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Journal of Applied Mathematics
Volume 2012, Article ID 165698, 15 pages
http://dx.doi.org/10.1155/2012/165698
Research Article

Fixed Point Theorem for Cyclic Chatterjea Type Contractions

1Department of Mathematics, Atılım University, Incek, 06836 Ankara, Turkey
2Department of Mathematics, Disha Institute of Management and Technology, Satya Vihar, Vidhansabha-Chandrakhuri Marg, Mandir Hasaud, Raipur 492101, India

Received 23 April 2012; Revised 1 August 2012; Accepted 1 August 2012

Academic Editor: Saeid Abbasbandy

Copyright © 2012 Erdal Karapınar and Hemant Kumar Nashine. 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 cyclic weakly Chatterjea type contraction and generalized cyclic weakly Chatterjea type contraction in metric spaces. We discussed the existence of fixed point theorems of (generalized) cyclic weakly Chatterjea type contraction mappings in the context of complete metric spaces. Our main theorems extend and improve some fixed point theorems in the literature.

1. Introduction and Preliminaries

Celebrated Banach’s contraction mapping principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the various branches of mathematics but also in economics, chemistry, biology, computer science, engineering, and others. In particular, such theorems are used to demonstrate the existence and uniqueness of a solution of differential equations, integral equations, functional equations, partial differential equations, and others. Due to the importance, generalizations of Banach fixed point theorem have been investigated heavily by many authors. (see, e.g., [1, 2]).

Following this trend, in 1972, Chatterjea [3] introduced the following definition.

Definition 1.1. Let be a metric space. A self-mapping is called a Chatterjea type contraction if there exists such that for all the following inequality holds:

In this interesting paper, Chatterjea [3] proved that every Chatterjea type contraction on a complete metric space has a unique fixed point. Later, Choudhury [4] introduced a generalization of Chatterjea type contraction as follows.

Definition 1.2. A self-mapping , on a metric space , is said to be a weakly -contractive (or a weak Chatterjea type contraction) if for all , where is a continuous function such that

In [4], the author proves that every weak Chatterjea type contraction on a complete metric space has a unique fixed point.

One of the interesting generalizations of a Banach’s contraction mapping principle was given by Kirk et al. [5] in 2003 by introducing the following notion of cyclic representation.

Definition 1.3 (see [5]). Let be a nonempty set and let be an operator. By definition, is a cyclic representation of with respect to if(a), are nonempty sets,(b), .

After the remarkable paper of Kirk et al. [5], some new fixed-point theorems for operators defined on a complete metric space with a cyclic representation of with respect to have appeared in the literature (see, e.g., [1, 2, 419]). Very recently, Karapınar [10] introduced the notion of the cyclic weak -contraction and proved fixed point theorems for these types of contractions.

Definition 1.4 (see [10]). Let be a metric space, closed nonempty subsets of , and . An operator is called a cyclic weak -contraction if(1) is a cyclic representation of with respect to ;(2)there exists a continuous, nondecreasing function with for and such that for any where .

Let denote all the continuous functions with for and .

Theorem 1.5 (see [10]). Let be a complete metric space, nonempty subsets of , and . Suppose that is a cyclic weak -contraction with . Then, has a fixed point .

In this paper, we introduce the notions of cyclic weakly Chatterjea type contractions and generalized cyclic weakly Chatterjea type contractions and then derive fixed point theorems on these cyclic contractions in the setup of complete metric spaces. Our results generalize fixed point theorems [2, 5, 20] in the sense of metric spaces.

2. Main Results

2.1. Fixed Point Theory for the Cyclic Weakly Chatterjea Type Contractions in Complete Metric Space

First we introduce the notion of cyclic weakly Chatterjea type contraction in metric space.

For convenience, we denote by the class of functions lower semicontinuous satisfying if and only if .

Definition 2.1. Let be a metric space, nonempty subsets of , and . An operator is called a cyclic weakly Chatterjea type contraction if(1) is a cyclic representation of with respect to ;(2)for any ,  ,   where , and .

Notice that the cyclic weak Chatterjea type contractions constitute a strictly larger class of mappings than cyclic weak -contractions.

The main result of this section is the following.

Theorem 2.2. Let be a complete metric space, ,   nonempty closed subsets of , and . Suppose that is a cyclic weakly Chatterjea type contraction. Then, has a unique fixed point .

Proof. Take and consider the sequence given by ,  . If there exists such that , then the existence of the fixed point is proved. Indeed, and is the desired point. Suppose that for all . Since , for any there exists such that and . Due to the fact that is a cyclic weakly Chatterjea type contraction, we have Therefore, If we take , then we have It follows that and In the sequel, we will prove that is a Cauchy sequence. First, we prove the following claim. For every there exists such that if with , then .
Suppose, to the contrary, that there exists such that for any we can find with satisfying Now, we take . Then, corresponding to we can choose in such a way that it is the smallest integer with satisfying and . Therefore, . Using the triangular inequality, we obtain Taking (2.5) into account and letting in the inequality above, we find Again, by the triangular inequality, we have Passing to the limit as in (2.8) and taking (2.5) and (2.8) into account, we get Since and lie in different adjacently labeled sets and for certain , using the fact that is a cyclic weakly Chatterjea type contraction, we have Taking (2.5) and (2.8) into account together with the lower semicontinuity of , the inequalities above yield as . Hence, . From the fact that , we have , which is a contradiction. Therefore, our claim is proved.
We prove that is a Cauchy sequence. Fix . By the claim, we find such that if with , Since , there exists such that for any . Suppose that with . Then, there exists such that . Therefore, we have for . So, we derive By (2.13) and (2.14) and the inequality above, we get This proves that is a Cauchy sequence. Since is a complete metric space, there exists such that .
Now, we prove that is a fixed point of . Since and, as is a cyclic representation of with respect to , the sequence has infinite terms in each for . Suppose that and we take a subsequence of with (the existence of this subsequence is guaranteed by the previously mentioned comment). By using the contractive condition, we obtain Letting and by using together with the lower semicontinuity of , we have which is a contradiction (since ) unless . Therefore, is a fixed point of .
Finally, to prove the uniqueness of the fixed point, we have with and being fixed points of . The cyclic character of and the fact that are fixed points of imply that . Using the contractive condition we obtain That is, This gives us , and, by our assumption about , , that is, . This finishes the proof.

Corollary 2.3. Let be a complete metric space, nonempty subsets of , and . Suppose that be an operator such that(1) is a cyclic representation of with respect to ;(2)there exists such that for any where . Then, has a fixed point .

Proof. Let . Here, it suffices to take the function as . It is clear that satisfies the following conditions:(1) if and only if ,(2). Hence, we apply Theorems 2.2 for and get the desired result.

The following corollary gives us a fixed point theorem with a contractive condition of integral type for cyclic contractions.

Corollary 2.4. Let be a complete metric space, ,   nonempty closed subsets of , and . Suppose that be an operator such that(i) is a cyclic representation of with respect to ;(ii)there exists such that for any where and is a Lebesgue-integrable mapping satisfying for . Then has unique fixed point .

Proof. It is easily proved that the function given by satisfies that . Therefore, Corollary 2.3 is obtained from Theorem 2.2, taking as the previously defined function and as the function .

If in Corollary 2.4, we take for , we obtain the following result.

Corollary 2.5. Let be a complete metric space and let be a mapping such that for any , where is a Lebesgue-integrable mapping satisfying for and the constant . Then has unique fixed point.

If in Theorem 2.2 we put for we get the main result of [4].

Corollary 2.6. Let be a complete metric space and a mapping such that for any , where and . Then has unique fixed point.

Example 2.7. Let with the usual metric. Suppose , and . Define such that for all . It is clear that is a cyclic representation of with respect to . Furthermore, if is defined by , then . Here is a generalized cyclic weakly Chatterjea type contraction for . To see this fact we examine three cases.(i)Suppose that . Then, the inequality (2.1) turns into If , then (2.25) becomes Hence, (2.1) is satisfied. If , then (2.25) becomes Thus, (2.1) is true.(ii)Suppose that . Then, the inequality (2.1) yields that So, (2.1) holds.(iii)Finally, suppose that . Then, the inequality (2.1) yields that Hence, (2.1) is true. Therefore, all conditions of Theorem 1.5 are satisfied, and so has a fixed point (which is .)

2.2. Fixed Point Theory for the Generalized Cyclic Weakly Chatterjea Type Contractions in Complete Metric Space

In this section we derive fixed point theorems for self-maps satisfying certain generalized cyclic weakly Chatterjea type contractions in a complete metric space.

For convenience, we denote by the class of functions lower semicontinuous satisfying if and only if .

We introduce the notion of generalized cyclic weakly Chatterjea type contraction in metric space.

Definition 2.8. Let be a metric space, nonempty subsets of and . An operator is called a generalized cyclic weakly Chatterjea type contraction if(1) is a cyclic representation of with respect to ; (2) for any where and

We state the main result of this section as follows.

Theorem 2.9. Let be a complete metric space, ,   nonempty closed subsets of and . Suppose that is a generalized cyclic weakly C-contraction. Then, has a unique fixed point .

Proof. As in the proof of Theorem 2.2, we take and construct a sequence by defining . Suppose that there exists such that . Then, since , the existence of the fixed point is proved. So, we assume that for any . Since , for any , there exists such that and . Regarding that is a generalized cyclic weakly Chatterjea type contraction, we have Therefore, If we take , then we have It follows that and We show that is a Cauchy sequence. For this purpose, first, we prove the following claim. For every there exists such that if with , then .
Assume the contrary. So there exists such that for any we can find with satisfying Now, we take . Then, corresponding to we can choose in such a way that it is the smallest integer with satisfying and . Therefore, by using the triangular inequality Passing to the limit as in the last inequality and taking into account that , we obtain Again, by the triangular inequality Taking (2.35) and (2.38) into account, we get as in (2.38).
Since and lie in different adjacently labeled sets and for certain , using the fact that is a generalized cyclic weakly Chatterjea type contraction, we have Taking into account (2.35) and (2.38) and the lower semicontinuity of , letting in the last inequality, we obtain and from the last inequality, . Therefore . From the fact that , we have , which is a contradiction. Therefore, our claim is proved.
In the sequel, we will prove that is a Cauchy sequence. Fix . By the claim, we find such that if with Since , we also find such that for any . Suppose that and . Then there exists such that . Therefore, for . So, we have By (2.43) and (2.44) and from the last inequality, we get This proves that is a Cauchy sequence. Since is a complete metric space, there exists such that . In what follows, we prove that is a fixed point of . In fact, since and as is a cyclic representation of with respect to , the sequence has infinite terms in each for . Suppose that and we take a subsequence of with (the existence of this subsequence is guaranteed by the previously mentioned comment). By using the contractive condition, we can obtain Passing to the limit as and using , lower semicontinuity of , we have which is a contradiction unless , and, therefore, is a fixed point of . Finally, to prove the uniqueness of the fixed point, we have with and being fixed points of . The cyclic character of and the fact that are fixed points of imply that . Using the contractive condition we obtain that is, This gives us , and, by our assumption about , , that is, . This finishes the proof.

Corollary 2.10. Let be a complete metric space, , nonempty subsets of , and . Suppose that be an operator such that(1) is a cyclic representation of with respect to ;(2)there exists such that for any , where . Then, has a fixed point .

Proof. Let . Here, it suffices to take the function defined by . Obviously, satisfies that if and only if , and . Then, we can apply Theorems 2.9.

The following corollary gives us a fixed point theorem with a contractive condition of integral type for cyclic contractions.

Corollary 2.11. Let be a complete metric space, nonempty closed subsets of and . Suppose that be an operator such that(1) is a cyclic representation of with respect to ;(2)there exists such that for any , where and is a Lebesgue-integrable mapping satisfying for . Then has unique fixed point .

Proof. It is easily proved that the function given by satisfies that . Therefore, Corollary 2.10 is obtained from Theorem 2.9, taking as the perviously defined function and as the function .

If in Corollary 2.11, we take for , we obtain the following result.

Corollary 2.12. Let be a complete metric space and let be a mapping such that for any , where is a Lebesgue-integrable mapping satisfying for and the constant . Then has unique fixed point.

If in Theorem 2.9 we put for , we have the generalized result of [4].

Corollary 2.13. Let be a complete metric space and let be a mapping such that for any , where and . Then has unique fixed point.

Example 2.14. Let with the usual metric. Suppose , , , and . Define such that for all . It is clear that is a cyclic representation of with respect to . Furthermore, if is defined by , then . As in Example 2.7, it can be easily shown that is a generalized cyclic weakly Chatterjea type contraction . Therefore, all conditions of Theorem 2.9 are satisfied, and so has a fixed point (which is ).

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