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Coupled fixed and coincidence points for monotone operators in partial metric spaces
Fixed Point Theory and Applications volume 2012, Article number: 173 (2012)
Abstract
In this paper, we prove some coupled fixed point results for -weakly contractive mappings in ordered partial metric spaces. As an application, we establish coupled coincidence results without any type of commutativity of the concerned maps. Consequently, the results of Luong and Thuan (Nonlinear Anal. 74:983-992, 2011), Alotaibi and Alsulami (Fixed Point Theory Appl. 2011:44, 2011) and many others are extended to the class of ordered partial metric spaces.
1 Introduction
The Banach contraction principle is the most celebrated fixed point theorem. Afterward many authors obtained various important extensions of this principle (see [1]). The concept of partial metric spaces was introduced by Matthews [2] in 1994. A partial metric space is a generalized metric space in which each object does not necessarily have to have a zero distance from itself. A motivation behind introducing the concept of a partial metric was to obtain appropriate mathematical models in the theory of computation and, in particular, to give a modified version of the Banach contraction principle [3, 4]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions on partial metric spaces (e.g., [5–7]).
Recently, Bhaskar and Lakshmikantham [8] presented coupled fixed point theorems for contractions in partially ordered metric spaces. Luong and Thuan [9] presented nice generalizations of these results. Alotaibi and Alsulami [10] further extended the work of Luong and Thuan to coupled coincidences. For more related work on coupled coincidences we refer the readers to recent work in [11–16]. Our main aim in this paper is to extend Luong and Thuan [9] results to ordered partial metric spaces. We shall also establish coupled coincidence results and show that main results in [10] hold in ordered partial metric spaces without the compatibility of maps.
2 Basic concepts
We start by recalling some definitions and properties of partial metric spaces.
Definition 2.1 A partial metric on a nonempty set X is a function 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.
From the above definition, if , then . But if , may not be 0 in general. A trivial example of a partial metric space is the pair , where is defined as . For some more examples of partial metric spaces, we refer to [4, 6].
Each partial metric p on X generates a topology on X which has as a base the family of open p-balls , where for all and . A sequence in X converges to a point , with respect to if and only if . A sequence in X is called a Cauchy sequence if exists and is finite.
If p is a partial metric on X, then the function given by
is a metric on X.
Let be a partial metric space. Then
-
(a)
is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space .
-
(b)
is complete if and only if the metric space is complete. Furthermore, if and only if
Let be a partial metric. We endow the product space with the partial metric q defined as follows:
A mapping is said to be continuous at if for each , there exists such that .
Definition 2.2 (Mixed monotone property)
Let be a partially ordered set and . We say that the mapping F has the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument. That is, for any ,
and
Definition 2.3 [11]
Let . We say that is a coupled fixed point of F if and .
Definition 2.4 (Mixed g-monotone property [11])
Let be a partially ordered set and . We say that the mapping F has the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument. That is, for any ,
and
Definition 2.5 [11]
Let and . We say that is a coupled coincidence point of F and g if and .
3 Coupled fixed point results
Let Φ denote all functions which satisfy
(Ï• 1) Ï• is continuous and non-decreasing,
(Ï• 2) if and only if ,
(Ï• 3) , ,
(Ï• 4) for ,
and let Ψ denote all functions which satisfy for all and .
Now, we state and prove our main result.
Theorem 3.1 Let be a partially ordered set and suppose there is a partial metric p on X such that is a complete partial metric space. Let be a mapping having the mixed monotone property on X. Assume that there exist two elements with
Suppose there exist and such that
for all with and . Suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence , then , for all n,
-
(ii)
if a non-increasing sequence , then , for all n.
Then there exist such that
that is, F has a coupled fixed point in X.
Proof Let be such that and . We construct sequences and in X as
We are to prove that
and
For this we shall use mathematical induction.
Let . Since and and as and , we have and . Thus (5) and (6) hold for .
Suppose now that (5) and (6) hold for some fixed , then, since and , we have
and
Using (7) and (8), we get
Hence, by the induction method we conclude that (5) and (6) hold for all . Therefore,
and
Since and , using (3) and (4), we have
Similarly, since and , using (3) and (4), we also have
Using (11) and (12), we have
By property (Ï• 3), we have
Using (13) and (14), we have
which implies, since ψ is a non-negative function,
Using the fact that Ï• is non-decreasing, we get
Set
Now, we show that as . It is clear that the sequence is decreasing. Therefore, there is some such that
We shall prove that . Suppose, to the contrary, that . Then taking the limit as (equivalently, ) of both sides of (15) and remembering for all and Ï• is continuous, we have
a contradiction. Thus , that is,
Let
for all . From the definition of , it is clear that for all . Using (17), we get
Now, we prove that and are Cauchy sequences in the partial metric space . From Lemma 2.1, it is sufficient to prove that and are Cauchy sequences in the metric space . Suppose, to the contrary, that at least one of or is not a Cauchy sequence. Then there exists an for which we can find subsequences , of and , of with such that
Further, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (18). Then
Using (18), (19) and the triangle inequality, we have
Letting and using (17), we get
By the triangle inequality,
Using the properties of Ï•, we have
Now, let
By the definition of , we have
In view of property (p2) and (17), we have
Therefore, letting in (22) and using (20), we get
Since and , we have
Similarly,
Adding (23) and (24), we get
Thus, from (21), we have
Letting , and using the properties of ϕ and ψ together with the inequalities established above, we have
which is a contradiction. Therefore, and are Cauchy sequences in the complete metric space . Thus, there are such that
which implies that
Therefore, from Lemma 2.1, using (17) and the property (p2), we have
We now show that and . Suppose that the assumption (a) holds.
As F is continuous at , so for any , there exists such that if with , meaning that
because . Then we have
Since , for , there exist such that, for , ,
Then for , , we have , so we get
Further, for any , by using (30), we have
On utilizing in (3), we get
which implies . Hence, for any , (31) implies that
Thus, we have . Similarly, we can show that .
Finally, suppose that (b) holds. By (5), (26) and (27), we have is a non-decreasing sequence, and is a non-increasing sequence, as . Hence, by the assumption (b), we have for all ,
By property (p4), we have
Therefore,
Taking limit as in the above inequality, using (31) and (29) and the properties of ϕ and ψ, we get , which implies . Hence, . Similarly, we can show that . Thus F has a coupled fixed point. □
Remark 3.1 Note that the property (Ï• 4) is utilized only to get the inequality (25). Thus the conclusion of Theorem 3.1 holds if we drop property (Ï• 4) and assume the additivity in (Ï• 3), i.e., , .
As an immediate consequence of the above theorem, by taking , we have:
Corollary 3.1 Let be a partially ordered set and suppose there is a partial metric p on X such that is a complete partial metric space. Let be a mapping having the mixed monotone property on X. Assume that there exist two elements with
Suppose there exist and such that
for all with and . Suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
Then there exist such that
that is, F has a coupled fixed point in X.
Moreover, if we take where in Corollary 3.1, we get:
Corollary 3.2 Let be a partially ordered set and suppose there is a partial metric p on X such that is a complete partial metric space. Let be a mapping having the mixed monotone property on X. Assume that there exist two elements with
Suppose there exist and such that
for all with and . Suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
Then there exist such that
that is, F has a coupled fixed point in X.
Recently, Alotaibi and Alsulami [10] extended Luong and Thuan’s [9] main result to coupled coincidences using the notion of compatible maps. Here we extend these results to partial metric spaces without the condition of compatible maps. We shall need the following lemma.
Let X be a nonempty set and be a mapping. Then there exists a subset such that and is one-to-one.
Theorem 3.2 Let be a partially ordered set and suppose there is a partial metric p on X such that is a partial metric space. Let and be a mapping having the mixed g-monotone property on X such that there exist two elements with
Suppose there exist and such that
for all with and . Suppose , g is continuous and is complete and also suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
Then there exist such that
that is, F and g have a coupled coincidence point in X.
Proof Using Lemma 3.1, there exists such that and is one-to-one. We define a mapping by
for all . As g is one-to-one on , so G is well defined. Thus, it follows from (33) and (34) that
for all for which and . Since F has the mixed g-monotone property, for all ,
and
which implies that G has the mixed monotone property. Also, there exist such that
This implies there exist such that
Suppose that the assumption (a) holds. Since F is continuous, G is also continuous. Using Theorem 3.1 with the mapping G, it follows that G has a coupled fixed point .
Suppose that the assumption (b) holds. We conclude similarly that the mapping G has a coupled fixed point . Finally, we prove that F and g have a coupled coincidence point. Since is a coupled fixed point of G, we get
Since , there exists a point such that
It follows from (38) and (39) that
Combining (34) and (40), we get
Thus, is a required coupled coincidence point of F and g. This completes the proof. □
The following coupled coincidence point theorems are obtained respectively from Corollaries 3.1 and 3.2 in a similar way as Theorem 3.2 from Theorem 3.1.
Theorem 3.3 Let be a partially ordered set, and suppose there is a partial metric p on X such that is a partial metric space. Let and be a mapping having the mixed g-monotone property on X such that there exist two elements with
Suppose there exist and such that
for all with and . Suppose , g is continuous and is complete and also suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
Then there exist such that
that is, F and g have a coupled coincidence point in X.
Theorem 3.4 Let be a partially ordered set and suppose there is a partial metric p on X such that is a complete partial metric space. Let and be a mapping having the mixed g-monotone property on X such that there exist two elements with
Suppose there exist and such that
for all with and . Suppose , g is continuous and is complete and also suppose either
-
(a)
F is continuous or
-
(b)
X has the following property:
-
(i)
if a non-decreasing sequence , then for all n,
-
(ii)
if a non-increasing sequence , then for all n.
Then there exist such that
that is, F and g have a coupled coincidence point in X.
Remark 3.2 From the proof of Theorem 3.2 we conclude that Theorems 3.3, 4.4 and 5.4 in [6] hold without the compatibility of the maps .
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Acknowledgements
This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. (113-130-D1432). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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Alsulami, S.M., Hussain, N. & Alotaibi, A. Coupled fixed and coincidence points for monotone operators in partial metric spaces. Fixed Point Theory Appl 2012, 173 (2012). https://doi.org/10.1186/1687-1812-2012-173
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DOI: https://doi.org/10.1186/1687-1812-2012-173