International Journal of Mathematics and Mathematical Sciences
Volume 2018 (2018), Article ID 4053478, 7 pages
https://doi.org/10.1155/2018/4053478

Research Article

Coincidence Point Theorems for -Contraction Mappings in Generalized Metric Spaces

Chaiporn Thangthong and Anchalee Khemphet

Center of Excellence in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand

Correspondence should be addressed to Anchalee Khemphet; anchalee.k@cmu.ac.th

Received 13 March 2018; Revised 29 April 2018; Accepted 13 May 2018; Published 11 June 2018

Academic Editor: Nawab Hussain

Copyright © 2018 Chaiporn Thangthong and Anchalee Khemphet. 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 result of our study is that a coincidence point of two mappings and can be achieved when the ordered pair is an -contraction with respect to a generalized metric space. Moreover, with some additional condition, a common fixed point can be obtained as a consequence of our main theorems. Further, we apply our findings to some examples and integral equation problems.

1. Introduction

There has been a wide range of research in discovering fixed points, or the only fixed point, of certain types of mappings that are contractions in the past. Many aspects have been used to accomplish the result. At the very beginning, Geraghty [1] generally developed the Banach contraction principle by considering the class whose elements are functions such that

In 2012, Samet et al. [2] studied the existing results for --contractions. His concept was given in the following definition. Suppose that and is a real-valued function on .

Definition 1 (see [2]). Let be a self-mapping on and . If whenever , then we say that is -admissible.

Later, Karapinar [3] added more conditions to Definition 1.

Definition 2 (see [3]). Let be an -admissible self-mapping on and . If and imply , then we say that is triangular -admissible.

Furthermore, another essential part in this topic is a metric space. There were a large number of literatures that worked not only on a metric space, but also on other topological spaces; for examples, see [4–6]. Three years ago, Jleli and Samet [7] defined a generalized metric, known as a JS-metric. The advantage of their idea is that many topological spaces are covered by the JS-metric space. With this reason, results of fixed point theorems on JS-metric spaces have been recently interesting (e.g., see [8]).

Let be a function and, for , denote a set of sequences in such that by .

Definition 3 (see [7]). Suppose that, for any ,

(D₁)if , then ;(D₂);(D₃)there is a so that, for each , if , then .

Then, we call a generalized metric, or a JS-metric on , and also a generalized metric space, or a JS-metric space.

Definition 4 (see [7]). Suppose that is a JS-metric space and .

(1)If for some , then we say that -converges to .(2)If , then we say that is -Cauchy.(3)If any -Cauchy sequence in -converges to some in , then we say that is -complete.

Proposition 5 (see [7]). Suppose that is a JS-metric space. Let and . If and , then .

Definition 6 (see [7]). Suppose that is a JS-metric space. Let be a self-mapping on and . If implies for some , then we say that is continuous at . Moreover, if is continuous at every , then we say is continuous.

Next, Martínez-Moreno et al. [9] had a new perspective to obtain common fixed points of particular contractive mappings on a space with two metrics.

Inspired by the above, we consider some existence results for a coincidence point of two functions when the ordered pair of these functions is an -contraction on a space with two JS-metrics. In addition, some examples and an application of an integral equation are presented.

2. Main Results

First, we assume throughout this section that all functions and are self-mappings defined on .

Definition 7. For , if

(1) implies ,(2) and imply ,

then we say that is triangular -admissible with respect to .

Next, we define as the class of mappings such that

Note that is defined at . Also, let be a nondecreasing continuous function satisfying

Denote the class of all such functions by .

Definition 8. Suppose that is a JS-metric space. If

(1) is triangular -admissible with respect to ,(2)there exist and ,

 for all ,

then we say that the pair is an -contraction with respect to .

Here, we are interested in the existence of a coincidence point of and , where is an -contraction with respect to some generalized metric on . This can be done under suitable relations between and .

Definition 9. Suppose that and are two JS-metric spaces and . If and are functions such that being -Cauchy in implies is -Cauchy in , then we say that is -Cauchy on .

Last but not least, we need the comparison notations for any two generalized metrics. If and are two generalized metrics on , the notation represents for every . If the inequality fails for some , we use the notation . All other inequality signs can be defined in the same fashion.

Theorem 10. Suppose that is a -complete JS-metric space and is a generalized metric on . If

(i) is an -contraction with respect to ,(ii) is a subspace of ,(iii)there is a , , and ,(iv) is -Cauchy on whenever ,(v) and commute,(vi) are continuous,

then there exists a coincidence point of and .

Proof. From assumption (iii), let such that and Since and , a sequence in can be constructed such that

for all . Observe that if for some , then , and so we are done. Assume that for each .
Since and is triangular -admissible with respect to , . Repeating this process inductively, we obtain that

for each .
Our task is now to prove that is -Cauchy.
Assume that this is not true. Equivalently, there is an so that, for each ,

for some . By inequality (6), together with the assumption that is triangular -admissible with respect to , we have that

for all . Then, by assumption (i),

Continuing to apply this concept totally times, we finally get the equality

Let such that

Denote
Notice that if , then , and so

which contradicts inequality (7).
Thus, . Then, there is a subsequence of which converges to 1. Without loss of generality, assume that

By the definition of ,

Therefore, there exists a such that

It follows that, also from inequality (7),

This is a contradiction since is nondecreasing. Thus, is -Cauchy.
The next goal is to show that is also -Cauchy. It can be observed that if , we are done. Assume that . Since is -Cauchy on , is -Cauchy. Consequently,

and so is -Cauchy.
Since is -complete, one can find a satisfying

That is,

By assumption (vi),

Since and commute, . This completes the proof.

As a consequence of Theorem 10, if , then we have the following theorem. Besides, we can replace properties (v) and (vi) by other conditions as stated in the theorem below.

Theorem 11. Suppose that is a -complete JS-metric space. Assume that

(i) is an -contraction with respect to ;(ii) is a subspace of ;(iii)there is a , , and ;(iv)either (a) or (b) holds:

(a) and are continuous mappings that commute.(b) is -complete and, for any satisfying for each , if there is a such that , then for all .

Then, there exists a coincidence point of and .

Proof. It is easy to see that if condition (a) is true, then applying Theorem 10 to the case yields the desired result. Assume that statement (a) does not hold. Thus, (b) must be valid. Let and for each . An argument similar to the one used in Theorem 10 shows that is -Cauchy and for all . From (b),

for some . That is,

Again, since condition (b) holds, for any , . From (4) and assumption (i), it follows that

Since is nondecreasing,

and so

Consider

for some . Therefore, . Hence, , completing the proof.

Adding some extra condition to Theorem 10, the coincidence point is actually a common fixed point. This can be shown in the following theorem. Denote

Theorem 12. Suppose that is a -complete JS-metric space and is a generalized metric on If all assumptions (i)-(vi) in Theorem 10 are satisfied and whenever , where , then there exists a common fixed point of and .

Proof. According to Theorem 10, . Then, we can let so that and .
Suppose that . By the assumption, . From the fact that is an -contraction with respect to , we have that

which leads to a contradiction. Therefore, .
Next, let . Since and commute, Thus, . Referring to the proof above, we can conclude that . Hence, the proof is complete.

We give examples to illustrate Theorems 10 and 11, respectively.

Example 13. Suppose that . Given the generalized metrics and on defined by

and

where and is a real number such that , we have that is -complete. Let be a function defined by

Given the self-mappings and on defined by

some tedious manipulation yields assumptions (ii), (v), and (vi) in Theorem 10. Further, notice that such that and .
Claim 1. is triangular -admissible with respect to .
Let . Assume that . Then, and or . Accordingly, , and or . It follows that , and or . Therefore, .
Next, assume that and . It can be observed that if , then , and if , then . That is, or . Therefore, . Thus, we have Claim 1.
Claim 2. is an -contraction with respect to , where and are as follows:

for .
Given , if , inequality (4) holds. Assume that . Similar as above, . Consider the following cases.
Case 1. . We have that

Case 2. . Then, . Consider

Therefore, is a -contraction with respect to .
Claim 3. is -Cauchy on .
Notice that . Suppose that such that is -Cauchy. Let There is a so that, for all , we have that

With this ,

for any . This completes Claim 3.
Thus, by Theorem 10, and have a coincidence point, precisely, 0.

Example 14. Let . Therefore, is -complete, where is the generalized metric as defined in Example 13. Suppose that is a function as follows:

Define self-mappings and on by

Note that and is -complete. Moreover, we have such that and .
Claim 1. is triangular -admissible with respect to .
Let . Assume that . Then, or . That is, or . Thus, or . Therefore, .
Similar as in the proof of the previous example, if and , then . Therefore, Claim 1 is obtained.
Claim 2. is an -contraction with respect to , where and defined by and for , respectively.
Suppose . If , then inequality (4) holds. Assume that . Consider the following cases.
Case 1. . We have that

Case 2. . Then, . Consider

Therefore, we have Claim 2.
Next, let such that, for any , . Assume that for some . By the definition of , for each , or . Fix . If , then . Assume that . Suppose that . Then, . This is a contradiction since . Thus, . Therefore, we get that . Since is arbitrary, this is true for every . Hence, by Theorem 11, is a coincidence point of and .

3. Application

We wish to apply our finding to the existence problem of a solution to the integral equation. This is one of the crucial uses of fixed point theorems that can be found in the literatures (see [10–13]).

for , where is a real number such that .

Suppose that and

for . We have that is a -complete JS-metric space. The following theorem shows when the equation (42) has a solution if the integral equation is homogeneous.

Theorem 15. Consider equation (42). Suppose that

(i) is continuous;(ii)for any , if , then and

 where ; (iii)there is a , , where ;(iv).

Then, the integral equation (42) has a solution.

Proof. Define the self-mappings and on as follows:

and for and . Suppose that is a function defined by

Clearly, , and and are continuous mappings that commute. Moreover, by assumptions and , it is straightforward to show that condition of Theorem 11 is satisfied. Our problem reduces to show that the pair is an -contraction with respect to for some and .
First, we establish that is triangular--admissible with respect to .
Assume that . Then, ; that is, for any . By assumption (ii), . Therefore,

That is, . It is simple to show that the second condition of Definition 7 holds. Thus, is triangular--admissible with respect to .
Finally, it remains to prove inequality (4).
If for some , we are done. Suppose that for all . From assumption , consider

and, then,

This gives us the desired inequality for and

for some , where . Thus, is an -contraction with respect to .
Hence, there exists a coincidence point of and which is a solution to the integral equation (42).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

I would like to thank the editor and the referees for their comments and suggestions on the manuscript. This research is supported by Chiang Mai University, Thailand.

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