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Some coupled coincidence and common fixed point results for a hybrid pair of mappings in 0-complete partial metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 145 (2013)
Abstract
In this paper we extend some coupled coincidence and common fixed point theorems for a hybrid pair of mappings obtained by Abbas et al. (Fixed Point Theory Appl. 2012:4, 2012, doi:10.1186/1687-1812-2012-4) from the complete metric space to 0-complete partial metric spaces. An example showing that this extension is proper is given.
MSC:47H10, 54H25.
1 Introduction
Let A be any nonempty subset of a metric space . For , define
Let denote the set of all nonempty closed bounded subset of X. For , define
Then H is a metric on and is called a Hausdorff metric.
Nadler [1] generalized the Banach contraction mapping principle to set-valued functions and proved the following fixed point theorem.
Theorem 1 Let be a complete metric space and let T be a mapping from X into such that for all ,
where . Then T has a fixed point.
Later, an interesting and rich fixed point theory was developed. On the other hand, Matthews [2] introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow networks, with the interesting property ‘non-zero self-distance’ in space. He showed that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification. Subsequently, several authors (see, e.g., [3–22]) derived fixed point theorems in partial metric spaces. Romaguera [17] introduced the notion of 0-Cauchy sequence, 0-complete partial metric spaces and proved some characterizations of partial metric spaces in terms of completeness and 0-completeness. Recently, Aydi et al. [9] introduced the notion of a partial Hausdorff metric and extended the Nadler’s theorem in partial metric spaces.
Bhaskar and Lakshmikantham [23] introduced the concept of a coupled fixed point and established some coupled fixed point theorems in partially ordered sets. As an application, they studied the existence and uniqueness of a solution for a periodic boundary value problem associated with a first-order ordinary differential equation. Recently Abbas et al. [24] extended these concepts to set-valued mappings and obtained coupled coincidence points and coupled common fixed point theorems involving a hybrid pair of single-valued and multi-valued maps satisfying generalized contractive conditions in the framework of a complete metric space (see also [25, 26]). The study of a coincidence point and common fixed points of a hybrid pair of mappings in Banach spaces and metric spaces is interesting and well developed. For applications of hybrid fixed point theory, we refer to [27–30].
In this paper, we extend and generalize the results of Abbas et al. [24] and Aydi et al. [9] for a hybrid pair of mappings in 0-complete partial metric spaces. Also, some new results are obtained. An example is included to support our results.
2 Preliminaries
Consistent with [2, 8, 9, 16, 17, 19], the following definitions and results will be needed in the sequel.
Definition 1 A partial metric on a nonempty set X is a function ( stands for nonnegative reals) such that for all ,
-
(P1)
,
-
(P2)
,
-
(P3)
,
-
(P4)
.
A partial metric space is a pair such that X is a nonempty set and p is a partial metric on X.
It is clear that if , then from (P1) and (P2) . But if , may not be 0. Also, every metric space is a partial metric space, with zero self-distance.
Example 1 If is defined by , for all , then is a partial metric space.
Some more examples of a partial metric space can be seen in [2, 9, 16].
Each partial metric on X generates a topology on X which has as a base the family of open p-balls , where for all and .
Theorem 2 [2]
For each partial metric , the pair , where for all , is a metric space.
Here is called an induced metric space and d is an induced metric. In further discussion, unless specified otherwise, will represent an induced metric space.
Let be a partial metric space.
-
(1)
A sequence in converges to a point if and only if .
-
(2)
A sequence in is called a Cauchy sequence if there exists (and is finite) .
-
(3)
is said to be complete if every Cauchy sequence in X converges with respect to to a point such that .
-
(4)
A sequence in is called 0-Cauchy sequence if . The space is said to be 0-complete if every 0-Cauchy sequence in X converges with respect to to a point such that .
Let be a partial metric space and be any sequence in X.
-
(i)
is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .
-
(ii)
is complete if and only if the metric space is complete. Furthermore, if and only if .
-
(iii)
Every 0-Cauchy sequence in is Cauchy in .
-
(iv)
If is complete, then it is 0-complete.
The converse assertions of (iii) and (iv) do not hold. Indeed, the partial metric space , where ℚ denotes the set of rational numbers and the partial metric p is given by for all , provides an easy example of a 0-complete partial metric space which is not complete. It is easy to see that every closed subset of a 0-complete partial metric space is 0-complete.
Let be a partial metric space. Let be the family of all nonempty, closed and bounded subsets of the partial metric space induced by the partial metric p. Note that closedness is taken from ( is the topology induced by p) and boundedness is given as follows: A is a bounded subset in if there exist and such that for all , we have , that is, .
For and , define
Lemma 2 [8]
Let be a partial metric space, . Then if and only if .
Proposition 1 [9]
Let be a partial metric space. For any , we have the following:
-
(i)
;
-
(ii)
;
-
(iii)
implies that ;
-
(iv)
.
Let be a partial metric space. For , define
Proposition 2 [9]
Let be a partial metric space. For , we have
-
(h1)
;
-
(h2)
;
-
(h3)
.
Corollary 1 [9]
Let be a partial metric space. For , the following holds:
In view of Proposition 2 and Corollary 1, we call the mapping a partial Hausdorff metric induced by p.
Lemma 3 [9]
Let be a partial metric space, and . For any , there exists such that .
The following lemma is crucial for the proof of our main result and its proof is similar to Lemma 3.
Lemma 4 Let be a partial metric space and , . Let be arbitrary, then there exists such that
Definition 2 [24]
Let X be a nonempty set, (collection of all nonempty subsets of X) and . An element is called
-
(i)
a coupled fixed point of F if and ;
-
(ii)
a coupled coincidence point of the hybrid pair if and ;
-
(iii)
a coupled point of coincidence if there exists such that and ;
-
(iv)
a coupled common fixed point of the hybrid pair if and .
Definition 3 Let X be a nonempty set, let and be two mappings. The hybrid pair is called weakly compatible if and whenever is a coupled coincidence point of the hybrid pair .
Now we can state our main results.
3 Main results
The following result extends and generalizes the main result of [24] in partial metric spaces.
Theorem 3 Let be a 0-complete partial metric space, let and be mappings satisfying
for all , where are nonnegative reals such that . If and is a closed subset of X, then F and g have a coupled point of coincidence and .
Proof Let be arbitrary, then . As , we can choose and for some . Again, as and , so by Lemma 4, for any , there exist and such that
Continuing this process, we obtain two sequences and in X such that
From the above inequalities and (1), we obtain
that is,
Interchanging the roles of and and using the symmetries of p and , we obtain
It follows from (2) and (3) that
Similarly, it can be obtained that
For simplicity, set , then it follows from (4) and (5) that
that is,
As was arbitrary, choose ; also, as , we have . Therefore, from (6) we have
From a successive application of the above inequality, we obtain
For with , using (7) we obtain
As , it follows from the above inequality that
So, and are 0-Cauchy sequences in ; therefore, by the closedness of , there exists such that
Using (1) we obtain
that is,
Using (8) and (9) and the fact that in the above inequality, we obtain
Therefore, by Lemma 2, . Similarly, . Thus is a coupled coincidence point and (say) is a point of coincidence of the mappings F and g with . □
The following is a coupled fixed point result for a set-valued mapping and it can be obtained by taking (that is an identity mapping of X) in the above theorem.
Corollary 2 Let be a 0-complete partial metric space, let be a mapping satisfying
for all , where are nonnegative reals such that . Then F has a coupled fixed point and .
With suitable values of control constants in Theorem 3, one can obtain the following corollaries.
Corollary 3 Let be a 0-complete partial metric space, let and be mappings satisfying
for all , where and are nonnegative reals such that . If and is a closed subset of X, then F and g have a coupled point of coincidence and .
Corollary 4 Let be a 0-complete partial metric space, let and be mappings satisfying
for all , where are nonnegative reals such that . If and is a closed subset of X, then F and g have a coupled point of coincidence and .
The following example illustrates the case when the results in partial metric spaces are applicable while the same results in usual metric spaces are not.
Example 2 Let , and let be defined by
Then the metric induced by p is given by for all and is not complete, therefore is not complete. Now, it is easy to see that is a 0-complete partial metric space and every singleton subset of X is closed with respect to p. Define and by
We shall show that F and g satisfy all the conditions of Corollary 3, with , while the metric versions of Corollary 3 are not applicable. We consider the following cases.
Case (i) If and , then suppose , so
where . Similarly, we obtain the same result for .
Case (ii) If and , then
where . Similarly, if any one of x, y, u, v is equal to 1, then we obtain the same result.
Case (iii) If any one of , is equal to , for example, let and , then we have
where . Similarly, the condition (10) is satisfied for in all possible cases and , that is, is a coupled coincidence point of F and g (here it is the unique common fixed point of F and g).
Note that, the metric spaces and (where ρ is usual and d is metric induced by p) are not complete, therefore metric versions of Corollary 3 are not applicable. Also, this example shows that F and g do not satisfy the metric versions of inequality (10). Indeed, if is the Hausdorff metric induced by the usual metric ρ, then for , , we have
and
Therefore, we cannot find the nonnegative reals , such that
for all with . So, F is not a contraction (in view of contraction condition (10)) with respect to the usual metric ρ. Similarly, one can see that F is not a contraction with respect to the induced metric d.
The following theorem provides a sufficient condition for the uniqueness of a coupled point of coincidence and a common fixed point of the hybrid pair .
Theorem 4 Let be a 0-complete partial metric space, let and be mappings such that all the conditions of Theorem 3 are satisfied and, for any coupled coincidence point of F and g, we have and , then F and g have a unique coupled point of coincidence. Suppose in addition that the hybrid pair is weakly compatible, then F and g have a unique coupled common fixed point.
Proof The existence of a coupled coincidence point and a point of coincidence follows from Theorem 3. Suppose that, for any coupled coincidence point of F and g, we have and . We shall show that the coupled point of coincidence is unique. Let be another coupled coincidence point and be the coupled point of coincidence of F and g, that is, , and , .
Using (1), we obtain
Again, using (1) we obtain
It follows from (11) and (12) that
As , it follows from the above inequality that , that is, , so and . Therefore, a coupled point of coincidence, that is, , of F and g is unique.
Suppose that F and g are weakly compatible, then we have
Therefore, is another coupled point of coincidence of F and g, and by uniqueness we have and . Thus is the unique coupled common fixed point of F and g. □
The following theorem is a new result for a hybrid pair of mappings in partial metric as well as in metric spaces.
Theorem 5 Let be a 0-complete partial metric space, let and be mappings satisfying
for all , where are nonnegative reals such that . If and is a closed subset of X, then F and g have a coupled point of coincidence and .
Proof By a similar process as used in Theorem 3, we can find two sequences and such that
where is arbitrary.
From the above inequality and (13), we obtain
that is,
Interchanging the roles of and and using the symmetries of p and , we obtain
It follows from (14) and (15) that
Similarly, it can be shown that
For simplicity, set , then it follows from (16) and (17) that
As was arbitrary, choose ; also, as , we have . Therefore
It follows from a successive application of the above inequality that
For with , using (18) we obtain
As , it follows from the above inequality that
So, and are 0-Cauchy sequences in , therefore by the closedness of , there exists such that
We shall show that and .
For all , we have
Using (19) and (20) in the above inequality, we obtain
Again, for all , we have
Using (20) and (19) in the above inequality, we obtain
Note that if or , then (21) and (22) give a contradiction. Therefore, we have and , and by Lemma 2, and . Thus is a coupled coincidence point and (say) is a point of coincidence of the mappings F and g with . □
The following is a coupled fixed point result for a set-valued mapping and can be obtained by taking (that is an identity mapping of X) in the above theorem.
Corollary 5 Let be a 0-complete partial metric space, let be a mapping satisfying
for all , where are nonnegative reals such that . Then F has a coupled fixed point and .
Theorem 6 Let be a 0-complete partial metric space, let and be mappings such that all the conditions of Theorem 5 are satisfied, and for any coupled coincidence point of F and g, we have and . Then F and g have a unique coupled point of coincidence. Suppose in addition that the hybrid pair is weakly compatible, then F and g have a unique coupled common fixed point.
Proof The proof of this theorem is followed by a similar process as used in Theorem 4. □
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Acknowledgements
Wei Long acknowledges support from the NSF of China, and the Research Project of Jiangxi Normal University (2012-114).
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Long, W., Shukla, S., Radenović, S. et al. Some coupled coincidence and common fixed point results for a hybrid pair of mappings in 0-complete partial metric spaces. Fixed Point Theory Appl 2013, 145 (2013). https://doi.org/10.1186/1687-1812-2013-145
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DOI: https://doi.org/10.1186/1687-1812-2013-145