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
for all A, B in
. If A is a singleton i.e.
, we write
and
If B is also a singleton i.e.
, we write
and
It is obvious that
. For all 
. The definition of 
yields the following:
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 
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 
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) 
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
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
Since
It follows that
(2.2)
Suppose that if
, for some positive integer n, then from (2.2), we have
which implies that 
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
Since
It follows that
(2.3)
Now if 
for each positive integer n, then from (2.3), we have
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
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 
Now,
i.e.
(2.6)
Taking the limit as 
in (2.6) and using (2.5), we have
(2.7)
Again
and
Taking the limit as 
and using (2.6) and (2.7), we have
(2.8)
Again we have
and
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
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 
is a subsequence of
, by the above 
is Cauchy and
, for some
.
We now show
. For suppose 
Since 
and 
therefore,
. But 
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
or
which is a contradiction. Consequently 
as
.
In the same manner, it follows that 
as 
We now show
. For this, in view of
, we have
implies
or
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
implies 
while 
. Therefore a contradiction arises. Hence
. But then
, which, by
, implies 
Therefore Fu is a singleton. Since 
and Fu is a singleton,
. Hence
Since the pair 
and 
are weakly compatible,
and
From the above, it is clear that Fp and Gp are singletons and 
We now show that
. For instance, suppose 
then from
, we have
Implies as above 
as
. Hence 
and therefore 
We now show
. For, suppose
. For this let 
in
, we have
or
which is a contradiction. Consequently 
as 
Therefore 
and hence
Let 
be any point satisfying
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)' 
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 
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
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)'' 
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
Example 2.1. Let 
be a sub set of 
with the order 
defined as for
if and only if
. Let 
be given as
for
.
The 
is a complete metric space with the required properties of Theorem 2.2.
Let
, be defined as follows:
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
2) If 
then
, and
3) If 
then
, and
4) If 
then
, and
5) If 
then 
and
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.