Journal of Applied Mathematics

Volume 2012 (2012), Article ID 867216, 15 pages

http://dx.doi.org/10.1155/2012/867216

## Fixed Point Theorems for Various Classes of Cyclic Mappings

^{1}Institut Supérieur d'Informatique et des Techniques de Communication de Hammam Sousse, Université de Sousse, Route GP1, 4011 Hammam Sousse, Tunisia^{2}Department of Mathematics, Atilim University, 06836 Incek, Ankara, Turkey^{3}Department 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 [1–11]. 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., [13–30]). 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. *
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. *
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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