Journal of Function Spaces
Volume 2018 (2018), Article ID 1378379, 13 pages
https://doi.org/10.1155/2018/1378379

Research Article

Some New Coupled Coincidence Point and Coupled Fixed Point Results in Partially Ordered Metric-Like Spaces and an Application

Min Liang, Chuanxi Zhu, Zhaoqi Wu, and Chunfang Chen

Department of Mathematics, Nanchang University, Nanchang 330031, China

Correspondence should be addressed to Chuanxi Zhu; chuanxizhu@126.com

Received 28 March 2018; Revised 9 July 2018; Accepted 15 July 2018; Published 1 August 2018

Academic Editor: Ismat Beg

Copyright © 2018 Min Liang 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.

Some new coupled coincidence point and coupled fixed point theorems are established in partially ordered metric-like spaces, which generalize many results in corresponding literatures. An example is given to support our main results. As an application, we discuss the existence of the solutions for a class of nonlinear integral equations.

1. Introduction and Preliminaries

As we all know, the fixed point theory is one of the most important tools in the field of nonlinear analysis. In particular, the coupled fixed point theorems in partially ordered metric-like spaces are very valuable for discussing the existence and uniqueness of solutions of any nonlinear problem in fields of mathematics and physics.

In order to solve the more complex nonlinear analysis problems, the concept of a metric space has been extended in many aspects. In 1992, the notion of a partial metric space was introduced by Matthews [1]. Considering may be greater than zero, he tried to extend the concept of a metric space. Since then, many authors proved various fixed point theorems in partial metric spaces (see [2–9]). In the interest of studying the existence of solutions of ordinary differential equations, the mixed monotone mapping was established. Afterward, Lakshmikantham and Ćirić [10] substituted the mixed monotone mapping for the mixed monotone mapping and proved the coupled common fixed point theorems. On the other hand, Hitzler and Seda [11] first studied the dislocated space in 2000, and then Amini-Harandi [12] named it as a metric-like space. Subsequently, some authors discussed the fixed point and coincidence point results in generalized metric spaces. And several applications to operator equations and integral equations are given in a line of research (see [13–19]). Recently, B. Hazarika et al. [20] generalized some previous results and gave some new common fixed point theorems in metric-like spaces.

In this paper, inspired by the above literatures, we propose some new coupled coincidence point and coupled fixed point theorems in partially ordered metric-like spaces, which extend the theorems of B. Hazarika et al. [20]. As an application, we discuss the existence of solutions for a system of nonlinear integral equations to illustrate our main results.

First, we review some concepts which are going to be used later.

Throughout this paper, let , , and Let be the set of all natural numbers and be the set of all positive integer numbers.

Definition 1 (see [12]). Let be a nonempty set. A function is said to be a dislocated (metric-like) metric on if, for all , the following conditions hold:
() .
() .
() .

Definition 2 (see [8]). Let be a metric-like space.
() A sequence in is said to be a Cauchy sequence if exists and is finite.
() is said to be complete if every Cauchy sequence in converges with respect to to a point such that

Definition 3 (see [6]). Let be two functions. A pair of functions is said to belong to the class if they satisfy the following conditions:
(i) For if , then .
(ii) For with , if for all , then

Definition 4 (see [7]). Let be the class of all functions satisfying the following conditions:
() for all .
() for all .
() is continuous.

Definition 5 (see [9]). Let be a partially ordered set and Then the map is said to have mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in ; that is,

Definition 6 (see [10]). Let be a partially ordered set and Then the map is said to have mixed -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in ; that is,

Definition 7 (see [9]). An element is called a coupled fixed point for the mapping if and

Definition 8 (see [10]). Let be a nonempty set. We say that the mappings and are commutative if , for all

Definition 9 (see [10]). An element is called a coupled coincidence point of the mapping and if and

2. Main Results

In this section, we put forward a new coupled coincidence point theorem in a partially ordered metric-like space, and if , which is a self-mapping, we get a new coupled fixed point theorem. Then, we establish a common coupled fixed point theorem in a partially ordered metric-like space and prove the uniqueness of the coupled fixed point.

Theorem 10. Let be a partially ordered set and be a complete metric-like space. Let and be two mappings such that the following conditions are satisfied:
(i) .
(ii) is closed.
(iii) has the mixed -monotone property.
(iv) There exist such that and .
(v)

where

with and , or and for all , is nondecreasing, and is a lower semicontinuous function.
Then and have a coupled coincidence point.

Proof. Let such that and . Since , we can find such that and . Similarly, there exist such that and . Repeating the above process, we can construct two sequences such that and . Since has the mixed -monotone property, we get and If for some and we have , then . This means and have a coupled coincidence point.
First, we show that if and have a coupled coincidence point, i.e., , then Indeed, we have , and supposing and using (4), we obtain

Similarly, since , we get

Noting that is nondecreasing and using (6) and (7), we have

By Definition 3, we obtain that if , then On the other hand, through the definition of , we can reach the following conclusions:

This is a contradiction. Hence, if there exist such that , we have

Now, suppose that for all , which means that either or . Since and , we obtain

where

Similarly, we have

where

Noting that is nondecreasing, using (11) and (13) we get

Using condition (i) of Definition 3, we get

which implies that , is a decreasing nonnegative sequence. Therefore, there exists , such that

Using condition (ii) of Definition 3, we get . Then

Thus

Next, we show that and Assume that this is not true; that is,

Then there exist , , and two subsequences of with , such that

With the definitions of metric-like spaces, we obtain

which yields that

Similarly, we have

Thus, using (19) and (21), we obtain

Since

taking the limit as , we get

Similarly, we have

Then it follows from the above inequalities that

Now, using (4), we have

where

Since

there exists with , such that

for all Hence, we get

for all Similarly, there exists with , such that

for all Letting and noting that is nondecreasing, we obtain

for all Taking the limit as and using (25) and (29), we have

According to the properties of , we get , which is a contradiction. So, and are -Cauchy sequences in , which is a complete metric-like space. Since and is closed, there exist , such that Correspondingly, we have

Since is a lower semicontinuous function, then We have By same arguments, we can derive that
Finally, we claim that and have a coupled coincidence point. It follows from (4) that

Similarly, we have

Letting and using (38), we obtain

Therefore, we get

We can take advantage of the properties of to get

By triangle inequality in metric-like space, we have

Letting , we get Similarly, we have , that is, Therefore, and have a coupled coincidence point.

Theorem 11. Define the partial order in by Add to the hypotheses of Theorem 10 the following conditions:
(i) and commute at their coincidence points.
(ii) For every and in , there exists such that is comparable to and .
Then and have a unique common coupled fixed point.

Proof. First, by Theorem 10, we know that the set of coupled coincidence points is nonempty. Let and be coupled coincidence points, i.e., and . Then we need to verify

From (ii), there exists such that is comparable to , , and . Letting , we can choose such that Using this method, we can choose such that Repeating the procedures, we have and such that Similarly, letting , we get , and , Since , we get And we have , so , that is, By condition (iii) of Theorem 10, we have Using (4), we obtain

where

Similarly,

Noting that is nondecreasing, we have

so , is a nonnegative decreasing sequence, and

which implies that

Through the same process, we can prove that

Using triangle inequality, we have

Letting , we get , which implies that
Denote We now show that is a coupled common fixed point. Since and commute at their coincidence points, then we have

Hence, () is a coupled coincidence point of and . Using (45), we have , and, therefore, , which means () is a coupled common fixed point.
Finally, we prove the existence and uniqueness of coupled common fixed point. If is also a coupled common fixed point, i.e., , then is also a coupled coincidence point. We have Therefore, and have a unique common coupled fixed point.

Corollary 12. Define the partial order in by Add to the hypotheses of Theorem 10 the following conditions:
(i) and commute at their coincidence points.
(ii) is a totally ordered subset of .
Then and have a unique common coupled fixed point.

Proof. It is easy to see that condition (ii) of Theorem 11 can be naturally established.

Theorem 13. Under the hypotheses of Theorem 11, add the following condition: suppose that Then and have a unique common coupled fixed point of the form .

Proof. Through the proof of Theorem 10, we construct the sequences such that , and and And is closed; then there exist , such that In Theorem 11, we prove that is the unique coupled common fixed point of and , i.e.,
Next, we shall show Owing to the supplementary condition, we have From (4), we obtain

By (10), we get , and thus we have

So (55) can be converted into

which implies that Then and have a unique common coupled fixed point of the form .

Example 14. Let and be defined by

Then () is a complete metric-like space, rather than a partial metric space. Indeed, if , we obtain Let be defined as

Define function by Let be defined by

Let be defined by and be the function defined by

It is clear that has the mixed -monotone property. Easily, we can know is closed and . Letting , then is true. And if , we have ; i.e., is established, so And , and then and are commutative. Next, we prove that (4) holds for all with We can distinguish three cases.

Case 1 (). If , we get and

If , we get or

Then we have

Hence, is proved.

Case 2 (). If , we have and

If , we have and

Then we have

Hence, is tenable.

Case 3 ().

Hence, (4) is established. And (4) still holds with

Note that all conditions of Theorems 10–13 are satisfied. Hence, and have a unique common coupled fixed point, which is

3. Consequences of the Main Result

By choosing suitable mappings , and , one can deduce subsequent corollaries.

Letting (the identity mapping on ) in Theorem 10, we get the following corollary.

Corollary 15. Let be a partially ordered set and be a complete metric-like space. Let be a mapping, satisfying the following conditions:
(i) has the mixed monotone property.
(ii) There exist such that and .
(iii) , where

with and , or and , for all , is nondecreasing, and is a lower semicontinuous function.
Then has a coupled fixed point.

Setting and in Theorem 10, we get another corollary.

Corollary 16. Let be a partially ordered set and be a complete metric-like space. Let and be two mappings such that the following conditions are satisfied:
(i) .
(ii) is closed.
(iii) has the mixed -monotone property.
(iv) There exist such that .
(v)

with and , or and , for all , and is nondecreasing.
Then and have a coupled coincidence point.

Corollary 17. Let be a partially ordered set and be a complete metric-like space. Let and be two mappings, such that the following conditions are satisfied:
(i) .
(ii) is closed.
(iii) has the mixed -monotone property.
(iv) There exist such that .
(v)

with and , or and , for all ,
Then and have a coupled coincidence point.

Proof. Denote , we can note that

Hence, the conditions of Corollary 16 can be satisfied.

Since metric spaces are metric-like spaces, corresponding theorems in metric spaces can be derived. On the other hand, according to the definition of , if , we obtain Now we have the following result in metric spaces.

Theorem 18. Let be a partially ordered set and be a complete metric space. Let and be two mappings such that the following conditions are satisfied:
(i) .
(ii) is closed.
(iii) has the mixed -monotone property.
(iv) There exist such that and .
(v) , where

with and , or and , for all , is nondecreasing, and is a lower semicontinuous function.
Then and have a coupled coincidence point.

4. An Application

In this section, we study the existence of a unique solution to a system of nonlinear integral equations by applying a coupled fixed point result in Section 3. Consider the following system of equations:

Now, we discuss the existence of the unique solution.

Theorem 19. Let be the set of all continuous functions defined on with a dislocated metric given by Assume that has the mixed monotone property, and is defined by

which satisfy the following conditions:
(i) is continuous.
(ii) and are continuous. There exists a constant such that And , where is a constant real number.
(iii) is nondecreasing and . There exists a constant such that for all ,

(iv)

If , or , and there exist such that and , then has a coupled fixed point; i.e., (83) has a unique solution in

Proof. Define Letting , or , and , we have

Hence, we get On the other hand, by the assumptions of the theorem, the other conditions of Corollary 16 are satisfied with Then has a coupled fixed point; i.e., (73) has a unique solution in

Example 20. Consider the following system of nonlinear integral equations:

where Then we can define the functions by

It is clear that and (i=1,2) are continuous functions, and Supposing , we obtain with and is nondecreasing. Since

we can get And we have

So, letting , we have

Through calculation, we can see Hence, all conditions of Theorem 19 are satisfied; then the integral equations have a unique solution in , where

Remark 21. Notice that the given , which is defined by , is a metric. Next, we consider an application in a metric-like space. Let be the set of all continuous functions with a dislocated metric given by

Clearly, is a complete metric-like space but not a metric space.

Theorem 22. Let be the set of all continuous functions defined on with a dislocated metric given by Assume that has the mixed monotone property, and is defined by

which satisfy the following conditions:
(i) and are continuous. There exists a constant such that And , where is a constant real number.
(ii) is nondecreasing and . There exists a constant such that, for all ,

(iii)
If , or , and there exist such that and , then has a coupled fixed point; i.e., (83) has a unique solution in

Proof. Denote Let , or , and , and we have

Hence, we get Then, all the conditions of Corollary 16 are satisfied with Then has a coupled fixed point; i.e., (83) has a unique solution in

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The research is supported by the National Natural Science Foundation of China (11771198, 11701259, 11661053, 11461045, 11361042, and 11071108).

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