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On Coincidence and Fixed-Point Theorems in Symmetric Spaces
Fixed Point Theory and Applications volume 2008, Article number: 562130 (2008)
Abstract
We give an axiom (C.C) in symmetric spaces and investigate the relationships between (C.C) and axioms (W3), (W4), and (H.E). We give some results on coinsidence and fixed-point theorems in symmetric spaces, and also, we give some examples for the results of Imdad et al. (2006).
1. Introduction
In [1], the author introduced the notion of compatible mappings in metric spaces and proved some fixed-point theorems. This concept of compatible mappings was frequently used to show the existence of common fixed points. However, the study of the existence of common fixed points for noncompatible mappings is, also, very interesting. In [2], the author initially proved some common fixed-point theorems for noncompatible mappings. In [3], the authors gave a notion (E-A) which generalizes the concept of noncompatible mappings in metric spaces, and they proved some common fixed-point theorems for noncompatible mappings under strict contractive conditions. In [4], the authors proved some common fixed-point theorems for strict contractive noncompatible mappings in metric spaces. Recently, in [5] the authors extended the results of [3, 4] to symmetric(semimetric) spaces under tight conditions. In [6], the author gave a common fixed-point theorem for noncompatible self-mappings in a symmetric spaces under a contractive condition of integral type.
In this paper, we give some common fixed-point theorems in symmetric(semimetric) spaces and give counterexamples for the results of Imdad et al. [5].
In order to obtain common fixed-point theorems in symmetric spaces, some axioms are needed. In [5], the authors assumed axiom (W3), and in [6] the author assumed axioms (W3), (W4), and (H.E); see Section 2 for definitions.
We give another axiom for symmetric spaces and study their relationships in Section 2. We give common fixed-point theorems of four mappings in symmetric spaces and give some examples which justifies the necessity of axioms in Section 3.
2. Axioms on Symmetric Spaces
A symmetric on a set is a function
satisfying the following conditions:
-
(i)
if and only if
for
,
-
(ii)
for all
Let be a symmetric on a set
. For
and
, let
. A topology
on
defined as follows:
if and only if for each
, there exists an
such that
. A subset
of
is a neighbourhood of
if there exists
such that
. A symmetric
is a semimetric if for each
and each
,
is a neighbourhood of
in the topology
.
A symmetric (resp., semimetric) space is a topological space whose topology
on
is induced by symmetric(resp., semi-metric)
.
The difference of a symmetric and a metric comes from the triangle inequality. Actually a symmetric space need not be Hausdorff. In order to obtain fixed-point theorems on a symmetric space, we need some additional axioms. The following axioms can be found in [7].
-
(W3)
for a sequence
in
,
and
imply
.
-
(W4)
for sequences
in
and
,
and
imply
Also the following axiom can be found in [6].
(H.E) for sequences in
and
,
and
imply
.
Now, we add a new axiom which is related to the continuity of the symmetric .
(C.C) for sequences in
and
,
implies
.
Note that if is a metric, then (W3), (W4), (H.E), and (C.C) are automatically satisfied. And if
is Hausdorff, then (W3) is satisfied.
Proposition 2.1.
For axioms in symmetric space , one has
-
(1)
(W4)
(W3),
-
(2)
(C.C)
(W3).
Proof.
Let be a sequence in
and
with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_IEq66_HTML.gif)
-
(1)
By putting
for each
, we have
By (W4), we have
.
-
(2)
By (C.C),
implies
The following examples show that other relationships in Proposition 2.1 do not hold.
Example 2.2.
(W4) (H.E) and (W4)
(C.C) and so (W3)
(H.E) and (W3)
(C.C) by Proposition 2.1 (1).
Let and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ1_HTML.gif)
Then, is a symmetric space which satisfies (W4) but does not satisfy (H.E) for
. Also
does not satisfy (C.C).
Example 2.3.
(H.E) (W3), and so (H.E)
(W4) and (H.E)
(C.C).
Let and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ2_HTML.gif)
and
Then, is a symmetric space which satisfies (H.E). Let
. Then,
But
and hence the symmetric space
does not satisfy (W3).
Example 2.4.
(C.C) (W4) and so (W3)
(W4) by Proposition 2.1 (2).
Let , and let
(
is odd),
(
is even) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ3_HTML.gif)
Then, the symmetric space satisfies (C.C) but does not satisfy (W4) for
and
.
Example 2.5.
(C.C) (H.E).
Let , and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ4_HTML.gif)
and . Then,
is a symmetric space which satisfies (C.C). Let
. Then,
But
Hence, the symmetric space
does not satisfy (H.E).
3. Common Fixed Points of Four Mappings
Let be a symmetric(or semimetric) space and let
be self-mappings of
. Then, we say that the pair
satisfies property (E-A) [3] if there exists a sequence
in
and a point
such that
A subset of a symmetric space
is said to be
-closed if for a sequence
in
and a point
,
implies
. For a symmetric space
,
-closedness implies
-closedness, and if
is a semimetric, the converse is also true.
At first, we prove coincidence point theorems of four mappings satisfying the property (E-A) under some contractive conditions.
Theorem 3.1.
Let be a symmetric(semimetric) space that satisfies (W3) and (H.E), and let
and
be self-mappings of
such that
-
(1)
and
,
-
(2)
the pair
satisfies property (E-A) (resp.,
satisfies property (E-A)),
-
(3)
for any
,
, where
(3.1) -
(4)
is a
-closed (
-closed) subset of
(resp.,
is a
-closed (
-closed) subset of
).
Then, there exist such that
.
Proof.
From (2), there exist a sequence in
and a point
such that
From , there exists a sequence
in
such that
and hence
. By (H.E),
From , there exists a point
such that
.
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ6_HTML.gif)
By taking , we have
By (W3), we get
Since , there exists a point
such that
.
We show that From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ7_HTML.gif)
Hence, and hence
.
For the existence of a common fixed point of four self-mappings of a symmetric space, we need an additional condition, so-called weak compatibility.
Recall that for self-mappings and
of a set, the pair
is said to be weakly compatible [8] if
, whenever
. Obviously, if
and
are commuting, the pair
is weakly compatible.
Theorem 3.2.
Let be a symmetric(semimetric) space that satisfies (W3) and (H.E), and let
and
be self-mappings of
such that
-
(1)
and
,
-
(2)
the pair
satisfies property (E-A) (resp.,
satisfies property (E-A)),
-
(3)
the pairs
and
are weakly compatible,
-
(4)
for any
-
(5)
is a
-closed (
-closed) subset of
(resp.,
is a
-closed (
-closed) subset of
).
Then, and
have a unique common fixed point in
.
Proof.
From Theorem 3.1, there exist such that
. From
,
,
and
If , then from (4) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ8_HTML.gif)
which is a contradiction.
Similarly, if , we have a contradiction. Thus,
and
is a common fixed point of
and
.
For the uniqueness, let be another common fixed point of
and
. If
, then from
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ9_HTML.gif)
which is a contradiction. Hence,
Remark 3.3.
In the case of and
in Theorem 3.1 (resp., Theorem 3.2), we can show that
and
have a coincidence point(resp.,
and
have a unique common fixed point) without making the assumption
.
Recently, R. P. Pant and V. Pant [4] obtained the existence of a common fixed point of the pair of in a metric space
satisfying the condition
(P.P) for any ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ10_HTML.gif)
where
Also in [5], the authors tried to extend the result of [4] to symmetric spaces which satisfy axiom (W3).
Now, we will extend R. P. Pant and V. Pant's result to symmetric spaces which satisfy additional conditions (H.E) and (C.C).
Theorem 3.4.
Let be a symmetric(semimetric) space that satisfies (H.E) and (C.C) and let
and
be self-mappings of
such that
-
(1)
and
,
-
(2)
the pair
satisfies property (E-A) (resp.,
satisfies property (E-A)),
-
(3)
for any
,
, where
-
(4)
is a
-closed (
-closed) subset of
(resp.,
is a
-closed (
-closed) subset of
).
Then, there exist such that
.
Proof.
As in the proof of Theorem 3.1, there exist sequences in
and a point
such that
and
. Hence,
.
From , there exists a point
such that
.
We show From
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ11_HTML.gif)
In the above inequality, we take , by (C.C) and (H.E), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ12_HTML.gif)
Since , we get
and hence
Since , there exists a point
such that
.
We show that From
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ13_HTML.gif)
Since , we get
and hence
. Therefore, we have
Theorem 3.5.
be a symmetric(semimetric) space that satisfies (H.E) and (C.C) and let
and
be self-mappings of
such that
-
(1)
and
,
-
(2)
the pair
satisfies property (E-A) (resp.,
satisfies property (E-A)),
-
(3)
the pairs
and
are weakly compatible,
-
(4)
for any
where
-
(5)
is a
-closed (
-closed) subset of
(resp.,
is a
-closed (
-closed) subset of
).
Then and
have a unique common fixed point in
.
Proof.
From Theorem 3.4, there exist points such that
,
and
We show that If
, then from (4) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ14_HTML.gif)
which is a contradiction.
Similarly, if , we have a contradiction. Thus
For the uniqueness, let be another common fixed point of
and
. If
, then from
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ15_HTML.gif)
which is a contradiction. Hence
Example 3.6.
Let and
. Define self-mappings
and
by
and
for all
. Then, we have the following:
-
(0)
is a symmetric space satisfying the properties (H.E) and (C.C),
-
(1)
and
,
-
(2)
the pair
satisfies property (E-A) for the sequence
-
(3)
the pairs
and
are weakly compatible,
-
(4)
for any
,
-
(5)
is a
-closed(
-closed) subset of
,
-
(6)
Remark 3.7.
In the case of and
in Theorem 3.4 (resp., Theorem 3.5), we can show that
and
have a coincidence point (resp.,
and
have a unique common fixed point) without the condition
, that is,
.
The following example shows that the axioms (H.E) and (C.C) cannot be dropped in Theorem 3.4.
Example 3.8.
Let be the symmetric space as in Example 2.2. Then, the symmetric
does not satisfy both (H.E) and (C.C).
Let and
be self-mappings of
defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ16_HTML.gif)
Then, the condition (resp.,
) of Theorem 3.4 (resp., Theorem 3.5) is satisfied for
.
To show this, let . We consider two cases.
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_IEq352_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ17_HTML.gif)
Case 2.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_IEq353_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F562130/MediaObjects/13663_2007_Article_1092_Equ18_HTML.gif)
Thus, the condition (resp.,
) of Theorem 3.4 (resp., Theorem 3.5) is satisfied. Note that
is a
-closed(
-closed) subset of
. Also, the pair
satisfies property (E-A) for
, but the pair
has no coincidence points, and also the pair
has no common fixed points.
Remark 3.9.
Example 3.6 satisfies all conditions of [5, Theorems 2.1 and 2.2] and satisfies also all conditions of [5, Theorem 2.3].
Let be a function such that
  is nondecreasing on
,
  for all
Note that from and
, we have
On the studying of fixed points, various conditions of have been studied by many different authors [3, 5, 6].
Remark 3.10.
The functions in Theorems 3.4 and 3.5 can be generalized to the compositions
for
.
Example 3.11.
Let be the symmetric space and
and
be the functions as in Example 3.8. Recall that
satisfies (W3) but does not satisfy both (H.E) and (C.C). Let
and
Then, for any
,
for
. Note that the pairs
and
satisfy property (E-A), and
,
and
are
-closed(
-closed).
Therefore, and
satisfy all conditions of [5, Theorem 2.4] and satisfy also all conditions of [5, Theorem 2.5]. But the pairs
and
have no points of coincidence, and also the pairs
and
have no common fixed points.
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Acknowledgments
The authors are very grateful to the referees for their helpful suggestions. The first author was supported by Hanseo University, 2007.
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Cho, SH., Lee, GY. & Bae, JS. On Coincidence and Fixed-Point Theorems in Symmetric Spaces. Fixed Point Theory Appl 2008, 562130 (2008). https://doi.org/10.1155/2008/562130
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DOI: https://doi.org/10.1155/2008/562130