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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 927457, 14 pages
http://dx.doi.org/10.1155/2013/927457
Research Article

Best Proximity Point Results for Modified --Proximal Rational Contractions

1Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Astara Branch, Islamic Azad University, Astara, Iran

Received 23 May 2013; Accepted 12 July 2013

Academic Editor: Bessem Samet

Copyright © 2013 N. Hussain 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

We first introduce certain new concepts of --proximal admissible and ---rational proximal contractions of the first and second kinds. Then we establish certain best proximity point theorems for such rational proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity point theory. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach's contraction principle and some of its generalizations as special cases. Moreover, some examples are given to illustrate the usability of the obtained results.

1. Introduction and Preliminaries

Let be a metric space and be a self-mapping defined on a subset of . Fixed point theory is an important tool for solving equations of the kind , whose solutions are the fixed points of the mapping . Many problems arising in different areas of mathematics, such as optimization, variational analysis, and differential equations, can be modeled as fixed point equations of the form . On the other hand, if is not a self-mapping, the equation could have no solutions and, in this case, it is of a certain interest to determine an element that is in some sense closest to . One of the most interesting results in this direction is due to Fan [1] and can be stated as follows.

Theorem F. Let be a nonempty compact convex subset of a normed space and be a continuous nonself-mapping. Then there exists an such that .

Many generalizations and extensions of this result appeared in the literature (see [26] and, references therein).

Let and be two nonempty subsets of a metric space . A best proximity point of a nonself-mapping is a point satisfying the equality , where . Though best approximation theorems ensure the existence of approximation solutions, such results need not yield optimal solutions. But best proximity point theorems provide sufficient conditions that assure the existence of approximate solutions which are optimal as well. For more details on this approach, we refer the reader to [5, 732].

The aim of this paper is to introduce certain new concepts of --proximal admissible and ---rational proximal contractions of the first and second kinds. Then we establish certain best proximity point theorems for such rational proximal contractions. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity point theory. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach's contraction principle and some of its generalizations as special cases. Moreover, some examples are given to illustrate the usability of the obtained results.

Now we give some basic notations and definitions that will be used in the sequel.

Let and be two nonempty subsets of a metric space . We denote by and the following sets: where . For the map , we define the set of all best proximity points of by

Definition 1. Let be a metric space and let and be two nonempty subsets of . Then is said to be approximatively compact with respect to if every sequence in , satisfying the condition for some in , has a convergent subsequence.

Obviously, every set is approximatively compact with respect to itself.

Very recently, Nashine et al. [22] introduced rational proximal contraction of the first and second kinds as follows.

Definition 2. Let and be nonempty subsets of a metric space . Then is said to be a rational proximal contraction of the first kind if there exist nonnegative real numbers , , , and with , such that, for all ,

Definition 3. Let and be nonempty closed subsets of a metric space . Then is said to be a rational proximal contraction of the second kind if there exist nonnegative real numbers , , , and with , such that, for all ,
Note that a rational proximal contraction of the second kind is not necessarily a rational proximal contraction of the first kind; for examples, see [22].

Definition 4 (see [28]). Let be a self-mapping on a metric space and be a function. We say that is -admissible mapping if

Recently, Jleli and Samet [15] introduced new concepts of -proximal admissible and --proximal contractive type mappings as follows.

Definition 5 (see [15]). Let , . We say that is -proximal admissible if for all .

Clearly, if , then -proximal admissible map reduces to -admissible map.

Definition 6 (see [15]). A nonself-mapping is said to be --proximal contraction if for all , , .

Salimi et al. [27] modified the concept of -admissible mappings as follows.

Definition 7. Let be a self-mapping on a metric space and be two functions. We say that is -admissible mapping with respect to if Note that if we take , then this definition reduces to Definition 4. Also, if we take , then we say that is -subadmissible mapping.

For the examples of -admissible mappings with respect to , we refer to [27] and the examples in the next section.

2. Best Proximity and Fixed Point Results in Metric Spaces

First we modify the notion of -proximal admissible mapping as follows.

Definition 8. Let and be functions. We say that is -proximal admissible with respect to if, for all , Note that if we take for all , then this definition reduces to Definition 5. In case for all , then we shall say that is -proximal subadmissible mapping.

Clearly, if , then the previous definition reduces to Definition 7.

Definition 9. Let and be nonempty subsets of a metric space . Then is said to be --rational proximal contraction of the first kind if there exist nonnegative real numbers , , , and with , such that, for all , In case for all , then is said to be -rational proximal contraction of the first kind.

Definition 10. Let and be nonempty closed subsets of a metric space . Then is said to be a --rational proximal contraction of the second kind if there exist nonnegative real numbers , , , and with , such that, for all , In case for all , then is said to be -rational proximal contraction of the second kind.

We are ready to prove the following best proximity point result for --rational proximal contraction of the first kind.

Theorem 11. Let and be nonempty closed subsets of a complete metric space and let be approximatively compact with respect to . Assume that are functions, and are nonempty, and is an --rational proximal contraction of the first kind which satisfies the following assertions:(i), (ii) is -proximal admissible with respect to ,(iii)there exist elements and in such that (iv)if is a sequence in such that and as , then for all .Then there exists , such that Moreover, if for all , then is unique best proximity point of .

Proof. By (iii) there exist elements and in , such that On the other hand, , then and there exists , such that Now, since, is --proximal admissible, then we have . That is, Again, since , there exists , such that Thus, together with is --proximal admissible imply that . Hence, Continuing this process, we get Since is --rational proximal contraction of the first kind, then we have which implies where . That is, is a Cauchy sequence in and since is a complete metric space and is closed, so there exists an element such that as . Also, we have Taking limit as in the previous inequality, we have As is approximatively compact with respect to , so the sequence has a subsequence that converges to some . Hence, and so . Now, since , then, for some . From (iv) and (66), we have for all . Therefore, we proved that for all . Since is a --rational proximal contraction of the first kind, so we have Taking limit as in the previous inequality, we get As , so . This implies that
Assume that is another best proximity point of such that . That is, Now, since is --rational proximal contraction of the first kind, so we have which implies that . As , so . That is, is a unique best proximity point of .

By taking in Theorem 11, we deduce the following corollary.

Corollary 12. Let and be nonempty closed subsets of a complete metric space such that is approximatively compact with respect to . Assume that , and are nonempty, and is an -rational proximal contraction of the first kind satisfying the following assertions: (i),(ii) is -proximal admissible,(iii)there exist elements and in , such that (iv)if is a sequence in such that and as , then for all . Then there exists , such that Moreover, if for all , then is unique best proximity point of .

If in the previous corollary we take , then we obtain the following result.

Corollary 13 (see [22, Theorem 3.1]). Let and be nonempty closed subsets of a complete metric space and let be approximatively compact with respect to . Assume that and are nonempty and is a rational proximal contraction of the first kind with . Then there exists a unique , such that Further, for any fixed , the sequence , defined by , converges to .

Example 14. Let and be metric on . Suppose and . Define by Also, define by Clearly, . Now, we have Also, . Let Then, Note that for all . Hence, . That is, . That is, is a -proximal admissible mapping. Also, assume that for all and as . Therefore, and then . That is, for all . Further, and .
Again, assume that Then and . Hence, Thus, all of the conditions of Corollary 12 (Theorem 11) hold and there exists a unique , such that

Example 15. Let and be metric on . Suppose and . Define by Also, define by Clearly, . Now, we have Also, . Let Then, and and . Note that for all ; Hence, that is, . That is, is a -proximal admissible mapping. Also, assume that for all and as . Therefore, and then . That is, for all . Further, and
Let , , and .
Again, assume that
Then and and . Hence, All of the conditions of Corollary 12 (Theorem 11) hold and there exists a unique , such that
Let imply Then, which is a contradiction. Therefore, Corollary 32 [22, Theorem 3.1] cannot be applied here.

If in Theorem 11 we take , then we obtain the following result.

Corollary 16. Let and be nonempty closed subsets of a complete metric space and let be approximatively compact with respect to . Assume that , and are nonempty, and is a nonself-mapping satisfying the following assertions: (i),(ii) is -proximal subadmissible,(iii)there exist elements and in , such that (iv)(v)if is a sequence in such that and as , then for all , where . Then there exists , such that, Moreover, if for all , then is unique.

The following are immediate consequences of Theorem 11.

Theorem 17. Let be a complete metric space and let be a mapping satisfying the following assertions: (i) is -admissible with respect to ,(ii)there exists element in , such that (iii)if is a sequence in such that and as , then for all ,(iv)where . Then has a unique fixed point in .

If in Theorem 17 we take , then we obtain the following result.

Theorem 18. Let be a complete metric space and let be a mapping satisfying the following assertions: (i) is -admissible,(ii)there exists element in , such that, (iii)if is a sequence in such that and as , then for all ,(iv) where . Then has a unique fixed point in .

If in Theorem 17 we take , then we obtain the following result.

Theorem 19. Let be a complete metric space and be a mapping satisfying the following assertions:(i) is -subadmissible,(ii)there exists element in , such that (iii)if is a sequence in such that and as , then for all ,(iv)where . Then has a unique fixed point in .

If in Theorem 18 we take , then we obtain the following fixed point result for rational contraction of first kind.

Theorem 20. Let be a complete metric space and let be a mapping satisfying the following rational inequality: where . Then has a unique fixed point in .

We now establish best proximity point result for --rational proximal contraction of the second kind.

Theorem 21. Let and be nonempty closed subsets of a complete metric space such that is approximatively compact with respect to . Assume that , and are nonempty, and is a continuous --rational proximal contraction of the second kind, such that(i),(ii) is -proximal admissible with respect to ,(iii)There exist elements and in , such that Then there exists and, for any fixed , the sequence , defined by , converges to , and for all when for all .

Proof. Following the same lines of the proof of Theorem 11, there exists a sequence , such that Since is a --rational proximal contraction of the second kind, we get which implies where . That is, is a Cauchy sequence and since is a complete metric space and is closed, so there exists an element such that as . Also, we have Taking limit as in the previous inequality, we have Since is approximatively compact with respect to , so the sequence, has a subsequence that converges to some . Now, by applying continuity of , we get That is, . Now, assume that is a another best proximity point of . That is, . Now, since, is a --rational proximal contraction of the second kind and for all , then This implies that And, hence, gives us .

Corollary 22. Let and be nonempty closed subsets of a complete metric space such that is approximatively compact with respect to . Assume that , and are nonempty, and is a continuous mapping, such that (i),(ii) is -proximal admissible,(iii)there exist elements and in such that (iv)where . Then there exists and, for any fixed , the sequence , defined by , converges to , and for all when for all .

If in the previous corollary we take , then we have the following result.

Corollary 23 (see [22, Theorem 3.2]). Let and be nonempty closed subsets of a complete metric space such that is approximatively compact with respect to . Assume that and are nonempty and is a continuous rational proximal contraction of the second kind, such that . Then there exists and, for any fixed , the sequence , defined by , converges to , and for all .

Corollary 24. Let and be nonempty closed subsets of a complete metric space such that is approximatively compact with respect to . Assume that , and are nonempty and is a continuous mapping, such that and is -proximal subadmissible, such that for all , where . Then there exists and, for any fixed , the sequence , defined by , converges to , and for all when for all .

The following are immediate consequences of Theorem 21.

Theorem 25. Let be a complete metric space and let be a continuous mapping satisfying the following assertions: (i) is -admissible with respect to , (ii)there exists element in , such that (iii)where . Then has a unique fixed point in .

If in Theorem 25 we take , then we obtain the following result.

Theorem 26. Let be a complete metric space and let be a continuous mapping satisfying the following assertions: (i) is -admissible,(ii)there exists element in , such that (iii) where . Then has a unique fixed point in .

If in Theorem 25 we take , then we obtain the following result.

Theorem 27. Let be a complete metric space and let be a continuous mapping satisfying the following assertions:(i) is -subadmissible(ii)there exists element in , such that (iii) where . Then has a unique fixed point in .

If in Theorem 26 we take , then we obtain the following result.

Theorem 28. Let be a complete metric space and let be a continuous mapping satisfying the following rational inequality: where . Then has a unique fixed point in .

Our next best proximity point result is about --rational proximal contraction of the first and second kinds where we consider only completeness of without assuming continuity of the mapping and approximative compactness of and .

Theorem 29. Let and be nonempty closed subsets of a complete metric space . Assume that , and are nonempty, and is --rational proximal contraction of the first and second kinds, such that (i),(ii) is -proximal admissible with respect to , (iii)there exists elements and in such that (iv)if is a sequence in such that and as , then for all . Then there exists unique . Also, for any fixed , the sequence , defined by , converges to , whenever for all .

Proof. As in proof of Theorem 11, there exists a sequence , such that and the sequence is a Cauchy sequence and so converges to some . Also, by proof of Theorem 21, we obtain that the sequence is a Cauchy sequence and converges to some . Hence, we have That is, . Since , so for some . Thus we have and and so by (iv) this implies that for all . Now, since is a --rational proximal contraction of the first kind, we get
Taking limit as in the previous inequality, we get which implies that . That is, . Hence, . Further, following similar proof of Theorem 11 we can deduce the uniqueness of best proximity point of .

If for all in the previous theorem, we obtain the following result.

Corollary 30. Let and be nonempty closed subsets of a complete metric space