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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 813614, 11 pages
http://dx.doi.org/10.1155/2014/813614
Research Article

Existence and Uniqueness of Best Proximity Points for Generalized Almost Contractions

1Department of Mathematics, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng Saen Campus, Nakhon Pathom 73140, Thailand
2Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

Received 5 August 2013; Revised 19 November 2013; Accepted 26 December 2013; Published 16 March 2014

Academic Editor: Adrian Petrusel

Copyright © 2014 Chirasak Mongkolkeha 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

The purpose of this paper is to elicit some interesting extensions of generalized almost contraction mappings to the case of non-self-mappings with -proximal admissible and prove best proximity point theorems for this classes. Moreover, we also give some examples and applications to support our main results.

1. Introduction

Many problems can be formulated as equations of the form , where is a self-mapping with some suitable domains. From the fact that fixed point theory plays an important role in furnishing a uniform treatment to solve various equations of the form However, in the case that is non-self-mapping, the aforementioned equation does not necessarily have a fixed point. In such case, it is worthy to determine an approximate solution such that the error is minimum. This is the idea behind best approximation theory. classical best approximation theorem was introduced by Fan [1]; that is, if is a nonempty compact convex subset of a Hausdorff locally convex topological vector space and is a continuous mapping, then there exists an element such that . Afterward, several authors, including Prolla [2], Reich [3], and Sehgal and Singh [4, 5], have derived extensions of Fan's Theorem in many directions. Moreover, for a detailed account of global optimization and the existence of a best proximity point, one can refer to [515]. In 2013, Samet [16] studied the existence and uniqueness of best proximity points for almost -contractive mappings in complete metric spaces. Recently, Jleli et al. [17] introduced a new class of non-self-contractive mappings with generalization of -proximal admissible defined by Samet et al. [18] which is called -proximal contractive type mappings and proved existence and uniqueness of best proximity points.

Motivated from the above results, we will study the best proximity point theorem for new classes as generalized almost contraction in metric spaces by using the -proximal admissible of Jleli et al. [17]. Also, we give some illustrative examples and applications to support our main results.

2. Preliminaries

Let and be nonempty subsets of a metric space ; we recall the following notations and notions that will be used in what follows:

If , then and are nonempty. Further, it is interesting to notice that and are contained in the boundaries of and , respectively, provided and are closed subsets of a normed linear space such that (see [19]).

Definition 1. A point is said to be a best proximity point of the mapping if it satisfies the following condition:
It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.

Definition 2 (see [13]). Let be a pair of nonempty subsets of with . Then the pair is said to have the - if and only if where and .
It is easy to see that, for any nonempty subset of , the pair has the .

Example 3 (see [13]). Let be two nonempty closed convex subsets of a Hilbert space . Then satisfies the .

Example 4 (see [20]). Let , be two nonempty, bounded, closed, and convex subsets of a uniformly convex Banach space . Then has the

Example 5 (see [20]). Let with the metric defined by Let and . Then satisfies the .

Definition 6 (see [18]). A self-mapping is said to be -admissible, where , if

Definition 7 (see [17]). Let and . One says that is proximal admissible, if for all .
Clearly, for self-mapping, being -proximal admissible implies that is -admissible.

Definition 8. One says the function is a (c)-comparison function if and only if the following conditions hold: is a nondecreasing function,for any , is a convergent series.One denotes the set of (c)-comparison function by .
It is easily proved that if is a (c)-comparison function, then for all .

Definition 9 (see [16]). Let satisfy the following conditions:(1) is continuous,(2) if and only if the product .One denotes the class of function by .

Example 10 (see [16]). The following functions belong to :(1), ;(2), ;(3), .

3. The Existence and Uniqueness of Best Proximity Points

In this section, we introduce the new class of the generalized Banach contraction for non-self-mappings so-called generalized almost contraction and we also study the best proximity theorems for these classes. First, we recall the notion of contraction defined by Samet [16] as follows

Definition 11 (see [16]). Let and be nonempty subsets of metric space . A mapping is said to be an almost contraction if and only if there exist and such that, for all ,

3.1. The Existence

Definition 12. Let and be nonempty subsets of metric space . A mapping is said to be a generalized almost contraction if and only if for all , where , ,and
Clearly, if we take for all and , the generalized almost contraction reduces to almost contraction.

Theorem 13. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions:(a) are -proximal admissible and generalized almost contraction;(b) is continuous;(c)there exist elements and in such that and ;(d).
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

Proof. By the hypothesis , there exist and in such that From the fact that , there exists an element such that By (12), (13), and the -proximal admissible, we get Since , we can find an element such that Again, by (13), (15), and the -proximal admissible, we have
By similar fashion, we can find in . Having chosen , one can determine an element such that
In view of the fact that the pair has and generalized almost contraction of , we have Since By (18) and (19), we get If there exists such that , by (17), we obtain the best proximity point. Suppose that for all ; then for all . If , by the property for all , we get which is a contradiction and hence = . That is, Again, since the pair has , is -proximal admissible, and generalized almost contraction of , we have and since By (23) and (24), we get By similar argument as above, we can conclude that and thus Using (22) and (26) and the nondecreasing of , we get Continuing this process, by induction we have that for all . Fix and let be a positive integer such that Let ; using the triangular inequality, (28) and (29), we obtain This shows that is a Cauchy sequence. Since is a closed subset of complete metric spaces , then there exists such that By (17), (31), and the continuity of , we get and the proof is complete.

Next, we remove condition is continuous in Theorem 13, by assuming the following condition which was defined by Jleli et al. [17] for proving the new best proximity point theorem.If is a sequence in such that for all and for some as , then there exists a subsequence of such that for all .

Theorem 14. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions:(a) are -proximal admissible and generalized almost contraction;(b) satisfies condition ;(c)there exist elements and in such that and ;(d).
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

Proof. As in the proof of Theorem 13, we have for all . Moreover, is a Cauchy sequence and converges to some point . By the and (28), we have for all . That is, and, by the same argument as proof of Theorem 13, we obtain that is a Cauchy sequence. Since is a closed subset of the complete metric space , there exists such that converges to . Therefore On the other hand, from the condition of , then there exists a subsequence of such that for all . The pair has and property of mapping ; we get Indeed, From the definition of , we get Since it follows that Suppose that Then for large enough, we have . Using the property for all , we get Combining (37) and (40) with (44) and the property of , we obtain that which is a contradiction and thus . Hence, and the proof is complete.

3.2. The Uniqueness

Next, we present an example where it can be appreciated that hypotheses in Theorems 13 and 14 do not guarantee uniqueness of the best proximity point.

Example 15. Let with the Euclidean metric. Consider and . Obviously, satisfies the - and ; furthermore and . Define by for all ; clearly is continuous. Let be defined by We can show that are -proximal admissible and generalized almost contraction with for all and for all . Furthermore, Therefore, and are a best proximity point of mapping .

Now, we need a sufficient condition to give uniqueness of the best proximity point as follows.

Definition 16 (see [17]). Let be a non-self-mapping and . One says that is regular if, for all , there exists such that

Theorem 17. Adding condition regular of to the hypotheses of Theorem 13, then one obtains the uniqueness of the best proximity point of  .

Proof. We will only prove the part of uniqueness. Suppose that there exist and in which are distinct best proximity points; that is,
Using the pair that has , we have
Case 1(if). By (50) and generalized almost contraction of , we have since Combining (51) with (52) and using the property for all , we get which is a contradiction and hence .
Case 2(if). By the regular of , there exists such that Since , there exists a point such that From , , and and by the -proximal admissible, we have Since , there exists a point such that By similar argument as above, we can conclude that . One can proceed further in a similar fashion to find in with such that for all . By (58), the pair has and property of mapping ; we get Using the property of mapping , we get since Thus If , for some . By (59), we get which implies that . Moreover, we obtain for all and thus as . Suppose that for all ; then for all . If , by the property for all , we get which is a contradiction and hence . That is, for all . By induction of (65), we have Taking , we obtain that as . So, in all cases, we have as . Similarly, we can prove that as . By the uniqueness of limit, we conclude that and this completes the proof.

Theorem 18. Adding condition regular of to the hypotheses of Theorem 14, then we obtain the uniqueness of the best proximity point of .

Proof. Combine the proofs of Theorems 17 and 14.

4. Consequences

4.1. Best Proximity Points Theorems

If we take , where and , then Theorem 13 and Theorem 14, we get the following.

Theorem 19. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions:(a) is -proximal admissible and for all ;(b) is continuous (or satisfies condition );(c) there exist elements and in such that and ;(d) .
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

If we add the condition that is regular in Theorem 19, therefore we can obtain the uniqueness of the best proximity point.

If we take , for all in Theorems 13 and 14, we get the following Theorems.

Theorem 20. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions:(a) for all ;(b) is continuous or satisfies condition ;(c).
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

If , then Theorem 20 includes the following.

Theorem 21. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions:(a)for all ;(b) is continuous (or satisfies condition );(c).
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

If we take and , for all in Theorem 21, we obtain the following theorem.

Theorem 22. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions:(a)for all ;(b) is continuous (or satisfies condition );(c).
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

If and putting in Theorem 22, we obtain the following.

Theorem 23. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions: (a)for all ;(b) is continuous (or satisfies condition );(c) there exist elements and in such that ;(d).
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

If and putting in Theorem 22, we obtain the following theorem.

Theorem 24. Let and be nonempty closed subsets of a complete metric space such that is nonempty and the pair has the . Let satisfy the following conditions:(a)for all ;(b) is continuous (or satisfies condition );(c)there exist elements and in such that ;(d).
Then there exists an element such that Moreover, for any fixed , the sequence , defined by converges to the element .

4.2. Fixed Points Theorem

It is easy to observe that, for self-mappings, our results include the following.

Theorem 25. Let be nonempty closed subsets of a complete metric space and such that for all , where . Then has a unique fixed point . Moreover, for any fixed , the sequence , defined by , converges to the element .

Theorem 26. Let be nonempty closed subsets of a complete metric space and such that Then has a unique fixed point . Moreover, for any fixed , the sequence , defined by , converges to the element .

Theorem 27. Let be nonempty closed subsets of a complete metric space and such that for all . Then has a unique fixed point . Moreover, for any fixed , the sequence , defined by , converges to the element .

5. Some Applications and an Example

We recall some preliminaries from (see, [6, 17] also) as follows.

Let be a metric space and a binary relation over . Denote this is the symmetric relation attached to . Clearly,

Definition 28 (see [17]). A mapping is said to be proximal comparative if and only if

Corollary 29. Let be a complete metric space, a binary relation over , and and two nonempty, closed subsets of such that are nonempty and the pair has the . Let such that the following conditions hold:(a) is a continuous proximal comparative mapping;(b) there exist elements and in such that and ;(c) there exist and such that implies that (d).Then there exists an element such that

Proof. Define the mapping by
Since is proximal comparative, we have for all . Using the definition of , we get for all and hence is -proximal admissible. Condition implies that and . By condition (c), we get that is, is generalized almost contraction. Therefore, all hypotheses of Theorem 13 are satisfied, and the desired result follows immediately.

Next, below we give an example to illustrate the main result of Theorem 13.

Example 30. Consider with the metric defined by for all . Let defined by