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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 313782, 6 pages
http://dx.doi.org/10.1155/2013/313782
Research Article

On Fixed Points of -Contractive Multivalued Mappings in Cone Metric Spaces

1Department of Mathematics, King Abdul Aziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan

Received 10 July 2013; Revised 13 September 2013; Accepted 19 September 2013

Academic Editor: Salvador Romaguera

Copyright © 2013 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 define the notion of -contractive mappings for cone metric space and obtain fixed points of multivalued mappings in connection with Hausdorff distance function for closed bounded subsets of cone metric spaces. We obtain some recent results of the literature as corollaries of our main theorem. Moreover, a nontrivial example of -contractive mapping satisfying all conditions of our main result has been constructed.

1. Introduction

Banach contraction principle is widely recognized as the source of metric fixed point theory. Also, this principle plays an important role in several branches of mathematics. For instance, it has been used to study the existence of solutions for nonlinear equations, systems of linear equations, and linear integral equations and to prove the convergence of algorithms in computational mathematics. Because of its importance for mathematical theory, Banach contraction principle has been extended in many directions.

In 2007, Huang and zhang [1] introduced cone metric space with normal cone, as a generalization of metric space. Rezapour and Hamlbarani [2] presented the results of [1] for the case of cone metric space without normality in cone. Many authors work out on it (see [3, 4]). Cho and Bae [5] introduced the Hausdorff distance function on cone metric spaces and generalized the result of [6] for multivalued mappings.

In 2012, Samet et al. [7] introduced the concept of --contractive type mappings. Their results generalized some ordered fixed point results (see [7]). In [8], Karapinar et al. introduced the notion of a -Meir-Keeler contractive mapping and established some fixed point theorems for the G-Meir-Keeler contractive mapping in the setting of G-metric spaces. For more details in fixed point theory related to our paper, we refer to the reader [919]. Asl et al. [20] introduced the notion of --contractive mappings and improved the concept of --contractive mappings along with some fixed point theorems in metric space. Consequently, Ali et al. [21], Mohammadi et al. [22] and Salimi et al. [23] studied the concept of --contractive mappings for proving fixed point results by using generalized contractive conditions in complete metric spaces.

In this paper, we first define the notion of --contractive mappings for cone metric spaces and then we use it to study fixed point theorems for multivalued mappings satisfying --contractive conditions in a complete cone metric space without the assumption of normality. We also furnish a nontrivial example to support our main result.

2. Preliminaries

In the following, we always suppose that is a real Banach space, is a cone in with nonempty interior, and is the partial ordering with respect to . By , we denote the zero element of . A subset is called a cone if and only if(i)is closed, nonempty, and ;(ii);(iii).

For a given cone we define a partial ordering with respect to by if and only if ; will stand for and , while stand for , where denotes the interior of .

Definition 1 (see [1]). Let be a nonempty set. A function is said to be a metric, if the following conditions hold:) for all and if and only if ;() for all ;() for all .
The pair is then called a metric space.

Lemma 2 (see [1]). Let be a metric space, , and let be a sequence in . Then (i) converges to whenever for every with there is a natural number such that , for all . We denote this by ;(ii) is a Cauchy sequence whenever for every with there is a natural number such that , for all ;(iii) is complete cone metric if every Cauchy sequence in is convergent.

Remark 3 (see [3]). The results concerning fixed points and other results, in case of cone spaces with nonnormal cones, cannot be provided by reducing to metric spaces, because in this case neither of the conditions of the lemmas 1–4, in [1] hold. Further, the vector cone metric is not continuous in the general case; that is, from , it need not follow that .

Let be a cone metric space. The following properties of cone metric spaces have been noticed [3]. If and , then . If and , then . If and , then . If for each , then . If , for each , then .() be a sequence in . If and (as ), then there exists such that for all , we have .

With some modifications, we have the following definition from [24].

Definition 4. Let be a family of nondecreasing functions, such that(i) and for ,(ii) implies ,(iii) for every .

3. Main Result

For a cone metric space , denote (see [5])

For we denote

Lemma 5. Let be a cone metric space, and let be a cone in Banach space .(1)Let . If , .(2)Let and . If , then .(3)Let and let and . If , then for all or for all .(4)Let and let , then .

Remark 6. Let be a metric space. If and , then is a metric space. Moreover, for , is the Hausdorff distance induced by .

Definition 7. Let be a complete cone metric space with cone , , and is known as --contractive multivalued mapping whenever for all , where . Also, we say that is -admissible whenever implies .

Note that an --contractive multivalued mappings for cone metric space is generalized --contractive. When is a strictly increasing mapping, --contractive is called strictly generalized --contractive.

Theorem 8. Let be a complete cone metric space with cone be a function, be a strictly increasing map and , and be -admissible and --contractive multivalued mapping on . Suppose that there exist such that . Assume that if is a sequence in such that for all and as then for all . Then, there exists a point in such that .

Proof. We may suppose that . Then and By Lemma 5(3), we have By definition, we can take such that By Lemma 5(4), we have So, Hence, and . Thus and . If , then is a fixed point of . Assume that . Then By Lemma 5(3), we have By definition, we can take such that By Lemma 5(4), we have So, Hence, It is clear that and . Thus, and .
If , then is a fixed point of . Assume that : By Lemma 5(3), we have By definition, we can take such that By Lemma 5(4), we have So Hence It is clear that and . Thus, and . By continuing this process, we obtain a sequence in such that , , and for all .

Fix . We choose a positive real number such that , where . By (iii) of Definition 4, there exists a natural number such that , for all . Then for all . Consequently, , for all . Fix . Now we prove

for all . Note that (24) holds when . Assume that (24) holds for some . Then, we have

Now by (22), we have Therefore, (24) holds when . By induction, we deduce that (24) holds for all . This is sufficient to conclude that is a Cauchy sequence. Choose such that . Since for all and is admissible, so for all . From (3), we have

for all . By Lemma 5(3), we have

By definition, we can take such that

By Lemma 5(4), we have

So

Hence

Moreover, for a given , we have Hence, according to Lemma 2(i), we have . Since is closed, .

Theorem 9. Let be a complete cone metric space with cone be a function, and be -admissible. If there exists a constant such that for all . Suppose that there exist such that . Assume that if is a sequence in such that for all and as ; then for all . Then, there exists a point in such that .

Proof. Take in Theorem 8.

Theorem 10. Let be a complete cone metric space with cone be a strictly increasing map, and be multivalued mapping such that for all . Then, there exists a point in such that .

Proof. Take in the Theorem 8.

Corollary 11. Let be a complete cone metric space with cone and let be a multivalued mapping. If there exists a constant such that for all , then, there exists a point in such that .

Proof. Take and in the Theorem 8.

Corollary 12 (see [20]). Let be a complete metric space be a function, be a strictly increasing map, and be -admissible such that for all . Suppose that there exist such that . Assume that if is a sequence in such that for all and as then for all . Then, there exists a point in such that .

By Remark 6, we have the following corollaries.

Corollary 13 (see [20]). Let be a complete metric space be a strictly increasing map, and be a multivalued mapping such that for all . Then, there exists a point in such that .

Proof. Take in the Corollary 12.

Corollary 14 (see [20]). Let be a complete metric space, be a function, and be -admissible. If there exists a constant such that for all . Suppose that there exist such that . Assume that if is a sequence in such that for all and as then for all . Then, there exists a point in such that .

Proof. Take in the Corollary 13.

Corollary 15 (see [25]). Let be a complete metric space and let be a multivalued mapping. If there exists a constant such that for all . Then, there exists a point in such that .

Proof. Take in the Corollary 14.

Example 16. Let , , , where for all . Define by and by for all and Then, . Then, clearly is -admissible. Now for and , we get which implies that So is --contractive multivalued mapping on where . Thus, all the conditions of main result are satisfied to obtain the fixed point of .

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contributions

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

Acknowledgments

The authors thank the editor and the referees for their valuable comments and suggestions which improved greatly the quality of this paper. The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

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