#### 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 [3โ32].

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 [34โ39].

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 2.1โTheorem 2.7 to get fixed points. Moreover, for , the analog of Theorem 7โTheorem 11 of Berinde and Borcut [2] yields fixed points.