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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 141489, 7 pages

http://dx.doi.org/10.1155/2014/141489
Research Article

An Approach to Existence of Fixed Points of Generalized Contractive Multivalued Mappings of Integral Type via Admissible Mapping

1Department of Mathematics, School of Natural Sciences, National University of Sciences and Technology, H-12, Islamabad 44000, Pakistan

2Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan

3Department of Mathematics, Atilim University, Incek, 06836 Ankara, Turkey

4Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah 21491, Saudi Arabia

Received 3 June 2014; Revised 4 July 2014; Accepted 4 July 2014; Published 21 July 2014

Academic Editor: Poom Kumam

Copyright © 2014 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 investigate the existence of a fixed point of certain contractive multivalued mappings of integral type by using the admissible mapping. Our results generalize the several results on the topic in the literature involving Branciari, and Feng and Liu. We also construct some examples to illustrate our results.

1. Preliminaries and Introduction

Fixed point theory is one of the most celebrated research areas that has an application potential not only in nonlinear but also in several branches of mathematics. As a consequence of this fact, several fixed point results have been reported. It is not easy to know, manage, and use all results of this reich theory to get an application. To overcome such problems and clarify the literature, several authors have suggested a more general construction in a way that a number of existing results turn into a consequence of the proposed one. One of the examples of this trend is the investigations of fixed point of certain operator by using the -admissible mapping introduced Samet et al. [1]. This paper has been appreciated by several authors and this trend has been supported by reporting several interesting results; see for example [212].

In this paper, we define ( )-contractive multivalued mappings of integral type and discuss the existence of a fixed point of such mappings. Our construction and hence results improve, extend, and generalize several results including Branciari [13] and Feng and Liu [14].

In what follows, we recall some basic definitions, notions, notations, and fundamental results for the sake of completeness. Let be a family of nondecreasing functions, such that for each , where is the th iterate of . It is known that, for each , we have for all and for [1]. We denote by the set of all Lebesgue integrable mappings, which is summable on each compact subset of and , for each .

Let be a metric space. We denote by the space of all nonempty subsets of , by the space of all nonempty bounded subsets of , and by the space of all nonempty closed subsets of . For and , For every , We denote by when . If, for , there exists a sequence in such that , then is said to be an orbit of at . A mapping is orbitally lower semicontinuous at , if is a sequence in and implies . Branciari [13] extended the Banach contraction principle [15] in the following way.

Theorem 1. Let be a complete metric space and let be a mapping such that for each , where and . Then has a unique fixed point.

Since then many authors used integral type contractive conditions to prove fixed point theorems in different settings; see for example [12, 1622]. Feng and Liu [14] extended the result of Branciari [13] to multivalued mappings as follows.

Theorem 2 (see [14]). Let be a complete metric space and let be a mapping. Assume that for each and , there exists such that where and . Then has a fixed point in provided is lower semicontinuous, with .

Definition 3 (see [3]). Let be a metric space and be a mapping. A mapping is -admissible if , where .

Definition 4 (see [3]). Let be a metric space. A mapping is called -contractive if there exist two functions and such that for all .

Theorem 5 (see [3]). Let be a complete metric space, let be a function, let be a strictly increasing map, and let be a closed-valued -admissible and -contractive multifunction on . Suppose that there exist and such that . Assume that if is a sequence in such that for all and , then for all . Then has a fixed point.

Definition 6 (see [2]). Let be a metric space and let be a mapping. We say that is a generalized -contractive if there exists such that for each and , where .

Theorem 7 (see [2]). Let be a complete metric space and let be a mapping such that for each and , we have where . Assume that there exist and such that . Moreover is an -admissible mapping. Then there exists an orbit of at and such that . Moreover, if and only if is lower semicontinuous at .

2. Main Results

In this section, we state and proof our main results. We first give the definition of the following notion.

Definition 8. Let be a metric space. We say that is an integral type -contractive mapping if there exist two functions and such that for each and , there exists satisfying where .

Example 9. Let be endowed with the usual metric . Define by and by Take and for all . Then, for each and , there exists such that Hence is an integral type -contractive mapping. Note that (4) does not hold at .

Definition 10. We say that is an integral subadditive if, for each , we have

We denote by the class of all integral subadditive functions .

Example 11. Let for all , for all ,  and for all . Then , where .

Definition 12. Let be a metric space. We say that is a subintegral type -contractive if there exist two functions and such that for each and , there exists satisfying where .

Example 13. Let be endowed with the usual metric . Define by and by Take and for all . Then, for each and , there exists such that Hence is an subintegral type -contractive mapping.

Theorem 14. Let be a complete metric space and let be an -admissible subintegral type -contractive mapping. Assume that there exist and such that . Then there exists an orbit of at and such that . Moreover, is a fixed point of if and only if is orbitally lower semicontinuous at .

Proof. By the hypothesis, there exist and such that . Since is -admissible, then . For and , there exists such that Since is nondecreasing, we have As by -admissibility of , we have . For and , there exists such that Since is nondecreasing, we have By continuing the same process, we get a sequence in such that , , and Letting in above inequality, we have Also, we have which implies that For any , we have Since , it can be shown by induction that From (21) and (26), we have Since it follows that is Cauchy sequence in . As is complete, there exists such that as . Suppose is orbitally lower semicontinuous at ; then By closedness of it follows that . Conversely, suppose that is a fixed point of then .

Example 15. Let be endowed with the usual metric . Define by and by Take and for all . Then, for each and , there exists such that Hence is a subintegral type -contractive mapping. Clearly, is -admissible. Also, we have and such that . Therefore, all the conditions of Theorem 14 are satisfied and has infinitely many fixed points. Note that Theorem 2 in Section 1 is not applicable here. For example, take and .

Definition 16. Let be a metric space. We say that is an integral type -contractive mapping if there exist two functions and such that for each and , where .

Definition 17. Let be a metric space. We say that is a subintegral type -contractive mapping if there exist two functions and such that for each and , where .

Theorem 18. Let be a complete metric space and let be an -admissible subintegral type -contractive mapping. Assume that there exist and such that . Then there exists an orbit of at and such that . Moreover, such that if and only if is orbitally lower semicontinuous at .

Proof. By the hypothesis, there exist and such that . Since is -admissible, then . For and , we have Since , then we have such that Since is nondecreasing, we have As by -admissibility of , we have . Thus, we have such that Since is nondecreasing, we have By continuing the same process, we get a sequence in such that , , and Letting in above inequality, we have which implies that For any , we have Since , it can be shown by induction that From (39) and (43), we have Since it follows that is Cauchy sequence in . As is complete, there exists such that as . Suppose is orbitally lower semicontinuous at ; then Hence, because implies . Conversely, suppose that . Then .

Example 19. Let be endowed with the usual metric . Define by and by Take and for all . Clearly, is an -admissible subintegral type -contractive mapping. Also, we have and such that . Therefore, all the conditions of Theorem 18 hold and has fixed points.

Example 20. Let be endowed with the usual metric . Define by and by Take and for all . Then it is easy to check that all the conditions of Theorem 18 hold. Therefore has infinitely many fixed points.

Remark 21. Let for all ; Theorem 18 reduces to Theorem 7 in Section 1.

Remark 22. Note that subadditivity of the integral was needed in the proofs of Theorems 14 and 18 in order to obtain inequalities (26) and (43). It is natural to ask wether the conclusions of Theorems 14 and 18 are valid if we replace subintegral contractive conditions (13) and (33) by integral contractive conditions (8) and (32), respectively. Looking at our proofs, we can say that it will be true if the inequalities (26) and (43) hold. Here we would like to mention that many authors (see for example [14, 23]) while proving the results on integral contractions have not assumed that the integral is subadditive but indeed they used the subadditivity of the integral in the proofs of their results while obtaining the inequalities comparable to inequalities (26) and (43).

3. Application

In this section, we obtain some fixed point results for partially ordered metric spaces, as consequences of aforementioned results. Moreover, we apply our result to prove the existence of solution for an integral equation.

Let and be subsets of a partially ordered set. We say that , if for each and , we have .

Theorem 23. Let be a complete ordered metric space and let be a mapping such that for each and with , there exists satisfying where and . Assume that there exist and such that . Also, assume that implies . Then there exists an orbit of at and such that . Moreover, is a fixed point of if and only if is orbitally lower semicontinuous at .

Proof. Define by By using hypothesis of corollary and definition of , we have . As implies , by using the definitions of and , we have that implies . Moreover, it is easy to check that is an integral type -contractive mapping. Therefore, by Theorem 14, there exists an orbit of at and such that . Moreover, is a fixed point of if and only if is orbitally lower semicontinuous at .

Considering and for each , Theorem 23 reduces to following result.

Corollary 24. Let be a complete ordered metric space and let be a nondecreasing mapping such that, for each with , we have where . Assume that there exists such that . Then there exists an orbit of at and such that . Moreover, is a fixed point of if and only if is orbitally lower semicontinuous at .

Consider an integral equation of the form where is continuous and nondecreasing.

Theorem 25. Assume that(i)for , with for each , we have for each , where ;(ii)for each , there exists such that Then there exists an iterative sequence , starting from , and   such that . Moreover, is a solution of (53) if and only if is lower semicontinuous at , where .

Proof. It is easy to see that is complete with respect to the metric . We define partial ordering on as follows if and only if for each . Define by , where , for each . From (ii), we have . For , let and ; that is, and , for each . Then, for each with , we have That is , for each with . Clearly, is nondecreasing. Therefore, all conditions of Corollary 24 hold and the conclusions follow from Corollary 24.

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.

Acknowledgment

The authors are grateful to the reviewers for their careful readings and useful comments.

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