Common Fixed Point Result of Multivalued and Singlevalued Mappings in Partially Ordered Metric Space ()
1. Introduction and Preliminaries
Throughout this paper, let
be a metric space unless mentioned otherwise and
is the set of all non-empty bounded subsets of
. Let
and
be the functions defined by
![](https://www.scirp.org/html/19-5300277\95cb5de1-31ee-4e11-8979-2cd3df3ad1f8.jpg)
![](https://www.scirp.org/html/19-5300277\00960280-882f-4efd-bb1c-5ac51dc4770f.jpg)
for all A, B in
. If A is a singleton i.e.
, we write
![](https://www.scirp.org/html/19-5300277\40f89f77-73e9-4b6c-a13d-04b3b4390e31.jpg)
and
![](https://www.scirp.org/html/19-5300277\e31411c0-d35b-42c3-88cf-6e34c4d3bba4.jpg)
If B is also a singleton i.e.
, we write
![](https://www.scirp.org/html/19-5300277\fcb5a7ba-9495-4716-b094-cf88ecdedaef.jpg)
and
![](https://www.scirp.org/html/19-5300277\cce8b9e1-fe1e-4a2a-a178-e10a9817110b.jpg)
It is obvious that
. For all
. The definition of
yields the following:
![](https://www.scirp.org/html/19-5300277\33437fb9-33f4-4f3d-b237-40cf24a51eb6.jpg)
![](https://www.scirp.org/html/19-5300277\67e86e72-2501-4082-8ff3-da2c07e74761.jpg)
![](https://www.scirp.org/html/19-5300277\a73094bf-73c3-4534-8eda-60690f647108.jpg)
and
.
Several authors used these concepts of weakly contraction, compatibility, weak compatibility to prove some common fixed point theorems for set valued mappings (see [2-8]).
Definition 1.1. [9] A sequence
of subsets of X is said to be convergent to a subset A of X if 1) Given
, there is a sequence
in X such that
for
and
converges to a.
2) Given
, there exists a positive integer N such that
for
where
is the union of all open spheres with centers in A and radius
.
Lemma 1.1. [9,10] If
and
are sequences in
converging to A and B in
, respectively, then the sequence
converges to
.
Lemma 1.2. [9] Let
be a sequence in
and y a point in X such that
. Then the sequence
converges to the set
in
.
In [11], Jungck and Rhoades extended definition of compatibility to set valued mappings setting as follows:
Definition 1.2. The mapping
and
are δ-compatible if
, whenever
is a sequence in X such the
for some
.
Recently, the following definition is given by Jungck and Rhoades [12].
Definition 1.3. The mapping
and
are weakly compatible if for each point u in X such that
, we have
.
It can be seen that any δ-compatible mappings are weakly compatible but the converse is not true as shown by an example in [13]. We will use the following relation between two nonempty subsets of a partially ordered set.
Definition 1.4. [3] Let A and B be two nonempty subsets of a partially ordered set
. The relation between A and B is denoted and defined as follows:
, if for every
there exists
such that
.
We will utilize the following control function which is also referred to as altering distance function.
Definition 1.5. [14] A function
is called an Altering distance function if the following properties are satisfied:
1)
is monotone increasing and continuous2)
if and only if ![](https://www.scirp.org/html/19-5300277\e963d5e8-0629-4f89-b06a-678960417d8e.jpg)
For the use of control function in metric fixed point theory see some recent references ([15,16]).
2. Main Result
Recently fixed point theory in partially ordered metric spaces has greatly developed. Choudhury and Metiya [17] proved certain fixed point theorems for multi valued and single valued mappings in partially ordered metric spaces. They proved the following:
Theorem 2.1. Let
be a partially ordered set and suppose that there exists a metric d on X such that
is a complete metric space. Let
be a multi valued mappings such that the following conditions are satisfied:
There exists
such that
1) For
implies ![](https://www.scirp.org/html/19-5300277\33c2b9d4-3f87-497a-94ac-463cb4b74fad.jpg)
2) If
is a non decreasing sequence in X, then
, for all n3)
for all comparable
, where
and
is an Altering distance function. Then T has a fixed point.
We prove the following theorem for four single-valued and multivalued mappings:
Theorem 2.2. Let
be a partially ordered set and suppose that there exists a metric d on X such that
is a complete metric space. Let
be single valued and
be multivalued mappings such that the following conditions are satisfied:
1) ![](https://www.scirp.org/html/19-5300277\cbd36a58-9359-44eb-a02e-1ec1542a0a12.jpg)
2)
and
are weakly compatible3) If
is a strictly decreasing sequence in X, then
, for all n4)
for all comparable
,
, where
and
is an Altering distance function and suppose that one of
or
is complete. Then there exists a unique point
such that
![](https://www.scirp.org/html/19-5300277\2a02e0ec-1fca-43c5-a86b-753970c328c9.jpg)
Proof: Let
be an arbitrary point of X. By 1) we choose a point
such that
. For this point
, there exists a point
such that
, and so on. Continuing in this manner we can define a sequence
as follows
(2.1)
We claim that
is a Cauchy sequence. For which two cases arise, either
for some n, or
, for each n.
Case I. If
for some n then,
for each
. For instance suppose
. Then
. Otherwise using 3), we get
![](https://www.scirp.org/html/19-5300277\ff544628-def6-4f51-9f49-5b37e910ae98.jpg)
Since
![](https://www.scirp.org/html/19-5300277\979a83f6-1c78-4c05-8dd9-a0d267fa36fe.jpg)
It follows that
(2.2)
Suppose that if
, for some positive integer n, then from (2.2), we have
![](https://www.scirp.org/html/19-5300277\df313336-83e4-483c-bdc8-4485af563315.jpg)
which implies that ![](https://www.scirp.org/html/19-5300277\0b625d91-75ae-45df-8797-6ec2d820b96d.jpg)
Hence
Similarly
implie
Proceeding in this manner, it follows that
for each
, so that
for each
, for some n, and
is a Cauchy sequence.
Case II. When
for each n. In this case, using 3), we obtain
![](https://www.scirp.org/html/19-5300277\4260574a-9fb5-4bed-8805-319c6e2d6f7f.jpg)
Since
![](https://www.scirp.org/html/19-5300277\95cc4790-7bcb-4a53-8b7b-99c77d7e9e99.jpg)
It follows that
(2.3)
Now if
for each positive integer n, then from (2.3), we have
![](https://www.scirp.org/html/19-5300277\c6be16ad-2fe5-471c-a0f4-cfce5ffcf032.jpg)
which implies that
contradicting our assumption that
, for each n. Therefore
for all
and
is strictly decreasing sequence of positive numbers and therefore tends to a limit
. If possible suppose r > 0. Then for given
, there exists a positive integer N such that for each
, we have
(2.4)
Taking the limit
in (2.3) and using the continuity of
, we have or
![](https://www.scirp.org/html/19-5300277\acc3d571-9ace-44c4-b5da-1b24d3153bbd.jpg)
which is a contradiction unless
. Hence
(2.5)
Next we show that
is a Cauchy sequence. Suppose it is not, then there exists an
and since
there exists two sequences of positive numbers
and
such that for all positive integers k,
and
. Assuming that
is the smallest positive integer, we get ![](https://www.scirp.org/html/19-5300277\51e15e0c-6da3-4a3c-ba84-36874b9cf1fd.jpg)
![](https://www.scirp.org/html/19-5300277\df301b58-79d9-49f7-9b37-6cd892e7a76f.jpg)
Now,
![](https://www.scirp.org/html/19-5300277\5f6b65cb-b632-44a9-bb16-96b779f5b7b9.jpg)
i.e.
(2.6)
Taking the limit as
in (2.6) and using (2.5), we have
(2.7)
Again
![](https://www.scirp.org/html/19-5300277\c5add9ec-2703-45ca-aae6-9c5a366cf8a8.jpg)
and
![](https://www.scirp.org/html/19-5300277\5d0d2be6-36a7-4a57-919d-d42864648eab.jpg)
Taking the limit as
and using (2.6) and (2.7), we have
(2.8)
Again we have
![](https://www.scirp.org/html/19-5300277\d2347579-232b-4920-ad80-92c323ec280b.jpg)
and
![](https://www.scirp.org/html/19-5300277\2a166e7f-fee7-492a-a8dc-accfd87ea82f.jpg)
Letting
and using (2.6) and (2.7), we have
(2.9)
Similarly, we have
.
For each positive integer k,
and
are comparable. Now using the monotone property of
in 4), we have
![](https://www.scirp.org/html/19-5300277\684094d4-b39e-4af3-a90a-f1c816f839c6.jpg)
Letting
and using (2.6)-(2.9), and the continuity of
, we have
, which is a contradiction by virtue of property of
. Therefore
and hence any subsequence thereof, is a Cauchy sequence.
Suppose
is complete. Since ![](https://www.scirp.org/html/19-5300277\29f3b68f-f2f8-44b8-8444-0dacbc323332.jpg)
is a subsequence of
, by the above
is Cauchy and
, for some
.
We now show
. For suppose ![](https://www.scirp.org/html/19-5300277\9197d454-2632-4d5d-89b5-feb64c33dd55.jpg)
Since
and
therefore,
. But ![](https://www.scirp.org/html/19-5300277\cf3c3997-512e-48ed-b4d4-6838c7591f39.jpg)
is a subsequence of the strictly decreasing sequence
which tends to the lim r = 0. Therefore
tends to limit r = 0 and hence
implying
. Thus
. Now using
, we have
![](https://www.scirp.org/html/19-5300277\be0112de-a486-4148-978d-25113844d6a9.jpg)
or
![](https://www.scirp.org/html/19-5300277\19561fe9-f820-4c29-a593-ff8b0fa79171.jpg)
which is a contradiction. Consequently ![](https://www.scirp.org/html/19-5300277\8d4f5f8f-db87-4665-b433-09f1211d93b8.jpg)
as
.
In the same manner, it follows that
as
We now show
. For this, in view of
, we have
![](https://www.scirp.org/html/19-5300277\d9410b3d-3e71-4e87-9d34-a3f89afbc13b.jpg)
implies
![](https://www.scirp.org/html/19-5300277\a50b6ade-aed4-4d85-8986-44a32f504b87.jpg)
or
![](https://www.scirp.org/html/19-5300277\d2fd3b15-b978-4493-8595-1b430503d645.jpg)
which is a contradiction. Consequently,
as
. Hence
. Since
there exists some
such that
. Hence
. We now show
. For this, first we prove
. Suppose
then
. Then in accordance with
such that
![](https://www.scirp.org/html/19-5300277\fd4872f8-d7e7-4fb1-a5ed-19c71dbc5d20.jpg)
implies
while
. Therefore a contradiction arises. Hence
. But then
, which, by
, implies ![](https://www.scirp.org/html/19-5300277\19232038-7d00-46a6-a154-f02c9a22bd90.jpg)
Therefore Fu is a singleton. Since
and Fu is a singleton,
. Hence
![](https://www.scirp.org/html/19-5300277\a9a021e0-dc10-4b55-a77d-6ee12ef3086a.jpg)
Since the pair
and
are weakly compatible,
![](https://www.scirp.org/html/19-5300277\4251315d-a659-42bf-bb18-9ff9a20c346c.jpg)
and
![](https://www.scirp.org/html/19-5300277\21ec787f-ee8d-426a-b989-11d1c57492ef.jpg)
From the above, it is clear that Fp and Gp are singletons and ![](https://www.scirp.org/html/19-5300277\bfe51498-927e-4969-9ee3-70d58e1a86f6.jpg)
We now show that
. For instance, suppose
then from
, we have
![](https://www.scirp.org/html/19-5300277\2920eb01-387f-426f-bc3d-1988134751b0.jpg)
Implies as above
as
. Hence
and therefore ![](https://www.scirp.org/html/19-5300277\e96166c9-36ab-452d-8df5-6dac7c46198f.jpg)
We now show
. For, suppose
. For this let
in
, we have
![](https://www.scirp.org/html/19-5300277\274b4a15-b603-45cd-aa8f-4bcd8426f53a.jpg)
or
which is a contradiction. Consequently
as
Therefore
and hence
![](https://www.scirp.org/html/19-5300277\9bc8503a-47bc-4f93-9f29-8dab18a2a7c8.jpg)
Let
be any point satisfying
![](https://www.scirp.org/html/19-5300277\5fbb0368-ffcd-47a3-b6f0-0ef334708ecd.jpg)
Suppose
then from
, we have
in view of
Hence
.
Corollary 2.1. Let I be a self mapping of a metric space
and
a set valued mapping satisfying 1)' ![](https://www.scirp.org/html/19-5300277\289d5e3f-4e13-4fd3-8a1d-9e66ca3613df.jpg)
2)'
are weakly compatible3)'
for all comparable
, where
and
is an altering distance function. If
is complete subspace of X, there exists a unique point
such that ![](https://www.scirp.org/html/19-5300277\00a71b5d-94fd-491d-9331-e3b317bd1ee0.jpg)
Proof: Taking I = J and
in Theorem 2.2.
Taking I = identity mapping in Corollary 2.1, we get the new corollary as follows:
Corollary 2.2. Let
be a complete metric space and
a set valued mapping satisfying
![](https://www.scirp.org/html/19-5300277\6c0265d0-2f7a-439b-a263-76f78094823d.jpg)
Then f has a unique fixed point in X.
Proof. Obvious.
Corollary 2.3. Let
be a partially ordered set and suppose that there exists a metric d on X such that
is a complete metric space. Let
be single valued and
be multivalued mappings such that the following conditions are satisfied:
1)'' ![](https://www.scirp.org/html/19-5300277\5f30c910-c47e-4c91-a13e-2a86188eaff7.jpg)
2)''
and
are weakly compatible3)'' if
is a strictly decreasing sequence in X, then
, for all n4)''
for all comparable
,
, where
and
is an Altering distance function and suppose that one of
or
is complete. Then there exists a unique point
such that
![](https://www.scirp.org/html/19-5300277\de2aa20c-3fb0-405b-839f-2f9f9dfc68b4.jpg)
Example 2.1. Let
be a sub set of
with the order
defined as for
![](https://www.scirp.org/html/19-5300277\865ed123-afe7-49c9-9fe8-59dea8e43147.jpg)
if and only if
. Let
be given as
![](https://www.scirp.org/html/19-5300277\ecee764b-7aa3-4a7f-879b-c6dd9a67ff19.jpg)
for
.
The
is a complete metric space with the required properties of Theorem 2.2.
Let
, be defined as follows:
![](https://www.scirp.org/html/19-5300277\a0bb4637-1a6c-4633-a50b-30a9aaa2e34b.jpg)
![](https://www.scirp.org/html/19-5300277\7e85db09-2332-481e-84ee-da4f46391bbf.jpg)
Let
defined as
, and
. Then all the conditions in the Theorem 2.2 satisfied. Without loss of generality, we assume that
, we discuss the following cases.
1) If
,
, then
and
![](https://www.scirp.org/html/19-5300277\fbb269e5-47af-4500-8981-a5688cc13df0.jpg)
2) If
then
, and
![](https://www.scirp.org/html/19-5300277\8b7f5230-ba2f-45ce-945f-b4c29485bc6f.jpg)
3) If
then
, and
![](https://www.scirp.org/html/19-5300277\3e41f133-6a6c-4346-b0d8-4308b97dd067.jpg)
4) If
then
, and
![](https://www.scirp.org/html/19-5300277\85421cab-cc33-44fa-bd58-9302fcb2adca.jpg)
5) If
then
and
![](https://www.scirp.org/html/19-5300277\fbfb0a9d-0286-4595-9af8-b068779848fd.jpg)
In all above cases, it is clearly shown that
Hence the conditions of Theorem 2.2 are satisfied and shown that
is a fixed point of I, J, F, and G.
3. Acknowledgements
Dedicated to Professor H. M. Srivastava on his 71st Birth Anniversary.