`ISRN Applied MathematicsVolume 2012 (2012), Article ID 539125, 16 pageshttp://dx.doi.org/10.5402/2012/539125`
Research Article

## Quadruple Fixed Point Theorems in Partially Ordered Metric Spaces Depending on Another Function

1Université de Sousse Institut Supérieur d'Informatique et des Technologies de Communication de Hammam Sousse, Route GP1, H. Sousse 4011, Tunisia
2Department of Mathematics, Atılım University, 06836 İncek, Ankara, Turkey

Received 27 February 2012; Accepted 30 April 2012

Copyright © 2012 Hassen Aydi 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

We prove quadruple fixed point theorems in partially ordered metric spaces depending on another function. Also, we state some examples showing that our results are real generalization of known ones in quadruple fixed point theory.

#### 1. Introduction and Preliminaries

Basic topological properties of an ordered set like convergence were introduced by Wolk [1]. In 1981, Monjardet [2] considered metrics on partially ordered sets. Ran and Reurings [3] proved an analog of Banach contraction mapping principle in partially ordered metric spaces. In their pioneering work, they also provide applications to matrix equations. As an extension, Nieto and Rodríguez-López [4] discovered further fixed point theorems in partially ordered metric spaces. For some other related results in ordered metric spaces, see, for example, [57].

Bhaskar and Lakshmikantham in [8] introduced the concept of coupled fixed point of a mapping and investigated the existence and uniqueness of a coupled fixed point theorem in partially ordered complete metric spaces. Lakshmikantham and Ćirić in [9] defined mixed -monotone property and coupled coincidence point in partially ordered metric spaces. They also proved related fixed point theorems. Later, various results on coupled fixed point have been obtained, see, for example, [920].

Following this trend, Berinde and Borcut [21] introduced the concept of tripled fixed point in ordered sets. The following two definitions are from [21].

Definition 1.1. Let be a partially ordered set and . The mapping is said to have the mixed monotone property if, for any ,

Definition 1.2. Let . An element is called a tripled fixed point of if

Also, Berinde and Borcut [21] proved the following theorem.

Theorem 1.3. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Suppose also that be a mapping such that it has the mixed monotone property and there exist with such that
for any for which , , and . Additionally suppose that either is continuous or has the following properties:(1)if a nondecreasing sequence , then for all ,(2)if a nonincreasing sequence , then for all .
If there exist such that , , and , then there exist such that
that is, has a tripled fixed point.

The notion of fixed point of order was first introduced by Samet and Vetro [22]. Very recently, Karapınar used the notion of quadruple fixed point and obtained some quadruple fixed point theorems [23] in partially ordered metric spaces. This work motivated the following studies [2427] which provide further fixed point theorems on quadruple fixed points.

From now on, we denote .

Definition 1.4 (see [24]). Let be a nonempty set and let be a given mapping. An element is called a quadruple fixed point of if
Let be a metric space. The mapping , given by defines a metric on , which will be denoted for convenience by .

Remark 1.5. In [23, 24, 27], the notion of quadruple fixed point is called quartet fixed point.

Definition 1.6 (see [24]). Let be a partially ordered set and be a mapping. We say that has the mixed monotone property if is monotone nondecreasing in and and is monotone nonincreasing in and ; that is, for any ,

In this paper, we prove some quadruple fixed point theorems in partially ordered metric spaces depended on another function .

#### 2. Main Results

Definition 2.1. Let be a metric space. A mapping is said to be ICS if is injective, continuous, and it has the property: for every sequence in , if is convergent then, is also convergent.

Let be the set of all functions such that(1) for all ,(2) for all .

Our first result is given by the following theorem.

Theorem 2.2. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Suppose also that is an ICS mapping and is such that has the mixed monotone property. Assume that there exists such that
for any for which , , , and . Additionally assume that either(a) is continuous, or(b) has the following properties:(i) if nondecreasing sequence (respectively, ), then (respectively, ) for all ,(ii) if nonincreasing sequence (respectively, ), then (respectively, ) for all .
If there exist such that , , , and , then there exist such that
that is, has a quadruple fixed point.

Proof. Let such that
Set
Then, , , , and . Again, define , , , and . Since has the mixed monotone property, we have , , , and . By continuing this process, we can construct four sequences , , , and in such that
Since has the mixed monotone property, by using a mathematical induction it is easy to see that
Assume that, for some , Then, by (2.6), is a quadruple fixed point of . Therefore, in the rest of the proof, for any we will assume that
Since is injective, for any ,
Due to (2.1), (2.6), and (2.7), we have
Using the fact that for all together with (2.11), we obtain that
It follows that
Thus, is a positive decreasing sequence. Hence, there exists such that
Suppose that . Letting in (2.12), we obtain that
which is a contradiction. Therefore, we deduce that
We will show that , , , and are Cauchy sequences. Assume the contrary, that is, either or or or is not a Cauchy sequence, consequently,
or or . This means that there exists for which we can find subsequences of integers and with such that
Furthermore, corresponding to , we can choose in such a way that it is the smallest integer with and satisfying (2.18). Then,
By the triangle inequality and (2.19), we have
Thus, by (2.16), we obtain
Similarly, we have
Again, by (2.19), we have
Letting and using (2.16), we get
Using (2.18) and (2.24), we have
Now, using inequality (2.1), we obtain
From (2.26), we deduce that
Letting in (2.27) and by using (2.25), we get that
which is a contradiction. Thus, , , , and are Cauchy sequences in . Since is a complete metric space, , , , and are convergent sequences.
Since is an ICS mapping, there exist such that
Since is continuous, we have
Suppose now the assumption holds, that is, is continuous. By (2.6), (2.29), and (2.30) we obtain
We have proved that has a quadruple fixed point.
Suppose now the assumption holds. Since and are nondecreasing with and and also and are nonincreasing, with and , we have
for all . Consider now
Taking and using (2.30), the right-hand side of (2.33) tends to , so we get that . Thus, , and since is injective, we get that . Analogously, one finds that
Thus, we proved that has a quadrupled fixed point. This completes the proof of Theorem 2.2.

Repeating the same proof of Theorem 2.2, we may state the following corollary.

Corollary 2.3. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Suppose also is an ICS mapping and is such that has the mixed monotone property. Assume that there exists such that
for any for which , , , and . Additionally suppose that either(a) is continuous, or(b) has the following property:(i) if non-decreasing sequence (resp., ), then (resp., ) for all ,(ii) if non-increasing sequence (resp., ), then (resp., ) for all .
If there exist such that , , and , then there exist such that
that is, has a quadruple fixed point.

Corollary 2.4. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Suppose also that is an ICS mapping and is such that has the mixed monotone property. Assume that there exists such that
for any for which , , , and . Suppose that either (a) is continuous, or(b) has the following property:(i)if nondecreasing sequence (resp., ), then (resp., ) for all ,(ii)if nonincreasing sequence (resp., ), then (resp., ) for all .
If there exist such that , , , and then, there exist such that
that is, has a quadruple fixed point.

Proof. It suffices to remark that in Theorem 2.2.

Corollary 2.5. Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Suppose also that is an ICS mapping and is such that has the mixed monotone property. Assume that there exists such that
for any for which , , and . Suppose that either(a) is continuous, or(b) has the following property:(i)if nondecreasing sequence (resp., ), then (resp., ) for all ,(ii)if nonincreasing sequence (resp., ), then (resp., ) for all .
If there exist such that , , and then there exist such that
that is, has a quadruple fixed point.

Proof. It suffices to take in Corollary 2.3.

Now, we shall prove the existence and uniqueness of a quadruple fixed point. For a product of a partial ordered set , we define a partial ordering in the following way: For all ,

We say that and are comparable if

Also, we say that is equal to if and only if .

Theorem 2.6. In addition to hypotheses of Theorem 2.2, suppose that that for all , there exists such that , , , is comparable to , , , and (, , , ). Then, has a unique quadruple fixed point .

Proof. The set of quadruple fixed points of is not empty due to Theorem 2.2. Assume, now, and are two quadrupled fixed points of , that is,
We shall show that and are equal. By assumption, there exists such that (, , , ) is comparable to (, , , ) and (, , , ).
Define sequences , , and such that
for all . Further, set , , , and , , , and on the same way define the sequences , , and and , , and . Then, it is easy that
for all . Since is comparable to , then it is easy to show . Recursively, we get that
By (2.46) and (2.1), we have
It follows from (2.47) that
Therefore, for each ,
It is known that and imply for each . Thus, from (2.49),
This yields that
Analogously, we show that
Combining (2.51) to (2.52) yields that and are equal. The fact that is injective gives us , , , and .

We state some examples showing that our results are effective.

Example 2.7. Let with the metric , for all and the usual ordering. Clearly, is a complete metric space.
Let and be defined by
It is clear that is an ICS mapping, has the mixed monotone property and continuous.
Set . Taking for which , , , and , we have which is the contractive condition (2.1). Moreover, taking , we have
Therefore, all the conditions of Theorem 2.2 hold and is the unique quadruple fixed point of , since also the hypotheses of Theorem 2.6 hold.
On the other hand, we can not apply Corollary 15 of Karapınar [27] to this example. Indeed, for , , and , we have for any .

Example 2.8. Let with and natural ordering. Let and be defined by and . It is clear that is an ICS mapping and has the monotone property and continuous. Set . It is clear that all conditions of Theorem 2.2 are satisfied and is the desired quadruple point.
Note that Corollary 15 of Karapınar [27] is not applicable. Indeed, for , and , we have for any .

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