Journal of Applied Mathematics

Volume 2013 (2013), Article ID 101439, 6 pages

http://dx.doi.org/10.1155/2013/101439

Research Article

Existence and Convergence Theorems of Best Proximity Points

Moosa Gabeleh¹ and Naseer Shahzad²

¹Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran

²Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 22 February 2013; Accepted 16 April 2013

Academic Editor: Gue Myung Lee

Copyright © 2013 Moosa Gabeleh and Naseer Shahzad. 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.

The aim of this paper is to prove some best proximity point theorems for new classes of cyclic mappings, called pointwise cyclic orbital contractions and asymptotic pointwise cyclic orbital contractions. We also prove a convergence theorem of best proximity point for relatively nonexpansive mappings in uniformly convex Banach spaces.

1. Introduction and Preliminaries

Let be a metric space, and let be subsets of . A mapping is said to be cyclic provided that and . In 2003, Kirk et al. [1] proved the following generalization of Banach contraction principle.

Theorem 1 (see [1]). Let and be nonempty closed subsets of a complete metric space . Suppose that is a cyclic mapping such that

for some and for all . Then has a unique fixed point in .

In [2] Eldred and Veeramani introduced the class of cyclic contractions as follows.

Definition 2 (see [2]). Let and be nonempty subsets of a metric space . A mapping is said to be a cyclic contraction if is cyclic and

for some and for all .

Let be a cyclic mapping. A point is said to be a best proximity point for provided that , where

Note that if , then the best proximity point is nothing but a fixed point of .

The next theorem ensures existence, uniqueness, and convergence of best proximity point for cyclic contractions in uniformly convex Banach spaces.

Theorem 3 (see [2]). Let and be nonempty closed convex subsets of a uniformly convex Banach space and let be a cyclic contraction map. For , define for each . Then there exists a unique such that and .

Recently, Suzuki et al. in [3] introduced the notion of property UC which is a kind of geometric property for subsets of a metric space .

Definition 4 (see [3]). Let and be nonempty subsets of a metric space . Then is said to satisfy property UC if the following holds.

If and are sequences in and is a sequence in such that and , then we have .

We mention that if and are nonempty subsets of a uniformly convex Banach space such that is convex, then satisfies the property UC. Other examples of pairs having the property UC can be found in [3]. Here, we state the following two lemmas of [3].

Lemma 5 (see [3]). Let and be nonempty subsets of a metric space . Assume that satisfies the property UC. Let and be sequences in and , respectively, such that either of the following holds:

Then is a Cauchy sequence.

Lemma 6 (see [3]). Let be a metric space and let and be nonempty subsets of such that satisfies the property UC. Let be a cyclic map such that

For a point , the following are equivalent:

(i)is a best proximity point of ;(ii) is a fixed point of .

Throughout this paper, stands for a nonempty pair in a metric space . When we say that a pair satisfies a specific property, we mean that both and satisfy the mentioned property. Also, we define and . Moreover, we use the following notations:

For a cyclic mapping and , we define the orbit setting at by

where for and . We set

for all . Note that if , then and . Also, the set of all best proximity points of the mapping in will be denoted by B.P.P.

We mention that a mapping is said to be relatively nonexpansive provided that is cyclic and satisfies the condition for each . Note that a relatively nonexpansive mapping need not be a continuous mapping. Also every nonexpansive self-map can be considered as a relatively nonexpansive mapping.

In 2005 Eldred et al. in [4] introduced a geometric concept called proximal normal structure. Using this notion they proved that if is a nonempty weakly compact convex pair in a Banach space and is a relatively nonexpansive mapping, then there exists such that . For more details on this subject, we refer the reader to [5–10].

2. Pointwise Cyclic Orbital Contractions

In [11], the notion of pointwise cyclic contractions was introduced as follows.

Definition 7 (see [11]). Let be a pair of subsets of a metric space . Let be a cyclic mapping. is said to be a pointwise cyclic contraction if for each there exist such that

The following result was proved in [11].

Theorem 8 (see [11]). Let be a nonempty weakly compact convex pair in a Banach space and suppose that is a pointwise cyclic contraction mapping. Then there exists such that .

In this section, we introduce a new class of cyclic mappings, called pointwise cyclic orbital contractions, which contains the pointwise cyclic contractions as a subclass. For such mappings, we study the existence of best proximity points in Banach spaces.

Definition 9. Let be a pair of subsets of a metric space . A cyclic mapping is said to be a pointwise cyclic orbital contraction if there exists such that for each

It is clear that the class of pointwise cyclic orbital contractions contains the class of pointwise cyclic contractions as a subclass. The following example shows that the converse need not be true. Moreover, it is interesting to note that a pointwise cyclic orbital contraction may not be relatively nonexpansive.

Example 10. Let with the usual metric. For , define by

Then is pointwise cyclic orbital contraction with for all .

Proof. If either or , then it is easy to see that relations (10) and (11) hold. Suppose that and . Thus,

Hence,

that is, (10) holds. Also, by the fact that then

which implies that (10) and (11) hold. Thus, is a pointwise cyclic orbital contraction. Now, we show that is not pointwise cyclic contraction. Indeed, if there exists a function such that for all , then for and we must have

and hence , which is a contradiction. Therefore, is not pointwise cyclic contraction. Moreover, we note that since is not continuous, is not relatively nonexpansive.

Let us state our main result of this section.

Theorem 11. Let be a nonempty weakly compact convex pair in a Banach space . If   is a pointwise cyclic orbital contraction, then the set of best proximity points of   is nonempty.

Proof. Let denote the collection of all nonempty weakly compact convex pairs which are subsets of and such that is cyclic on . Then is nonempty, since . is partially ordered by reverse inclusion; that is, . It is easy to check that every increasing chain in is bounded above. Hence by Zorn's lemma we can get a minimal element say . We have

Moreover

and also

Now, by the minimality of , we have ,  . Suppose that . Then for each we have

which implies that . Hence,

Thus, for each we must have

which ensures that

Similarly, we can see that if , then

Assume that is a fixed element in . Let . Set and

Obviously, . Also, from (23) and then is a nonempty pair. Besides, it is easy to see that

Now, let . Then and by (24), which implies that . Hence, . Similarly, by relation (23) we conclude that . That is, is cyclic on . By the minimality of we must have and . Therefore,

for each . Then for all we have

Particularly, . Thus,

Similar argument implies that if , then relation (29) is to be achieved. Therefore, (29) holds for all . To complete the proof of the theorem, we consider the following cases.

Case  1.  If , then we have

that is, is a best proximity point of .

Case  2.  If , it now follows from (23) and (29) that

which is a contradiction. Hence, each point of is a best proximity point of and so . Similarly, we can see that . Thus, for each we must have

3. Asymptotic Pointwise Cyclic Orbital Contractions

Definition 12. Let be a pair of subsets of a metric space . A cyclic mapping is said to be an asymptotic pointwise cyclic orbital contraction if for each ,

where for each , and for some and for all .

The following theorem establishes existence and convergence of a best proximity point for asymptotic pointwise cyclic orbital contractions in metric spaces with the property UC.

Theorem 13. Let be a nonempty closed pair in a complete metric space such that satisfies the property UC. Assume that is an asymptotic pointwise cyclic orbital contraction such that is continuous on . If there exists such that the orbit of at is bounded, then has a best proximity point in . Moreover, if and , then converges to the best proximity point of .

Proof. Let . We note that the sequence is decreasing and bounded below by . Let . We claim that . For all with we have

Taking the supremum with respect to and and then letting we obtain

Besides, for each we have

Now, if we obtain

and hence . We now conclude that

Since has the property UC, by Lemma 5   is a Cauchy sequence. Suppose that . Continuity of on implies that . Thus, . That is, is a best proximity point of the mapping in .

The next corollary is a direct result of Theorem 13.

Corollary 14 (compare to Theorem 3). Let be a nonempty closed pair in a uniformly convex Banach space such that is convex. Assume that is an asymptotic pointwise cyclic orbital contraction such that is continuous on . If there exists such that the orbit of at is bounded, then has a best proximity point in . Moreover, if and , then converges to the best proximity point of .

4. A Convergence Theorem

In this section, we give a convergence theorem of best proximity point for cyclic mappings which is derived from Ishikawa's convergence theorem ([12]). We begin with the following proposition which is an inequality characterization of uniformly convex Banach spaces.

Proposition 15 (see [13]). Let be a uniformly convex Banach space. Then for each , there exists a strictly increasing, continuous and convex function such that and

for all and all such that and .

Definition 16. Let be a nonempty pair of subsets of a normed linear space . Suppose that is a cyclic mapping on . We say that is hemicompactness on provided that each sequence in with has a convergent subsequence.

It is clear that if is compact set, then each cyclic mapping defined on is hemicompactness, where is a nonempty subset of .

Theorem 17. Let be a nonempty, bounded, closed, and convex pair in a uniformly convex Banach space . Assume that is a cyclic relatively nonexpansive mapping such that is hemicompactness on and is continuous and satisfies the condition

for all with . Define a sequence in by and

for , where is a real number belonging to . Then converges strongly to a best proximity point of   in .

Proof. Since is a bounded, closed, and convex pair in a uniformly convex Banach space , the relatively nonexpansive mapping has a best proximity point in ([4]). Also, we note that both of the and have the property UC. So, by Lemma 6 a point is a best proximity point of the mapping if and only if is a fixed point of the mapping . We now have

Therefore, is a decreasing sequence and hence is convergent. So is bounded. From the uniform convexity of a Banach space and by Proposition 15, there exists a strictly increasing, continuous and convex function such that and

Thus

which implies that . Since is strictly increasing and continuous at 0, it follows that .

On the other hand, since is hemicompactness on , there exists a subsequence of the sequence such that . By the continuity of the mapping on , we have . Since , we obtain . Hence is a fixed point of the mapping in and again by Lemma 6, is a best proximity point of in and strongly.

Acknowledgement

The research of N. Shahzad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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