Journal of Mathematics
Volume 2019 (2019), Article ID 8485412, 7 pages
https://doi.org/10.1155/2019/8485412

Research Article

Related Results to Hybrid Pair of Mappings and Applications in Bipolar Metric Spaces

G. N. V. Kishore,¹ K. P. R. Rao,² A. Sombabu,³ and R. V. N. S. Rao⁴

¹Department of Mathematics, SRKR Engineering College, Bhimavaram 534 204, A.P., India
²Department of Mathematics, Acharya Nagarjuna University, Nagarjuna Nagar 522 510, A.P., India
³Department of Mathematics, Lingayas Institute of Management and Technology, Madalavarigudem 521212, Krishna Dt., A.P., India
⁴Department of Mathematics, College of Natural and Computational Sciences, Wollega University, East Wollega, Nekemte, Ethiopia

Correspondence should be addressed to R. V. N. S. Rao; rvnrepalle@wollegauniversity.edu.et

Received 14 November 2018; Accepted 5 May 2019; Published 15 May 2019

Academic Editor: Ji Gao

Copyright © 2019 G. N. V. Kishore et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we introduce the concept of multivalued contraction mappings in partially ordered bipolar metric spaces and establish the existence of unique coupled fixed point results for multivalued contractive mapping by using mixed monotone property in partially ordered bipolar metric spaces. Some interesting consequences of our results are obtained.

1. Introduction and Preliminaries

Fixed point theory has been playing a vital role in the study of nonlinear phenomena. The Banach fixed point theorem or contraction mapping principle was proved by Banach [1] in 1922. Turinici [2] extended the Banach contraction principle in the setting of partially ordered sets and laid the foundation of a new trend in fixed point theory.

The theory of mixed monotone multivalued mappings in ordered Banach spaces was extensively investigated by Y. Wu [3]. Existence of fixed points in ordered metric spaces was initiated by Ran and Reurings [4], and later on several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions in the framework of partially ordered metric spaces ([1–27] and references therein).

In [19], Bhaskar and Lakshmikantham introduced the concept of coupled fixed point and proved some coupled fixed point theorems in partially ordered metric spaces (see also [1–27] for more works). The study of fixed points for multivalued contraction mappings using the Hausdorff metric was initiated by Markin [20]. Later, many authors established hybrid fixed point theorems and gave applications of their results (see also [21–24]).

Very recently, in 2016 Mutlu and Grdal [25] introduced the notion of bipolar metric spaces. Also they investigated some fixed point and coupled fixed point results on this space (see [25, 26]).

This paper aims to introduce some coupled fixed point theorems for a multivalued mappings satisfying various contractive conditions defined on partially ordered bipolar metric spaces. We have illustrated the validity of the hypotheses of our results.

First we recall some basic definitions and results.

Definition 1 ([25]). Let and be two nonempty sets. Suppose that is a mapping satisfying the following properties:

(B₀) if and only if for all ,(B₁), for all ,(B₂), for all , .

Then the mapping is called a bipolar metric on the pair and the triple is called a bipolar metric space.

Definition 2 ([25]). Assume and as two pairs of sets.
The function is said to be a covariant map, if and and denote this as .
The mapping is said to be a contravariant map, if and and denote this as .
In particular, and are bipolar metrics in and , respectively. Sometimes we use the notations and .

Definition 3 ([25]). Let be a bipolar metric space. A point is said to be left point if , a right point if , and a central point if both.
Similarly, a sequence on the set and a sequence on the set are called a left and right sequence, respectively.
In a bipolar metric space, sequence is the simple term for a left or right sequence.
A sequence is convergent to a point if and only if is a left sequence, is a right point, and ; or is a right sequence, is a left point, and .
A bisequence on is sequence on the set . If the sequences and are convergent, then the bisequence is said to be convergent. is Cauchy sequence, if . In a bipolar metric space, every convergent Cauchy bisequence is biconvergent.
A bipolar metric space is called complete, if every Cauchy bisequence is convergent, hence biconvergent.

Now we give our main results.

2. Main Results

The following definitions and results will be needed in the sequel.

Let be a bipolar metric space. For points and the subsets , , consider the bipolar metric and . We denote by and a class of all nonempty closed and bounded subsets of and , respectively. Also denote and . Let be the Hausdorff bipolar metric induced by the bipolar metric on ; that is,

for every and

Definition 4. Let be given the mapping; an element is called a coupled fixed point of a set valued mapping if and .

Lemma 5 ([21]). Let . If with , then, for each , there exists an element such that .

Definition 6. Let be a partially ordered set and let be covariant map. We say that has the mixed monotone property if is monotone-nondecreasing in its first argument and is monotone-nonincreasing in its second argument , that is, for any .

Note that if , , and has mixed monotone property, by Definition 6, we obtain and

Theorem 7. Let be a partially ordered set such that there exists a bipolar metric on with being complete bipolar metric spaces. Consider the covariant mapping satisfying the following condition:

for all , , and are nonnegative constants with and And

(7.1) has a mixed monotone property(7.2)There exists and, for some , , we have and (7.3)If a nondecreasing sequence is convergent to for , then , for all and if a nonincreasing sequence is convergent to for , then , for all

Then has a coupled fixed point.

Proof. Let and . Consider the sequences and such that , and , . By (7.2), we have that and and and , where and .

Applying this in inequality (3), we have

and

On adding (4) and (5), we get

On the other hand

and

On adding (7) and (8), we get

Moreover,

and

On adding (10) and (11), we get

Also, if , then . If , then . If , then .

It follows that is a coupled fixed point of .

Assume that either or and or ; also or .

Since , , then from (6) and Lemma 5 there exist , such that

and since , , then from (9) and Lemma 5 there exist , such that

Also since , and , , from (12) and Lemma 5 then we have

Since, , and , , , and , , by assumption (7.1), we get

Similarly from (3) and above, we have

and

and also

Since, we have , and , , , and , . Again, applying our assumption (7.1), we get

Continuing similarly this process, we have , with

such that

and

and also

Put for any ; then

Put for any ; then

Put for any ; then

Therefore , , and are nonincreasing sequences. From (25), (26), and (27) we have that

which implies that

Using the property , we have

and

Next, we show that and are Cauchy bisequence in for each such that From (25), (26), (27), (30), and (31), we have

and

From above, it is clear that and are Cauchy bisequences in Since, is complete, and such that

Now we will show that and . As is a nondecreasing bisequence and is a nonincreasing bisequence in (A, B),

By assumption (7.1), we get and for all . If and for some , then and , implies and ; therefore, and .

So is coupled fixed point of .

Suppose that and for all

From (3), we have

and

Therefore,

Letting , we have that

Since and , we have and since and , we have .

Similarly, we can prove and .

On the other hand,

and

Therefore, and and hence has a coupled fixed point.

Theorem 8. Let be a partially ordered set such that there exists a bipolar metric on with being complete bipolar metric spaces. Consider a covariant set valued mapping, such that

for all , , and with and Suppose also that

(8.1) has a mixed monotone property(8.2)there exist and for some , we have and (8.3)if a nondecreasing sequence is convergent to for , then , for all and if a nonincreasing sequence is convergent to for , then , for all

Then has a coupled fixed point; that is, there exist such that and .

Example 9. Let be upper triangular matrices over and let be lower triangular matrices over with the bipolar metric for all and . On the set , consider the following relation:

where is usual ordering. Then, clearly, is a complete bipolar metric space and is a partially ordered set. Let be defined as for all .
Then obviously has mixed monotone property; also there exist and such that

and

Taking with , , that is, , , we have

Therefore, all the conditions of Theorem 8 hold and is a coupled fixed point of .

Definition 10. Let be bipolar metric spaces, , and let be a covariant multivalued map. An element is called a coupled fixed point of if and

Theorem 11. Let be a partially ordered set such that there exists a bipolar metric on with being complete bipolar metric spaces. Consider a covariant set valued mapping, such that

for all , and with and Suppose also that

(11.1) has a mixed monotone property(11.2)there exist and for some , we have and (11.3)if a nondecreasing sequence is convergent to for , then , for all and if a nonincreasing sequence is convergent to for , then , for all

Then has a coupled fixed point; that is, there exist such that and .

Proof. The proof will follow when we replace and in place of , , respectively, in Theorem 7.

Theorem 12. Let be a partially ordered set such that there exists a bipolar metric on with being complete bipolar metric spaces. Consider a covariant set valued mapping, such that

for all , and with and Suppose also that

(12.1) has a mixed monotone property(12.2)there exist and for some , we have and (12.3)if a nondecreasing sequence is convergent to for , then , for all and if a nonincreasing sequence is convergent to for , then , for all

Then has a coupled fixed point; that is, there exist such that and .

3. Conclusions

In the present research, we introduced and proved a coupled fixed point theorem for a multivalued mapping, satisfying various contractive conditions, defined on a partially ordered bipolar metric space, and gave suitable example that supports our main result.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no competing interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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