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`Journal of Function SpacesVolume 2018, Article ID 1378379, 13 pageshttps://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

Department of Mathematics, Nanchang University, Nanchang 330031, China

Correspondence should be addressed to Chuanxi Zhu; moc.621@uhzixnauhc

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

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 [29]). 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 [1319]). 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)wherewith 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 obtainSimilarly, since , we getNoting 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 haveNow, suppose that for all , which means that either or . Since and , we obtainwhere Similarly, we havewhere 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 ThusNext, we show that and Assume that this is not true; that is, Then there exist , , and two subsequences of with , such thatWith the definitions of metric-like spaces, we obtain which yields that Similarly, we have Thus, using (19) and (21), we obtainSince taking the limit as , we get Similarly, we have Then it follows from the above inequalities thatNow, 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 haveSince 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,