#### Abstract

The notion of coupled fixed point is introduced in by Gnana Bhaskar and Lakshmikantham (2006). Very recently, the concept of tripled fixed point is introduced by Berinde and Borcut (2011). In this paper, quadruple fixed point is introduced, and some new fixed point theorems are obtained.

#### 1. Introduction and Preliminaries

Very recently, Berinde and Borcut [1] introduced the concept of triple fixed point. Their contributions are inspired from the remarkable paper of Gnana Bhaskar and Lakshmikantham [2] in which they introduced the notion of coupled fixed point and proved some fixed point theorems under certain condition. Later, Lakshmikantham and Ljubomir Ćirić in [3] extended these results by defining of -monotone property. Many authors focused on coupled fixed point theory and proved remarkable results (see, e.g., [4–9]).

Here we recall the basic definitions and results from which triple and quadruple fixed point [10] notions are inspired. Let be a metric space and . Then the mapping such that forms a metric on . A sequence is said to be a double sequence of .

*Definition 1.1 (see [2]). *Let be partially ordered set and . is said to have mixed monotone property if is monotone nondecreasing in and is monotone nonincreasing in , that is, for any ,

*Definition 1.2 (see [2]). *An element is said to be a couple fixed point of the mapping if

Throughout this paper, let be partially ordered set and a metric on such that is a complete metric space. Further, the product spaces satisfy the following:

The following two results of Gnana Bhaskar and Lakshmikantham in [2] were extended to class of cone metric spaces in [7].

Theorem 1.3. *Let be a continuous mapping having the mixed monotone property on . Assume that there exists a with
**
If there exists such that and , then, there exists such that and . *

Theorem 1.4. *Let be a mapping having the mixed monotone property on . Suppose that has the following properties: *(i)*if a nondecreasing sequence , then , for all n; *(ii)*if a nonincreasing sequence , then , for all n. ** Assume that there exists a with
**
If there exists such that and , then, there exists such that and . *

Inspired by Definition 1.1, Berinde and Borcut [1] introduced the following partial order on the product space : where . Regarding this partial order, we state the definition of the following mapping.

*Definition 1.5 (see [1]). *Let be partially ordered set and . We say that has the mixed monotone property if is monotone nondecreasing in and , and it is monotone nonincreasing in , that is, for any

*Definition 1.6 (see [1]). *An element is called a quadruple fixed point of if

For a metric space , the function , given by forms a metric space on ; that is, is a metric induced by .

Theorem 1.7. *Let be partially ordered set, and let be a complete metric space. Let be a continuous mapping having the mixed monotone property on . Assume that there exist constants such that for which
**
for all . If there exist such that
**
then there exist such that
*

The aim of this paper is to introduce the concept of quadruple fixed point and prove the related fixed point theorems.

#### 2. Quadruple Fixed Point Theorems

Let be partially ordered set and a complete metric space. We consider the following partial order on the product space : where . Regarding this partial order, we state the definition of the following mapping.

*Definition 2.1. *Let be partially ordered set and . We say that has the mixed monotone property if is monotone nondecreasing in and , and it is monotone nonincreasing in and , that is, for any

*Definition 2.2. *An element is called a triple fixed point of if

For a metric space , the function , given by forms a metric space on ; that is, is a metric induced by . Let denote the all functions which satisfies that for all and

The aim of this paper is to prove the following theorem.

Theorem 2.3. *Let be partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property on . Assume that for all ,
**
where . Suppose there exist such that
**
Suppose 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 . *

*Then there exist such that*

* Proof. *We construct a sequence in the following way. Set
and by the mixed monotone property of , for , inductively we get
Due to (2.5) and (2.9), we have
Regarding (2.5) together with (2.14) we have

Inductively we have
Set . Due to (2.15), we conclude that is a nonincreasing sequence. Since it is bounded below, there is some such that
We shall show that . Suppose, to the contrary, that .

Again by (2.15) and (2.9) together with (2.5), we have

Letting in (2.17) and having in mind that we suppose for all and , we have
which is a contradiction. Thus, , that is,

Now, we shall prove that ,,, and are Cauchy sequences. Suppose, to the contrary, that at least one of ,,, and is not Cauchy. So, there exists an for which we can find subsequences , of and , of and , of and , of with such that
Additionally, corresponding to , we may choose such that it is the smallest integer satisfying (2.20) and . Thus,
By using triangle inequality and having (2.20), (2.21) in mind
Letting in (2.22) and using (2.16)
Again by triangle inequality,
Since , then
Hence from (2.25), (2.9) and (2.5), we have
Combining (2.24) with (2.26), we obtain that
Letting and having in mind (2.19) we get a contradiction. This shows that , , , and are Cauchy sequences. Since is complete metric space, there exists such that

Suppose now the assumption holds. Then by (2.9) and (2.28), we have
Analogously, we also observe that
Thus, we have

Suppose now the assumption holds. Since are nondecreasing and , , and also are nonincreasing and , then by assumption we have
for all . Consider now
Taking in (2.33) and using (2.28), we get that . Thus, . Analogously, we get that
Thus, we proved that has a quadruple fixed point.

Corollary 2.4. *Let be partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property on . Assume that there exists such that
**
for all , , . Suppose there exist such that
**
Suppose 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 , *

*then there exist such that*

* Proof. *It is sufficient to take in previous theorem.

#### 3. Uniqueness of Quadruple Fixed Point

In this section we shall prove the uniqueness of 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 is equal if and only if , and .

Theorem 3.1. *In addition to hypothesis of Theorem 2.3, suppose that for all ; there exists that is comparable to and ; then has a unique quadruple fixed point.*

* Proof. *The set of quadruple fixed point of is not empty due to Theorem 2.3. Assume, now, that and are the quadruple fixed point of , that is,
We shall show that and are equal. By assumption, there exists that is comparable to and . Define sequences , and such that
for all . Since is comparable with , we may assume that . Recursively, we get that
By (3.4) and (2.5), we have
Set . Then, due to (3.5), we have

Hence, the sequence is decreasing and bounded below. Thus, there exists such that

Now, we shall show that . Suppose, to the contrary, that . Again by (3.5), we have Letting in (3.8), we obtain that

which is a contradiction. Therefore, . That is,

Consequently, we have Analogously, we show that Combining (3.11) and (3.12) yields that and are equal.

*Example 3.2. *Let with the metric , for all and the usual ordering.

Let be given by
and let be given by for all .

It is easy to check that all the conditions of Theorem 2.3 are satisfied and is the unique quadruple fixed point of .