#### Abstract

We prove a number of quadruple fixed point theorems under -contractive conditions for a mapping in ordered metric spaces. Also, we introduce an example to illustrate the effectiveness of our results.

#### 1. Introduction and Preliminaries

The notion of coupled fixed point was initiated by Gnana Bhaskar and Lakshmikantham [1] in 2006. In this paper, they proved some fixed point theorems under a set of conditions and utilized their theorems to prove the existence of solutions to some ordinary differential equations. Recently, Berinde and Borcut [2] introduced the notion of tripled fixed point and extended the results of Gnana Bhaskar and Lakshmikantham [1] to the case of contractive operator , where is a complete ordered metric space. For some related works in coupled and tripled fixed point, we refer readers to [3โ32].

For simplicity we will denote the cross product of copies of the space by .

*Definition 1.1 (see [2]). *Let be a nonempty set and a given mapping. An element is called a tripled fixed point of if
Let be a metric space. The mapping , given by
defines a metric on , which will be denoted for convenience by .

*Definition 1.2 (see [2]). *Let be a partially ordered set and a mapping. One says that has the mixed monotone property if is monotone nondecreasing in and and is monotone nonincreasing in ; that is, for any ,

Let us recall the main results of [2] to understand our motivation toward our results in this paper.

Theorem 1.3 (see [2]). *Let be a partially ordered set and a complete metric space. Let be a continuous mapping such that has the mixed monotone property. Assume that there exist with such that
**
for all with , , and . If there exist such that , , and , then has a tripled fixed point.*

Theorem 1.4 (see [2]). *Let be a partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property. Assume that there exist with such that
**
for all with , , and . Assume that has the following properties: *)* if a nondecreasing sequence , then for all , *()* if a nonincreasing sequence , then for all . ** If there exist such that , , and , then has a tripled fixed point.*

Very recently, Karapฤฑnar introduced the notion of quadruple fixed point and obtained some fixed point theorems on the topic [33]. Extending this work, quadruple fixed point is developed and related fixed point theorems are proved in [34โ39].

*Definition 1.5 (see [34]). *Let be a nonempty set and 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 1.6. *In [33, 34, 38], the notion of *quadruple fixed point* is called *quartet fixed point*.

*Definition 1.7 (see [34]). *Let be a partially ordered set and a mapping. One says that has the mixed monotone property if is monotone nondecreasing in and and is monotone nonincreasing in and ; that is, for any ,

By following Matkowski [40], we let be the set of all nondecreasing functions such that for all . Then, it is an easy matter to show that(1) for all ,(2).

In this paper, we prove some quadruple fixed point theorems for a mapping satisfying a contractive condition based on some .

#### 2. Main Results

Our first result is the following.

Theorem 2.1. *Let be a partially ordered set and a complete metric space. Let be a continuous mapping such that has the mixed monotone property. Assume that there exists such that
**
for all with , , , and . If there exist such that , , and , then has a quadruple fixed point.*

* Proof. *Suppose are such that , , , and . Define
Then, , , , and . Again, define , , , and . Since has the mixed monotone property, we have , , , and . Continuing this process, we can construct four sequences , , , and in such that
If, for some integer , we have , then , , , and ; that is, is a quadruple fixed point of . Thus, we will assume that for all ; that is, we assume that ,, or or . For any , we have
From (2.4), it follows that
By repeating (2.5) times, we get that
Now, we will show that , , , and are Cauchy sequences in . Let . Since
and , there exist such that
This implies that
For , we will prove by induction on that
Since , then by using (2.9) we conclude that (2.10) holds when . Now suppose that (2.10) holds for . For , we have
Similarly, we show that
Hence, we have
Thus, (2.10) holds for all . Hence, , , , and are Cauchy sequences in .

Since is a complete metric space, there exist such that , , and converge to , , , and , respectively. Finally, we show that is a quadruple fixed point of . Since is continuous and , we have . By the uniqueness of limit, we get that . Similarly, we show that , , and . So, is a quadruple fixed point of .

By taking , where , in Theorem 2.1, we have the following.

Corollary 2.2. *Let be a partially ordered set and a complete metric space. Let be a continuous mapping such that has the mixed monotone property. Assume that there exists such that
**
for all with , , , and . If there exist such that , , , and , then has a quadruple fixed point.*

As a consequence of Corollary 2.2, we have the following.

Corollary 2.3. *Let be a partially ordered set and a complete metric space. Let be a continuous mapping such that has the mixed monotone property. Assume that there exist with such that
**
for all with , , , and . If there exist such that , , and , then has a quadruple fixed point.*

By adding an additional hypothesis, the continuity of in Theorem 2.1 can be dropped.

Theorem 2.4. *Let be a partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property. Assume that there exists such that
**
for all with , , , and . Assume also that has the following properties: *()* if a nondecreasing sequence , then for all , *()* if a nonincreasing sequence , then for all . ** If there exist such that , , , and , then has a quadruple fixed point.*

* Proof. *By following the same process in Theorem 2.1, we construct four Cauchy sequences , , , and in with
such that , , , and . By the hypotheses on , we have , , , and for all . From (2.16), we have
From (2.18), we have
Letting in (2.19), it follows that , , , and . Hence, is a quadruple fixed point of .

By taking , where , in Theorem 2.4, we have the following result.

Corollary 2.5. *Let be a partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property. Assume that there exists such that
**
for all with , , , and . Assume also that has the following properties: *()* if a nondecreasing sequence , then for all , *()* if a nonincreasing sequence , then for all . ** If there exist such that , , , and , then has a quadruple fixed point.*

As a consequence of Corollary 2.5, we have the following.

Corollary 2.6. *Let be a partially ordered set and a complete metric space. Let be a mapping having the mixed monotone property. Assume that there exist with such that
**
for all with , , , and . Assume that has the following properties: *()* if a nondecreasing sequence , then for all , *()* if a nonincreasing sequence , then for all . ** If there exist such that , , , and , then has a quadruple fixed point.*

Now we prove the following result.

Theorem 2.7. *In addition to the hypotheses of Theorem 2.1 (resp., Theorem 2.4), suppose that
**
Then, .*

* Proof. *Without loss of generality, we may assume that , , , and . By the mixed monotone property of , we have , , , and for all . Thus, by (2.1), we have
By (2.23) and (2.26), we have
By letting in (2.27) and using the property of and the fact that is continuous on its variable, we get that . Hence, .

Corollary 2.8. *In addition to the hypotheses of Corollary 2.3 (resp., Corollary 2.5), suppose that
**
Then, .*

*Example 2.9. * Let with usual order. Define by . Define by
Then, (a) is a complete ordered metric space, (b)for with , , , and , we have that
(c)holds for all , , , and ,(d) has the mixed monotone property.

* Proof. *To prove , given with , , , and , we examine the following cases.*Case *1*. *If , and . Here, we have
*Case *2*. *If and . This case is impossible since
So,
*Case *3*. *If and .

This case will have different possibilities.

(i) Let and . Suppose that ; then and hence
Therefore,
Suppose that ; then and hence
Therefore,

(ii) Let and . Suppose that ; then and (since ) hence
Therefore,

Suppose that ; then and (since ) hence
Therefore,

(iii) Let and . Suppose that ; then , but , and hence
Therefore,

Suppose that ; then and hence
Therefore,

(iv) Let and . Suppose that ; then and hence
Therefore,

Suppose that ; then and hence
Therefore,
*Case *4*.* (i) If and .

Since and , then , and also since and , then . Thus,

(ii) If and , then and . Thus,

(iii) If and , then and , hence

(iv) If and , then and , and hence

(v) If and , then and , and hence

To prove (c), let . To show that is monotone nondecreasing in , let with .

If , then . If , then
Therefore, is monotone nondecreasing in . Similarly, we may show that is monotone nondecreasing in .

To show that is monotone nonincreasing in , let with . If , then . If , then
Therefore, is monotone nonincreasing in . Similarly, we may show that is monotone nonincreasing in .

Thus, by Theorem 2.1 (let ), has a unique quadruple fixed point, namely, . Since the condition of Theorem 2.7 is satisfied, is the unique quadruple fixed point of .

*Remark 2.10. *We notice that for, , it is very natural to consider the analog of Theorem 2.1โTheorem 2.7 to get fixed points. Moreover, for , the analog of Theorem 7โTheorem 11 of Berinde and Borcut [2] yields fixed points.