Abstract
In this paper we establish some coincidence point results for generalized weak contractions with discontinuous control functions. The theorems are proved in metric spaces with a partial order. Our theorems extend several existing results in the current literature. We also discuss several corollaries and give illustrative examples. We apply our result to obtain some coupled coincidence point results which effectively generalize a number of established results.
MSC: 54H10, 54H25.
Keywords:
partially ordered set; control function; coincidence point; coupled coincidence point1 Introduction
In this paper we prove certain coincidence point results in partially ordered metric spaces for functions which satisfy a certain inequality involving three control functions. Two of the control functions are discontinuous. Fixed point theory in partially ordered metric spaces is of relatively recent origin. An early result in this direction is due to Turinici [1], in which fixed point problems were studied in partially ordered uniform spaces. Later, this branch of fixed point theory has developed through a number of works, some of which are in [26].
Weak contraction was studied in partially ordered metric spaces by Harjani et al.[3]. In a recent result by Choudhury et al.[2], a generalization of the above result to a coincidence point theorem has been done using three control functions. Here we prove coincidence point results by assuming a weak contraction inequality with three control functions, two of which are not continuous. The results are obtained under two sets of additional conditions. A fixed point theorem is also established. There are several corollaries and two examples. One of the examples shows that the corollaries are properly contained in their respective theorem. The corollaries are generalizations of several existing works.
We apply our result to obtain some coupled coincidence point results. Coupled fixed theorems and coupled coincidence point theorems have appeared prominently in recent literature. Although the concept of coupled fixed points was introduced by Guo et al.[7], starting with the work of Gnana Bhaskar and Lakshmikantham [8], where they established a coupled contraction principle, this line of research has developed rapidly in partially ordered metric spaces. References [920] are some examples of these works. There is a viewpoint from which coupled fixed and coincidence point theorems can be considered as problems in product spaces [21]. We adopt this approach here. Specifically, we apply our theorem to a product of two metric spaces on which a metric is defined from the metric of the original spaces. We establish a generalization of several results. We also discuss an example which shows that our result is an actual improvement over the results it generalizes.
2 Mathematical preliminaries
Let be a partially ordered set and . The mapping T is said to be nondecreasing if for all , implies and nonincreasing if for all , implies .
Definition 2.1 ([22])
Let be a partially ordered set and and . The mapping T is said to be Gnondecreasing if for all , implies and Gnonincreasing if for all , implies .
Definition 2.2 Two selfmappings G and T of a nonempty set X are said to be commutative if for all .
Definition 2.3 ([23])
Two selfmappings G and T of a metric space are said to be compatible if the following relation holds:
whenever is a sequence in X such that for some is satisfied.
Definition 2.4 ([24])
Two selfmappings G and T of a nonempty set X are said to be weakly compatible if they commute at their coincidence points; that is, if for some , then .
Definition 2.5 ([8])
Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument; that is, if
and
Definition 2.6 ([17])
Let be a partially ordered set, and . We say that F has the mixedgmonotone property if
and
Definition 2.7 ([8])
An element is called a coupled fixed point of the mapping if and .
Definition 2.8 ([17])
An element is called a coupled coincidence point of the mappings and if and .
Definition 2.9 ([17])
Let X be a nonempty set. The mappings g and F, where and , are said to be commutative if for all .
Definition 2.10 ([12])
Let be a metric space. The mappings g and F, where and , are said to be compatible if the following relations hold:
whenever and are sequences in X such that and for some are satisfied.
Definition 2.11 Let X be a nonempty set. The mappings g and F, where and , are said to be weakly compatible if they commute at their coupled coincidence points, that is, if and for some , then and .
Definition 2.12 ([25])
A function is called an altering distance function if the following properties are satisfied:
(i) ψ is monotone increasing and continuous;
In our results in the following sections, we use the following classes of functions.
We denote by Ψ the set of all functions satisfying
(i_{ψ}) ψ is continuous and monotone nondecreasing,
and by Θ we denote the set of all functions such that
3 Main results
Let be an ordered metric space. X is called regular if it has the following properties:
(i) if a nondecreasing sequence in , then for all ;
(ii) if a nonincreasing sequence in , then for all .
Theorem 3.1Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatGis continuous and nondecreasing, , TisGnondecreasing with respect to ⪯ and the pairis compatible. Suppose that there existandsuch that
Also, suppose that
(a) Tis continuous, or
(b) Xis regular.
If there existssuch that, thenGandThave a coincidence point inX.
Proof Let be such that . Since , we can choose such that . Again, we can choose such that . Continuing this process, we construct a sequence in X such that
Since and , we have , which implies that . Now, , that is, implies that . Again, , that is, implies that . Continuing this process, we have
and
Since , from (3.3) and (3.4), we have
that is,
which, in view of the fact that , yields , which by (3.1) implies that for all positive integer n, that is, is a monotone decreasing sequence. Hence there exists an such that
Taking limit supremum on both sides of (3.7), using (3.8), the property (i_{α}) of φ and θ, and the continuity of ψ, we obtain
Since , it follows that
that is,
which by (3.2) is a contradiction unless . Therefore,
Next we show that is a Cauchy sequence.
Suppose that is not a Cauchy sequence. Then there exists an for which we can find two sequences of positive integers and such that for all positive integers k, and . Assuming that is the smallest such positive integer, we get
Now,
that is,
Letting in the above inequality and using (3.9), we have
Again,
and
Letting in the above inequalities, using (3.9) and (3.10), we have
As , , from (3.3) and (3.4), we have
Taking limit supremum on both sides of the above inequality, using (3.10), (3.11), the property (i_{α}) of φ and θ, and the continuity of ψ, we obtain
Since , it follows that
that is,
which is a contradiction by (3.2). Therefore, is a Cauchy sequence in X. From the completeness of X, there exists such that
Since the pair is compatible, from (3.12), we have
Let the condition (a) hold.
By the triangular inequality, we have
Taking in the above inequality, using (3.12), (3.13) and the continuities of T and G, we have , that is, , that is, x is a coincidence point of the mappings G and T.
Next we suppose that the condition (b) holds.
By (3.5) and (3.12), we have for all . Using the monotone property of G, we obtain
As G is continuous and the pair is compatible, by (3.12) and (3.13), we have
Then
Since ψ is continuous, from the above inequality, we obtain
which, by (3.3) and (3.14), implies that
Using (3.15) and the property (ii_{α}) of φ and θ, we have
which, by the property of ψ, implies that , that is, , that is, x is a coincidence point of the mappings G and T. □
Next we discuss some corollaries of Theorem 3.1. By an example, we show that Theorem 3.1 properly contains all its corollaries.
Every commuting pair is also a compatible pair. Then considering to be the commuting pair in Theorem 3.1, we have the following corollary.
Corollary 3.1Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatGis continuous and nondecreasing, , TisGnondecreasing with respect to ⪯ and the pairis commutative. Suppose that there existandsuch that (3.1), (3.2) and (3.3) are satisfied. Also, suppose that
(a) Tis continuous, or
(b) Xis regular.
If there existssuch that, thenGandThave a coincidence point inX.
Considering ψ to be the identity mapping and for all in Theorem 3.1, we have the following corollary.
Corollary 3.2Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatGis continuous and nondecreasing, , TisGnondecreasing with respect to ⪯ and the pairis compatible. Suppose that there existssuch that for any sequenceinwith,
Also, suppose that
(a) Tis continuous, or
(b) Xis regular.
If there existssuch that, thenGandThave a coincidence point inX.
Considering for all and with in Theorem 3.1, we have the following corollary.
Corollary 3.3Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatGis continuous and nondecreasing, , TisGnondecreasing with respect to ⪯ and the pairis compatible. Suppose that there existsandsuch that for allwith,
Also, suppose that
(a) Tis continuous, or
(b) Xis regular.
If there existssuch that, thenGandThave a coincidence point inX.
Considering φ to be the function ψ in Theorem 3.1, we have the following corollary.
Corollary 3.4Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatGis continuous and nondecreasing, , TisGnondecreasing with respect to ⪯ and the pairis compatible. Suppose that there existandsuch that for any sequenceinwith,
Also, suppose that
(a) Tis continuous or
(b) Xis regular.
If there existssuch that, thenGandThave a coincidence point inX.
If ψ and φ are the identity mappings in Theorem 3.1, we have the following corollary.
Corollary 3.5Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatGis continuous and nondecreasing, , TisGnondecreasing with respect to ⪯ and the pairis compatible. Suppose that there existssuch that for any sequenceinwith, and for allwith,
Also, suppose that
(a) Tis continuous, or
(b) Xis regular.
If there existssuch that, thenGandThave a coincidence point inX.
Considering ψ and φ to be the identity mappings and , where in Theorem 3.1, we have the following corollary.
Corollary 3.6Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatGis continuous and nondecreasing, , TisGnondecreasing with respect to ⪯ and the pairis compatible. Assume that there existssuch that for allwith,
Also, suppose that
(a) Tis continuous, or
(b) Xis regular.
If there existssuch that, thenGandThave a coincidence point inX.
The condition (i), the continuity and the monotone property of the function G, and (ii), the compatibility condition of the pairs , which were considered in Theorem 3.1, are relaxed in our next theorem by taking to be closed in .
Theorem 3.2Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letbe two mappings such thatandTisGnondecreasing with respect to ⪯ andis closed inX. Suppose that there existandsuch that (3.1), (3.2) and (3.3) are satisfied. Also, suppose thatXis regular.
If there existssuch that, thenGandThave a coincidence point inX.
Proof We take the same sequence as in the proof of Theorem 3.1. Then we have (3.12), that is,
Since is a sequence in and is closed in X, . As , there exists such that . Then
Now, is nondecreasing and converges to Gz. So, by the order condition of the metric space X, we have
Putting and in (3.3), by the virtue of (3.24), we get
Taking in the above inequality, using (3.23), the property (ii_{α}) of φ and θ and the continuity of ψ, we have
which, by the property of ψ, implies that , that is, , that is, z is a coincidence point of the mappings G and T. □
In the following, our aim is to prove the existence and uniqueness of the common fixed point in Theorems 3.1 and 3.2.
Theorem 3.3In addition to the hypotheses of Theorems 3.1 and 3.2, in both of the theorems, suppose that for everythere existssuch thatTuis comparable toTxandTy, and also the pairis weakly compatible. ThenGandThave a unique common fixed point.
Proof From Theorem 3.1 or Theorem 3.2, the set of coincidence points of G and T is nonempty. Suppose x and y are coincidence points of G and T, that is, and . Now, we show
By the assumption, there exists such that Tu is comparable with Tx and Ty.
Put and choose so that . Then, similarly to the proof of Theorem 3.1, we can inductively define sequences where for all . Hence and are comparable.
Suppose that (the proof is similar to that in the other case).
In fact, we will use mathematical induction. Since , our claim is true for . We presume that holds for some . Since T is Gnondecreasing with respect to ⪯, we get
and this proves our claim.
Let . Since , using the contractive condition (3.3), for all , we have
that is, , which, in view of the fact that , yields , which by (3.1) implies that for all positive integer n, that is, is a monotone decreasing sequence.
Then, as in the proof of Theorem 3.1, we have
Similarly, we show that
By the triangle inequality, using (3.26) and (3.27), we have
Hence . Thus (3.25) is proved.
Since , by weak compatibility of G and T, we have
Denote
Then from (3.28) we have
Thus z is a coincidence point of G and T. Then from (3.25) with it follows that
By (3.29) it follows that
From (3.29) and (3.30), we get .
Therefore, z is a common fixed point of G and T.
To prove the uniqueness, assume that r is another common fixed point of G and T. Then by (3.25) we have . Hence the common fixed point of G and T is unique. □
Example 3.1 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for . Then is a complete metric space.
Let be given respectively by the formulas and for all . Then T and G satisfy all the properties mentioned in Theorem 3.1.
Let be given respectively by the formulas
Then ψ, φ and θ have the properties mentioned in Theorem 3.1.
It can be verified that (3.3) is satisfied for all with . Hence the required conditions of Theorem 3.1 are satisfied and it is seen that 0 is a coincidence point of G and T. Also, the conditions of Theorem 3.3 are satisfied and it is seen that 0 is the unique common fixed point of G and T.
Remark 3.1 In the above example, the pair is compatible but not commuting so that Corollary 3.1 is not applicable to this example and hence Theorem 3.1 properly contains its Corollary 3.1.
Remark 3.2 In the above example, ψ is not the identity mapping and for all t in . Let us consider the sequence in , where for all n. Then for all n. Now , but . Therefore, Corollary 3.2 is not applicable to this example, and hence Theorem 3.1 properly contains its Corollary 3.2.
Remark 3.3 The above example for all , and hence Corollary 3.3 is not applicable to the example, and so Theorem 3.1 properly contains its Corollary 3.3.
Remark 3.4 In the above example, φ is not identical to the function ψ, and also for any sequence in with . Therefore, Corollaries 3.4 and 3.5 are not applicable to this example, and hence Theorem 3.1 properly contains its Corollaries 3.4 and 3.5.
Remark 3.5 In the above example, ψ and φ are not the identity functions and with . Therefore, Corollary 3.6 is not applicable to the above example. Hence Theorem 3.1 properly contains its Corollary 3.6.
Remark 3.6 Theorem 3.1 generalizes the results in [24,6,2529].
Example 3.2 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for . Then is a metric space with the required properties of Theorem 3.2.
Let be given respectively by the formulas
Then T and G have the properties mentioned in Theorem 3.2.
Let be given respectively by the formulas
Then ψ, φ and θ have the properties mentioned in Theorem 3.2.
All the required conditions of Theorem 3.2 are satisfied. It is seen that every rational number is a coincidence point of G and T. Also, the conditions of Theorem 3.3 are satisfied and it is seen that 1 is the unique common fixed point of G and T.
Remark 3.7 In the above example, the function g is not continuous. Therefore, Theorem 3.1 is not applicable to the above example.
4 Applications to coupled coincidence point results
In this section, we use the results of the previous section to establish new coupled coincidence point results in partially ordered metric spaces. Our results are extensions of some existing results.
Let be a partially ordered set. Now, we endow the product space with the following partial order:
Let be a metric space. Then given by the law
Let and be two mappings. Then we define two functions and respectively as follows:
Lemma 4.1Letbe a partially ordered set, and. IfFhas the mixedgmonotone property, thenTisGnondecreasing.
Proof Let such that . Then, by the definition of G, it follows that , that is, and . Since F has the mixed gmonotone property, we have
and
It follows that , that is, . Therefore, T is Gnondecreasing. □
Lemma 4.2LetXbe a nonempty set, and. IfgandFare commutative, then the mappingsGandTare also commutative.
Proof Let . Since g and F are commutative, by the definition of G and T, we have
which shows that G and T are commutative. □
Lemma 4.3Letbe metric space and, . IfgandFare compatible, then the mappingsGandTare also compatible.
Proof Let be a sequence in such that for some . By the definition of G and T, we have , which implies that
Now
Since g and F are compatible, we have
It follows that G and T are compatible. □
Lemma 4.4LetXbe a nonempty set, and. IfgandFare weak compatible, then the mappingsGandTare also weak compatible.
Proof Let be a coincidence point G and T. Then , that is, , that is, and . Since g and F are weak compatible, by the definition of G and T, we have
which shows that G and T commute at their coincidence point, that is, G and T are weak compatible. □
Theorem 4.1Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letandbe two mappings such thatgis continuous and nondecreasing, , Fhas the mixedgmonotone property onXand the pairis compatible. Suppose that there existandsuch that (3.1) and (3.2) are satisfied and for allwithand,
Also, suppose that
(a) Fis continuous, or
(b) Xis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
Proof We consider the product space , the metric on and the functions and as mentioned above. Denote . Then is a complete metric space. By the definition of G and T, we have that
(i) G is continuous and nondecreasing; and T is continuous,
(iii) T is Gnondecreasing with respect to ⪯,
Let such that and , that is, , that is, . Then (4.1) reduces to
Now, the existence of such that and implies the existence of a point such that , that is, . Therefore, the theorem reduces to Theorem 3.1, and hence there exists such that , that is, , that is, , that is, and , that is, is a coupled coincidence point of g and F. □
The following corollary is a consequence of Corollary 3.1.
Corollary 4.1Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letandbe two mappings such thatgis continuous and nondecreasing, , Fhas the mixedgmonotone property onXand the pairis commutative. Suppose that there exist, such that (3.1), (3.2) and (4.1) are satisfied. Also, suppose that
(a) Fis continuous, or
(b) Xis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
The following corollary is a consequence of Corollary 3.2.
Corollary 4.2Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letandbe two mappings such thatgis continuous and nondecreasing, , Fhas the mixedgmonotone property onXand the pairis compatible. Suppose that there existssuch that for any sequenceinwith,
Also, suppose that
(a) Fis continuous, or
(b) Xis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
The following corollary is a consequence of Corollary 3.3.
Corollary 4.3Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letandbe two mappings such thatgis continuous and nondecreasing, , Fhas the mixedgmonotone property onXand the pairis compatible. Suppose that there existandsuch that for allwithand,
Also, suppose that
(a) Fis continuous, or
(b) Xis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
The following corollary is a consequence of Corollary 3.4.
Corollary 4.4Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letandbe two mappings such thatgis continuous and nondecreasing, , Fhas the mixedgmonotone property onXand the pairis compatible. Suppose that there existandsuch that for any sequenceinwith, and for allwithand,
Also, suppose that
(a) Fis continuous, or
(b) Xis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
Remark 4.1 The above result is also true if the arguments of ψ and θ in (4.2) are replaced by their half values, that is, when (4.2) is replaced by
In this case, we can write and and proceed with the same proof by replacing ψ, θ by , respectively. Then we obtain a generalization of Theorem 2 of Berinde in [10].
The following corollary is a consequence of Corollary 3.5.
Corollary 4.5Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letandbe two mappings such thatgis continuous and nondecreasing, , Fhas the mixedgmonotone property onXand the pairis compatible. Suppose that there existssuch that for any sequenceinwith, and for allwithand,
Also, suppose that
(a) Fis continuous, or
(b) Xis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
The following corollary is a consequence of Corollary 3.6.
Corollary 4.6Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Letandbe two mappings such thatgis continuous and nondecreasing, , Fhas the mixedgmonotone property onXand the pairis compatible. Assume that there existssuch that for allwith,
Also, suppose that
(a) Fis continuous, or
(b) Xis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
The following theorem is a consequence of Theorem 3.2.
Theorem 4.2Letbe a partially ordered set and suppose that there is a metricdonXsuch thatis a complete metric space. Consider the mappingsandsuch that, Fhas the mixedgmonotone property onXandis closed inX. Suppose that there existandsuch that (3.1), (3.2) and (4.1) are satisfied. Also, suppose thatXis regular.
If there existsuch thatand, then there existsuch thatand; that is, gandFhave a coupled coincidence point inX.
The following theorem is a consequence of Theorem 3.3.
Theorem 4.3In addition to the hypotheses of Theorems 4.1 and 4.2, in both of the theorems, suppose that for every, there exists asuch thatis comparable toand, and also the pairis weakly compatible. ThengandFhave a unique coupled common fixed point; that is, there exists a uniquesuch thatand.
Example 4.1 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for . Then is a complete metric space.
Let be given by for all . Also, consider
which obeys the mixed gmonotone property.
Let and be two sequences in X such that
Then it follows that
Hence, the pair is compatible in X.
Let and be two points in X. Then
Let be given respectively by the formulas
Then ψ, φ and θ have the properties mentioned in Theorem 4.1.
We now verify inequality (4.1) of Theorem 4.1.
We take such that and , that is, and .
The following are the four possible cases.
Case4: The case ‘ and ’ is not possible. Under this condition, and . Then by the condition , we have , which contradicts that .
In all the above cases, it can be verified that (4.1) is satisfied. Hence the required conditions of Theorem 4.1 are satisfied, and it is seen that is a coupled coincidence point of g and F in X. Also, the conditions of Theorem 4.3 are satisfied, and it is seen that is the unique coupled common fixed point of g and F in X.
Remark 4.2 In the above example, the pair is compatible but not commuting so that Corollary 4.1 is not applicable to this example, and hence Theorem 4.1 properly contains its Corollary 4.1.
Remark 4.3 As discussed in Remarks 3.23.5, Theorem 4.1 properly contains its Corollaries 4.24.6.
Remark 4.4 Theorem 4.1 properly contains its Corollary 4.6, which is an extension of Theorem 3 of Berinde [9], and Theorems 2.1 and 2.2 of Bhaskar and Lakshmikantham [8]. Therefore, Theorem 4.1 is an actual extension over Theorem 3 of Berinde [9] and Theorems 2.1 and 2.2 of Bhaskar and Lakshmikantham [8].
Example 4.2 Let . Then is a partially ordered set with the natural ordering of real numbers. Let for . Then is a metric space with the required properties of Theorem 4.2.
Let , for . Let be given by the formula
Then F and g have the properties mentioned in Theorems 4.2.
Let be given respectively by the formulas
Then ψ, φ and θ have the properties mentioned in Theorem 4.2.
All the required conditions of Theorem 4.2 are satisfied. It is seen that every , where both x and y are rational, is a coupled coincidence point of g and F in X. Also, the conditions of Theorem 4.3 are satisfied and it is seen that is the unique coupled common fixed point of g and F in X.
Remark 4.5 In the above example, the function g is not continuous. Therefore, Theorem 4.1 is not applicable to the above example.
Remark 4.6 In some recent papers [30,31] it has been proved that some of the contractive conditions involving continuous control functions are equivalent. Here two of our control functions are discontinuous. Therefore, the contraction we use here is not included in the class of contractions addressed by Aydi et al.[30] and Jachymski [31].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
The work is partially supported by the Council of Scientific and Industrial Research, India (No. 25(0168)/09/EMRII). Professor BS Choudhury gratefully acknowledges the support.
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