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.