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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 549518, 11 pages
http://dx.doi.org/10.1155/2013/549518
Research Article

Common Fixed Point Results for Mappings with Rational Expressions

1Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
2Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan
3Department of Mathematics, International Islamic University, Sector H-10, Islamabad 44000, Pakistan

Received 2 March 2013; Accepted 14 May 2013

Academic Editor: Irena Rachůnková

Copyright © 2013 M. A. Kutbi 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 obtain some common fixed point results for single as well as set valued mappings involving certain rational expressions in complete partial metric spaces. In the process, we generalize various results of the literature. Two examples are also included to illustrate the fact that our results cannot be obtained from the corresponding results in metric spaces.

1. Introduction and Preliminaries

In 1994, Matthews [1] introduced the concept of a partial metric space and obtained a Banach type fixed point theorem on a complete partial metric space. Later on, several authors (see, e.g., [128]) proved fixed point theorems in partial metric spaces. After the definition of the Partial Hausdorff metric, Aydi et al. [9] proved Banach type fixed point result for set valued mappings in complete partial metric space. Here, we prove some common fixed point results for single as well as set valued mappings involving certain rational expressions in complete partial metric spaces and show by examples that the results proved in this paper cannot be deduced from the corresponding results in metric spaces (see Example 10, Remark 13).

We start with recalling some basic definitions and lemmas on partial metric space. The definition of a partial metric space is given by Matthews (see [1]) as follows.

Definition 1. A partial metric on a nonempty set is a function such that for all : (P1) if and only if ,(P2) ,(P3) ,(P4) .

The pair is then called a partial metric space.

If is a partial metric space, then the function given by , , is a metric on .

A basic example of a partial metric space is the pair , where for all .

Lemma 2 (see [1]). Let be a partial metric space; then one has the following.(1)A sequence in a partial metric space converges to a point if and only if .(2)A sequence in a partial metric space is called a Cauchy sequence if the exists and is finite.(3)A partial metric space is said to be complete if every Cauchy sequence in converges to a point ; that is, .(4)A partial metric space is complete if and only if the metric space is complete. Furthermore, if and only if .

Remark 3 (see [1]). Let be a partial metric space and let be a nonempty set in ; then if and only if where denotes the closure of with respect to the partial metric . Note is closed in if and only if .

Definition 4 (see [24]). Two families of self-mappings and are said to be pairwise commuting if(1) ;(2) ;(3) , .

Now we recall the following definitions and results from [9].

Let be the collection of all nonempty, closed, and bounded subsets of with respect to the partial metric . For , we define For , For ,

Proposition 5 (see [9]). Let be a partial metric space. For any , one has(i) ;(ii) ;(iii)   implies that   ; (iv) .

Proposition 6 (see [9]). Let be a partial metric space. For any , one has(h1) ;(h2) ;(h3) .

Lemma 7 (see [9]). Let and be nonempty, closed, and bounded subsets of a partial metric space and . Then, for every , there exists such that .

Lemma 8 (see [10]). Let and be nonempty, closed, and bounded subsets of a partial metric space and . Then, for every , there exists such that .

2. Results for Single Valued Mappings

The following result, regarding the existence of the common fixed point of the mappings satisfying a contractive condition on the closed ball, is very useful in the sense that it requires the contractiveness of the mappings only on the closed ball instead of the whole space.

Theorem 9. Let be mappings on a complete PMS and and . Suppose that there exist nonnegative reals , , and such that . If and satisfy for all , where . Then there exists a unique point such that . Also .

Proof. Let be an arbitrary point in and define We will prove that for all by mathematical induction. Using inequality (6) and the fact that , we have It implies that . Let for some . If , where , so using inequality (5), we obtain as , and so which implies that If where , one can easily prove that Thus from inequality (11) and (12), we have Now gives . Hence for all . One can easily prove that for all . We now show that is a Cauchy sequence. Without loss of generality assume that . Then, using and the triangle inequality for partial metrics we have Inductively, we have Thus, By the definition of , we get for any , Hence the sequence is a Cauchy sequence in . By Lemma 2(4), is a Cauchy sequence in . Therefore there exists a point with . Also . Again from Lemma 2(4), we have By the triangle inequality , we have Letting and using (19), we obtain By , we concluded that . It follows similarly that . To prove the uniqueness of common fixed point, let be another common fixed point of and , that, is . Then so that because . Therefore which is a contradiction so that (as ). Hence and have a unique common fixed point in .

Example 10. Let endowed with the usual partial metric defined by with . Clearly, is a partial metric space. Now we define as for all . Taking , , , , and , then . Also, we have , with Also if , then So the contractive condition does not hold on whole of . Now if , then Therefore, all the conditions of Theorem 9 are satisfied. Thus is the common fixed point of and and . Moreover, note that for any metric on Therefore common fixed points of and cannot be obtained from a metric fixed point theorem.

Corollary 11. Let be mappings on a complete PMS . Suppose that there exist nonnegative reals , , and such that . If and satisfy for all . Then there exists a unique point such that . Also . Further and have no fixed point other than .

By choosing in Corollary 11, we get the following corollary.

Corollary 12. Let , be a mappings on complete PMS . If and satisfy for all ( is a nonnegative real). Then and have a common fixed point and .

Remark 13. If we impose Banach type contractive condition for a pair of mappings on a metric space ; that is, for all , and then it follows that , for all (i.e., and are equal). Therefore the above condition fails to find common fixed points of and . This can be seen as However the same condition in partial metric space does not assert that . This can be seen as by taking the partial metric same as in Example 10, for any . Hence Corollary 12 cannot be obtained from a metric fixed point theorem.

Remark 14. By equating , , to in all possible combinations, one can derive a host of corollaries which include Matthews theorem for mappings defined on a complete partial metric space.

By taking in the Theorem 9, we get the following corollary.

Corollary 15. Let be a mapping on a complete PMS and and . Suppose that there exist nonnegative reals , and such that . If satisfies for all , where . Then there exists a unique point such that . Also . Further has no fixed point other than .

By taking in Corollary 11, we get the following corollary.

Corollary 16. Let be a mapping on a complete PMS . Suppose that there exist nonnegative reals , and such that . If satisfies for all . Then there exists a unique point such that . Also . Further has no fixed point other than .

Now we give an example in favour of Corollary 16.

Example 17. Let endowed with the usual partial metric defined by . Clearly, is a complete partial metric space. Now we define as follows: for all . Now, let . If (and so ). Then , , , , , . Taking , , , we can prove that all the conditions of Corollary 16 are satisfied. Now if , then , , , , , and taking , , , one can verify the condition of the above corollary. Thus all the conditions of Corollary 16 are satisfied and is a fixed point of the mapping .

As an application of Theorem 9, we prove the following theorem for two finite families of mappings.

Theorem 18. If and are two pairwise commuting finite families of self-mapping defined on a complete partial metric space such that the mappings and (with and satisfy the contractive condition (5), then the component maps of the two families and have a unique common fixed point.

Proof. From Theorem 9, we can say that the mappings and have a unique common fixed point ; that is, . Now our requirement is to show that is a common fixed point of all the component mappings of both the families. In view of pairwise commutativity of the families and , (for every ) we can write and which show that (for every ) is also a common fixed point of and . By using the uniqueness of common fixed point, we can write (for every ) which shows that is a common fixed point of the family . Using the same argument one can also show that (for every ) . Thus component maps of the two families and have a unique common fixed point.

By setting and , in Theorem 18, we get the following corollary.

Corollary 19. Let , be two commuting self-mappings defined on a complete PMS satisfying the condition for all , ( , , and are nonnegative reals). Then F and G have a unique common fixed point.

By setting and in Corollary 19, we deduce the following corollary.

Corollary 20. Let be a mapping defined on a complete PMS satisfying the condition for all , ( , , and are nonnegative reals). Then F has a unique fixed point.

By setting , we draw following corollary which can be viewed as an extension of Bryant's theorem [15] for a mapping on a complete PMS .

Corollary 21. Let be a mapping on a complete PMS . If satisfies for all . Then has a unique fixed point.

The following example demonstrates the superiority of Bryant's theorem over Matthews theorem on complete partial metric space.

Example 22. Let . Define the partial metric by Then is a complete partial metric space. Let be defined as follows: Then for and , we get because . However, satisfies the requirement of Bryant’s theorem and is the unique fixed point of .

3. Results for Set Valued Mappings

Theorem 23. Let be a complete partial metric space and let be mappings such that for all , with . Then and have a common fixed point.

Proof. Assume that . Let be arbitrary but fixed element of and choose . By Lemma 8 we can choose such that So we get Since , so it further implies that By Lemma 8 we can choose such that So we get Continuing in this manner, one can obtain a sequence in as and such that where for all . Without loss of generality assume that . Then, using (48) and the triangle inequality for partial metrics , we have
By the definition of , we get, This yields that is a Cauchy sequence in . Since is complete, then from Lemma 2(4), is a complete metric space. Therefore, the sequence converges to some with respect to the metric ; that is, . Again, from Lemma 2(4), we get taking limit as and using (51), we get Now gives that which implies that On the other hand by , we have Taking limit as and using (51) and (55), we obtain . Therefore, from (51) ( ), we obtain which from Remark 14 implies that . Similarly one can easily prove that . Thus and have a common fixed point.

Remark 24. For and , Theorem 23 reduces to the following result of Aydi et al. [9].

Corollary 25 (see [9, Theorem 3.2]). Let be a partial metric space. If is a multivalued mapping such that for all , one has where . Then has a fixed point.

Theorem 26. Let be a complete partial metric space and be multivalued mappings such that for all , with , and . Then and have a common fixed point.

Proof. Assume that . Let be arbitrary but fixed element of and choose .
When . By Lemma 8 we can choose such that because and . Thus we get It further implies that By Lemma 8 we can choose such that because and . Thus we get It further implies that Continuing in this manner, one can obtain a sequence in as and such that where for all . Without loss of generality assume that . Then using (66) and the triangle inequality for partial metrics , one can easily prove that
By the definition of , This yields that is a Cauchy sequence in . Since is complete, then from Lemma 2(4), is a complete metric space. Therefore, the sequence converges to some with respect to the metric ; that is, . Again, from Lemma 2(4), we get therefore Now gives that which implies that On the other hand, we have Taking limit as and using (69) and (73), we obtain . Therefore, from (69) ( ), we obtain which from Remark 14 implies that . It follows similarly that . Thus and have a common fixed point.

Now we give an example which illustrates our Theorem 26.

Example 27. Let be endowed with usual order and let be a partial metric on defined as Define the mappings by Note that and are closed and bounded for all with respect to the partial metric . To show that for all in , (59) is satisfied with , , we consider the following cases: if , then, and condition (59) is satisfied obviously.

If , then

If , , then

If , , then

If , , then

If , , then Thus, all the conditions of Theorem 26 are satisfied. Here is a common fixed point of and .

On the other hand, the metric induced by the partial metric is given by Note t