#### Abstract

We prove coupled coincidence point and coupled fixed point results of and involving Meir-Keeler type contractions on the class of partially ordered metric spaces. Our results generalize some recent results in the literature. Also, we give some illustrative examples and application.

#### 1. Introduction and Preliminaries

Fixed point theory has wide applications in many areas. In economics it has applications in the study of market stability, in dynamic systems it is used to deterministic timed systems on feedback semantics, and in the theory of differential and integral equations to demonstrate the existence and uniqueness of solutions; see, for example, [1โ5]. On the other hand, fixed point theory, in particular fixed point iteration, has also numerous applications in engineering. For example, use of the fixed point iteration in image retrieval provides much better accuracy [6]. Fixed point algorithms proved to be very successful in practical optimization of the contrast functions in independent component analysis in neural-network research, as well as in statistics and signal processing [7]. These algorithms optimize the contrast functions very fast and reliably. Relaxation in linear systems and relaxation and stability in neural networks are also analyzed by means of fixed point iteration [8].

The problem of existence and uniqueness of fixed points in partially ordered sets has been studied thoroughly because of its interesting nature. The first result in this direction was given by Turinici [9], where he extended the Banach contraction principle in partially ordered sets. Ran and Reurings [10] presented some applications of Turiniciโs theorem to matrix equations. The result of Turinici was further extended and refined in [11โ25]. In particular, Gnana Bhaskar and Lakshmikantham in [12] introduced the concept of coupled fixed point of a mapping and investigated some coupled fixed point theorems in partially ordered sets. They also discussed an application of their result by investigating the existence and uniqueness of solution of the periodic boundary value problem: where the function satisfies certain conditions. Following this trend, Harjani et al. [4] studied the existence and uniqueness of solutions of a nonlinear integral equation as an application of coupled fixed points. Very recently, motivated by [5], Jleli and Samet [13] discussed the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem: where is a real number, is the Riemann-Liouville fractional derivative and is continuous, ( is singular at ) for all , is nondecreasing with respect to first component and decreasing with respect to its second and third components.

On the other hand, Lakshmikantham and ฤiriฤ [19] proved coupled coincidence and common coupled fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces which extend the coupled fixed point theorem given in [12]. Recently, Samet [23] proved some coupled fixed point theorems under a generalized Meir-Keeler contractive condition.

In this paper, we introduce the definition of weak generalized -Meir-Keeler type contractions and prove some coupled coincidence point theorems for such contractions. The theorems presented here generalize, enrich, and improve the previous results. Moreover, they have application potential in the theory of existence and uniqueness of solutions of boundary value problems.

Hereafter, we assume that and we use the notation Let be the set of real numbers.

*Definition 1.1 (see [12]). *Let be a partially ordered set and . The mapping is said to have the mixed monotone property if is monotone nondecreasing in and monotone nonincreasing in ; that is, for any ,

*Definition 1.2 (see [12]). *An element is said to be a coupled fixed point of the mapping if

The following result of Gnana Bhaskar and Lakshmikantham [12] was also proved in the context of cone metric spaces in [16].

Theorem 1.3 (see [12]). *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let be a given mapping having the mixed monotone property on . Assume that there exists with
**
Assume either is continuous, or satisfies the following property:*(i)*if a nondecreasing sequence converges to , then ;*(ii)*if a nonincreasing sequence converges to , then .**
If there exist such that and , then, there exist such that and .*

Inspired by Definition 1.1, Lakshmikantham and ฤiriฤ [19] introduced the concept of the mixed -monotone property.

*Definition 1.4 (see [19]). *Let be a partially ordered set. Let and let . The mapping is said to have the mixed -monotone property if is monotone -nondecreasing in and is monotone -nonincreasing in ; that is, for any ,

It is clear that Definition 1.4 reduces to Definition 1.1 when is the identity map.

*Definition 1.5 (see [19]). *An element is called a coupled coincidence point of the mappings and if
Moreover, is called a common coupled fixed point of and if

*Definition 1.6 (see [19]). *Let and let . The mappings and are said to commute if

In 2009, Lakshmikantham and ฤiriฤ [19] also proved a common coupled fixed point on partially ordered complete metric spaces.

Theorem 1.7 (see [19]). *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let and let such that has the mixed -monotone property. Suppose that there exists such that
**
for all for which and . Suppose , is continuous and commutes with . Also suppose that either is continuous or has the following property:
**
If there exist such that and , then there exist such that and ; that is, and have a coupled coincidence point.*

In 2010, Samet [23] introduced the mixed strict monotone property.

*Definition 1.8 (see [23]). *Let be a partially ordered set and let . is said to have mixed strict monotone property if is monotone increasing in and is monotone decreasing in ; that is, for any ,

Also, Samet [23] defined generalized Meir-Keeler contractions as follows.

*Definition 1.9 (see [23]). *Let be a partially ordered set, and suppose that there is a metric on . Let . The mapping is said to be a generalized Meir-Keeler type contraction if for any there exists such that
for all with , .

The existence and uniqueness of common coupled coincidence points via generalized Meir-Keeler type contractions was investigated by Samet [23].

Theorem 1.10 (see [23]). *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let be a map satisfying the following conditions:*(i)* has the mixed strict monotone property,*(ii)* is a generalized Meir-Keeler type contraction,*(iii)*there exist such that
**Assume either is continuous or satisfies the following property:*(i)*if a nondecreasing sequence converges to , then ,*(ii)*if a nonincreasing sequence converges to , then .**
Then has a coupled fixed point in ; that is, there exist such that
*

Very recently, Gordji et al. [26] replaced the *mixed **-monotone property* by the *mixed strict **-monotone property. *

*Definition 1.11 (see [26]). *Let be a partially ordered set. Let and let . is said to have the mixed strict -monotone property if is monotone -increasing in and is monotone -decreasing in ; that is, for any ,

If we replace with identity map in (1.17), we get Definition 1.8 of the mixed strict monotone property of .

Gordji et al. [26] gave also the following definition.

*Definition 1.12 (see [26]). *Let be a partially ordered metric space and , . The operator is said to be a generalized -Meir-Keeler type contraction if for any there exists such that
for all with , .

Note that if we replace with the identity in (1.18), we get Definition 1.9 of generalized Meir-Keeler type contraction.

Gordji et al. [26] proved the following theorem.

Theorem 1.13 (see [26]). *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let and let be mappings such that , is continuous and commutes with . Suppose also that satisfies the following conditions:*(i)* is continuous,*(ii)* has the mixed strict -monotone property,*(iii)* is a generalized -Meir-Keeler type contraction,*(iv)*there exist such that
** Then and have a coupled coincidence point in ; that is, there exist such that
*

In this paper, we proved coupled coincidence point results in the setting of partially ordered metric spaces. Also, the existence and uniqueness of a common coupled fixed point of and is studied. Our results improve the results of Berinde [15] and Gordji et al. [26]. We give two examples and an application that illustrate our results.

#### 2. Existence of Coupled Fixed Point

We start this section with the following definition which is modification of Definition 1.12.

*Definition 2.1. *Let be a partially ordered set, and suppose that there is a metric on such that is a metric space. Let and . The mapping is said to be a weak generalized -Meir-Keeler type contraction if for any there exists such that
for all with and .

*Remark 2.2. *If we replace with the identity in (2.1), we get the definition of a weak Meir-Keeler type contraction; that is, for all there exists such that
for all with and .

Note that (2.2) corresponds to a Meir-Keeler contraction type studied very recently by Berinde [15].

The following fact can be derived easily from Definition 2.1.

Lemma 2.3. *Let be a partially ordered set, and suppose that there is a metric on such that is a metric space. Let and . If is a weak generalized -Meir-Keeler type contraction, then we have
**
for all with or .*

*Proof. *Without loss of generality, suppose that where . It is clear that . Set . Since is a weak generalized -Meir-Keeler type contraction, then for this , there exits such that
for all with . The result follows by choosing and , that is:

Next, we state an existence theorem of a coupled coincidence point for and .

Theorem 2.4. *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let and be mappings such that . Moreover, assume that is continuous and commutes with . Suppose also that the following conditions hold:*(i)* is continuous,*(ii)* has the mixed strict -monotone property,*(iii)* is a weak generalized -Meir-Keeler type contraction,*(iv)*there exist such that
**Then and have a coupled coincidence point; that is, there exist such that
*

*Proof. *Let be a point satisfying (iv); that is, and . We define the sequences and in the following way. Because of the assumption , we can choose such that and . By the same argument, we can take in such a way that and . Inductively, we define

We claim that the the sequence is increasing and the sequence is decreasing, that is:
We will use mathematical induction to show (2.9). By assumption (iv), we have
Assume that (2.9) holds for some . Regarding the mixed strict -monotone property of , we have
By repeating the same arguments, we observe that
Combining the previous inequalities, together with (2.8), we get
We conclude that (2.9) holds for all . Set
Making use of Lemma 2.3 and (2.8), we obtain
Thus, we have . Hence, the sequence is monotone decreasing and clearly bounded below by . Therefore, for some .

We show that . Suppose the contrary; that is, . Then, for some positive integer , we have for all
where we choose . In particular, for
Regarding the assumption (iii) and (2.17), we have
which by (2.8) is equivalent to
Hence, we obtain
which contradicts (2.16) for . Thus, we deduce that , that is:
This implies that
We claim that the sequences and are Cauchy sequences. Take an arbitrary . It follows from (2.21) that there exists such that
Without loss of the generality, assume that and define the following set:
Take . We claim that
Take . Then, by (2.23) and the triangle inequality we have
We distinguish two cases.*First Case.*

By Lemma 2.3 and the definition of ฮ , (2.26) turns into
*Second Case.*.

In this case, we have
Since and , by (ii), we get
Thus, combining (2.26) and (2.29), we obtain
On the other hand, using (i), it is obvious that
We conclude that . Since , so
that is, (2.25) holds. By (2.23), we have . This implies with (2.25) that
Then, for all , we have . This implies that for all , we have
Thus, the sequences and are Cauchy in .

Since is complete, so there exist such that
Finally, by continuity of and , the commutativity of and , and using exactly the same argument of Lakshmikantham and ฤiriฤ [19], we get that and , which completes the proof.

*Remark 2.5. *Theorem 2.4 holds if we replace (iv) by the following: there exist such that

Theorem 2.6. *Let be a partially ordered set, and suppose that there is a metric on such that is a metric space. Let and let be mappings such that . Assume that satisfies the following property:*(a)*if is a sequence such that for each and , then for each ,*(b)*if is a sequence such that for each and , then for each .**
Suppose the following conditions hold:*(i)* has the mixed strict -monotone property,*(ii)* is a weak generalized -Meir-Keeler type contraction,*(iii)* is a complete subspace of ,*(iv)*there exist such that
** Then and have a coupled coincidence point; that is, there exist such that
*

*Proof. *Proceeding exactly as in Theorem 2.4, we have that and are Cauchy sequences in the complete metric space . Then, there exist such that and . Since is increasing and is decreasing, using the assumptions (a) and (b), we have
for each . Using triangle inequality together with (2.8), we find
Similarly,
Taking side-by-side sum of the above mentioned inequalities and having in mind (2.39), the fact that , and Lemma 2.3, we get
as . Hence, we end up with , that is, and , which completes the proof.

As a particular case of Theorems 2.4 and 2.6, we state the following corollary where the function is taken as the identity function.

Corollary 2.7. *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let . Suppose that satisfies the following conditions:*(i)* has the mixed strict monotone property,*(ii)* is a weak Meir-Keeler type contraction,*(iii)*there exist such that
** Assume either is continuous or satisfies the following property:*(a)*if is a sequence such that for each and , then for each ,*(b)*if is a sequence such that for each and , then for each .**
Then has a coupled fixed point; that is, there exist such that
*

We give the following examples.

*Example 2.8. *Let and . Set and let be defined as and . Then, the mapping has the strict mixed monotone property. We claim that condition (2.1) holds, but the condition (1.18) is not satisfied.

Note that in order to guarantee (1.18), we must have
for with . This means that
Choosing for simplicity (so ), we get
Hence for , (2.46) implies that
Combining (2.47) and(2.48), we get that
which is a contradiction.

On the other hand, and satisfy (2.1). Indeed, if we take the sum of
and divide by 2, we obtain for and let
Choosing , we get the desired result. Note also that and satisfy (2.6).

So Theorem 2.4 can be applied to ad in this example to conclude that and have a coupled coincidence point , while Theorem 1.13 cannot be applied since (1.18) is not satisfied.

*Example 2.9. *Let and . Set and let be defined as and . Then, the mapping has the strict mixed monotone property. We claim that condition (2.1) holds for and . Indeed,
Choosing , we get the desired result. Note also that and satisfy (2.6).

All hypotheses of Theorem 2.4 are satisfied. Here, and have a coupled coincidence point .

#### 3. Uniqueness of Common Coupled Fixed Point

In this section we will prove the uniqueness of a common coupled fixed point. We endow the product space with the following partial order: Note that a pair is comparable with if either or . We next state the conditions for the existence and uniqueness of a common coupled fixed point of maps and .

Theorem 3.1. *In addition to the hypotheses of Theorem 2.4 (resp., Theorem 2.6), assume that for all , , there exists such that is comparable to both and . Then, and have a unique common coupled fixed point, that is:
*

*Proof. *The set of coupled coincidence points of and is not empty due to Theorem 2.4 (resp., Theorem 2.6). We suppose that are two coupled coincidence points of and . We distinguish the following two cases.*First Case. * is comparable to with respect to the ordering in , where
Without loss of the generality, we may assume that
By Lemma 2.3, we have
which is a contradiction. Therefore, we have and .*Second Case.* Suppose that and are not comparable. By assumption there exists such that is comparable to both and .

Setting , as in the proof of Theorem 2.4, we define the sequences and as follows:
Since and are comparable, we may assume without loss of generality that and . Inductively, we observe that and for all . Thus, by Lemma 2.3, we get that
Set . Hence, for each
Therefore, the sequence is decreasing and bounded below by . Hence, it converges to some . Assume that . Then, for some positive integer , we have for all
where we choose . In particular, for
Having in mind (3.10) and the fact that is a weak generalized -Meir-Keeler contraction, we get that
which is equivalent to
Hence, we obtain
which contradicts (3.9) for . Thus, we deduce that , that is:
In a similar manner, we can show that
By the triangle inequality, we have
Combining all observations mentioned previously, we get and . Hence, we have
Last, we show that and . Let and . By the commutativity of and and the fact that and , we have
Thus, is a coupled coincidence point of and . However, according to (3.17), we must have
Hence, we deduce
that is, the pair is the coupled common fixed point of and .

We claim that is the unique coupled common fixed point of and . Assume that is another coupled common fixed point of and . But,
follows from (3.17).

The particular case in which is the identity function can be given as a corollary.

Corollary 3.2. *In addition to the hypotheses of Corollary 2.7, assume that for all , , there exists such that is comparable to both and . Then, has a unique coupled fixed point.*

#### 4. Application

In this section we give an application of the main theorems relevant to weak generalized -Meir-Keeler type contractions. For this, we need the following theorem.

Theorem 4.1. *Let be a partially ordered set, and suppose that there is a metric on . Let and let be two given mappings. Let also be a function satisfying the following:*(i)* and for all ,*(ii)* is nondecreasing and right continuous,*(iii)*for any there exists such that for all with and **Then the mapping is a weak generalized -Meir-Keeler contraction.*

*Proof. *By the condition (i) for any . Then according to (iii), for there exists such that, for all with and
Since is right continuous, so there exists such that
Now, fix satisfying , and
Since is nondecreasing, so we have
From (4.2),
Regarding the nondecreasing behavior of the function , we get
Consequently, is a weak generalized -Meir-Keeler type contraction.

If is the identity function, we derive the following special case of the Theorem 4.1.

Corollary 4.2. *Let be a partially ordered set, and suppose that there is a metric on . Let . Let also be a function satisfying the following:*(i)* and for all ,*(ii)* is nondecreasing and right continuous,*(iii)*for any there exists such that for all with and ,
**Then, the mapping is a weak Meir-Keeler contraction.*

The subsequent results are particular cases of Theorems 2.4 and 4.1.

Corollary 4.3. *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let and let be two given mappings such that , is continuous and commutes with . Also, suppose the following conditions:*(i)* is continuous,*(ii)* has the mixed strict -monotone property,*(iii)*for any , there exists such that for all satisfying and ,
where is locally integrable and for all *(iv)*there exist such that
**Then, and have a coupled coincidence point; that is, there exist such that
**
If, in addition, for all , , there exists such that is comparable to both and , then and have a unique common coupled fixed point.*

Corollary 4.4. *Let be a partially ordered set, and suppose that there is a metric on such that is a complete metric space. Let be a mapping satisfying the following conditions:*(i)* is continuous,*(ii)* has the mixed strict monotone property,*(iii)*for any there exists such that for all satisfying and ,
where is locally integrable and for all *(iv)*there exist such that
**Then has a coupled fixed point; that is, there exist such that
*