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Journal of Function Spaces
Volume 2018, Article ID 4034535, 10 pages
https://doi.org/10.1155/2018/4034535
Research Article

On Strong Coupled Coincidence Points of -Couplings and an Application

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2Department of Mathematics, College of Education-Jubail, Imam Abdulrahman Bin Faisal University, P.O. 12020, Industrial Jubail 31961, Saudi Arabia
3Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Correspondence should be addressed to Hassen Aydi; as.ude.dou@idyamh

Received 29 March 2018; Revised 4 June 2018; Accepted 28 June 2018; Published 12 July 2018

Academic Editor: Tomonari Suzuki

Copyright © 2018 Tawseef Rashid 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

The purpose of this paper is to prove the existence and uniqueness of a strong coupled coincidence point of and , involving Banach and Chatterjea type -couplings. We also give some examples and an application in support of the given concepts and our main results.

1. Introduction and Preliminaries

The concept of a coupled fixed point was first introduced by Guo et al. [1]. Later, many coupled fixed and coupled coincidence point results were given. For more details, see [222]. Kirk et al. [23] gave the concept of cyclic mappings. Recently, Choudhury et al. [24] introduced the concept of couplings, which are actually coupled cyclic mappings with respect to two given subsets of a metric space. In [24], they also proved the existence of strong coupled fixed points for Banach and Chatterjea couplings. Very recently, Aydi et al. [25, 26] proved some existence and uniqueness results of a strong coupled fixed point for nonlinear couplings in (partial) metric spaces. In this paper, we extend the concept of the Banach (resp., Chatterjea) type coupling to the Banach (resp., Chatterjea) type -couplings. We generalize the results of Choudhury et al. [24].

First, we recall some known definitions.

Definition 1 (coupled fixed point, [27]). Let be a nonempty set. An element is called a coupled fixed point of the mapping if and .

Definition 2 (strong coupled fixed point, [28]). Let be a nonempty set. An element is called a strong coupled fixed point of the mapping if is a coupled fixed point and , that is, if .

Definition 3 (coupled Banach contraction mapping, [27]). Let be a metric space. A mapping is called a coupled Banach contraction if there exists such that, for all , the following inequality is satisfied:

Definition 4 (cyclic mapping, [23]). Let and be two nonempty subsets of a given set . The mapping is said to be cyclic (with respect to and ) if

Definition 5 (coupling, [28]). Let be a metric space and let , be two nonempty subsets of . Then a map is said to be a coupling with respect to and if whenever and .

Definition 6 (Banach type coupling, [24]). Let and be two nonempty subsets of a metric space . A coupling is called a Banach type coupling with respect to and if it satisfies the following inequality: where , , and .

Definition 7 (coupled coincidence point of and , [14]). An element is called a coupled coincidence point of the mappings and if and .

Definition 8. Let be a nonempty subset of a set . is said invariant by if

2. Main Results

First of all, we introduce some definitions.

Definition 9 (strong coupled coincidence point of and ). A coupled coincidence point of and is said to be a strong coupled coincidence point of and if , i.e., .

Definition 10 (-coupling). Let be a metric space and let , be two nonempty subsets of . Given and , then is said to be a -coupling (with respect to and ) if, for and , one has

Now, we give an example illustrating the concept of -couplings.

Example 11. Let . and are two subsets of . Let be defined by , for all . Define by . Then and . Further, and . Now we show is -coupling with respect to and . For and , we have This shows that is a -coupling with respect to and .

Remark 12. Every -coupling is a coupling. For convenience, let be a -coupling. Then by Definition 14, we have . Then is a coupling (with respect to and ). But, the converse is not true in general.

Now, we give an example which shows that every coupling need not be a -coupling.

Example 13. Let . Consider and . We define by , for all . We first show that is a coupling with respect to and . For and , we have and Thus, is a coupling with respect to and . Now, if we define by . Then and . Further, and . This proves that is not a -coupling with respect to and . Indeed, for and ,

Definition 14 (Banach type -coupling). Let and be two nonempty subsets of a metric space . Then a -coupling is said to be Banach type -coupling (with respect to and ) if whenever and , where .

Note that if = (the identity mapping), then the Banach type -coupling becomes a Banach type coupling. Our main result is as follows.

Theorem 15. Let and be any two subsets of a complete metric space . If there exists a Banach type -coupling, (with respect to and ). Suppose that and are invariant by , and , are closed subsets of . Then(i),(ii) and have a coupled coincidence point in . If, in addition, is one to one on , then and have a unique strong coupled coincidence point in .

Proof. Recall thatwhere and .
Also, as and are invariant by , so Let and , then by definition of a -coupling, we have and . In particular, and Hence, there exist and such that Continuing in this way, we get sequences and in and , respectively, such thatNow, using (12) and (15), we get and From the above two inequalities, we have that is,Inductively, we haveandUsing (12) and (15), we have that is,From (12), (15), and (23), we have Inductively,Now, by (20), (21), and (25) and by triangular inequality, we have, for all , Since , it follows that . Obviously, the sequences and are Cauchy in and , respectively. As and are closed subsets in the complete metric space , and are convergent in and , respectively. Therefore, there exist and such thatUsing (25), we have Hence, from (27)As and , now, since and , there exist and such that and . By (27) and (29), we haveandTherefore, Part (i) is completed. Now, by (12), (15), (30), (31), and triangular inequality, we have Thus, we haveSimilarly, We deduce thatUsing (34) and (36), we get and . Thus is a coupled coincidence point of and . (ii) is completed.
In view of (31) and the fact that is one to one, we have , so and have a strong coupled coincidence point; that is, . Suppose there exist two strong coupled coincidence points of and , thenFrom (12), we have Since , we deduce that , , so due to the fact that is one to one. Hence and have a unique strong coupled coincidence point in .

We present the following examples.

Example 16. Let be endowed by , where . Let and . We define by , where and is defined by . Then and , so and are closed subsets of . Also and , so and are invariant by . Now, we show that is a -coupling.
As and , so for and , we have and This shows that is a -coupling with respect to and . Again, for and , we have This shows that is a Banach type -coupling (with respect to and ). Thus all the conditions of Theorem 15 are satisfied; therefore there exists such that and . Here, .

Example 17. Let be endowed with the metric . Take and . Let be defined as , where . Consider as . Then and , so and are closed subsets of . Also, and , so and are invariant by . Now, we shall show that is a -coupling. As and , so for all and , we have and ; that is, and . Thus, is a -coupling with respect to and . Again, for and , we have This shows that is a Banach type -coupling (with respect to and ). Thus all the conditions of Theorem 15 are satisfied, so there exists such that and , i.e.,Then . The mapping is one to one, so is the unique strong -coupled coincidence point of and .

Corollary 18. Let be a Banach contraction. Then every Banach type -coupling (with respect to and ) is a Banach type coupling (with respect to and ), where and are subsets of .

Proof. Since is a Banach contraction, there exists such thatLet be a Banach type -coupling (with respect to and ), ,where , , and . By Definition 5,   is a coupling (with respect to and ). Now, using (44) in (45), we get Since , we have , so where , , and . Hence, is a Banach type coupling (with respect to and ).

Now, we give some examples where Theorem 15 works, but the result of Choudhury et al. [24] is not applicable.

Example 19. Let be endowed with the metric . Take and . Let be defined as Consider by Since and are not closed in , we can not apply the result of Choudhury et al. [24]. Clearly, and , so and are closed subsets of . Also, and ; hence and are invariant by . Now, we shall show that is a -coupling. As and , so for all and , we have and ; that is, and . Thus, is a -coupling with respect to and . Again, for and , we have and , so . Thus, for all and , with , we have This shows that is a Banach type -coupling (with respect to and ). Thus all the conditions of Theorem 15 are satisfied, so there exists such that and . Clearly, for and , we have and Thus is a coupled coincidence point of and in .

Example 20. Let be endowed with the metric . Take and . Let be defined as We define by Since and are not closed in , we can not apply the result of Choudhury et al. [24]. Clearly, and , so and are closed subsets of . Also, and ; hence and are invariant by . Now, we shall show that is a -coupling. As , so for all and , we have ; that is, and . Thus, is a -coupling with respect to and . Again, for and , we have , so . This shows that is a Banach type -coupling (with respect to and ). Thus all the conditions of Theorem 15 are satisfied, so there exists such that and . Clearly, for and , we have and Thus, is a coupled coincidence point of and in .

Example 21. Let be endowed with the metric . Take and . Let be defined as and is defined as . Since and are not closed in , we can not apply the result of Choudhury et al. [24]. We have . Also, and ; hence and are invariant by . Now, we shall show that is a -coupling. As , so for all and , we have and Thus, is a -coupling with respect to and . Again, for and , we have where . This shows that is a Banach type -coupling (with respect to and ). Thus all the conditions of Theorem 15 are satisfied, so there exists such that and . Note that is a (strong) coupled coincidence point of and in .

Now, we introduce the concept of Chatterjea type -couplings and we prove existence of strong coupled coincidence points.

Definition 22. Let and be any two nonempty subsets of a metric space and be a self-mapping. Let be a -coupling with respect to and . Then is said to be a Chatterjea type -coupling with respect to and , if it satisfieswhere and , where .

Theorem 23. Let and be any two nonempty subsets of a complete metric space and let be a self-mapping. Let be a Chatterjea type -coupling with respect to and . Further, suppose that and are invariant by and , are closed subsets of . Then(i),(ii) and have a coupled coincidence point in . If, in addition, is one to one, then and have a unique strong coupled coincidence point in .

Proof. Let and . Following the same lines as proof of Theorem 15, we construct and in and , respectively, such that, for all ,Now, by using (60) and (61), we get Therefore,where . Similarly, We deduceAgain, Using (65),Also, Using (63),By induction, as (63), (65), (67), and (69), we haveBy (60), we have Using (70) and (71), we getBy triangular inequality, using (70), (71), and (73), Since , it follows that This implies that and are Cauchy sequences in and , respectively. Since and are closed subsets of a complete metric space, and are convergent in and , respectively. Thus there exist and , such thatBy (73), we haveFrom (76) and (77), we getProceeding similarly to the proof of Theorem 15, we get that and . Hence is a coupled coincidence point of and in .
Again, if is one to one, it is obvious that is the unique strong coupled coincidence point of and .

Now, we report some examples where again the result of Choudhury et al. [24] can not be applied.

Example 24. Let be endowed with the metric . Take and . Let be defined as , where denotes the first integer less than or equal to and is defined as Since and are not closed in , we can not apply the result of Choudhury et al. [24]. We have . Clearly, and are invariant by . Now, we shall show that is a -coupling. As , so for all