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International Journal of Analysis

Volume 2013 (2013), Article ID 654659, 7 pages

http://dx.doi.org/10.1155/2013/654659

## Common Fixed Point Theorems for --Contractive Type Mappings

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

Received 8 November 2012; Revised 5 March 2013; Accepted 11 March 2013

Academic Editor: Leo G. Rebholz

Copyright © 2013 Jalal Hassanzadeasl. 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

Recently, Samet et al. (2012) introduced the notion of --contractive type mappings. They established some fixed point theorems for these mappings in complete metric spaces. In this paper, we introduce the notion of a coupled --contractive mapping and give a common fixed point result about the mapping. Also, we give a result of common fixed points of some coupled self-maps on complete metric spaces satisfying a contractive condition.

#### 1. Introduction

We know fixed point theory has many applications and was extended by several authors from different views (see, e.g., [1–33]). Recently, Samet et al. introduced the notion of --contractive type mappings [3]. Denote with the family of nondecreasing functions such that for all , where is the th iterate of . It is known that for all and [3]. Let be a metric space, a self map on , and . Then, is called a --contractive mapping whenever for all . Also, we say that is -admissible whenever implies for all [3]. Also, we say that has the property () if is a sequence in such that for all and , then for all . Let be a complete metric space and let a -admissible --contractive mapping on . Suppose that there exists such that . If is continuous or has the property (), then has a fixed point (see [3]; Theorems 2.1 and 2.2). Finally, we say that has the property () whenever for each there exists such that and . If has the property () in the Theorems 2.1 and 2.2, then has a unique fixed point ([3]; Theorem 2.3). It is considerable that the results of Samet et al. generalize similar ordered results in the literature (see the results of the third section [3]). The aim of this paper is introducing the notion of generalized coupled --contractive mappings and give a common fixed point result about the mappings.

*Definition 1. *Let the family of functions satisfy:(i) and for all ;(ii) is continuous;(iii) is nondecreasing on ;(iv) for all .

*Definition 2. *Let the family of functions satisfy is nondecreasing;, for all .

These functions are known in the literature as ()-comparison functions. It is easily proved that if is a ()-comparison function, then for all .

*Definition 3. *Let be a metric space, and let with given coupled mappings. Let , , , and let
for all coupled mappings and . One says that , are generalized coupled -contractive mappings whenever
for all .

*Definition 4. *Let , and let . One says that , are coupled -admissible if
for all .

*Definition 5. *Let be a compete metric space. For two subsets , of , one marks , if for all , there exists such that .

*Definition 6. *A partial metric on a nonempty set is a function such that for all :;;;.

A partial metric space is a pair such that is a nonempty set, and is a partial metric on . It is clear that if , then from and , . But if , may not be . A basic example of a partial metric is the pair , where for all . If is a partial metric on , then the function given by is a metric on .

*Example 7. *Let endowed with the standard metric for all . Define the coupled mappings by
We define the mapping by
If
Similarly, . This shows that , are coupled -admissible.

Lemma 8. *Let be a metric space. Suppose that are generalized coupled --contractive mappings. Then, . *

*Proof. *We first show that any fixed point of is also a fixed point of and conversely. Define for all . Since , we may assume there exists such that , but . Since , we have
This contradiction establishes that . A similar argument establishes the reverse containment, and therefore .

#### 2. Main Results

Now, we are ready to state and prove our main results.

Theorem 9. *Let be a complete metric space. Suppose that are generalized coupled --contractive mappings and satisfy the following conditions:**, are coupled -admissible;** there exists such that ; or** there exists such that ;** or is continuous.**Then , have common fixed point . Further, for each , the iterated sequence with and converges to the common fixed point of . *

* Proof. *By Lemma 8, we have . Let such that . Define the sequence in by and for all . If for some , then are a common fixed point for , . So, we can assume that and for all . Since , are coupled -admissible, we have
Inductively, we have
for all . We obtain
Now,
if
So, in general,
which is a contradiction since . Thus,
Similarly, if
we have
for all . By induction, we get
for all . Fix , and let such that
Let with . Using the triangle inequality, we obtain
Thus we proved that is a Cauchy sequence in the metric space .

Since is a complete metric space, there exists such that as . From the continuity of , it follows that as , then . Similarly if is continuous, we have .

Corollary 10. *Let be a complete metric space. Suppose that is a generalized --contractive mapping and satisfies the following conditions:** is -admissible;** there exists such that ; or** there exists such that ;** is continuous.**Then, has a fixed point . Further, for each , the iterated sequence with converges to the fixed point of . *

*Example 11. *Let endowed with the standard metric for all . Define the coupled mappings by
We define the mapping by
If and for all , we have
for all . Thus, , are generalized coupled --contractive mappings. Moreover, there exists such that . In fact, for , we have . Obviously, is continuous, and so it remains to show that , are coupled -admissible. To do so, let such that . This implies that by the definition of . We then have , and . Then, , are coupled -admissible. Now, all the hypotheses of Theorem 9 are satisfied.

Consequently, , have common fixed points. In this example, is at least one common fixed point of and .

Now, we omit the continuity hypothesis of and .

Theorem 12. *Let be a complete metric space. Suppose that are generalized coupled --contractive mappings and satisfy the following conditions:**, are coupled -admissible;** there exists such that ; or** there exists such that ;** if is a sequence in such that for all and as , then there exists a subsequence of such that for all ; or** if is a sequence in such that for all and as , then there exists a subsequence of such that for all .**Then, , have common fixed point . Further, for each , the iterated sequence with and converges to the common fixed point of . *

*Proof. *Following the proof of Theorem 9, we know that is a Cauchy sequence in the complete metric space . Then, there exists such that as . From Theorem 9 and condition (iii), there exists a subsequence of such that for all .

Applying Theorem 9, for all , we get that
On the other hand, we have
Letting , in the above equality, we get that
Suppose that . From (25), for large enough, we have , which implies that
Thus, from (23), we have . Letting in the above inequality, and using (25), we obtain that which is a contradiction. Thus, we have ; that is, . Similarly, it can be shown that .

Corollary 13. *Let be a complete metric space. Suppose that is generalized --contractive mapping and satisfies thefollowing conditions:** is -admissible;** there exists such that ; or** there exists such that ;** if is a sequence in such that for all and as , then there exists a subsequence of such that for all ; or** if is a sequence in such that for all and as , then there exists a subsequence of such that for all .**Then, has a fixed point . Further, for each , the iterated sequence with converges to the fixed point of . *

*Example 14. *Let endowed with the standard metric for all . Define the coupled mappings by
We define the mapping by
If and for all , we have
for all . Thus, , are generalized coupled --contractive mappings. Moreover, there exists such that . In fact, for , we have . Let , and for all . By the definition of we have, for all , so and . It remains to show that , are coupled -admissible. In doing so, let such that . This implies that by the definition of . We have , , and . Then , are coupled -admissible. Now, all the hypotheses of Theorem 12 are satisfied. Consequently, , have common fixed points. In this example, is at least one common fixed point of and .

#### 3. Fixed Point Theorems on Ordered Metric Space

Theorem 15. *Let be a complete ordered metric space, , , and , be coupled mappings on such that and for all with . Suppose that there exists such that or , and if is a sequence in such that or for all and , then or for all . If implies or or or , then have common fixed points. *

*Proof. *Define by whenever , and define whenever *⋠*. It is easy to check that , are a coupled -admissible and generalized coupled --contractive mappings on . Now, by using Theorem 9, , have common fixed points.

Corollary 16. *Let be a complete ordered metric space, , , and a mapping on such that for all with . Suppose that there exists such that or and if is a sequence in such that or for all and , then or for all . If implies or , then has a fixed point. *

#### 4. Fixed Point Theorems on Metric Spaces Endowed with Partial Metric

If we substitute a partial metric instead the metric in Theorem 9, it is easy to check that a similar result holds for the partial metric space case as follows. We define for all coupled mappings and .

Theorem 17. *Let be a complete partial metric space, a function, , , and , self maps on such that
**
for all . Suppose that , are coupled -admissible and there exists such that or . Assume that if is a sequence in such that or for all and as , then there exists a subsequence of such that or for all . Then have common fixed points. *

Corollary 18. *Let be a complete partial metric space, a function, , , and a self map on such that
**
for all . Suppose that is -admissible and there exists such that or . Assume that if is a sequence in such that or for all and as , then there exists a subsequence of such that or for all . Then has a fixed point. *

*Example 19. *Let endowed with the partial metric , for all . Define the mapping , by
We define the mapping by
since is -admissible. If , we have
Suppose that is a sequence in such that for all and as ; by definition of , we have , and , on the other hand ; then there exists a subsequence of such that for all . There exists such that . This show that all conditions of Corollary 18 are satisfied, and so has a fixed point in .

#### Acknowledgment

The author would like to thank Tehran Science and Research Branch, Islamic Azad University, for the financial support of this research, which is based on a research project contract.

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