1  x
  0, .
2   3 
Thus all
1
conditions of Theorem 2.1 are satisfied to obtain 0 2 T (0),
while previously known result [4, Theorem 2.1] is not applicable to
obtain it.
g : X  R by g ( x)  d ( x, ( x ) T ( x)) .
It follows that
g ( x)  d ( x, ( x ) T ( x))  d ( x, y)  d ( y, ( x ) T ( x))
 d ( x, y)  d ( y,
 ( y)
T ( y))  d H (
 ( x)
T ( x),
 ( y)
Corollary 2.2 Let ( X , d ) be a compact metric space and
T ( y)) .
 d ( x, y)  g ( y)  d H ( ( x ) T ( x), ( y ) T ( y))
g ( x)  g ( y)  d ( x, y)  d H ( ( x ) T ( x), ( y ) T ( y)) .
g ( x)  g ( y)  d ( x, y)  d H ( ( x)T ( x), ( y )T ( y)) .
is continuous. By
compactness, this function attains a minimum, say at
compactness of
such that
x 
*
 ( x*)
T ( x* ) ,
we can choose
*  ( x*)
d ( x , x1 )  d ( x ,
*
 ( z *)
*
T (x )
d  (T ( x), T ( y))  d ( x, y)
Then there exists x
By symmetry, we obtained
g ( x)  d ( x, ( x ) T ( x))
x
*
. Now, by
Proof. Let
*
X
*
.
such that {x}  T ( x
x  X , by hypothesis 1T ( x)
*
)
is nonempty compact
subset of X for each x. Thus
d H (1T ( x),1T ( y ))  D1 (T ( x), T ( y ))
 d  (T ( x), T ( y ))  d ( x, y ).
x1   ( x*)T ( x * )
T ( x )  g ( x ) . Then
*
be a fuzzy mapping such that for each
x, y  X , x  y
That is
It follows that
T : X  EC ( X )
Apply theorem 3.1 to obtain
hence {x}  T ( x
*
)
x*  X
such that x
*
1 T ( x * ) ,
.
, otherwise,
g ( x1 )  d ( x1, ( x1 )T ( x1 ))  dH ( ( x )T ( x* ), ( x1 )T ( x1 ))
 d ( x * , x1 )  d ( x * , ( x*)T ( x * )  g ( x * ) .
*
Which is a contradiction to the minimality of
g ( x) at x * . It
completes the proof.
Example 2.2 Let X
er
x, y  X
and
3
FUZZY LOCALLY CONTRACTIVE MAPS
In this section we established fuzzy fixed point theorem
for locally contractive fuzzy mappings. The following lemma is
recorded from [27].
Lemma 3.1 [27] Let ( X , d ) be a compact connected metric
 [0, ), d ( x, y)  x  y
A : (0, )  F ( X )
, whenev-
be defined as
follows:
1 if 0  t  8x
1
if 8x  t  4x

A( x)(t )   12
x
 3 if 4  t  x
0 if x  t  .
Now, define T : X  F ( X ) as follows:
 {0} if x  0
T ( x)  
 A( x) if x  0.
1
Then, if x  0, T ( x)  [0, 8x ), which is not compact and
space. Then for each
 0
and
chain from x to y and the mapping
x, y  X
there exists an
-
d : X  X  R
defined by
 n-1
 d ( x j , x j 1 ) : x0 , x1 , x2 ,, xn
d  ( x, y )  inf  j 0
is an  - chain from x to y






3
Azam A. and Arshad M.
is a metric on X equivalent to d. Furthermore, for
x, y  X
and
  0 there exists an  -chain x  x0 , x1 , x2 ,, xn  y
dH (
n -1
d  ( x, y)   d ( x j , x j 1 )
such that
be a fuzzy mapping such that the follow-
ing conditions are satisfied:
( i ) For each
 ( x)
x X
T ( x j ),
 ( x j 1 )
d ( x j , x j 1 )  d H (
Theorem 3.2 Let ( X , d ) be a compact connected metric space
T : X  F(X )
(xj )
belongs to an open set U such that for each
y , z  U , y  z.
equivalent to
d
d H* ( ( x ) T ( x), ( y ) T ( y))  d ( x, y) and there exists
x X
such that x

(z )
p, q  X
 (xj )
T ( x j ),
 ( x j 1 )
0
d ( y0 , y1 )  d ( x0 , x1 ) 
such that
y2
d ( y1 , y 2 )  d ( x1 , x2 ) 
and each pair of
produce a set of points
p to q . Next use compact-
  0 such that if x  y and d ( x, y )  
(xj )
y j
T (x j )
M1
2
, then
Now let
d d

2
that is for
Obviously
p, q  X

 d ( x j , x j 1 ) : x0 , x1 , x2 ,, xn
*
d ( p, q)  inf  j 0
is an  - chain from p to q
2

d * is a metric on X

there exists a
2
-chain
equivalent to
d * ( p, q)   d ( x j , x j 1 ).
j 0
4
Now,





.
d and
p  x0 , x1 , x2 ,, xn  q from
n -1
q such that
.
M0
2
.
such that
. Continuing in this fashion we
where
such that
y0 , y1 , y2 , , yn is a
M j 1
2

2
for j  0.1,2, , n  1.
- chain from
y 0 to y n
.Thus
n -1
By lemma 3.1
T ( x2 )
y0 , y1 , y 2 ,, y n
d ( y j 1 , y j )  d ( x j 1 , x j ) 
d H ( ( x ) T ( x), ( y ) T ( y))  d ( x, y) .
*
Mj
2
y 0  ( x0 ) T ( x0 ) .In the view of
ness of X
to find
T ( x j 1 ))
For j  0,1,2,..., n  1 (2)
Similarly, we may choose
 -chain
p  x0 , x1 , x2 ,, xn  q from
 ( x j 1 )
and
 ( x2 )
T (x ) .
there exists an
T ( x j ),
T ( x j 1 ))  d ( x j , x j 1 ) 
*
Proof. First, by Lemma 3.1 for each 
points
dH (
y1 ( x1 ) T ( x1 )
*
T ( x j 1 ))  0 . As-
such that
x, y  X
*
 ( x j 1 )
inequality (2) along with Lemm1.2 we may choose
d * for X
*
 (xj )
Mj 0
Consider an arbitrary element
d H ( ( y ) T ( y), ( z ) T ( z ))  d ( y, z ) .
for each
T ( x j ),
For j  0,1,2,..., n  1
there exists  ( x )  (0,1] such that
Then there is a new metric
 (xj )
M J  d ( x j , x j 1 )  d H (
is nonempty , compact and
x of X
( ii ) each
T ( x j 1 ))  d ( x j , x j 1 )   .
sume that
It further implies that
T ( x)
implies that
It follows that
.
j 0
and
d ( x j , x j1 )   2  
p to
 n -1
 d ( x j , x j 1 ) : x0 , x1 , x2 ,, xn
d * ( y 0 , y n )  inf  j 0
is an  - chain from y to y
2
0
n

n -1
  d ( y j , y j 1 )
j 0
n 1
Mj

   d ( x j , x j 1 ) 
2
j 0 

.






Corollary 3.3 Let ( X , d ) be a compact connected metric space
n -1
d * ( p, q)   d ( x j , x j 1 ).
j 0
Since
n 1 M
 j
d * ( y 0 , y n )  d * ( p, q)   
j 0  2
Assume that

.

n 1 M
 j
k  d * ( p, q)   
j 0  2

y 0  N d k ,  ( xn ) T ( x n )
*
 ( x0 )
T ( x0 )  N
d*
and
Therefore,
k ,
set U such that for each

 ,

then
k  0 and

(3)
M n1
2
*

 ( xn )
T ( xn )  N
d*
k ,
Let
(X ,d)
{x * }  T ( x * ) .
be a compact metric space and
S : X  C( X )
be a set valued mapping such that either for each
or X is connected and each

2
such that for each
- chain
x, y  X
,

  k.

x of X
belongs to an open set U
y , z  U , y  z.
d H (S ( y), S ( z))  d ( y, z).
where,
Thus
Then there exists x
*
X
such that x  S ( x
*
)
.
Proof. Consider a fuzzy mapping T : X  F ( X ) defined by
as follows:

109
T ( x)(t )   1
10
T ( x0 ) (4) .
In the view of inequalities (3) and (4), it follows that
k  E d( x0 )T ( x
such that
x y
such that
 , hence
 ( x0 )
X
d H (S ( x), S ( y))  d ( x, y)
n1 M
 j
d * ( z0 , z n )  d * ( p, q)   
j 0  2
z n  N d k ,  ( x0 ) T ( x 0 )
*
.
Then by the same procedure we obtain a
z 0 to z n
y , z  U , y  z.
Theorem 3.5
z n  ( xn ) T ( x n ) . Again in
z n 1 ( xn 1 ) T ( x n 1 )
from
belongs to an open
d  (T ( y), T ( z))  d ( y, z)
Then there exists x
the view of inequality (2) along with Lemma1.2, we
z 0 , z1 , z 2 ,, z n
x of X
sion of Edelstein Theorems.
T ( xn ) .
d ( z n1 , z n )  d ( x0 , x1 ) 
each
Here by providing following theorem, we achieve set-valued ver-
Now consider an arbitrary element
may choose
be a fuzzy mapping such that the fol-
lowing condition is satisfied:
.Hence
 ( xn )
T : X  EC ( X )
t  S ( x)
t  S ( x).
*
 ( xn )
0 ),
T ( xn )
Then
. Thus
d H* (  ( x0 ) T ( x0 ),  ( xn ) T ( x n ))  k
9
10
exists x
.
T ( x)  S ( x)
*
X
hence by Theorem 2.1 and 3.2 there
such that x
*
9
 10T ( x * )  S ( x * )
.
It further implies that
n1 M
 j
d H* ( ( p )T ( p), ( q ) T (q)  d * ( p, q)   
j 0  2
Hence for all x, y,
d H* ( ( p )T ( x), ( q ) T ( y)  d * ( x, y) . Now
by Theorem 3.1 there exists x
x*   ( z )T ( x* ) .
*



*
X
ACKNOWLEDGEMENTS
The authors would like to thank HEC, Pakistan, for providing travel grant to attend the conference.
The authors also would like to thank Professor Muhammed Syam,
the organizer of ICM-2012, for his generous and kind cooperation.
such that
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