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Some fixed point results for multi-valued mappings in partial metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 175 (2013)
Abstract
In this paper, we obtain some fixed point results for multi-valued mappings in partial metric spaces. Our results unify, generalize and complement various known comparable results from the current literature. An example is also included to illustrate the main result in the paper.
MSC:46S40, 47H10, 54H25.
1 Introduction and preliminaries
The fixed point theory is one of the most powerful and fruitful tools in nonlinear analysis. Its core subject is concerned with the conditions for the existence of one or more fixed points of a mapping T from a topological space X into itself, that is, we can find such that . The Banach contraction principle [1] is the simplest and one of the most versatile elementary results in fixed point theory. Moreover, being based on an iteration process, it can be implemented on a computer to find the fixed point of a contractive mapping. It produces approximations of any required accuracy, and, moreover, even the number of iterations needed to get a specified accuracy can be determined. Recently, Samet et al. [2] introduced a new concept of α-contractive type mappings and established various fixed point theorems for such mappings in complete metric spaces. The presented theorems extend, generalize and improve several results on the existence of fixed points in the literature.
In 1994, Matthews [3] introduced the concept of a partial metric space and obtained a Banach-type fixed point theorem on complete partial metric spaces. Later on, several authors (see, for example, [4–31]) proved fixed point theorems in partial metric spaces. After the definition of the partial Hausdorff metric, Aydi et al. [28] proved the Banach-type fixed point result for set-valued mappings in complete partial metric spaces.
The aim of this paper is to generalize various known results proved by Nadler [32], Kikkawa and Suzuki [33], Mot and Petrusel [34], Dhompongsa and Yingtaweesittikul [35] to the case of partial metric spaces and give one example to illustrate our main results.
We start with recalling some basic definitions and lemmas on partial metric spaces. The definition of a partial metric space is given by Matthews [3] (see also [7, 29, 30]) as follows.
Definition 1 A partial metric on a nonempty set X is a function such that the following conditions hold: for all ,
(P1) if and only if ,
(P2) ,
(P3) ,
(P4) .
The pair is then called a partial metric space.
If is a partial metric space, then the function given by for all is a metric on X.
A basic example of a partial metric space is the pair , where for all .
Lemma 1 Let be a partial metric space, then we have the following:
-
(1)
A sequence in a partial metric space converges to a point if and only if .
-
(2)
A sequence in a partial metric space is called a Cauchy sequence if the exists and is finite.
-
(3)
A partial metric space is said to be complete if every Cauchy sequence in X converges to a point , that is, .
-
(4)
is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .
-
(5)
A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if
Remark 1 ([7], Lemma 1)
Let be a partial metric space and let A be a nonempty set in , then if and only if
where denotes the closure of A with respect to the partial metric p. Note A is closed in if and only if .
Now, we state the following definitions and propositions of a very recent paper of Aydi et al. [28].
Let be a collection of all nonempty closed and bounded subsets of X with respect to the partial metric p. For any , we define
On the other hand, for any , we define
and
Proposition 1 [28]
Let be a partial metric space. For any , we have
-
(1)
.
-
(2)
.
-
(3)
implies that .
-
(4)
.
Proposition 2 [28]
Let be a partial metric space. For any , we have
-
(1)
.
-
(2)
.
-
(3)
.
Lemma 2 [28]
Let A and B be nonempty closed and bounded subsets of a partial metric space and . Then, for all , there exists such that .
The following result was proved by Aydi et al. in [28].
Theorem 1 Let be a partial metric space. If is a multi-valued mapping such that, for all ,
where . Then T has a fixed point.
2 Main results
Now, we characterize the celebrated theorem of Kikkawa and Suzuki [33] in the framework of partial metric spaces.
Theorem 2 Define a strictly decreasing function Θ from onto by . Let be a complete partial metric space and be a multi-valued mapping. Assume that there exists such that
for all . Then there exists such that .
Proof Let be arbitrarily chosen. For all , we have
and, by the condition (2.1), we get
Let , by Lemma 2, there exists such that . Using the previous inequality, we obtain
Now, we have
and, by the condition (2.1), we get
By Lemma 2, there exists such that
Continuing in this way, we can generate a sequence in X such that and
for all , where .
Now, we show that is a Cauchy sequence. Using (2.2) and the triangle inequality for partial metrics , for all , we have
Inductively, we have
as since . By the definition of , we get
as , which implies that is a Cauchy sequence in . Since is complete, by Lemma 1, the corresponding metric space is also complete. Therefore, the sequence converges to some with respect to the metric , that is, . Again, by Lemma 1, we have
Next, we show that
for all . Since as , there exists such that
for all with . Then we have
and hence . Since
letting , we obtain
for all .
Next, we prove that
for all with . For all , we choose such that
Then, using (2.4) and the previous inequality, we get
for all . As , we obtain . From the assumption, we have
Finally, if, for some , we have , then is a fixed point of F. Assume that for all . This implies that there exists an infinite subset J of ℕ such that for all . From
letting with , we get
By Remark 1, we deduce that and hence u is a fixed point of F. This completes the proof. □
It is obvious that Theorem 1 of Aydi et al. follows directly from Theorem 2.
The following theorem is a result of Reich type [36] as well as a generalization of Kikkawa and Suzuki type in the framework of partial metric spaces.
Theorem 3 Let be a complete partial metric space and let be a multi-valued mapping satisfying the following:
for all , nonnegative numbers a, b, c with and . Then F has a fixed point.
Proof Let and be arbitrary. Let . By Lemma 2, there exists such that
Since , we have
Continuing in a similar way, we can obtain a sequence of successive approximations for F, starting from , satisfying the following:
-
(a)
for all ;
-
(b)
for all ,
where . Now, proceeding as in the proof of Theorem 2, we deduce that the sequence converges to some with respect to the metric , that is, . Moreover, (2.3) holds by Lemma 2.
First, we show that
for all . Since as under the metric p, there exists such that
for each . Then we have
which implies that
for all . Thus we have
for all . Letting , we get
for all .
Next, we show that
for all with . Now, for all , there exists such that
From
for all , it follows that, as ,
and so . Thus we have
for all .
Finally, if, for some , we have , then is a fixed point of F. Assume that for all . This implies that there exists an infinite subset J of ℕ such that for all . Now, for all , we have
Letting with , we get
By Remark 1, we deduce that and hence u is a fixed point of F. This completes the proof. □
The following theorem is a generalization of a result of Dhompongsa and Yingtaweesittikul [35] to the setting of partial metric space.
Theorem 4 Let be a complete partial metric space and let be a multi-valued mapping such that
for all , where with λ, μ nonnegative real numbers and . Then F has a fixed point.
Proof Let and be arbitrary. Following the same proof of Theorem 3, by replacing in the proof by , we can obtain a sequence such that
-
(a)
for all ;
-
(b)
for all ,
where .
Now, proceeding as in the proof of Theorem 2, we deduce that the sequence converges to some with respect to the metric , that is, . Again, from Lemma 2, we have
Next, we show that
for all . Since as , there exists such that for all . We have
Now, using the conditions (2.6) and (2.7), we obtain
for all . Letting , we get
as desired.
Next, we show that
for all . By Lemma 2, for all , there exists such that
Clearly, we have
for all . Hence, as , we get
and so since . Now, using the condition (2.6), we obtain
for all .
Finally, if, for some , we have , then is a fixed point of F. Assume that for all . This implies that there exists an infinite subset J of ℕ such that for all . From
letting with , we get
By Remark 1, we deduce that and hence u is a fixed point of F. This completes the proof. □
Now, we give one example to illustrate Theorem 3.
Example 1 Let and be a partial metric on X defined by
Let be defined by
It is easy to see that and are closed in X with respect to the partial metric p. Now, we have
If we choose , and , the multi-valued mapping F satisfies the hypotheses of Theorem 3 and so has a fixed point. To such end, it is enough to show that (2.5) is satisfied in the following cases.
Case 1. and . Now, , where and
Case 2. and . Now, and
Case 3. and . Now, and
Case 4. and . Now, and
Thus all the conditions of Theorem 3 are satisfied. Here is a fixed point of F.
On the other hand, the metric induced by the partial metric p is given by
Note that, in the case of an ordinary Hausdorff metric, the given mapping does not satisfy the condition (2.5). Indeed, for and , the condition is satisfied. But the condition is not satisfied.
In fact, we have
and
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Acknowledgements
The second author was supported by Università degli Studi di Palermo, Local University Project R. S. ex 60% and the third author was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012-0008170).
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Ahmad, J., Di Bari, C., Cho, Y.J. et al. Some fixed point results for multi-valued mappings in partial metric spaces. Fixed Point Theory Appl 2013, 175 (2013). https://doi.org/10.1186/1687-1812-2013-175
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DOI: https://doi.org/10.1186/1687-1812-2013-175