Journal of Applied Mathematics

Volume 2012 (2012), Article ID 951912, 17 pages

doi:10.1155/2012/951912

Research Article

Quadruple Fixed Point Theorems under Nonlinear Contractive Conditions in Partially Ordered Metric Spaces

Erdal Karapınar,¹ Wasfi Shatanawi,² and Zead Mustafa²

¹Department of Mathematics, Atılım University, 06836 İncek, Turkey

²Department of Mathematics, The Hashemite University, Zarqa 13115, Jordan

Received 17 March 2012; Accepted 7 May 2012

Academic Editor: Debasish Roy

Copyright © 2012 Erdal Karapınar 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 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 [3–32].

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, Karapınar 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 [34–39].

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

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