Research Article | Open Access

Volume 2013 |Article ID 654659 | https://doi.org/10.1155/2013/654659

Jalal Hassanzadeasl, "Common Fixed Point Theorems for --Contractive Type Mappings", International Journal of Analysis, vol. 2013, Article ID 654659, 7 pages, 2013. https://doi.org/10.1155/2013/654659

# Common Fixed Point Theorems for --Contractive Type Mappings

Revised05 Mar 2013
Accepted11 Mar 2013
Published11 Apr 2013

#### 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., [133]). 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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