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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 867216, 15 pages
http://dx.doi.org/10.1155/2012/867216
Research Article

Fixed Point Theorems for Various Classes of Cyclic Mappings

1Institut Supérieur d'Informatique et des Techniques de Communication de Hammam Sousse, Université de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia
2Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey
3Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia

Received 29 June 2012; Accepted 8 September 2012

Academic Editor: Francis T. K. Au

Copyright © 2012 Hassen Aydi 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 new classes of cyclic mappings and we study the existence and uniqueness of fixed points for such mappings. The presented theorems generalize and improve several existing results in the literature.

1. Introduction

The Banach contraction principle is one of the most important results on fixed point theory. Several extensions and generalizations of this result have appeared in the literature and references therein, see [111]. One of the most interesting extensions was given by Kirk et al. in [12].

Theorem 1.1 (see [12]). Let and be two nonempty closed subsets of a complete metric space . Suppose that satisfies (i), ; (ii)there exists a constant such that Then has a unique fixed point that belongs to .

A mapping satisfying (i) is called cyclic. Recently, many results dealing with cyclic contractions have appeared in several works (see, e.g., [1330]). Karpagam and Agrawal [31, 32] introduced the notion of a cyclic orbital contraction, and obtained a unique fixed point theorem for such class of mappings (see also [33, 34]).

In what follows, is the set of positive integers.

Definition 1.2 (see [31]). Let and be nonempty subsets of a metric space , let be a cyclic mapping such that for some , there exists a such that for all and . Then is called a cyclic orbital contraction.

Theorem 1.3 (see [31]). Let and be two nonempty closed subsets of a complete metric space , and let be a cyclic orbital contraction. Then has a unique fixed point that belongs to .

Very recently, Chen [19] introduced the class of cyclic orbital stronger Meir-Keeler contraction.

Definition 1.4 (see [19]). Let be a metric space. A mapping is called a stronger Meir-Keeler type mapping in if the following condition holds for all , there exist and such that for all , we have

Definition 1.5 (see [19]). Let and be nonempty subsets of a metric space . Suppose that is a cyclic map such that for some , there exists a stronger Meir-Keeler type mapping such that for all and . Then is called a a cyclic orbital stronger Meir-Keeler -contraction.

Clearly, if is a cyclic orbital contraction, then is a cyclic orbital stronger Meir-Keeler -contraction, where for all .

Theorem 1.6 (see [19]). Let and be two nonempty closed subsets of a complete metric space , and let be a cyclic orbital stronger Meir-Keeler -contraction, for some . Then has a unique fixed point that belongs to .

In this work, we introduce new classes of cyclic contractive mappings and we study the existence and uniqueness of fixed points for such mappings. Our obtained results extend and generalize several existing fixed point theorems in the literature, including Theorems 1.3 and 1.6.

2. Main Results

We denote by the class of stronger Meir-Keeler type mappings (see Definition 1.4).

The following lemma will be useful later.

Lemma 2.1. Let be a metric space, , . If as , then there exist and such that

Proof. Since , there exist and such that for all , we have Since as , there exists such that It follows from (2.2) that This makes end to the proof.

Our first result is the following.

Theorem 2.2. Let and be two nonempty closed subsets of a complete metric space . Let be a cyclic mapping. Suppose that for some there exists such that for all and . Then has a unique fixed point that belongs to .

Proof. At first, since is a cyclic mapping and , we have . So, taking in (2.5), for all , we get Thus for all , we have Again, taking in (2.5), for all , we get Thus for all , we have It follows from (2.7) and (2.9) that is a decreasing sequence. Then there exists such that Applying Lemma 2.1 with , we obtain that there exist and such that Denote , where is the integer part of . By (2.11) and (2.5), for all , we have This implies that Note that . Similarly, from (2.11) and (2.5), for all , we have This implies that Now, it follows from (2.13) and (2.15) that Denote . From (2.16), for all , we have This implies that is a Cauchy sequence. Since is complete, and are closed, and , there exists a such that as . On the other hand, since is a cyclic mapping and , we have and . Since and are closed, this implies that . On the other hand, since as , by Lemma 2.1, there exist and such that From (2.5) and (2.18), for all , we have Letting in the above inequality, we get which implies (since ) that , that is, is a fixed point of . To show the uniqueness of the fixed point, suppose that is also a fixed point of . Clearly, since is a cyclic mapping, we have . Now, from (2.5), for all , we have Thus we have as . From the uniqueness of the limit, we have . Thus is the unique fixed point of .

Example 2.3. Let with the usual metric for all . Consider and . We have . Define by Fix any in . For all and , we have that is, (2.49) holds. Applying Theorem 2.2, the mapping has a unique fixed point which is .

Our second main result is the following.

Theorem 2.4. Let and be two nonempty closed subsets of a complete metric space . Let be a cyclic mapping. Suppose that for some there exists such that for all and . Then has a unique fixed point that belongs to .

Proof. Taking in (2.24), for all , we get Thus for all , we have Again, taking in (2.24), for all , we get Thus for all , we have It follows from (2.26) and (2.28) that is a decreasing sequence. Then there exists such that Applying Lemma 2.1 with , we obtain that there exist and such that Denote . By (2.30) and (2.24), for all , we have This implies that where . Similarly, we can show that Now, it follows from (2.32) and (2.33) that As in the proof of Theorem 2.2, we obtain from the above inequality that is a Cauchy sequence in , which implies that there exists a such that as . On the other hand, by Lemma 2.1, there exist and such that From (2.24) and (2.35), for all , we have Letting in the above inequality, we get which implies (since ) that , that is, is a fixed point of . Suppose now that is also a fixed point of . From (2.24), for all , we have This implies that Then there exists such that as . On the other hand, we have Thus we have as . By Lemma 2.1, there exist and such that From (2.41) and (2.24), for all , we have Letting , we get which implies (since ) that . Thus we have as . By the uniqueness of the limit, we get that . Thus is the unique fixed point of .

The following result is an immediate consequence of Theorem 2.2.

Theorem 2.5. Let and be two nonempty closed subsets of a complete metric space . Let be a cyclic mapping. Suppose that for some there exists such that for all and . Then has a unique fixed point that belongs to .

The following result is an immediate consequence of Theorem 2.4.

Theorem 2.6. Let and be two nonempty closed subsets of a complete metric space . Let be a cyclic mapping. Suppose that for some there exists such that for all and . Then has a unique fixed point that belongs to .

Now, we present a data dependence result.

Theorem 2.7. Let and be nonempty closed subsets of a complete metric space and are two mappings satisfying (i) is a cyclic mapping; (ii) satisfies (2.5); (iii) has a fixed point ; (iv)there exists such that for all . Then , where is the unique fixed point of .

Proof. From Theorem 2.2, conditions (i) and (ii) imply that has a unique fixed point . Using (2.5), we have Letting and using (iv), we get that

We introduce now the following class of cyclic contractive mappings. At first define the following set Let be the set of functions satisfying the following conditions () is lower semicontinuous; () if and only if .

Definition 2.8. Let and be nonempty subsets of a metric space . A cyclic map is said to be a cyclic weakly orbital contraction if for some , there exists such that

We have the following result.

Theorem 2.9. Let and be two nonempty closed subsets of a complete metric space . Let be a cyclic weakly orbital contraction. Then has a unique fixed point that belongs to .

Proof. Since is a cyclic weakly orbital contraction, there exist and such that (2.48) is satisfied. Taking in (2.48), for all , we get Thus for all , we have Again, taking in (2.48), for all , we get Thus for all , we have Then is a decreasing sequence, and there exists such that Letting in (2.49), using (2.53) and the lower semicontinuity of , we obtain which implies from condition () that . Then we have Now, we shall prove that is Cauchy. From (2.55), it is sufficient to show that is Cauchy. We argue by contradiction, suppose that is not a Cauchy sequence. Then there exists and two subsequences and of such that for all , and We can take the smallest integer such that the above inequality is satisfied. So, for all , we have Using (2.56) and (2.57), we obtain Letting and using (2.55), we have On the other hand, for all , we have Letting , using (2.55) and (2.59), we have Similarly, for all , we have Letting , using (2.55) and (2.59), we have Now, applying (2.48) with , for all , we have Letting in the above inequality, using (2.59), (2.63), and the lower semi-continuity of , we obtain which implies from () that , a contradiction. Thus we proved that is a Cauchy sequence. Since is complete, and are closed, and , there exists a such that as . On the other hand, since is a cyclic mapping and , we have and . Since and are closed, this implies that . Now, we will prove that is a fixed point of . From (2.48), for all , we have Letting and using the properties of , we get which implies that , that is, . Then is a fixed point of . Suppose now that is a fixed point of . From (2.48), for all , we have Letting , we get which implies that , that is, . Then has a unique fixed point.

Denote now by the set of functions satisfying () is locally integrable on ; () for all , we have . The following result is an immediate consequence of Theorem 2.9.

Theorem 2.10. Let and be two nonempty closed subsets of a complete metric space . Let be a cyclic mapping. Suppose that for some there exists such that for all and . Then has a unique fixed point that belongs to .

Remark 2.11. Clearly, any cyclic orbital contraction is a cyclic weakly orbital contraction. So Theorem 1.3 is a particular case of our Theorem 2.9.

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