#### 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 -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.

#### 1. Introduction and Preliminaries

One of the most important results in fixed point theory is the Banach Contraction Principle ( for short) proposed by Banach . 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 .

For the sake of convenience, we recall Geraghty’s theorem. Let be the family of all functions satisfying the condition: Geraghty  proved the following unique fixed point theorem in complete metric spaces.

Theorem 1 (see ). Let be a complete metric space and let be an operator. Suppose that there exists such that for all . Then has a unique fixed point .

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

Theorem 2 (see ). Let be a complete partially ordered metric space and let be an increasing self-mapping such that there exists such that . Suppose that there exists such that for all satisfying or . Then, in each of the following two cases, the mapping has at least one fixed point in : (1)is continuous or,(2)for any nondecreasing sequence in , if as , then for all .If, moreover, for all , there exists comparable with and , then the fixed point of is unique.

For more generalizations of Theorems 1 and 2, see .

On the other hand, several authors have studied fixed point theory in generalized metric spaces. For details, we refer readers to . In 2012, Sedghi et al.  have introduced the notion of an -metric space and proved that this notion is a generalization of a metric space. Also, they have proved some properties of -metric spaces and some fixed point theorems for a self-map on an -metric space. An interesting work is that we can naturally transport certain results in metric spaces and known generalized metric spaces to -metric spaces. After that, Sedghi and Dung  proved a general fixed point theorem in -metric spaces which is a generalization of [14, Theorem 3.1] and obtained many analogues of fixed point theorems in metric spaces for -metric spaces. In , 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 -metric spaces. In 2013, Dung  used the notion of a mixed weakly monotone pair of maps to state a coupled common fixed point theorem for maps on partially ordered -metric spaces and generalized the main results of  into the structure of -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  in partially ordered -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 , , and will denote the set of all real numbers, the set of all nonnegative real numbers, and the set of all positive integers, respectively.

Let be a partially ordered set. Then is a partially ordered set with partial order defined by

Definition 3 ([14, Definition  2.1]). Let be a nonempty set. An -metric on is a function that satisfies the following conditions for all :(1) if and only if (2)The pair is called an -metric space.

The following is an intuitive geometric example for -metric spaces.

Example 4 ([14, Example  2.4]). Let and let be an ordinary metric on . Put for all ; that is, is the perimeter of the triangle given by . Then is an -metric on .

Lemma 5 ([17, Lemma  1.4]). Let be an -metric space. Then for all

Lemma 6 ([14, Lemma  2.5]). Let be an -metric space. Then , for all

Lemma 7 (see ). Let be a metric space. Then is a metric space with metric given by for all .

Lemma 8. Let be an -metric space. Then is an -metric space with -metric given by for all .

Proof. For all , it is obvious that the first condition of -metric for holds true.
We only need to check the second condition of -metric: By the above, is an -metric on .

Definition 9 ([16, Definition  1.5]). Let be a partially ordered set and let be two maps. We say the pair has the mixed weakly monotone property on if for all , we have

Example 10 ([16, Example  1.6]). Let be two functions given by Then the pair has the mixed weakly monotone property.

Definition 11 ([16, Definition  1.1]). Let be a partially ordered set and let be a map. We say the pair has the mixed monotone property on if for all , we have

Remark 12 ([17, Remark  1.20]). Let be a partially ordered set; let be a map with the mixed monotone property on . Then, for all , the pair has the mixed weakly monotone property on .

Definition 13. An element is called a(1)coupled fixed point of a mapping if and ;(2)coupled common fixed point of two mappings if and .

#### 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 and be any two sequences of nonnegative real numbers. Define with the set of all functions which, satisfying , implies .

Some examples of such a function are as follows.

Example 14. Let be defined by

Example 15. Let be defined by

Example 16. Let be defined by

Example 17. Let be defined by

Theorem 18. Let be a partially ordered -metric space; let be two maps such that(1) is complete;(2)the pair has the mixed weakly monotone property on :(3)assume that there exists such thatfor all with , ;(4) or is continuous.Then and have a coupled common fixed point in .

Proof.
Step  1. We construct two Cauchy sequences in .
Let be such that , .
Put
From the choice of and the fact that has mixed weakly monotone property we haveThus,Continuing this way, we obtainfor all .
Therefore, the sequences and are monotone:Assume that there exists a nonnegative integer such that It follows that From the definition of -metric space, we have , . It follows from (21) that is a coupled common fixed point of and .
Now, we suppose that for all nonnegative Using (18) and (21), for , we havewhich implies thatFor all , write and then the sequence is monotone decreasing. Therefore, there exists such that We claim that . On the contrary, suppose that , and we have from (26) thatLetting , we get Using the property of the function , we have So, we havewhich contradicts the assumption . Thus, .
Analogously to , we also have Thus, we haveNow, we have to prove that and are two Cauchy sequences in the -metric space .
For all with , by using Lemma 5, we have that Taking the limit as and using (35), we obtain Therefore, By interchanging the roles of and and proceeding along the arguments discussed above, we also obtain that Hence, for all with , we get It implies that Therefore, and are two Cauchy sequences in the -metric space . Since is a complete -metric space, hence and are -convergent. Then there exist such that and , respectively.
Step  2. We prove that is a coupled common fixed point of and .
We consider the following two cases.
Case  1 ( is continuous). We have Now using (18), we have That is,Since , we get ; that is, , .
Therefore, is a coupled common fixed point of and .
Case  2 ( is continuous). We also prove that is a coupled common fixed point of and similarly as in Case  1.

Theorem 19. Let be a partially ordered -metric space; let be two maps such that(1) is complete;(2)the pair has the mixed weakly monotone property on :(3)assume that there exists such that for all with , ;(4) has the following properties:(a)If is an increasing sequence with , then for all .(b)If is a decreasing sequence with , then for all .Then, and have a coupled common fixed point in .

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

Corollary 20. Let be a partially ordered set and let be an -metric on such that is a complete -metric space. Suppose that are two maps having the mixed weakly monotone property and assume that there exists such thatfor all with , .
Suppose that either(1) or is continuous;(2) has the following property:(a)If is an increasing sequence with , then for all .(b)If is a decreasing sequence with , then for all .If there exist such that , or , , then and have a coupled common fixed point in .

Proof. For all , writeAdding (49) and (50), we get where , for all .
It is easy to verify that . Applying Theorems 18 and 19, we get desired result.

Remark 21. Taking in Corollary 20 for all and , 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 with , , and some , inequality (49) in Corollary 20 is replaced byThen and have a coupled common fixed point in .

By choosing in Theorems 18 and 19 and using Remark 12, we get coupled fixed point theorem of written by the following corollary.

Corollary 23. Let be a partially ordered -metric space and let be a map such that(1) is complete;(2) has the mixed monotone property on ;(3)assume that there exists such that for all with , ;(4) is continuous or has the following properties:(a)If is an increasing sequence with , then for all .(b)If is a decreasing sequence with , then for all .Then has a coupled common fixed point in .

Theorem 24. In addition to the hypotheses of Theorem 18, suppose that, for all , there exists that is comparable with and . Then and have a unique coupled fixed point in .

Proof. By Theorem 18, and have a coupled common fixed point . Let be another coupled common fixed point of and .
By assumption, there exists that is comparable to and . Put , and choose such that , . Using the same construction as in the proof of Theorem 18, we have two sequences and such that satisfyingSince is comparable to , we can assume that .
Then it is easy to show that and are comparable; that is, , for all .
For , from (18) we have which impliesWe see that the sequence is decreasing, and there exist such thatNow, we have to show that . On the contrary, suppose that . Following the same arguments as in the proof of Theorem 18, we obtain It follows that This implies which is not possible in virtue of (30). Hence, . Therefore, (59) becomesSimilarly, we can get thatUsing (63)-(64), the second condition of -metric, and taking the limit , we obtain thatThus, we conclude that , .
Analogous to , by interchanging the roles of and , (65) holds true for .
Therefore, we conclude that and 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 , there exists that is comparable with and . Then and have a unique coupled fixed point in .

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

Example 26. Let , with the -metric defined by and the natural ordering of real numbers . Then is a totally ordered, complete -metric space.
Let be defined by For all , put .
The pair has the mixed weakly monotone property and with , , we have that Then the contractive condition (18) in Theorem 18 holds, and is the unique coupled common fixed point.

Example 27. Let , with the -metric defined by and the natural ordering of real numbers . Then is a totally ordered, complete -metric space.
Let be defined by For all , put .
The pair has the mixed weakly monotone property.
with , , we have that Then the contractive condition (18) in Theorem 18 holds, and is the unique coupled common fixed point.

#### Conflict of Interests

The authors declare that they have no competing interests.

#### Authors’ Contribution

Both authors contributed equally and significantly to writing this paper. All authors read and approved the final paper.

#### Acknowledgments

The authors thank Dr. Dolićanin-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).