Advances in Pure Mathematics
Vol.3 No.1(2013), Article ID:27396,7 pages DOI:10.4236/apm.2013.31019
Common Fixed Point Result of Multivalued and Singlevalued Mappings in Partially Ordered Metric Space
Department of Mathematical Sciences and Computer Applications, Bundelkhand University, Jhansi, India
Email: rksaini03@yahoo.com
Received July 19, 2012; revised September 28, 2012; accepted October 8, 2012
Keywords: Multi-Valued Mapping; Single-Valued Mapping; Partial Ordering; Control Function; Fixed Point Theorem
ABSTRACT
In recent times the fixed point resulting in partially ordered metric spaces has greatly developed. In this paper we prove common fixed point results for multivalued and singlevalued mappings in partially ordered metric space. Our theorems generalized the theorem in [1] and extended much more recent results in such spaces.
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.
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