On Coupled Common Fixed Point Theorems for Nonlinear Contractions with the Mixed Weakly Monotone Property in Partially Ordered S -Metric Spaces

Abstract

The main aim of this paper is to establish some coupled common fixed point theorems under a Geraghty-type contraction using mixed weakly monotone property in partially ordered S -metric space. Also, we give some sufficient conditions for the uniqueness of a coupled common fixed point. Some examples are provided to demonstrate the validity of our results.

Mi Zhou 1 and Xiao-Lan Liu 2,3

Academic Editor:Hugo Leiva

1, School of Polytechnics, Sanya University, Sanya, Hainan 572000, China
2, Department of Mathematics, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China
3, Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing, Zigong, Sichuan 643000, China

Received 18 November 2015; Accepted 29 December 2015

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.

1. Introduction and Preliminaries

One of the most important results in fixed point theory is the Banach Contraction Principle ( [figure omitted; refer to PDF] for short) proposed by Banach [1]. After that, there were many authors who have studied and proved the results for fixed point theory by generalizing the Banach Contraction Principle in several directions. One of the celebrated results was given by Geraghty [2].

For the sake of convenience, we recall Geraghty's theorem. Let [figure omitted; refer to PDF] be the family of all functions [figure omitted; refer to PDF] satisfying the condition: [figure omitted; refer to PDF] Geraghty [2] proved the following unique fixed point theorem in complete metric spaces.

Theorem 1 (see [2]).

Let [figure omitted; refer to PDF] be a complete metric space and let [figure omitted; refer to PDF] be an operator. Suppose that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] has a unique fixed point [figure omitted; refer to PDF] .

Later, Amini-Harandi and Emami [3] generalized this result to the setting of partially ordered metric spaces as follows.

Theorem 2 (see [3]).

Let [figure omitted; refer to PDF] be a complete partially ordered metric space and let [figure omitted; refer to PDF] be an increasing self-mapping such that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Suppose that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . Then, in each of the following two cases, the mapping [figure omitted; refer to PDF] has at least one fixed point in [figure omitted; refer to PDF] :

(1) [figure omitted; refer to PDF] is continuous or,

(2) for any nondecreasing sequence [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

If, moreover, for all [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] comparable with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then the fixed point of [figure omitted; refer to PDF] is unique.

For more generalizations of Theorems 1 and 2, see [4-7].

On the other hand, several authors have studied fixed point theory in generalized metric spaces. For details, we refer readers to [8-13]. In 2012, Sedghi et al. [14] have introduced the notion of an [figure omitted; refer to PDF] -metric space and proved that this notion is a generalization of a metric space. Also, they have proved some properties of [figure omitted; refer to PDF] -metric spaces and some fixed point theorems for a self-map on an [figure omitted; refer to PDF] -metric space. An interesting work is that we can naturally transport certain results in metric spaces and known generalized metric spaces to [figure omitted; refer to PDF] -metric spaces. After that, Sedghi and Dung [15] proved a general fixed point theorem in [figure omitted; refer to PDF] -metric spaces which is a generalization of [14, Theorem 3.1] and obtained many analogues of fixed point theorems in metric spaces for [figure omitted; refer to PDF] -metric spaces. In [16], Gordji et al. have introduced the concept of a mixed weakly monotone pair of maps and proved some coupled common fixed point theorems for contractive-type maps using the mixed weakly monotone property in partially ordered metric spaces. These results are of particular interest to state coupled common fixed point theorems for maps with mixed weakly monotone property in partially ordered [figure omitted; refer to PDF] -metric spaces. In 2013, Dung [17] used the notion of a mixed weakly monotone pair of maps to state a coupled common fixed point theorem for maps on partially ordered [figure omitted; refer to PDF] -metric spaces and generalized the main results of [16-18] into the structure of [figure omitted; refer to PDF] -metric spaces.

In this paper, motivated by the developments discussed above, we state some coupled common fixed point theorems for a pair of mappings with the mixed weakly monotone property satisfying a generalized contraction by using the ideas of Geraghty [2] in partially ordered [figure omitted; refer to PDF] -metric spaces. Also, we give some sufficient conditions for the uniqueness of a coupled common fixed point. Some examples are provided to illustrate our main theorems.

In the sequel, the letters [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] will denote the set of all real numbers, the set of all nonnegative real numbers, and the set of all positive integers, respectively.

Let [figure omitted; refer to PDF] be a partially ordered set. Then [figure omitted; refer to PDF] is a partially ordered set with partial order [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF]

Definition 3 ([14, Definition 2.1]).

Let [figure omitted; refer to PDF] be a nonempty set. An [figure omitted; refer to PDF] -metric on [figure omitted; refer to PDF] is a function [figure omitted; refer to PDF] that satisfies the following conditions for all [figure omitted; refer to PDF] :

(1) [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF]

(2) [figure omitted; refer to PDF]

The pair [figure omitted; refer to PDF] is called an [figure omitted; refer to PDF] -metric space.

The following is an intuitive geometric example for [figure omitted; refer to PDF] -metric spaces.

Example 4 ([14, Example 2.4]).

Let [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be an ordinary metric on [figure omitted; refer to PDF] . Put [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] is the perimeter of the triangle given by [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -metric on [figure omitted; refer to PDF] .

Lemma 5 ([17, Lemma 1.4]).

Let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -metric space. Then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF]

Lemma 6 ([14, Lemma 2.5]).

Let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -metric space. Then [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF]

Lemma 7 (see [16]).

Let [figure omitted; refer to PDF] be a metric space. Then [figure omitted; refer to PDF] is a metric space with metric [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Lemma 8.

Let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -metric space. Then [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -metric space with [figure omitted; refer to PDF] -metric [figure omitted; refer to PDF] given by [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Proof.

For all [figure omitted; refer to PDF] , it is obvious that the first condition of [figure omitted; refer to PDF] -metric for [figure omitted; refer to PDF] holds true.

We only need to check the second condition of [figure omitted; refer to PDF] -metric: [figure omitted; refer to PDF] By the above, [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -metric on [figure omitted; refer to PDF] .

Definition 9 ([16, Definition 1.5]).

Let [figure omitted; refer to PDF] be a partially ordered set and let [figure omitted; refer to PDF] be two maps. We say the pair [figure omitted; refer to PDF] has the mixed weakly monotone property on [figure omitted; refer to PDF] if for all [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF]

Example 10 ([16, Example 1.6]).

Let [figure omitted; refer to PDF] be two functions given by [figure omitted; refer to PDF] Then the pair [figure omitted; refer to PDF] has the mixed weakly monotone property.

Definition 11 ([16, Definition 1.1]).

Let [figure omitted; refer to PDF] be a partially ordered set and let [figure omitted; refer to PDF] be a map. We say the pair [figure omitted; refer to PDF] has the mixed monotone property on [figure omitted; refer to PDF] if for all [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF]

Remark 12 ([17, Remark 1.20]).

Let [figure omitted; refer to PDF] be a partially ordered set; let [figure omitted; refer to PDF] be a map with the mixed monotone property on [figure omitted; refer to PDF] . Then, for all [figure omitted; refer to PDF] , the pair [figure omitted; refer to PDF] has the mixed weakly monotone property on [figure omitted; refer to PDF] .

Definition 13.

An element [figure omitted; refer to PDF] is called a

(1) coupled fixed point of a mapping [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ;

(2) coupled common fixed point of two mappings [figure omitted; refer to PDF] if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

2. Main Results

In this section, we establish some coupled common fixed point theorems by considering mappings on generalized metric spaces endowed with partial order. Before proceeding further, first, we define the following function which will be used in our results.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be any two sequences of nonnegative real numbers. Define with [figure omitted; refer to PDF] the set of all functions [figure omitted; refer to PDF] which, satisfying [figure omitted; refer to PDF] , implies [figure omitted; refer to PDF] .

Some examples of such a function are as follows.

Example 14.

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF]

Example 15.

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF]

Example 16.

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF]

Example 17.

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF]

Theorem 18.

Let [figure omitted; refer to PDF] be a partially ordered [figure omitted; refer to PDF] -metric space; let [figure omitted; refer to PDF] be two maps such that

(1) [figure omitted; refer to PDF] is complete;

(2) the pair [figure omitted; refer to PDF] has the mixed weakly monotone property on [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

(3) assume that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(4) [figure omitted; refer to PDF] or [figure omitted; refer to PDF] is continuous.

Then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a coupled common fixed point in [figure omitted; refer to PDF] .

Proof.

Step 1 . We construct two Cauchy sequences in [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Put [figure omitted; refer to PDF]

From the choice of [figure omitted; refer to PDF] and the fact that [figure omitted; refer to PDF] has mixed weakly monotone property we have [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Continuing this way, we obtain [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Therefore, the sequences [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are monotone: [figure omitted; refer to PDF] Assume that there exists a nonnegative integer [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] From the definition of [figure omitted; refer to PDF] -metric space, we have [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . It follows from (21) that [figure omitted; refer to PDF] is a coupled common fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Now, we suppose that for all nonnegative [figure omitted; refer to PDF] [figure omitted; refer to PDF] Using (18) and (21), for [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] For all [figure omitted; refer to PDF] , write [figure omitted; refer to PDF] and then the sequence [figure omitted; refer to PDF] is monotone decreasing. Therefore, there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] We claim that [figure omitted; refer to PDF] . On the contrary, suppose that [figure omitted; refer to PDF] , and we have from (26) that [figure omitted; refer to PDF] Letting [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] Using the property of the function [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] So, we have [figure omitted; refer to PDF] which contradicts the assumption [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] .

Analogously to [figure omitted; refer to PDF] , we also have [figure omitted; refer to PDF] Thus, we have [figure omitted; refer to PDF] Now, we have to prove that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two Cauchy sequences in the [figure omitted; refer to PDF] -metric space [figure omitted; refer to PDF] .

For all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , by using Lemma 5, we have that [figure omitted; refer to PDF] Taking the limit as [figure omitted; refer to PDF] and using (35), we obtain [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] By interchanging the roles of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and proceeding along the arguments discussed above, we also obtain that [figure omitted; refer to PDF] Hence, for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] It implies that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are two Cauchy sequences in the [figure omitted; refer to PDF] -metric space [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is a complete [figure omitted; refer to PDF] -metric space, hence [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -convergent. Then there exist [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

Step 2 . We prove that [figure omitted; refer to PDF] is a coupled common fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

We consider the following two cases.

Case 1 ( [figure omitted; refer to PDF] is continuous). We have [figure omitted; refer to PDF] Now using (18), we have [figure omitted; refer to PDF] That is, [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Therefore, [figure omitted; refer to PDF] is a coupled common fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Case 2 ( [figure omitted; refer to PDF] is continuous). We also prove that [figure omitted; refer to PDF] is a coupled common fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] similarly as in Case 1.

Theorem 19.

Let [figure omitted; refer to PDF] be a partially ordered [figure omitted; refer to PDF] -metric space; let [figure omitted; refer to PDF] be two maps such that

(1) [figure omitted; refer to PDF] is complete;

(2) the pair [figure omitted; refer to PDF] has the mixed weakly monotone property on [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

(3) assume that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(4) [figure omitted; refer to PDF] has the following properties:

(a) If [figure omitted; refer to PDF] is an increasing sequence with [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

(b) If [figure omitted; refer to PDF] is a decreasing sequence with [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Then, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a coupled common fixed point in [figure omitted; refer to PDF] .

Proof.

Proceeding along the same steps as in Theorem 18, we obtain a nondecreasing sequence [figure omitted; refer to PDF] converging to [figure omitted; refer to PDF] and a nonincreasing sequence [figure omitted; refer to PDF] converging to [figure omitted; refer to PDF] , for some [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , then by construction, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] is a coupled common fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . So we assume either [figure omitted; refer to PDF] or [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Then by using (18) and Lemma 5, we have [figure omitted; refer to PDF] Letting [figure omitted; refer to PDF] in the above inequality, we get [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . By interchanging the roles of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and using the same method mentioned above, we also get [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Hence, [figure omitted; refer to PDF] is a coupled common fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Corollary 20.

Let [figure omitted; refer to PDF] be a partially ordered set and let [figure omitted; refer to PDF] be an [figure omitted; refer to PDF] -metric on [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] is a complete [figure omitted; refer to PDF] -metric space. Suppose that [figure omitted; refer to PDF] are two maps having the mixed weakly monotone property and assume that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Suppose that either

(1) [figure omitted; refer to PDF] or [figure omitted; refer to PDF] is continuous;

(2) [figure omitted; refer to PDF] has the following property:

(a) If [figure omitted; refer to PDF] is an increasing sequence with [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

(b) If [figure omitted; refer to PDF] is a decreasing sequence with [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

If there exist [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a coupled common fixed point in [figure omitted; refer to PDF] .

Proof.

For all [figure omitted; refer to PDF] , write [figure omitted; refer to PDF] Adding (49) and (50), we get [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] .

It is easy to verify that [figure omitted; refer to PDF] . Applying Theorems 18 and 19, we get desired result.

Remark 21.

Taking [figure omitted; refer to PDF] in Corollary 20 for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we get the following corollary coinciding with [17, Corollary 2.4].

Corollary 22.

In addition to the hypotheses of Corollary 20, suppose that for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and some [figure omitted; refer to PDF] , inequality (49) in Corollary 20 is replaced by [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a coupled common fixed point in [figure omitted; refer to PDF] .

By choosing [figure omitted; refer to PDF] in Theorems 18 and 19 and using Remark 12, we get coupled fixed point theorem of [figure omitted; refer to PDF] written by the following corollary.

Corollary 23.

Let [figure omitted; refer to PDF] be a partially ordered [figure omitted; refer to PDF] -metric space and let [figure omitted; refer to PDF] be a map such that

(1) [figure omitted; refer to PDF] is complete;

(2) [figure omitted; refer to PDF] has the mixed monotone property on [figure omitted; refer to PDF] ; [figure omitted; refer to PDF]

(3) assume that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

: for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(4) [figure omitted; refer to PDF] is continuous or [figure omitted; refer to PDF] has the following properties:

(a) If [figure omitted; refer to PDF] is an increasing sequence with [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

(b) If [figure omitted; refer to PDF] is a decreasing sequence with [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Then [figure omitted; refer to PDF] has a coupled common fixed point in [figure omitted; refer to PDF] .

Theorem 24.

In addition to the hypotheses of Theorem 18, suppose that, for all [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] that is comparable with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a unique coupled fixed point in [figure omitted; refer to PDF] .

Proof.

By Theorem 18, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a coupled common fixed point [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be another coupled common fixed point of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

By assumption, there exists [figure omitted; refer to PDF] that is comparable to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Put [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and choose [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Using the same construction as in the proof of Theorem 18, we have two sequences [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is comparable to [figure omitted; refer to PDF] , we can assume that [figure omitted; refer to PDF] .

Then it is easy to show that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are comparable; that is, [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] .

For [figure omitted; refer to PDF] , from (18) we have [figure omitted; refer to PDF] which implies [figure omitted; refer to PDF] We see that the sequence [figure omitted; refer to PDF] is decreasing, and there exist [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Now, we have to show that [figure omitted; refer to PDF] . On the contrary, suppose that [figure omitted; refer to PDF] . Following the same arguments as in the proof of Theorem 18, we obtain [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] This implies [figure omitted; refer to PDF] which is not possible in virtue of (30). Hence, [figure omitted; refer to PDF] . Therefore, (59) becomes [figure omitted; refer to PDF] Similarly, we can get that [figure omitted; refer to PDF] Using (63)-(64), the second condition of [figure omitted; refer to PDF] -metric, and taking the limit [figure omitted; refer to PDF] , we obtain that [figure omitted; refer to PDF] Thus, we conclude that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Analogous to [figure omitted; refer to PDF] , by interchanging the roles of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , (65) holds true for [figure omitted; refer to PDF] .

Therefore, we conclude that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a unique coupled common fixed point.

Similarly, we can prove the following theorem.

Theorem 25.

In addition to the hypotheses of Theorem 19, suppose that, for all [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] that is comparable with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] have a unique coupled fixed point in [figure omitted; refer to PDF] .

Finally, we give some examples to demonstrate the validity of our results.

Example 26.

Let [figure omitted; refer to PDF] , with the [figure omitted; refer to PDF] -metric defined by [figure omitted; refer to PDF] and the natural ordering of real numbers [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a totally ordered, complete [figure omitted; refer to PDF] -metric space.

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF] For all [figure omitted; refer to PDF] , put [figure omitted; refer to PDF] .

The pair [figure omitted; refer to PDF] has the mixed weakly monotone property and [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have that [figure omitted; refer to PDF] Then the contractive condition (18) in Theorem 18 holds, and [figure omitted; refer to PDF] is the unique coupled common fixed point.

Example 27.

Let [figure omitted; refer to PDF] , with the [figure omitted; refer to PDF] -metric defined by [figure omitted; refer to PDF] and the natural ordering of real numbers [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a totally ordered, complete [figure omitted; refer to PDF] -metric space.

Let [figure omitted; refer to PDF] be defined by [figure omitted; refer to PDF] For all [figure omitted; refer to PDF] , put [figure omitted; refer to PDF] .

The pair [figure omitted; refer to PDF] has the mixed weakly monotone property.

[figure omitted; refer to PDF] with [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , we have that [figure omitted; refer to PDF] Then the contractive condition (18) in Theorem 18 holds, and [figure omitted; refer to PDF] is the unique coupled common fixed point.

Acknowledgments

The authors thank Dr. Dolicanin-Djekic Diana (Faculty of Technical Science, University of Pristina-Kosovska Mitrovica, Serbia) for her editing and polishing work for the revised paper. This work is partially supported by Natural Science Foundation of China (Grant no. 61573010), Natural Science Foundation of Hainan Province (Grant no. 114014), Opening Project of Sichuan Province University Key Laboratory of Bridge Non-Destruction Detecting and Engineering Computing (2015QZJ01), Artificial Intelligence of Key Laboratory of Sichuan Province (2015RZJ01), Scientific Research Fund of Sichuan Provincial Education Department (14ZB0208 and 16ZA0256), and Scientific Research Fund of Sichuan University of Science and Engineering (2014RC01 and 2014RC03).

Conflict of Interests

The authors declare that they have no competing interests.

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