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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 363925, 7 pages

http://dx.doi.org/10.1155/2014/363925
Research Article

Generalized Common Fixed Point Results with Applications

1Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

2Department of Mathematics & Statistics, International Islamic University H-10, Islamabad, Pakistan

3Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan

Received 10 February 2014; Accepted 23 March 2014; Published 27 April 2014

Academic Editor: Ljubomir B. Ćirić

Copyright © 2014 Marwan Amin 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 obtained some generalized common fixed point results in the context of complex valued metric spaces. Moreover, we proved an existence theorem for the common solution for two Urysohn integral equations. Examples are presented to support our results.

1. Introduction and Preliminaries

Since the appearance of the Banach contraction mapping principle, a number of papers were dedicated to the improvement and generalization of that result. Most of these deal with the generalizations of the contractive condition in metric spaces.

Gähler [1] generalized the idea of metric space and introduced a 2-metric space which was followed by a number of papers dealing with this generalized space. Plenty of material is also available in other generalized metric spaces, such as, rectangular metric spaces, semimetric spaces, pseudometric spaces, probabilistic metric spaces, fuzzy metric spaces, quasimetric spaces, quasisemi metric spaces, -metric spaces, -metric space, partial metric space, and cone metric spaces (see [214]). Azam et al. [15] improved the Banach contraction principle by generalizing it in complex valued metric space involving rational inequity which could not be handled in cone metric spaces [3, 5, 11, 15] due to limitations regarding product and quotient. Rouzkard and Imdad [16] extended the work of Azam et al. [15]. Sintunavarat and Kumam [17] obtained common fixed point results by replacing constant of contractive condition to control functions. Recently, Klin-eam and Suanoom [12] extend the concept of complex valued metric spaces and generalized the results of Azam et al. [15] and Rouzkard and Imdad [16]. In this paper we continue the study of complex valued metric spaces and established some fixed point results for mappings satisfying a rational inequality. The idea of complex valued metric spaces can be exploited to define complex valued normed spaces and complex valued Hilbert spaces and then it will bring wonderful research activities in nonlinear analysis.

In this paper we continue our investigations initiated by Azam et al. [15] and prove a common fixed point result for two mappings and applied it to get the coincidence and common fixed points of three and four mappings.

We begin with listing some notations, definitions, and basic facts on these topics that we will need to convey our theorems. Let be the set of complex numbers and . Define a partial order on as follows: It follows that if one of the following conditions is satisfied: In particular, we will write if and one of (i), (ii), and (iii) is satisfied and we will write if only (iii) is satisfied. Note that

Definition 1. Let be a nonempty set. Suppose that the self-mapping satisfies:(1) , for all and if and only if ;(2) for all ;(3) , for all .

Then is called a complex valued metric on , and is called a complex valued metric space. A point is called interior point of a set whenever there exists such that

A point is called a limit point of whenever for every ,

is called open whenever each element of is an interior point of . Moreover, a subset is called closed whenever each limit point of belongs to . The family is a subbasis for a Hausdorff topology on .

Let be a sequence in and . If for every with there is such that for all , , then is said to be convergent, converges to , and is the limit point of . We denote this by , or , as . If for every with there is such that for all , , then is called a Cauchy sequence in . If every Cauchy sequence is convergent in , then is called a complete complex valued metric space. Let be a nonempty set and . The mappings , are said to be weakly compatible if they commute at their coincidence point (i.e., whenever ). A point is called point of coincidence of and if there exists a point such that . We require the following lemmas.

Lemma 2 (see [15]). Let be a complex valued metric space and let be a sequence in . Then converges to if and only if as .

Lemma 3 (see [15]). Let be a complex valued metric space and let be a sequence in . Then is a Cauchy sequence if and only if as .

Definition 4 (see [18]). Two families of self-mappings and are said to be pairwise commuting if:(1) , ;(2) , ;(3) , , .

Lemma 5 (see [19]). Let be a nonempty set and a function. Then there exists a subset such that and is one to one.

Lemma 6 (see [20]). Let be a nonempty set and the mappings have a unique point of coincidence in . If and are weakly compatible, then is a unique common fixed point of , , .

2. Main Results

Theorem 7. Let be a complete complex valued metric space and . If the self-mappings satisfy for all , where then and have a unique common fixed point.

Proof. We will first show that fixed point of one map is a fixed point of the other. Suppose that . Then from (8) Case 1 Case 2 which  yields  that   .

Case 3 Case 4 which implies that , and hence . In a similar manner it can be shown that any fixed point of is also the fixed point of . Let and define We will assume that for each . Otherwise, there exists an such that . Then and is a fixed point of , hence a fixed point of . Similarly, if for some , then is common fixed point of and hence of . From (8) Case 1 Case 2 Case 3 which implies that a contradiction to our assumption.

Case 4 That is a contradiction to our assumption.

Thus, . Similarly, one can show that . It follows that, for all , Now for any , and so This implies that is a Cauchy sequence. Since is complete, there exists such that . It follows that ; otherwise and we would then have Case 1 That is, , a contradiction and hence .

Case 2 That is, , a contradiction and hence .

Case 3 This in turn gives us , a contradiction and hence .

Case 4 That is, and hence . It follows similarly that . We now show that and have unique common fixed point. For this, assume that in is another common fixed point of and . Then .

Case 1 Case 2 This gives us .

Case 3 Case 4 Hence, in all cases . This completes the proof of the theorem.

Corollary 8 (see [15]). Let be a complete complex valued metric space and let and . If the self-mappings , satisfy for all , where then and have a unique common fixed point.

Corollary 9. Let be a complete complex valued metric space and . If the self-mapping satisfies for all , where then has a unique fixed point.

Corollary 10 (see [15]). Let be a complete complex valued metric space and let and . If the self-mapping satisfies for all , where then has a unique fixed point.

As an application of Theorem 7, we prove the following theorem for two finite families of mappings.

Theorem 11. If and are two finite pairwise commuting finite families of self-mapping defined on a complete complex valued metric space such that the mappings and (with and ) satisfy the contractive condition (8), then the component maps of the two families and have a unique common fixed point.

Proof. From theorem 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 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 11, we get the following corollary.

Corollary 12. If and are two commuting self-mappings defined on a complete complex valued metric space satisfying the condition for all and , where then and have a unique common fixed point.

Corollary 13. Let be a complete complex valued metric space and let be a self-mapping satisfying for all and , where Then has a unique fixed point.

Corollary 14 (see [15]). Let be a complete complex valued metric space and and . The self-mapping satisfies for all , where Then has a unique fixed point.

Our next example exhibits the superiority of Corollary 13 over Corollary 9.

Example 15. Let and and let . Then with , set and define as follows: Consider a complex valued metric as follows: where , . Then is a complete complex valued metric space. By a routine calculation, one can verify that the map satisfies condition (43) with . It is interesting to notice that this example cannot be covered by Corollary 9 as , implies a contradiction for every choice of which amounts to say that condition (37) is not satisfied. Notice that the point remains fixed under and and is indeed unique.

3. Application

By providing the following result, we establish an existence theorem for the common solution for two Urysohn integral equations.

Theorem 16. Let , , and is defined as follows: Consider the Urysohn integral equations where , .

Suppose that are such that for each , where, If there exists such that for every where then the system of integral equations and has a unique common solution.

Proof. Define by Then It is easily seen that , where for every . By Theorem 7, the Urysohn integral equations and have a unique common solution.

Remark 17. Now we will apply techniques of [6] to obtain the common fixed points of three and four mappings by using a common fixed point result for two mappings.

Theorem 18. Let be a complete complex valued metric space and . Let by the self-mappings such that . Assume that the following holds: for all , where If and are weakly compatible and is closed, then , , and have a unique common fixed point in .

Proof. By Lemma 5, there exists such that and is one to one. Now define the self-mappings by and , respectively. Since is one to one on , then , are well defined. Note that where By Theorem 7 as is complete, we deduce that there exists a unique common fixed point of and ; that is, . Thus, is a coincidence point of , , and . Now we show that , , and have unique point of coincidence. Now let such that , . Then is another common fixed point of and , which is a contradiction, which implies that , , and have a unique point of coincidence. Since and are weakly compatible by Lemma 6, we deduce that is a unique common fixed point of , , and .

Theorem 19. Let be a complete complex valued metric space and . Let by the self-mappings such that . Assume that the following holds: for all , where If and are weakly compatible and is closed in , then , , , and have a unique common fixed point in .

Proof. By Lemma 5, there exists such that , are one to one. Now define the mappings by and , respectively. Since , are one to one on and , respectively, then the mappings , are well-defined. Now where for all . By Theorem 7, as is complete subspace of , we deduce that there exists a unique common fixed point of and ; that is, . This implies that ; let such that . We have . We show that and have a unique point of coincidence. If then is a fixed point of . By the proof of Theorem 7 is another common fixed point of and which is a contradiction. Hence, and have a unique point of coincidence. By Lemma 6, it follows that is a unique common fixed point of and . Similarly, is the unique common fixed point for and . This proves that is the unique common fixed point for , , , and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

This paper is dedicated to Professor Miodrag Mateljević on the occasion of his 65th birthday. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The third author gratefully acknowledges the support from the Higher Education Commission of Pakistan.

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