Available online through
www.ijma.info
UNIQUE FIXED POINT THEOREMS ON HAUSDORFF SPACES
1
Zaheer K. Ansari*,
2Rajesh Shrivastava and
3Arun Garg
1
Department of Applied Mathematics, JSS Academy of Technical education, C – 20/1, Sector 62, Noida – 201301, U.P. (India)
E-mail: [email protected]
2
Department of Mathematics, Govt. Science & Commerce College Benazir, Bhopal, M.P. (India)
E-mail: [email protected]
3
Department of Mathematics, Institute of Technology & Management, ITM Campus, N. H. – 75, Jhansi Road, Gwalior, M. P. – 474001 (India)
E-mail: [email protected]
(Received on: 11-06-11; Accepted on: 30-06-11)
--- ABSTRACT
I
n present paper, some fixed point theorems in Hausdorff spaces have been proved, which generalize the theorem of YEH, C.C. [2].X
y
x
y
x
d
Ty
Tx
d
(
,
)
<
(
,
);
∀
,
∈
.Keywords and Phrases: Result on Hausdorff spaces, fixed point theorem.
MSC (2000): 54H25, 28A78.
---
INTRODUCTION:
In 1962, M. Edelstein [1] proved the following result:
Theorem 1: Let (X,d) be a metric space, T be a contractive self-mapping of X. If for some
x
∈
X
, the sequence of iterates{
T
nx
}
has a convergent subsequence{
T
nix
}
converging to a point
x
0∈
X
, thenx
0=
lim
{
T
nx
}
is unique fixed point.Singh, S.P. and Zorzitto [3] generalized the result of M. Edelstein [1].
Here, we prove the following theorems, which generalized the result of YEH, C.C. [2]:
Theorem 2: Let T be a continuous mapping of a Housdorff space X into itself and let f be a continuous mapping of X×X into non-negative reals such that
X
x
x
x
f
(
,
)
=
0
;
∀
∈
andf
(
x
,
y
)
≠
0
,
∀
x
,
y
∈
X
,
x
≠
y
(2.1))
,
(
)
,
(
)
,
(
x
z
f
x
y
f
y
z
f
≤
+
, for all x, y, z in X (2.2)≤
(
,
)
2
1
),
,
(
),
,
(
),
,
(
),
,
(
max
)
,
(
Tx
Ty
q
f
x
y
f
x
Tx
f
y
Ty
f
y
Tx
f
x
Ty
f
,)
,
(
)
,
(
)
,
(
y
x
f
Ty
y
f
Tx
x
f
,
)
,
(
)
,
(
)
,
(
y
x
f
Tx
y
f
Ty
x
f
,
)
,
(
)
,
(
)
,
(
)
,
(
Ty
Tx
f
y
x
f
Ty
x
f
y
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
Ty
Tx
f
y
x
f
Ty
Tx
f
Ty
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
Ty
Tx
f
y
x
f
Ty
x
f
Tx
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
Ty
Tx
f
y
x
f
Ty
x
f
Ty
y
f
+
,
+
+
)
,
(
)
,
(
2
)]
,
(
)
,
(
)
,
(
)
,
(
[
Ty
Tx
f
y
x
f
Ty
y
f
Ty
x
f
Ty
Tx
f
y
x
f
(2.3)
--- 1
July-2011, Page: 1040-1045
© 2011, IJMA. All Rights Reserved 1041
hold for all
x
,
y
∈
X
whereq
∈
[
0
,
1
)
. Then T has a unique fixed point in X.Proof: Let
x
0 be an arbitrary point in X and a sequence{
x
n}
defined byTx
n=
x
n+1, n a positive integer. If for some n,Tx
n=
x
n thenx
n is a fixed point. If notTx
n≠
x
n then by (2.3), we have.)
,
(
)
,
(
x
1x
2f
Tx
0Tx
1f
=
≤
)
,
(
)
,
(
)
,
(
),
,
(
2
1
)
,
(
),
,
(
),
,
(
),
,
(
max
1 0
2 1 1 0 2 0 1
1 2 1 1 0 1 0
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
q
,)
,
(
)
,
(
)
,
(
1 0
1 1 2 0
x
x
f
x
x
f
x
x
f
,
)
,
(
)
,
(
)
,
(
)
,
(
2 1 1
0
2 0 1 0
x
x
f
x
x
f
x
x
f
x
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
2 1 1
0
2 1 2 0
x
x
f
x
x
f
x
x
f
x
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
2 1 1
0
2 0 1 0
x
x
f
x
x
f
x
x
f
x
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
2 1 1
0
2 0 2 1
x
x
f
x
x
f
x
x
f
x
x
f
+
,+
+
)
,
(
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
2 1 1
0
2 1 2 0 2
1 1 0
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
+
≤
{
(
,
)
(
,
)},
(
,
),
0
,
(
,
)
2
1
,
0
),
,
(
)
,
(
),
,
(
max
f
x
0x
1f
x
0x
1f
x
1x
2f
x
0x
1f
x
1x
2f
x
1x
2f
x
0x
1q
,
f
(
x
1,
x
2),
f
(
x
0,
x
1),
f
(
x
1,
x
2),
f
(
x
1,
x
2)
}
i.e.
≤
[
(
,
)
+
(
,
)]
2
1
)
,
(
),
,
(
max
f
x
0x
1f
x
1x
2f
x
0x
1f
x
1x
2q
Then we have either
f
(
x
1,
x
2)
<
qf
(
x
1,
x
2)
which is a contradiction as q<1or
f
(
x
1,
x
2)
<
qf
(
x
0,
x
1)
<
f
(
x
0,
x
1)
as q<1and
(
,
)
(
,
)
2
)
,
(
1 2f
x
0x
1qf
x
0x
1q
q
x
x
f
≤
−
≤
<
f
(
x
0,
x
1)
as q<1 Proceeding in the same manner we have...
)
,
(
)
,
(
)
,
(
x
0x
1>
f
x
1x
2>
f
x
2x
3>
f
Thus we have a monotone sequence of positive reals which converges with all it subsequences to some
x
∈
X
, x being real.Again
{
x
n}
has a convergent subsequence{
}
k n
x
in X which converges to somex
∈
X
. From the continuity of T,we have
{
x
n}
Tx
nT
Tx
=
lim
=
lim
=
lim
(
x
n+1)
{
1}
2
lim
)
(
)
(
x
=
T
Tx
=
T
x
nk+T
=
lim
{
Tx
nk+1}
=
x
n+2Now we are to prove that x is a fixed point for T.
(
lim
,
lim
1)
)
,
(
x
Tx
=
f
x
nx
n+f
July-2011, Page: 1040-1045
Again by (2.3) we have
≤
(
,
)
2
1
),
,
(
),
,
(
),
,
(
),
,
(
max
)
,
(
Tx
T
2x
q
f
x
Tx
f
x
Tx
f
Tx
T
2x
f
Tx
Tx
f
x
T
2x
f
,,
)
,
(
)
,
(
)
,
(
2Tx
x
f
x
T
Tx
f
Tx
x
f
)
,
(
)
,
(
)
,
(
2Tx
x
f
Tx
Tx
f
x
T
x
f
,)
,
(
)
,
(
)
,
(
)
,
(
2 2x
T
Tx
f
Tx
x
f
x
T
x
f
Tx
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
2 2 2x
T
Tx
f
Tx
x
f
x
T
Tx
f
x
T
x
f
+
, ,)
,
(
)
,
(
)
,
(
)
,
(
2 2 2x
T
Tx
f
Tx
x
f
x
T
x
f
x
T
Tx
f
+
,+
+
)
,
(
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
2 2 2 2x
T
Tx
f
Tx
x
f
x
T
Tx
f
x
T
x
f
x
T
Tx
f
Tx
x
f
≤
{
(
,
)
+
(
,
)
}
2
1
,
0
),
,
(
),
,
(
),
,
(
max
f
x
Tx
f
x
Tx
f
Tx
T
2x
f
x
Tx
f
Tx
T
2x
q
,
f
(
Tx
,
T
2x
),
0
,
f
(
x
,
Tx
),
f
(
Tx
,
T
2x
),
f
(
x
,
Tx
),
f
(
Tx
,
T
2x
),
f
(
Tx
,
T
2x
)
}
≤
[
(
,
)
+
(
,
)]
2
1
),
,
(
),
,
(
max
f
x
Tx
f
Tx
T
2x
f
x
Tx
f
Tx
T
2x
q
which gives)
,
(
)
,
(
Tx
T
2x
q
f
x
Tx
f
≤
≤
f
(
x
,
Tx
)
as q<1So we arrive at a contradiction hence
x
=
Tx
i.e. x is a fixed point for T.Uniqueness: Let y (
≠
x
) be another fixed point. Then by (2.3) we have≤
(
,
)
2
1
),
,
(
),
,
(
),
,
(
),
,
(
max
)
,
(
x
y
q
f
x
y
f
x
x
f
y
y
f
y
x
f
x
y
f
)
,
(
)
,
(
)
,
(
y
x
f
y
y
f
x
x
f
,)
,
(
)
,
(
)
,
(
y
x
f
x
y
f
y
x
f
,)
,
(
)
,
(
)
,
(
)
,
(
y
x
f
y
x
f
y
x
f
y
x
f
+
,(
,
)
(
,
)
)
,
(
),
,
(
y
x
f
y
x
f
y
x
f
y
x
f
,)
,
(
)
,
(
)
,
(
)
,
(
y
x
f
y
x
f
y
x
f
y
x
f
+
,(
,
)
(
,
)
)
,
(
)
,
(
y
x
f
y
x
f
y
x
f
y
y
f
+
,+
+
)
,
(
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
y
x
f
y
x
f
y
y
f
y
x
f
y
x
f
y
x
f
)
,
(
)
,
(
)
,
(
x
y
q
f
x
y
f
x
y
f
≤
<
which is a contradiction as q<1Hence x = y i.e. x is a unique fixed point of T.
This completes the proof of the theorem.
Theorem 3: Let
T
1 andT
2 be two commuting continuous mapping of a Hausdorff space X into it self and let f be a symmetric continuous mapping of X×X into non-negative reals such thatX
x
x
x
f
(
,
)
=
0
∀
∈
andf
(
x
,
y
)
≠
0
∀
x
,
y
∈
X
,
x
≠
y
(3.1))
,
(
)
,
(
)
,
(
x
z
f
x
y
f
y
z
f
≤
+
(3.2)≤
(
,
),
(
,
),
(
,
)
2
1
)
,
(
),
,
(
max
)
,
(
T
1x
T
2y
q
f
x
y
f
y
T
1x
f
x
T
2y
f
y
T
2y
f
y
T
1x
f
,)
,
(
)
,
(
)
,
(
1 2July-2011, Page: 1040-1045
© 2011, IJMA. All Rights Reserved 1043
hold for all
x
,
y
∈
X
whereq
∈
[
0
,
1
)
. If for somex
0∈
X
, the sequence{
x
n}
whereT
1x
2n=
x
2n+1 and 22 1 2
2
x
n+=
x
n+T
for n = 0, 1, 2, 3, ... has a convergent. Subsequence of the type{
x
(2p+1)n}
wherep
∈
N
, is fixed andn
∈
N
, thenT
1 andT
2 have a unique fixed point.Proof: Using (3.3) we have
)
,
(
)
(
x
1,x
2f
T
1x
0T
2x
1f
=
(
,
)
,
2
1
),
,
(
),
,
(
),
,
(
),
,
(
max
0 1 1 2 1 2 1 1 0 2≤
q
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
2 1 1 0 2 0 1 0 1 0 2 0 1 1 1 0 2 1 1 0x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
+
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
)
,
(
2 1 1 0 2 0 2 1 2 1 1 0 2 0 1 0 2 1 1 0 2 0 2 1x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
+
+
+
+
+
)
,
(
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
2 1 1 0 2 1 2 0 2 1 1 0x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
x
x
f
≤
{
(
,
)
+
(
,
)
}
2
1
,
0
),
,
(
),
,
(
max
f
x
0x
1f
x
1x
2f
x
0x
1f
x
1x
2q
Then we have either
f
(
x
1,
x
2)
≤
q
f
(
x
1,
x
2)
which is a contradiction as q<1.or
f
(
x
1,
x
2)
≤
qf
(
x
0,
x
1)
as q<1. and(
,
)
2
)
,
(
1 2f
x
0x
1q
q
x
x
f
−
≤
as q<1.Repeating this argument we have
...
)
,
(
)
,
(
)
,
(
x
0x
1>
f
x
1x
2>
f
x
2x
3>
f
Thus, the sequence
{
f
(
x
n,
x
n+1)
}
is converging to some z. Again sequence{
x
n}
has a subsequence{
x
(2p+1)2n}
.From the continuity of
T
1 andT
2, we haveT
1x
=
T
1lim
x
(2p+1)2n=
lim
x
(2p+1)2n+12 2 ) 1 2 ( 1 2 ) 1 2 ( 2 1
2
T
x
=
T
lim
x
p+ n+=
lim
x
p+ n+T
Now, we have to show that x is a fixed point for
T
1 andT
2, we get(
(2 1)2 (2 1)2 1)
1
)
lim
,
,
(
x
T
x
=
f
x
p+ nx
p+ n+f
=
f
(
lim
x
(2p+1)2n,
x
(2p+1)2n+1)
=
f
(
lim
x
(2p+1)2n+1,
x
(2p+1)2n+2)
=
f
(
T
1x
,
T
2T
1x
)
Suppose
x
≠
T
1x
then by (3.1), we have,
)
,
(
2
1
),
,
(
)
,
(
),
,
(
),
,
(
max
)
,
(
T
1x
T
2T
1x
≤
q
f
x
T
1x
f
x
T
1x
f
T
1x
T
2T
1x
f
T
1x
T
1x
f
x
T
2T
1x
f
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
1 2 1 1 1 2 1 2 1 1 1 1 1 2 1 1 2 1 1x
T
T
x
T
f
x
T
x
f
x
T
T
x
f
x
T
T
x
T
f
x
T
x
f
x
T
x
T
f
x
T
T
x
f
x
T
x
f
x
T
T
x
T
f
x
T
x
f
+
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
)
,
(
1 2 1 1 1 2 1 1 2 1 2 1 1 1 2 1 1 2 1 1 1 2 1x
T
T
x
T
f
x
T
x
f
x
T
T
x
T
f
x
T
T
x
f
x
T
T
x
T
f
x
T
x
f
x
T
T
x
f
x
T
x
f
x
T
T
x
T
f
x
T
x
f
x
T
T
x
f
x
T
x
f
+
+
+
+
+
)
,
(
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
1 2 1 1 1 2 1 1 2 1 2 1 1x
T
T
x
T
f
x
T
x
f
x
T
T
x
T
f
x
T
T
x
f
x
T
T
x
T
f
x
T
x
f
Implies)
,
(
)
,
(
)
,
(
T
1x
T
2T
1x
qf
x
T
1x
f
x
T
1x
July-2011, Page: 1040-1045
Thus
x
=
T
1x
, similarly if{
x
(2p+1)2n+1}
be subsequence of{
x
(2p+1)n}
then we can get easilyx
=
T
2x
. Therefore x is a fixed point forT
1 andT
2.Uniqueness: Let
y
(
≠
x
)
be another fixed point ofT
1 andT
2. By applying (3.3) we can prove thatf
(
T
1x
,
T
2y
)
≤
f
(
x
,
y
)
, a contradiction as q<1.Hence x is a unique fixed point of
T
1 andT
2.This completes the proof of the theorem.
Theorem 4: Let
T
1,
T
2,
T
3,...
T
n be continuous mappings of a Hausdorff space X into itself and let f be a continuous mapping of X x X into the non-negative reals such thatX
x
x
x
f
(
,
)
=
0
∀
∈
andf
(
x
,
y
)
≠
0
∀
x
,
y
∈
X
,
x
≠
y
(4.1))
,
(
)
,
(
)
,
(
x
z
f
x
y
f
y
z
f
≤
+
(4.2)),
,
(
2
1
),
,
(
),
,
(
),
,
(
),
,
(
max
)
,
(
T
x
T
1y
q
f
x
y
f
x
T
x
f
x
T
1y
f
y
T
x
f
x
T
1y
f
r r+≤
r r+ r r+
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
1 1 1
1
x
T
x
T
f
y
x
f
y
T
x
f
y
x
f
y
x
f
y
T
x
f
x
T
y
f
y
x
f
y
T
y
f
x
T
x
f
r r
r r
r r
r
+ + +
+
+
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
)
,
(
,
)
,
(
)
,
(
)
,
(
)
,
(
1 1 1
1 1 1
1 1
y
T
T
f
y
x
f
y
T
y
f
y
T
x
f
y
T
x
T
f
y
x
f
y
T
x
f
y
T
x
T
f
y
T
x
T
f
y
x
f
y
T
x
f
x
T
x
f
r r
r r
r r
r r
r
r r
r r
+ + +
+ + +
+ +
+
+
+
+
+
+
+ +
+
)
,
(
)
,
(
2
)
,
(
)
,
(
)
,
(
)
,
(
1
1 1
1
y
T
x
T
f
y
x
f
y
T
y
f
y
T
x
f
y
T
x
T
f
y
x
f
r r
r r
r
r (4.3)
hold for all distinct
x
,
y
∈
X
whereq
∈
[
0
,
1
)
andT
kx
=
x
k+1.If for some
x
0∈
X
, the sequence{
x
n}
where1 2
3 3 1 2 2 0 1
1
=
T
x
,
x
=
T
x
,
x
=
T
x
,...
...
...
x
k=
T
kx
r−x
... ...
1 2 2
1 2 2 1
1
,
+ +,...
...
...
.
−+
=
k k=
k k=
k kk
T
x
x
T
x
x
T
x
x
1 ) 1 ( ) 1 ( 1
2 2 1
1
,
+ +...
...
+ + −+
=
n n=
n n=
k n knk
T
x
kx
kT
x
kx
k kT
x
kx
for n = 0, 1, 2, 3, ... has a convergent subsequence of the type
{
x
mk+1}
wherem
∈
N
is fixed andn
∈
N
. Thenk
T
T
T
T
1,
2,
3,...
has a unique fixed point.Proof: Using (4.3) we have
[
+
]
≤
=
(
,
)
(
,
)
2
1
,
0
),
,
(
),
,
(
),
,
(
max
)
,
(
)
,
(
x
1x
2f
T
1x
0T
2x
1q
f
x
0x
1f
x
0x
1f
x
1x
2f
x
0x
1f
x
1x
2f
f
(
x
1,
x
2),
0
,
f
(
x
0,
x
1),
f
(
x
1,
x
2),
f
(
x
0,
x
1),
f
(
x
1,
x
2)
f
(
x
1,
x
2)
≤
[
(
,
)
+
(
,
)
]
2
1
),
,
(
,
,
(
max
f
x
0x
1f
x
1x
2f
x
0x
1f
x
1x
2q
July-2011, Page: 1040-1045
© 2011, IJMA. All Rights Reserved 1045
Analogously
)
,
(
)
,
(
x
2x
3f
x
1x
2f
<
...f
(
x
k−1,
x
k)
<
f
(
x
k−2,
x
k−1)
By (4.3) we have)
,
(
)
,
(
x
kx
k 1f
T
kx
k 1T
kx
kf
+=
−≤
− +[
(
−,
)
+
(
,
+)
]
2
1
)
,
(
)
,
(
max
f
x
k 1x
kf
x
kx
k 1f
x
k 1x
kf
x
kx
k 1q
implies
)
,
(
)
,
(
)
,
(
x
kx
k 1qf
x
k 1x
kf
x
k 1x
kf
+≤
−<
− as q<1Hence
)
,
(
)
,
(
x
nx
n1f
x
n1x
nf
+<
− , n = 0, 1, 2, 3,…The uniqueness of
T
1,
T
2,
T
3,....
T
k is similar as of theorem-3.This completes the proof of theorem.
REFERENCE:
[1] Edelstein, M : On fixed and periodic points under contractive mappings. J. London Math Soc. 37, 74 – 79 (1962).
[2] Yeh, C.C. : Some fixed point theorems. Boll Greek Math Soc. 21 (1980), 47–58.
[3] Singh, S.P. and Zorzitto : On fixed point theorem in metric spaces. Ext. Des Annales De Lu Soc. Sci. Bruzelles, 85 (1971), 117 – 123.
