Journal of Function Spaces
Volume 2018 (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

Tawseef Rashid,¹ Qamrul Haq Khan,¹ and Hassen Aydi2,3

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; hmaydi@uod.edu.sa

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 [2–22]. 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 that

where 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 that

Now, using (12) and (15), we get

and

From the above two inequalities, we have

that is,

Inductively, we have

and

Using (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 that

Using (25), we have

Hence, from (27)

As and , now, since and , there exist and such that and . By (27) and (29), we have

and

Therefore,

Part (i) is completed. Now, by (12), (15), (30), (31), and triangular inequality, we have

Thus, we have

Similarly,

We deduce that

Using (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 , then

From (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 that

Let 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 satisfies

where 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 deduce

Again,

Using (65),

Also,

Using (63),

By induction, as (63), (65), (67), and (69), we have

By (60), we have

Using (70) and (71), we get

By 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 that

By (73), we have

From (76) and (77), we get

Proceeding 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 and , we have . Also, and . Thus, is a -coupling with respect to and . Also, is a Chatterjea type -coupling with respect to and . Thus, all the conditions of Theorem 23 are satisfied, so there exists such that and . For example, take and .

Example 25. Let be endowed with the Euclidean metric . It is known that is complete. Let and . Consider as

where denotes the first integer less than or equal to . Since and are not closed in , we can not apply the result of Choudhury et al. [24]. Now, we define by . We have and ; , and are the sets of lattice points in and with nonnegative integers, respectively. Clearly, and are closed subsets in . Also and are invariant by . As , so for all and , we have . Also, and . Thus, is a -coupling with respect to and . Also, is a Chatterjea type -coupling with respect to and . Thus all the conditions of Theorem 23 are satisfied, so there exists such that and . For example, take and .

3. Application to Integral Equations

In this section, we study an existence and uniqueness problem of a solution for a class of nonlinear integral equations by using our obtained results. First, let be the set of all continuous functions defined on and equipped by the metric for all . Clearly, is complete.

Consider the following system of nonlinear integral equations:

where , , and .

Consider two nonempty subsets . Let and , where . Suppose there exist such that, for each , we have

Theorem 26. Under conditions (82)-(84), system (81) has a unique common solution in .

Proof. We define the mappings and as

and

where is a closed subset in . Taking , we have and . Also, and are closed subsets in . Now, we show that is a -coupling. For and , we have

and

This shows that is a -coupling with respect to and . We shall show that is a Banach type -coupling. For this, let and . Using (82), (83), and (84),

Thus all the conditions of Theorem 15 are satisfied. Thus and have a unique common coupled fixed point , satisfying in . So is the unique solution of the system of integral equations (81).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

References

D. Guo and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 11, no. 5, pp. 623–632, 1987.

M. Abbas, B. Ali, and Y. I. Suleiman, “Generalized coupled common fixed point results in partially ordered -metric spaces,” Fixed Point Theory and Applications, vol. 64, 24 pages, 2015.

T. Abdeljawad, H. Aydi, and E. Karapinar, “Coupled fixed points for Meir-Keeler contractions in ordered partial metric spaces,” Mathematical Problems in Engineering, vol. 2012, Article ID 327273, 20 pages, 2012.

A. H. Ansari, H. Aydi, M. A. Barakat, and M. S. Khan, “On coupled coincidence point results in partially ordered metric spaces via generalized compatibility and C-class functions,” Journal of Inequalities and Special Functions, vol. 8, no. 3, pp. 125–138, 2017.

H. Aydi, E. Karapınar, and Z. Mustafa, “Coupled coincidence point results on generalized distance in ordered cone metric spaces,” Positivity, vol. 17, no. 4, pp. 979–993, 2013.

W. Shatanawi, E. Karapinar, and H. Aydi, “Coupled coincidence points in partially ordered cone metric spaces with a -distance,” Journal of Applied Mathematics, vol. 2012, Article ID 312078, 15 pages, 2012.

H. Aydi, B. Samet, and C. Vetro, “Coupled fixed point results in cone metric spaces fo w-compatible mappings,” Fixed Point Theory and Applications, vol. 27, 15 pages, 2011.

H. Aydi, B. Damjanović, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered -metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2443–2450, 2011.

V. Berinde, “Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 18, pp. 7347–7355, 2011.

N. Bilgili, I. M. Erhan, E. Karapinar, and D. Turkoglu, “A note on coupled fixed point theorems for mixed g-monotone mappings in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2014, no. 120, 6 pages, 2014.

B. S. Choudhury and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 8, pp. 2524–2531, 2010.

B. S. Choudhury and A. Kundu, “Two coupled weak contraction theorems in partially ordered metric spaces,” Revista de la Real Academia de ciencias Exactas, Fisicas y Naturales. Series A. Mathematicas, vol. 108, no. 2, pp. 335–351, 2014.

J. Harjani, B. Lopéz, and K. Sadarangani, “Fixed point theorems for mixed monotone operators and applications to integral equations,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 74, no. 5, pp. 1749–1760, 2011.

V. Lakshmikantham and L. Ćirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009.

N. V. Luong and N. X. Thuan, “Coupled fixed point theorems in partially ordered metric spaces depended on another function,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 3, pp. 129–140, 2011.

N. Van and N. Xuan, “Coupled points in ordered generalized metric spaces and application to integro-differential equations,” Analele Universitatii "Ovidius" Constanta Seria Matematica, vol. 21, no. 3, pp. 155–180, 2013.

S. Moradi, E. Karapınar, and H. Aydi, “Existence of solutions for a periodic boundary value problem via generalized weakly contractions,” Abstract and Applied Analysis, vol. 2013, Article ID 704160, 7 pages, 2013.

H. Kumar Nashine, “Coupled common fixed point results in ordered -metric spaces,” Journal of Nonlinear Sciences and Applications. JNSA, vol. 5, no. 1, pp. 1–13, 2012.

H. K. Nashine and W. Shatanawi, “Coupled common fixed point theorems for a pair of commuting mappings in partially ordered complete metric spaces,” Computers & Mathematics with Applications. An International Journal, vol. 62, no. 4, pp. 1984–1993, 2011.

S. Rasouli and M. Bahrampour, “A remark on the coupled fixed point theorems for mixed monotone operators in partially ordered metric spaces,” Journal of Mathematical and Computational Science, vol. 3, pp. 246–261, 2011.

B. Samet, E. Karapinar, H. Aydi, and V. Ć. Rajić, “Discussion on some coupled fixed point theorems,” Fixed Point Theory and Applications, vol. 2013, article 50, 2013.

W. Shatanawi, H. K. Nashine, and N. Tahat, “Generalization of some coupled fixed point results on partial metric spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 686801, 10 pages, 2012.

W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,” Fixed Point Theory and Applications, vol. 4, no. 1, pp. 79–89, 2003.

B. S. Choudhury, P. Maity, and P. Konar, “Fixed point results for couplings on metric spaces,” University of Bucharest. Scientific Bulletin. Series A. Applied Mathematics and Physics, vol. 79, no. 1, pp. 1–12, 2017.

H. Aydi, M. Barakat, A. Felhi, and S. Radenovic, “On strong coupled fixed points for nonlinear contractive type couplings,” Journal of Mathematical Analysis, vol. 8, no. 5, pp. 86–96, 2017.

H. Aydi, M. Barakat, A. Felhi, and H. Isik, “On ϕ-contraction type couplings in partial metric spaces,” Journal of Mathematical Analysis, vol. 8, no. 4, pp. 78–89, 2017.

T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006.

B. S. Choudhury and P. I. Maity, “Cyclic Coupled Fixed Point Result Using Kannan Type Contractions,” Journal of Operators, vol. 2014, Article ID 876749, 5 pages, 2014.