In recent times the fixed point results 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 extends the many more recent results in such spaces.
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. [
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. [
In [
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 [
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 [
Definition 1.4. [
We will utilize the following control function which is also referred to as altering distance function.
Definition 1.5. [
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]).
Recently fixed point theory in partially ordered metric spaces has greatly developed. Choudhury and Metiya [
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 that1) 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
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
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
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
Taking the limit in (2.3) and using the continuity of, we have or
which is a contradiction unless. Hence
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.
Taking the limit as in (2.6) and using (2.5), we have
Again
and
Taking the limit as and using (2.6) and (2.7), we have
Again we have
and
Letting and using (2.6) and (2.7), we have
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.
Dedicated to Professor H. M. Srivastava on his 71st Birth Anniversary.