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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 534127, 10 pages
http://dx.doi.org/10.1155/2013/534127
Research Article

Best Proximity Points for Generalized -Proximal Contractive Type Mappings

1Department of Mathematics, King Saud University, Riyadh, Saudi Arabia
2Department of Mathematics, Atılım University, İncek, 06836 Ankara, Turkey

Received 19 February 2013; Accepted 7 April 2013

Academic Editor: Lai-Jiu Lin

Copyright © 2013 Mohamed Jleli 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 introduce a new class of non-self-contractive mappings. For such mappings, we study the existence and uniqueness of best proximity points. Several applications and interesting consequences of our obtained results are derived.

1. Introduction and Preliminaries

Let and be two nonempty subsets of a metric space . An element is said to be a fixed point of a given map if . Clearly, is a necessary (but not sufficient) condition for the existence of a fixed point of . If , then for all that is, the set of fixed points of is empty. In a such situation, one often attempts to find an element which is in some sense closest to . Best proximity point analysis has been developed in this direction.

An element is called a best proximity point of if where Because of the fact that for all , the global minimum of the mapping is attained at a best proximity point. Clearly, if the underlying mapping is a self-mapping, then it can be observed that a best proximity point is essentially a fixed point. The goal of best proximity point theory is to furnish sufficient conditions that assure the existence of such points. For more details on this approach, we refer the reader to [112] and references therein.

Recently, Samet et al. [13] introduced a new class of contractive mappings called --contractive type mappings. Let be a metric space.

Definition 1. A self-mapping is said to be an --contraction, where and is a (c)-comparison function, if

Definition 2. A self-mapping is said to be -admissible, where , if

The main results obtained in [13] are the following fixed point theorems.

Theorem 3. Let be a complete metric space and be an --contractive mapping satisfying the following conditions: (i)is -admissible; (ii)there exists such that ; (iii) is continuous. Then, has a fixed point; that is, there exists such that .

Theorem 4. Let be a complete metric space and be an --contractive mapping satisfying the following conditions: (i) is -admissible; (ii)there exists such that ; (iii)if is a sequence in such that for all and as , then for all . Then, has a fixed point.

Theorem 5. In addition to the hypotheses of Theorem 3 (resp., Theorem 4), suppose that for all , there exists such that and . Then we have a unique fixed point.

It was shown in [13, 14] that various types of contractive mappings belong to the class of --contractive type mappings (classical contractive mappings, contractive mappings on ordered metric spaces, cyclic contractive mappings, etc.). For other works in this direction, we refer the reader to [15, 16].

In a very recent paper, Jleli and Samet [17] established some best proximity point results for --contractive type mappings. Before presenting the main results obtained in [17], we need to fix some notations and recall some definitions.

Let and , two nonempty subsets of a metric space . We will use the following notations:

Definition 6. An element is said to be a best proximity point of the non-self-mapping if it satisfies the condition that

The following concept was introduced in [11].

Definition 7. Let be a pair of nonempty subsets of a metric space with . Then, the pair is said to have the -property if and only if where and .

The following concepts were introduced in [17].

Definition 8. Let and . We say that is -proximal admissible if for all .

Clearly, if , is -proximal admissible implies that is -admissible.

Definition 9. A non-self-mapping is said to be an --proximal contraction, where and is a (c)-comparison function, if

The main results obtained in [17] are the following.

Theorem 10. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Let and be a (c)-comparison function. Suppose that is a non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii) is -proximal admissible; (iii)there exist elements and in such that (iv) is a continuous --proximal contraction. Then, there exists an element such that

Theorem 11. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Let and be a (c)-comparison function. Suppose that is a non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii) is -proximal admissible; (iii) there exist elements and in such that (iv) is an --proximal contraction; (v)if is a sequence in such that for all and as , then there exists a subsequence of such that for all . Then, there exists an element such that

Theorem 12. In addition to the hypotheses of Theorem 10 (resp., Theorem 11), suppose that for all , there exists such that and . Then, has a unique best proximity point.

In this paper, we extend and generalize the above results by introducing a new family of non-self-contractive mappings that will be called the class of generalized --proximal contractive type mappings. For such mappings, we discuss the existence and uniqueness of best proximity points. Various applications and interesting consequences are derived from our main results.

2. Main Results

All the notations presented in the previous section will be used through this paper.

We denote by the set of nondecreasing functions such that where is the th iterate of . These functions are known in the literature as (c)-comparison functions. It is easily proved that if is a (c)-comparison function, then for all .

We introduce the following concept.

Definition 13. A non-self-mapping is said to be a generalized --proximal contraction, where and , if where

Our first main result is the following best proximity point theorem.

Theorem 14. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Let and . Suppose that is a non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii) is -proximal admissible; (iii)there exist elements and in such that (iv) is a continuous generalized --proximal contraction. Then, there exists an element such that

Proof. From condition (iii), there exist elements and in such that Since , there exists such that Now, we have Since is -proximal admissible, this implies that . Thus, we have Again, since , there exists such that Now, we have Since is -proximal admissible, this implies that . Thus, we have Continuing this process, by induction, we can construct a sequence such that Since satisfies the -property, we conclude from (26) that From condition (iv), that is, is a generalized --proximal contraction, for all , we have On the other hand, using (26) and (27), we have Thus, we proved that Using the above inequality, (26), (27), and (28), and taking in consideration that is a nondecreasing function, we get that If for some , we have , from (26), we get that ; that is, is a best proximity point. So, we can suppose that Suppose that . Using (32) and since for all , we have which is a contradiction. Thus, we have Now, from (31), we get that Using the monotony of , by induction, it follows from (35) that Now, we shall prove that is a Cauchy sequence in the metric space . Let be fixed. Since , there exists some positive integer such that Let , using the triangular inequality, (36) and (37), we obtain Thus, is a Cauchy sequence in the metric space . Since is complete and is closed, there exists some such that as . On the other hand, is a continuous mapping. Then, we have as . The continuity of the metric function implies that as . Therefore, . This completes the proof of the theorem.

In the next result, we remove the continuity hypothesis, assuming the following condition in :(H) If is a sequence in such that for all and as , then there exists a subsequence of such that for all .

Theorem 15. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Let and . Suppose that is a non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii) is -proximal admissible; (iii)there exist elements and in such that (iv) (H) holds, and is a generalized --proximal contraction. Then, there exists an element such that

Proof. Following the proof of Theorem 14, there exists a Cauchy sequence such that (26) holds, and as . From the condition (H), there exists a subsequence of such that for all . Since is a generalized --proximal contraction, we get that where On the other hand, from (26), for all , we have Thus, we have Combining (41) with (44), we get that From (26), for all , we have Since is a nondecreasing function, we get from (45) that Suppose that . In this case, we have that is, Since for large enough, we have . On the other hand, we have for all . Then, from (47), we get that Using (49) and letting in the above inequality, we obtain that which is a contradiction. Thus, we deduce that is a best proximity point of ; that is, .

The next result gives us a sufficient condition that assures the uniqueness of the best proximity point. We need the following definition.

Definition 16. Let be a non-self-mapping and . We say that is regular if for all , there exists such that

Theorem 17. In addition to the hypotheses of Theorem 14 (resp., Theorem 15), suppose that is regular. Then, has a unique best proximity point.

Proof. From the proof of Theorem 14 (resp., Theorem 15), we know that the set of best proximity points of is nonempty ( is a best proximity point). Suppose that is another best proximity point of , that is, Using the -property and (54), we get that We distinguish two cases.
Case  1. If .
Since is a generalized --proximal contraction, using (55), we obtain that where from (54) and (55), we have This equality with (56) imply that Since for all , the above inequality holds only if , that is, .
Case  2. If .
By hypothesis, there exists such that and . Since , there exists such that Now, we have Since is -proximal admissible, we get that . Thus, we have Continuing this process, by induction, we can construct a sequence in such that Using the -property and (62), we get that Since is a generalized --proximal contraction, we have Combining the above inequality with (63), we get that This implies from (62) that On the other hand, from (63), for all , we have Combining the above inequality with (71), we get that Suppose that for some , we have . From (63), we get that for all . This implies that as . Now, suppose that for all . Since for all , the inequality (68) holds only if Now, we have By induction, we then derive Letting in (71), we obtain that as . So, in all cases, we have as . Similarly, we can prove that as . By uniqueness of the limit, we obtain that .

3. Applications

3.1. Standard Best Proximity Point Results

We have the following best proximity point result.

Corollary 18. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Let and suppose that is a non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii), for all . Then, there exists a unique element such that

Proof. Consider the mapping defined by: From the definition of , clearly is -proximal admissible and also it is an --proximal contraction. On the other hand, for any , since , there exists such that . Moreover, from condition (ii), is a continuous mapping. Now, all the hypotheses of Theorem 14 are satisfied and the existence of the best proximity point follows from Theorem 14. The uniqueness is an immediate consequence of the definition of and Theorem 17.

Taking in Corollary 18  , where , we obtain the following best proximity point result.

Corollary 19. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Suppose that is a non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii)there exists such that , for all . Then, there exists a unique element such that

3.2. Best Proximity Points on a Metric Space Endowed with an Arbitrary Binary Relation

Before presenting our results, we need a few preliminaries.

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

Definition 20. We say that is a proximal comparative mapping if for all .

We have the following best proximity point result.

Corollary 21. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Let be a binary relation over . Suppose that is a continuous non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii) is a proximal comparative mapping; (iii)there exist elements and in such that (iv)there exists such that Then, there exists an element such that

Proof. Define the mapping by: Suppose that for some . By the definition of , we get that Condition (ii) implies that , which gives us from the definition of that . Thus, we proved that is -proximal admissible. Condition (iii) implies that Finally, condition (iv) implies that that is, is a generalized --proximal contraction. Now, all the hypotheses of Theorem 14 are satisfied, and the desired result follows immediately from this theorem.

In order to remove the continuity assumption, we need the following condition: () if the sequence in and the point are such that for all and , then there exists a subsequence of such that for all .

Corollary 22. Let and be nonempty closed subsets of a complete metric space such that is nonempty. Let be a binary relation over . Suppose that is a non-self-mapping satisfying the following conditions: (i), and satisfies the -property; (ii) is a proximal comparative mapping; (iii)there exist elements and in such that (iv)there exists such that (v)() holds. Then, there exists an element such that

Proof. The result follows from Theorem 15 by considering the mapping given by (81), and by observing that, condition () implies condition (H).

Corollary 23. In addition to the hypotheses of Corollary 21 (resp., Corollary 22), suppose that the following condition holds: for all with , there exists such that and . Then, has a unique best proximity point.

Proof. The result follows from Theorem 17 by considering the mapping given by (81).

3.3. Related Fixed Point Theorems
3.3.1. Fixed Points for Generalized - Contractive Type Mappings

The concept of generalized - contractive type mappings was introduced recently in [14].

Definition 24. Let be a nonempty subset of a metric space and be a self-mapping. We say that is a generalized - contractive mapping if there exist two functions and such that for all , we have

Taking in Theorems 1417, we obtain the following fixed point results established in [14].

Corollary 25. Let be a nonempty closed subset of a complete metric space . Let be a generalized - contractive mapping satisfying the following conditions: (i) is -admissible; (ii)there exists such that ; (iii) is continuous. Then, has a fixed point.

Corollary 26. Let be a nonempty closed subset of a complete metric space . Let be a generalized - contractive mapping satisfying the following conditions: (i) is -admissible; (ii)there exists such that ; (iii)condition (H) holds. Then, has a fixed point.

Corollary 27. In addition to the hypotheses of Corollary 25 (resp., Corollary 26), suppose that for all , there exists such that Then, has a unique fixed point.

3.3.2. Fixed Points on a Metric Space Endowed with an Arbitrary Binary Relation

We recall the following concept introduced in [18].

Let be a nonempty closed subset of a complete metric space . Suppose that is endowed with an arbitrary binary relation . We denote by the symmetric relation attached to . Let be a given mapping.

Definition 28. We say that is a comparative mapping if   maps comparable elements into comparable elements, that is,

We have the following fixed point theorem.

Corollary 29. Assume that is a continuous comparative map, and where . Suppose also that there exists such that . Then, has a fixed point.

Proof. It follows from Corollary 21 by taking and remarking that if , a comparative map is a proximal comparative map.

Remark that a self-mapping satisfying the property (23) is not necessarily continuous (see Example  2.2 in [18]).

Similarly, Taking in Corollary 22, we obtain the following fixed point result.

Corollary 30. Assume that is a comparative map satisfying (23) for some . Suppose also that there exists such that . If () holds, then has a fixed point.

The uniqueness of the fixed point follows from Corollary 23 by taking .

Corollary 31. In addition to the hypotheses of Corollary 29 (resp., Corollary 30), suppose that the following condition holds: for all with , there exists such that and . Then, has a unique fixed point.

Acknowledgment

This work is supported by the Research Center, College of Science, King Saud University.

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