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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 418798, 4 pages
http://dx.doi.org/10.1155/2013/418798
Research Article

Mizoguchi-Takahashi’s Fixed Point Theorem with , Functions

1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12, Islamabad, Pakistan
2Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan
3Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand
4Division of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

Received 6 September 2013; Accepted 13 December 2013

Academic Editor: Hassen Aydi

Copyright © 2013 Muhammad Usman Ali 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 introduce the notion of generalized -admissible mappings. By using this notion, we prove a fixed point theorem. Our result generalizes Mizoguchi-Takahashi’s fixed point theorem. We also provide some examples to show the generality of our work.

1. Introduction and Preliminaries

Let be a metric space. For each and , . We denote by the class of all nonempty compact subset of , by the class of all nonempty closed and bounded subsets of , and by the class of all nonempty closed subsets of . For every , let Such a map is called generalized Hausdorff metric induced by .

Nadler [1] extended the Banach contraction principle to multivalued mappings as follows.

Theorem 1 (see [1]). Let be a complete metric space and is a mapping from into such that where . Then has a fixed point.

Reich [2] extended the above result in the following way.

Theorem 2 (see [2]). Let be a complete metric space and is a mapping satisfying where and is a function from into such that Then has a fixed point.

Reich [2] raised the question: whether the range of , can be replaced by or . Mizoguchi and Takahashi [3] gave a positive answer to the conjecture of Reich [2], when the inequality holds also for ; in particular they proved the following.

Theorem 3 (see [3]). Let be a complete metric space and is a mapping satisfying where and is a function from into such that Then has a fixed point.

The other proofs of Theorem 3 have been given by Daffer and Kaneko [4] and Chang [5]. Eldred et al. [6] claimed that Theorem 3 is equivalent to Theorem 1. Suzuki produced an example [7, page 753] to disprove their claim and showed that Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s fixed point theorem. Reader can find some more results related to Mizoguchi-Takahashi’s fixed point theorem in [814].

Samet et al. [15] introduced the notion of --contractive and -admissible self-mappings and proved some fixed point results for such mappings in complete metric spaces. Karapinar and Samet [16] generalized these notions and obtained some fixed point results. Asl et al. [17] extended these notions to multifunctions by introducing the notions of --contractive and -admissible mappings and proved some fixed point results. Some results in this direction are also given by the authors [18, 19]. Ali and Kamran [20] further generalized the notion of --contractive mappings and obtained some fixed point theorems for multivalued mappings.

Recently, Salimi et al. [21] modified the notions of --contractive and -admissible self-mappings by introducing another function and established some fixed point theorems for such mappings in complete metric spaces. Hussain et al. [22] extended the result of Asl et al. and introduced the following definition.

Definition 4 (see [22]). Let be a mapping on a metric space . Let be two functions, where is bounded. We say that is an -admissible mapping with respect to if we have where and . In case when for all , then is -subadmissible mapping. In case when for all , then is -admissible.

Definition 5 (see [17]). Let be a metric space and let be a mapping. A mapping is -admissible if , where .

In this paper, we generalize Definition 4 and provide some examples to show generality of such concept. We also establish a fixed point theorem which generalizes Mizoguchi-Takahashi’s fixed point theorem. Some illustrative examples to claim that our results properly generalize some results in the literature are presented. Furthermore, at the end of this paper, we give an open problem for further investigation.

2. Main Results

We begin this section by generalizing Definition 4.

Definition 6. Let be a mapping on a metric space . Let be two functions. We say that is generalized -admissible mapping with respect to if we have When for all , then is a generalized -subadmissible mapping. When for all , then is -admissible.

Remark 7. Note that inequality (8) is weaker than (7). Moreover, involved in inequality (8) is not necessarily bounded.

Example 8. Let be endowed with the usual metric . Define by for all , by and by for each . Then for with , we have for all and . Therefore is generalized -admissible mapping with respect to but it is not -admissible mapping with respect to .

Theorem 9. Let be a complete metric space and let be a generalized -admissible mapping with respect to such that where satisfying for all . Assume that the following conditions hold:(i)there exist and such that ;(ii)either(1) is continuousor(2)if is a sequence in with as and for each , then one has for each .Then has a fixed point.

Proof. By hypothesis, there exist and such that . If , then we have nothing to prove. Let . Then from (10), we have There exists such that Consider for all . Then for all . From (12), we have Since is an -admissible mapping with respect to , then . If , then we have nothing to prove. Let . Then from (10), we have There exists such that Continuing the same process, we get a sequence in such that , , , and It follows from for all that is a strictly decreasing sequence in . Hence it converges to some nonnegative real number . Since and , there exists and such that for all . We can find such that for all with . Then for each . Also, we have Hence is a Cauchy sequence in . Since is complete, then there exists such that . If we suppose that is continuous, then On the other hand, since for each and as , then we have for each . Then from (10), we have Therefore . This completes the proof.

The following example shows that Theorem 9 properly generalizes Theorem 3, in Section 1.

Example 10. Let be endowed with the usual metric . Define by by and by for each . Take for all . Then for with , we have Also, is generalized -admissible mapping with respect to . For and we have . Further, for any sequence in with as and for each , we have for each . Therefore, all conditions of Theorem 9 are satisfied and has infinitely many fixed points.

Corollary 11. Let be a complete metric space and let be an -admissible mapping with respect to such that where satisfying for all . Assume that the following conditions hold:(i)there exist and such that ;(ii)either(1) is continuousor(2)if is a sequence in with as and for each , then we have for each .Then has a fixed point.

Proof. We can prove this result by using Theorem 9 and the fact that inequality (8) is weaker than (7).

Acknowledgments

This research was supported by the Commission on Higher Education, the Thailand Research Fund, and the Rajamangala University of Technology Lanna Tak (Grant no. MRG5580233).

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