Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 934758, 4 pages

http://dx.doi.org/10.1155/2013/934758

Research Article

A Note on the Fixed Point of Generalized Eventual Cyclic Gross Contractive Mappings

Jin Liang and Sheng-Hua Yan

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 5 August 2013; Accepted 7 September 2013

Academic Editor: T. Diagana

Copyright © 2013 Jin Liang and Sheng-Hua Yan. 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.

Based on the work of Agarwal et al., in press, we introduce a concept of generalized eventual cyclic gross contractive mapping in metric spaces, which is a generalization of the eventual cyclic gross contractions. Furthermore, we analyze the existence of a fixed point for this type of contractive mapping, and present a new and general fixed point theorem.

1. Introduction

The following classical Banach contraction principle is well-known.

Theorem 1. If is a self-mapping in a complete metric space satisfying the following condition:

then has a unique fixed point.

Based on this principle and the idea of its proof, many researchers have presented numerous extensions of the principle and new types of fixed point theorems (cf, e.g., [1–3]).

In [3], Karpagam and Agrawal proved the following unique fixed point for cyclic mappings.

Theorem 2 (see [3]). Let and be two nonempty closed subsets of a complete metric space . Suppose is a cyclic mapping such that for some , there exists a such that

Then, is nonempty and has a unique fixed point in .

Very recently, Agarwal et al. [1] defined a new type of cyclic mappings, “eventual cyclic gross contraction” in complete metric spaces, and established a fixed point theorem for the eventual cyclic gross contraction.

Theorem 3 (see [1]). Let and be two nonempty closed subsets of a complete metric space , and let be an eventual cyclic gross contraction. Then is nonempty and has a unique fixed point in .

Inspired by the work in [1], in this paper, we introduce a concept of generalized eventual cyclic gross contractive mapping in metric spaces, which is a generalization of the eventual cyclic gross contractions. Furthermore, we obtain a fixed point theorem for this type of contractive mappings, which extends Theorem 3.

2. Generalized Eventual Cyclic Gross Contractive Mappings

Let be the set of all nonnegative real numbers and the set of all natural numbers. Let be a metric space, and let and be nonempty subsets of .

Now, we introduce the concept of generalized eventual cyclic gross contractive mappings.

Definition 4. A mapping is called a generalized eventual cyclic gross contractive mapping if the following are satisfied:

 (1),  (2)for some ,

where , is a monotone increasing and continuous function, is a lower semicontinuous mapping such that if and only if , and is sufficiently large.

Lemma 5. Let be a generalized eventual cyclic gross contractive mapping and . Then is a Cauchy sequence for every .

Proof. For every , let

Then, (3) and the monotone increasing property of imply that

since

Hence,

Thus,

Since , we deduce that exists. Set

If , then (5) implies that

So, we have

This is impossible since . Therefore,

This means that, for any , there exists a natural number such that for any natural number ,

Moreover, for any natural number , we have

This implies that

as , since

Therefore, is a Cauchy sequence.

Lemma 6. Let be a generalized eventual cyclic gross contractive mapping and . Then is a Cauchy sequence for every .

Proof. For every , let

Then,

since

So,

Hence,

Since , we see that for all , the nonnegative sequence is decreasing. Let

Then we obtain

This implies that

if . Clearly, is impossible. Moreover, is also impossible by the proof of [1, Theorem 2.4].

Now, we show that is a Cauchy sequence. It follows from

that for any , there exists a natural number such that for any natural number ,

For and any natural number , we have

So

as , since

Therefore, is a Cauchy sequence.

For , by the proof of [1, Theorem 2.4], we know that also is a Cauchy sequence.

The following is our central theorem about cycle mapping in complete metric spaces.

Theorem 7. Assume that is a complete metric space, and are closed, and is a generalized eventual cyclic gross contractive mapping. Then, is nonempty and has a unique fixed point in .

Proof. By the virtue of Lemmas 5 and 6, we know that for every , is a Cauchy sequence. Since is a complete metric space, and are closed, there exists some such that

Therefore,

So, ; that is, .

On the other hand, we obtain

in view of

If , then is a fixed point of .

Otherwise, if , then we have

Equation (32) implies that

Hence,

This is impossible since .

According to previous discussions, it is concluded that

and therefore, is a fixed point of .

If there is such that , then ; that is, the fixed point of is unique, since

in view of

The proof is completed.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the NSF of China (11171210).

References

R. P. Agarwal, J. Liang, and J. Zhang, “A fixed point theorem for cyclic mappings in complete metric spaces,” Journal of Nonlinear and Convex Analysis. In press

M. Jleli and B. Samet, “Best proximity points for α-ψ-proximal contractive type mappings and applications,” Bulletin des Sciences Mathématiques. In press

S. Karpagam and S. Agrawal, “Best proximity point theorems for cyclic orbital MeirKeeler contraction maps,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 4, pp. 1040–1046, 2011.