Journal of Applied Mathematics

Volume 2012 (2012), Article ID 643729, 14 pages

doi:10.1155/2012/643729

Research Article

Best Proximity Point Theorems for Some New Cyclic Mappings

Chi-Ming Chen¹ and Chao-Hung Chen²

¹Department of Applied Mathematics, National Hsinchu University of Education, Taiwan

²Department of Applied Mathematics, Chung Yuan Christian University, Taiwan

Received 26 February 2012; Accepted 16 June 2012

Academic Editor: Pablo González-Vera

Copyright © 2012 Chi-Ming Chen and Chao-Hung Chen. 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.

By using the stronger Meir-Keeler mapping, we introduce the concepts of the sMK-G-cyclic mappings, sMK-K-cyclic mappings, and sMK-C-cyclic mappings, and then we prove some best proximity point theorems for these various types of contractions. Our results generalize or improve many recent best proximity point theorems in the literature (e.g., Elderd and Veeramani, 2006; Sadiq Basha et al., 2011).

1. Introduction and Preliminaries

Let and be nonempty subsets of a metric space . Consider a mapping , is called a cyclic map if and . is called a best proximity point of in if is satisfied, where . In 2005, Eldred et al. [1] proved the existence of a best proximity point for relatively nonexpansive mappings using the notion of proximal normal structure. In 2006, Eldred and Veeramani [2] proved the following existence theorem.

Theorem 1.1 (see Theorem 3.10 in [2]). Let and be nonempty closed convex subsets of a uniformly convex Banach space. Suppose is a cyclic contraction, that is, and , and there exists such that

Then there exists a unique best proximity point in . Further, for each , converges to the best proximity point.

Later, best proximity point theorems for various types of contractions have been obtained in [3–7]. Particularly, in [8], the authors prove some best proximity point theorems for -cyclic mappings and -cyclic mappings in the frameworks of metric spaces and uniformly convex Banach spaces, thereby furnishing an optimal approximate solution to the equations of the form , where is a non-self--cyclic mapping or a non-self--cyclic mapping.

Definition 1.2 (see [8]). A pair of mappings and is said to form a -cyclic mapping between and if there exists a nonnegative real number such that

for and .

Definition 1.3 1.3 (see [8]). A pair of mappings and is said to form a -cyclic mapping between and if there exists a nonnegative real number such that

for and .

In this paper, we also recall the notion of Meir-Keeler mapping (see [9]). A function is said to be a Meir-Keeler mapping if, for each , there exists such that, for with , we have . Generalization of the above function has been a heavily investigated branch of research. In this study, we introduce the below notion of the stronger Meir-Keeler function .

Definition 1.4. We call a stronger Meir-Keeler mapping if the mapping satisfies the following condition:

The following provides two example of a stronger Meir-Keeler mapping.

Example 1.5. Let be defined by

Then is a stronger Meir-Keeler mapping which is not a Meir-Keeler function.

Example 1.6. Let be defined by

Then is a stronger Meir-Keeler mapping.

In this paper, by using the stronger Meir-Keeler mapping, we introduce the concepts of the sMK--cyclic mappings, sMK--cyclic mappings and sMK--cyclic mappings, and then we prove some best proximity point theorems for these various types of contractions. Our results generalize or improve many recent best proximity point theorems in the literature (e.g., [2, 8]).

2. sMK-G-Cyclic Mappings

In this section, we prove the best proximity point theorems for the sMK--cyclic non-self mappings.

Definition 2.1. Let be a metric space, and let and be nonempty subsets of . A pair of mappings and is said to form an sMK--cyclic mapping between and if there is a stronger Meir-Keeler function in such that for and ,

where .

Lemma 2.2. Let and be nonempty subsets of a metric space . Suppose that the mappings and form an sMK--cyclic mapping between and . Then there exists a sequence in such that

Proof. Let be given and let and for each . Taking into account (2.1) and the definition of the stronger Meir-Keeler function , we have that for each

where

Taking into account (2.3) and (2.4), we have that for each

and so we conclude that

and, for each ,

where

Taking into account (2.7) and (2.8), we have that for each

and so we conclude that

Generally, by (2.6) and (2.10), we have that for each

Thus the sequence is decreasing and bounded below and hence it is convergent. Let . Then there exists and such that for all with

Taking into account (2.12) and the definition of stronger Meir-Keeler function , corresponding to use, there exists such that

Thus, we can deduce that for each with

and so

Since , we get

that is, .

Lemma 2.3. Let and be nonempty closed subsets of a metric space . Suppose that the mappings and form an sMK--cyclic mapping between and . For a fixed point , let and . Then the sequence is bounded.

Proof. It follows from Lemma 2.2 that is convergent and hence it is bounded. Since and form an sMK--cyclic mapping between and , there is a stronger Meir-Keeler function in such that

where

Taking into account (2.17) and (2.18), we get

Therefore, the sequence is bounded. Similarly, it can be shown that is also bounded. So we complete the proof.

Theorem 2.4. Let and be nonempty closed subsets of a metric space. Let the mappings and form an sMK--cyclic mapping between and . For a fixed point , let and . Suppose that the sequence has a subsequence converging to some element in . Then, is a best proximity point of .

Proof. Suppose that a subsequence converges to in . It follows from Lemma 2.2 that converges to . Since and form an sMK--cyclic mapping between and and taking into account (2.13), we have that for each with

where

Following from (2.20) and (2.21), we obtain that

that is, we have that

letting . Then we conclude that

Therefore, , that is, is a best proximity point of .

3. sMK-K-Cyclic Mappings

In this section, we prove the best proximity point theorems for the sMK--cyclic non-self mappings.

Definition 3.1. Let be a metric space, and let and be nonempty subsets of . A pair of mappings and is said to form an sMK--cyclic mapping between and if there is a stronger Meir-Keeler function in such that, for and ,

where .

Lemma 3.2. Let and be nonempty subsets of a metric space . Suppose that the mappings and form an sMK--cyclic mapping between and . Then there exists a sequence in such that

Proof. Let be given and let and for each . Taking into account (3.1) and the definition of the stronger Meir-Keeler function , we have that

where

Taking into account (3.3) and (3.4), we have that

Similarly, we can conclude that

Generally, by (3.5) and (3.6), we have that for each

Thus the sequence is decreasing and bounded below and hence it is convergent. Let . Then there exists and such that for all with

Taking into account (3.8) and the definition of stronger Meir-Keeler function , corresponding to use, there exists such that

Thus, we can deduce that for each with

that is,

since . Therefore we get that for each with

Since , we get

that is, .

Lemma 3.3. Let and be nonempty closed subsets of a metric space . Suppose that the mappings and form an sMK--cyclic mapping between and . For a fixed point , let and . Then the sequence is bounded.

Proof. It follows from Lemma 3.2 that is convergent and hence it is bounded. Since and form an sMK--cyclic mapping between and , there is a stronger Meir-Keeler function in such that, for and ,

where . So we get that

Therefore, the sequence is bounded. Similarly, it can be shown that is also bounded. So we complete the proof.

Theorem 3.4. Let and be nonempty closed subsets of a metric space. Let the mappings and form an sMK--cyclic mapping between and . For a fixed point , let and . Suppose that the sequence has a subsequence converging to some element in . Then, is a best proximity point of .

Proof. Suppose that a subsequence converges to in . It follows from Lemma 2.2 that converges to . Since and form an sMK--cyclic mapping between and and taking into account (3.9), we have that for each with

where

Following from (3.16) and (3.17), we obtain that for each with

Letting . Then we conclude that , that is, is a best proximity point of .

4. sMK-C-Cyclic Mappings

In this section, we prove the best proximity point theorems for the sMK--cyclic non-self mappings.

Definition 4.1. Let be a metric space, and let and be nonempty subsets of . A pair of mappings and is said to form an sMK--cyclic mapping between and if there is a stronger Meir-Keeler function in such that, for and ,

where .

Lemma 4.2. Let and be nonempty subsets of a metric space . Suppose that the mappings and form an sMK--cyclic mapping between and . Then there exists a sequence in such that

Proof. Let be given and let and for each . Taking into account (4.1) and the definition of the stronger Meir-Keeler function , we have that

where

Taking into account (4.3) and (4.4), we conclude that

Similarly, we can conclude that

Generally, by (4.5) and (4.6), we have that for each

Thus the sequence is decreasing and bounded below and hence it is convergent. Let . Then there exists and such that for all with

Taking into account (4.5) and the definition of stronger Meir-Keeler function , corresponding to use, there exists such that

Thus, we can deduce that for each with

that is,

since . Therefore we get that for each with

Since , we obtain that .

Lemma 4.3. Let and be nonempty closed subsets of a metric space . Suppose that the mappings and form an sMK--cyclic mapping between and . For a fixed point , let and . Then the sequence is bounded.

Proof. It follows from Lemma 4.2 that is convergent and hence it is bounded. Since and form an sMK--cyclic mapping between and , there is a stronger Meir-Keeler function in such that for and ,

where

So we get that

Therefore, the sequence is bounded. Similarly, it can be shown that is also bounded. So we complete the proof.

Theorem 4.4. Let and be nonempty closed subsets of a metric space. Let the mappings and form an sMK--cyclic mapping between and . For a fixed point , let and . Suppose that the sequence has a subsequence converging to some element in . Then, is a best proximity point of .

Proof. Suppose that a subsequence converges to in . It follows from Lemma 2.2 that converges to . Since and form an sMK--cyclic mapping between and and taking into account (4.9), we have that, for each with ,

where

Following from (4.16) and (4.17), we obtain that

that is, we have that

Letting . Then we conclude that

Therefore, , that is, is a best proximity point of .

Acknowledgment

The authors would like to thank the referee(s) for many useful comments and suggestions for the improvement of the paper.

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