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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 713252, 10 pages
http://dx.doi.org/10.1155/2013/713252
Research Article

Best Proximity Points for Some Classes of Proximal Contractions

1Department of Mathematics, King Abdulaziz University, Science Faculty for Girls, P.O. Box 4087, Jeddah 21491, Saudi Arabia
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia
3DEIM, Università Degli Studi di Palermo, Viale Delle Scienze, 90128 Palermo, Italy

Received 9 May 2013; Revised 11 August 2013; Accepted 11 August 2013

Academic Editor: E. Karapinar

Copyright © 2013 Maryam A. Alghamdi 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

Given a self-mapping and a non-self-mapping , the aim of this work is to provide sufficient conditions for the existence of a unique point , called g-best proximity point, which satisfies . In so doing, we provide a useful answer for the resolution of the nonlinear programming problem of globally minimizing the real valued function , thereby getting an optimal approximate solution to the equation . An iterative algorithm is also presented to compute a solution of such problems. Our results generalize a result due to Rhoades (2001) and hence such results provide an extension of Banach's contraction principle to the case of non-self-mappings.

1. Introduction

A fundamental result in the fixed point theory is the Banach contraction principle, which has various nontrivial implications in many branches of pure and applied sciences.

Let and be nonempty subsets of a metric space . We say that a non-self-mapping is a contraction if there exists such that, for all ,

The Banach contraction principle asserts that if a self-mapping is a contraction and is complete, then has a unique fixed point . This result was extended to other important classes of mappings and has numerous applications. For some important and interesting generalizations of Banach contraction principle, one can refer to [1, 2]. The following notion of weakly contractive self-mapping was introduced by Alber and Guerre-Delabriere in [3].

Definition 1 (see [3]). Let be a metric space and let be a nonempty subset of . A self-mapping is said to be weakly contractive if for all , where is a continuous and nondecreasing function such that is positive on , and . If is bounded, then the infinity condition can be omitted.

Since all contractions are weakly contractive with the function , the above theorem extends Banach contraction principle. In fact, the class of weakly contractive mappings lies between the classes of mappings called contraction ones and contractive ones (, for all with ).

Generally, the solution of the equation , where is a non-self-mapping, is called a fixed point of . Hence, the condition is necessary for the existence of a fixed point of . Clearly, when , we have , for all . In such a situation it is natural to search for a point such that the is closest to in some sense. The following well-known best approximation theorem, due to Fan [4], explores the existence of an approximate solution to the equation .

Theorem 2 (see [4]). Let be a nonempty compact convex subset of a normed linear space and let be a continuous mapping. Then there exists such that .

The point in Theorem 2 is called a best approximant of in . Again, let be nonempty subsets of a metric space and let be a non-self-mapping. A point is called a best proximity point of if . Some interesting results in approximation theory can be found in [423].

The aim of this paper is to prove some best proximity point theorems for proximal contractions which are extensions of Banach contraction principle to the case of non-self-mappings. Precisely, given a self-mapping and a non-self-mapping , this work focuses on -best proximity point theorems for some classes of proximal contractions and a new family of mappings known as -weak contractions. In fact, we provide sufficient conditions for the existence of a unique point , called -best proximity point, which satisfies the condition . Further, an iterative algorithm is furnished to determine an optimal approximate solution in the guise of a -best proximity point. As a consequence, one can compute an optimal approximate solution to some coincidence point equations.

2. Preliminaries

Let denote the set of all positive real numbers and denote the set of all positive integers. Let be two nonempty subsets of a metric space . Let us fix the following notation which will be needed throughout this paper: where . In [11], the authors discussed sufficient conditions which guarantee the nonemptiness of and . Also, in [20], the authors proved that is contained in the boundary of .

We denote by the set of nondecreasing functions satisfying the following condition:, for all , where is the th iterate of .

Note that if , then the following conditions hold:, for all ; ;   is continuous at .

We denote by the set of nondecreasing functions such that if and only if and with .

Definition 3 (see [21]). Let and be two nonempty subsets of a metric space . A non-self-mapping is said to be a proximal -contraction of the first kind if for all , where . If for some , then is said to be a proximal contraction of the first kind.

Definition 4 (see [21]). Let and be two nonempty subsets of a metric space . A non-self-mapping is said to be a proximal -contraction of the second kind if for all , where . If for some , then is said to be a proximal contraction of the second kind.

Definition 5 (see [14]). Let and be two nonempty subsets of a metric space . A non-self-mapping is said to be a weak proximal -contraction of the first kind if for all , where .

Definition 6 (see [14]). Let and be two nonempty subsets of a metric space . A non-self-mapping is said to be a weak proximal -contraction of the second kind if for all , where .

An example of a non-self-mapping that is weak proximal -contraction of the first and second kinds can be found in [14].

The following result is a best proximity point theorem for weak proximal -contraction of the first and second kinds.

Theorem 7 (see [14, Theorem 3.1]). Let and be closed subsets of a complete metric space such that and are nonvoid. Suppose that the mappings and satisfy the following conditions:(a) is a weak proximal -contraction of the first and second kinds;(b) is an isometry;(c);(d);(e) preserves the isometric distance with respect to .
Then, there exists a unique element in such that . Further, for any fixed element in , the iterative sequence , defined by for every , converges to the element .

Note that in Theorem 7, Sadiq Basha assumes that the function is continuous such that .

Let us define the notion of non-self--weakly contractive mappings as follows.

Definition 8. Let be a metric space, let be two nonempty subsets of , and let . A non-self-mapping is said to be a -weakly contractive mapping if there exists such that for all .

Note that if with ; that is, is a -contractive mapping.

Sankar Raj, in [22], introduced the notion called -property, which was used to prove an extended version of Banach contraction principle.

Definition 9. Let be a pair of nonempty subsets of a metric space with . (i)The pair is said to have the -property if and only if implies , where and (see [22]). (ii)The pair is said to have the weak -property if and only if implies , where and (see [24]).
It is easy to see that, for any nonempty subset of , the pair has the -property.

Definition 10. Let and be two nonempty subsets of a metric space . Let be a self-mapping and a non-self-mapping. Then(i) if is continuous and , for all ;(ii) if for all ;(iii) is said to preserve (isometric) distance with respect to if , for every (see [9]).

3. Best Proximity Point Theorems for Proximal Contractions

In this section, we establish some results of best proximity point for proximal -contractions and weak proximal -contractions.

Theorem 11. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Assume also that the mappings and satisfy the following conditions:(a) is a proximal -contraction of the first kind;(b);(c);(d).
Then there exists a unique point such that . Moreover, for every there exists a sequence such that for every and .

Proof. Let . Since and , there exists such that
Again, for , there exists such that
By repeating this process, for , we can find such that
Since is a proximal -contraction of the first kind and , we have for every . Since is nondecreasing, we get by induction that
By the definition of , letting , we obtain that
We now prove that is a Cauchy sequence. Given that there exists such that
Now, fix and we prove that
Note that (17) holds if , by (16). Assume that (17) holds for some . Since is a proximal -contraction of the first kind,
This implies that (17) holds, for all , and hence That is, is a Cauchy sequence. By the completeness of and since is closed, we have . Moreover, by the continuity of , we have and thus , since , for all . On the other hand, since and , there exists such that Clearly . Again, since is a proximal -contraction of the first kind, we get for all . Letting , we obtain that and then . This implies that
To prove the uniqueness, let be another point in such that
If , since and is a proximal -contraction of the first kind, we get which is a contradiction; thus we have .

Remark 12. If in Theorem 11 we assume , then we get that there exists a unique such that .

From Theorem 11 and the above remark, we obtain the following corollary.

Corollary 13 (see [9, Theorem 3.1]). Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Assume also that the mappings and satisfy the following conditions:(a) is a proximal contraction of the first kind;(b) is an isometry;(c); (d).
Then there exists a unique point such that . Moreover, for every there exists a sequence such that for every and .

If in Theorem 11 the mapping is the identity on , then we get the following corollary.

Corollary 14. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Let satisfy the following conditions:(a) is a proximal -contraction of the first kind;(b).
Then there exists a unique point such that . Moreover, for every there exists a sequence such that for every and .

The following theorem is our main result for proximal -contractions of the second kind.

Theorem 15. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Assume also that the mappings and satisfy the following conditions:(a) is a proximal -contraction of the second kind;(b);(c); (d).
Then there exists a point such that . Moreover, if is injective, then the point such that is unique.

Proof. Similar to the proof of Theorem 11, we can find a sequence such that
Since is a proximal -contraction of the second kind, we have for every . Since , we get for every . Since is nondecreasing, we get by induction that
By definition of , letting , we obtain that
Similar to the proof of Theorem 11, we prove that is a Cauchy sequence. By the completeness of and since is closed, we have . Moreover, there exists such that
Since , we obtain that for some , and then
Again, since is a proximal -contraction of the second kind, we get
Letting , we obtain that and hence . This implies that
To prove the uniqueness, let be another point in such that
If , since is injective, we deduce which is a contradiction; thus we have and hence .

From Theorem 15, we deduce the following corollary.

Corollary 16 (see [15, Theorem 3.2]). Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Assume also that the mappings and satisfy the following conditions:(a) is a proximal contraction of the second kind;(b) is an isometry;(c) preserves isometric distance with respect to ;(d); (e).
Then there exists a point such that . Moreover, if is another point for which , then .

If in Theorem 15 the mapping is the identity on , then we get the following corollary.

Corollary 17. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Let satisfy the following conditions:(a) is a proximal -contraction of the second kind;(b).
Then there exists a point such that . Moreover, if is injective on , then the point such that is unique.

The following is a theorem for weak proximal -contractions of the first kind.

Theorem 18. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Assume also that the mappings and satisfy the following conditions:(a) is a weak proximal -contraction of the first kind;(b);(c);(d).
Then there exists a unique point such that . Moreover, for every there exists a sequence such that for every and .

Proof. Let . Since and , there exists such that
Again, for , there exists such that
By repeating this process, for , we can find such that
Since is a weak proximal -contraction of the first kind and , we have for every . Let ; then is a bounded nonincreasing sequence of nonnegative real numbers. Therefore, converges to , where . Now let us claim that . Suppose that . Since , we get , for all . Then, we have
Inductively we obtain , which is a contradiction for large enough. Therefore and hence .
Now let us claim that is a Cauchy sequence. Suppose it is not. Then there exist and subsequences of such that and , for all . Therefore,
By letting , we have
Since and is a weak proximal -contraction of the first kind, we obtain that
Thus,
Letting , we have , which is a contradiction. Therefore, is a Cauchy sequence. By the completeness of and since is closed, we have . Moreover, by the continuity of , we have and thus , since , for all .
On the other hand, since and , there exists such that
Again, since is a weak proximal -contraction of the first kind, we get
Letting , we obtain that and then . This implies that
To prove the uniqueness, let be another point in such that
If , since and is a weak proximal -contraction of the first kind, we get which is a contradiction; thus we have .

Remark 19. If in Theorem 18 we assume , then we get that there exists a unique such that .

If we take as the identity mapping on in Theorem 18, then we get the following corollary, which extends a result of Rhoades [25] to non-self-mappings.

Corollary 20. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Let satisfy the following conditions:(a) is a weak proximal -contraction of the first kind;(b).
Then there exists a unique point such that . Moreover, for every there exists a sequence such that for every and .

The following theorem is our main result for weak proximal -contractions of the second kind.

Theorem 21. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Assume also that the mappings and satisfy the following conditions:(a) is a weak proximal -contraction of the second kind;(b);(c);(d).
Then there exists a point such that . Moreover, if is injective on , then the point such that is unique.

Proof. Similar to the proof of Theorem 18, we can find a sequence such that
Since is a weak proximal -contraction of the second kind, we have for every . Since , we get for every . Let ; then is a bounded nonincreasing sequence of nonnegative real numbers. Therefore, converges to , where . Now let us claim that . Suppose that . Since , we get , for all . Then, we have
Inductively we obtain , which is a contradiction for large enough. Therefore and hence .
Now let us claim that is a Cauchy sequence. Suppose it is not. Then there exist and subsequences of such that and , for all . Therefore, we get
By letting , we have
Since and is a weak proximal -contraction of the second kind, we obtain that
Thus,
Letting , we have , which is a contradiction. Therefore, is a Cauchy sequence. By the completeness of and since is closed, we have . Moreover, there exists such that
Since , we obtain that for some , and then
Again, since is a weak proximal -contraction of the second kind, we get
Letting , we obtain that and hence . This implies that
To prove the uniqueness, let be another point in such that
If , since is injective on , we have which is a contradiction; thus we have and hence .

If in Theorem 21 the mapping is the identity on , we get the following corollary.

Corollary 22. Let and be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Let satisfy the following conditions:(a) is a weak proximal -contraction of the second kind;(b).
Then there exists a point such that . Moreover, if is injective on , then the point such that is unique.

4. Best Proximity Point Theorem for -Weak Contractions

The following result is a best proximity point theorem for -weak contractions. Recall that a non-self-mapping is -weakly contractive if there exists such that , for all , where .

Theorem 23. Let and be closed subsets of a complete metric space such that and the pair has the weak -property. Suppose that the mappings and satisfy the following conditions:(a) is a -weak contraction;(b);(c).
Then, there exists an element such that . Further, if is one to one then we have a unique element such that .

Proof. Let be an element of . In light of the fact that and , it is ensured that there exists an element such that
Again, in view of the fact that and , it is guaranteed that there exists an element such that
Continuing this process, we can find a sequence in such that
Since has the weak -property, we conclude that
Now, as is a -weak contraction, we get where (see Definition 8). If we set , then is a nonincreasing sequence of nonnegative real numbers and hence converges. Let be the limit of the sequence . Now let us claim that . Suppose that . Since is a nondecreasing function, we deduce that , for all . Then for any positive integer , by (72), we get that
Now, for all , by (73), we obtain that a contradiction. Therefore and hence the sequence converges to . As we deduce that the sequence converges to . Now, let us prove that is a Cauchy sequence. Let be given and we choose a positive integer such that for all . Fix and let
Now, it is asserted that if and is such that , then . First, we note that as , then by the weak -property . Two cases will be considered to establish this fact. Precisely, if , then it follows that
On the other hand if , then it follows that
So, . Now, we prove that for all . From and , we deduce that ; that is (80) holds for . Now, we assume that (80) holds for some . From, and , we deduce that ; that is (80) holds for and hence for all . Thus, it follows that is a Cauchy sequence. From the completeness of the space , the sequence converges to some element . From , we deduce that is also a Cauchy sequence. As is a complete subspace of , then there exists such that . Therefore, we have and so . In light of the fact that is contained in , there is such that . Since , there exists an element such that
In view of the fact that is a -weak contraction and has the weak -property and the continuity of at , we get
Letting , it follows that . Thus, we conclude that .
To assert the uniqueness, let us assume that is another element such that . Then from which it follows that and hence . If is one to one then we deduce the uniqueness.

Remark 24. From the proof of Theorem 23, we obtain that the method for getting the sequence , that is the relation , also gives an iterative algorithm for computing solutions of coincidence equations.

If in Theorem 23 the mapping is the identity on , then yields the following result which is a generalization of a result due to Rhoades [25] to non-self-mappings.

Corollary 25. Let and be closed subsets of a complete metric space such that and the pair has the weak -property. Suppose that the mapping satisfies the following conditions:(i) is a -weak contraction;(ii).
Then, there exists a unique element such that . Further, for any fixed element , the iterative sequence , defined by , converges to the element .

Example 26. Consider with the usual metric. Let us define
Then and are nonempty closed subsets of and and . Note that . Let and be defined as and . Define by , for all . Then, is a -weak contraction. As has the weak -property and is one to one, we obtain that is the unique -best proximity point of ; that is, .

The following example shows that the weak -property in Theorem 23 cannot be relaxed; that is, a -weakly contractive mapping may not have a -best proximity point in if the pair does not have the weak -property, where and are nonempty closed subsets of a complete metric space .

Example 27. Consider with the usual metric, and . Then and are nonempty closed subsets of with and . Note that . Let be a mapping given by and . It is easy to see that is a contraction mapping with and hence it is -weakly contractive, where is the identity mapping. Since , for all , then has no -best proximity points. It is worth noting that the pair does not have the weak -property.

Acknowledgment

The authors thank the referees for providing useful comments and suggestions that improved the paper. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first and second authors acknowledge with thanks DSR for financial support.

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