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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 896912, 19 pages
http://dx.doi.org/10.1155/2012/896912
Research Article

Generalized Proximal -Contraction Mappings and Best Proximity Points

1Department of Mathematics, Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng-Saen Campus, Nakhonpathom 73140, Thailand
2Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand
3Department of Mathematics, Faculty of Science, King Mongkut's, University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Received 19 July 2012; Accepted 24 September 2012

Academic Editor: Haiyun Zhou

Copyright © 2012 Winate Sanhan 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 generalized the notion of proximal contractions of the first and the second kinds and established the best proximity point theorems for these classes. Our results improve and extend recent result of Sadiq Basha (2011) and some authors.

1. Introduction

The significance of fixed point theory stems from the fact that it furnishes a unified treatment and is a vital tool for solving equations of form where is a self-mapping defined on a subset of a metric space, a normed linear space, topological vector space or some suitable space. Some applications of fixed point theory can be found in [112]. However, almost all such results dilate upon the existence of a fixed point for self-mappings. Nevertheless, if is a non-self-mapping, then it is probable that the equation has no solution, in which case best approximation theorems explore the existence of an approximate solution whereas best proximity point theorems analyze the existence of an approximate solution that is optimal. A classical best approximation theorem was introduced by Fan [13]; 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 [14], Reich [15], Sehgal, and Singh [16, 17], have derived extensions of Fan’s theorem in many directions. Other works of the existence of a best proximity point for contractions can be seen in [1821]. In 2005, Eldred et al. [22] have obtained best proximity point theorems for relatively nonexpansive mappings. Best proximity point theorems for several types of contractions have been established in [2336].

Recently, Sadiq Basha in [37] gave necessary and sufficient to claimed that the existence of best proximity point for proximal contraction of first kind and the second kind which are non-self mapping analogues of contraction self-mappings and also established some best proximity and convergence theorem as follow.

Theorem 1.1 (see [37, Theorem 3.1]). Let be a complete metric space and let and be nonempty, closed subsets of . Further, suppose that and are nonempty. Let , and satisfy the following conditions.(a) and are proximal contractions of first kind.(b) is an isometry.(c) The pair is a proximal cyclic contraction.(d).(e) and .Then, there exists a unique point and there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the element . For any fixed , the sequence , defined by converges to the element .
On the other hand, a sequence in converges to if there is a sequence of positive numbers such that where satisfies the condition that .

Theorem 1.2 (see [37, Theorem 3.4]). Let be a complete metric space and let and be nonempty, closed subsets of . Further, suppose that and are nonempty. Let and satisfy the following conditions.(a) is proximal contractions of first and second kinds.(b) is an isometry.(c) preserves isometric distance with respect to .(d).(e).Then, there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the element .
On the other hand, a sequence in converges to if there is a sequence of positive numbers such that where satisfies the condition that .

The aim of this paper is to introduce the new classes of proximal contractions which are more general than class of proximal contraction of first and second kinds, by giving the necessary condition to have best proximity points and we also give some illustrative examples of our main results. The results of this paper are extension and generalizations of main result of Sadiq Basha in [37] and some results in the literature.

2. Preliminaries

Given nonvoid subsets and 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 [31]).

Definition 2.1 ([37, Definition 2.2]). A mapping is said to be a proximal contraction of the first kind if there exists such that for all .

It is easy to see that a self-mapping that is a proximal contraction of the first kind is precisely a contraction. However, a non-self-proximal contraction is not necessarily a contraction.

Definition 2.2 (see [37, Definition 2.3]). A mapping is said to be a proximal contraction of the second kind if there exists such that for all .

Definition 2.3. Let and . The pair is said to be a proximal cyclic contraction pair if there exists a nonnegative number such that for all and .

Definition 2.4. Leting and an isometry , the mapping is said to preserve isometric distance with respect to if for all .

Definition 2.5. A point is said to be a best proximity point of the mapping if it satisfies the condition that

It can be observed that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.

Definition 2.6. is said to be approximatively compact with respect to if every sequence in satisfies the condition that for some has a convergent subsequence.

We observe that every set is approximatively compact with respect to itself and that every compact set is approximatively compact. Moreover, and are nonempty set if is compact and is approximatively compact with respect to .

3. Main Results

Definition 3.1. A mapping is said to be a generalized proximal -contraction of the first kind, if for all satisfies where is an upper semicontinuous function from the right such that for all .

Definition 3.2. A mapping is said to be a generalized proximal -contraction of the second kind, if for all satisfies where is a upper semicontinuous from the right such that for all .

It is easy to see that if we take , where , then a generalized proximal -contraction of the first kind and generalized proximal -contraction of the second kind reduce to a proximal contraction of the first kind Definition 2.1 and a proximal contraction of the second kind Definition 2.2, respectively. Moreover, it is easy to see that a self-mapping generalized proximal -contraction of the first kind and the second kind reduces to the condition of Boy and Wong' s fixed point theorem [3].

Next, we extend the result of Sadiq Basha [37] and the Banach's contraction principle to the case of non-self-mappings which satisfy generalized proximal -contraction condition.

Theorem 3.3. Let be a complete metric space and let and be nonempty, closed subsets of such that and are nonempty. Let , , and satisfy the following conditions:(a) and are generalized proximal -contraction of the first kind;(b) is an isometry;(c) The pair is a proximal cyclic contraction;(d);(e) and .Then, there exists a unique point and there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the element . For any fixed , the sequence , defined by converges to the element .
On the other hand, a sequence in converges to if there is a sequence of positive numbers such that where satisfies the condition that .

Proof. Let be a fixed element in . In view of the fact that and , it is ascertained that there exists an element such that Again, since and , there exists an element such that By similar fashion, we can find in . Having chosen , one can determine an element such that Because of the facts that and , by a generalized proximal -contraction of the first kind of , is an isometry and property of , for each , we have This means that the sequence is nonincreasing and bounded. Hence there exists such that If , then which is a contradiction unless . Therefore, We claim that is a Cauchy sequence. Suppose that is not a Cauchy sequence. Then there exists and subsequence of such that with for . Thus It follows from (3.13) that On the other hand, by constructing the sequence , we have Sine is a generalized proximal -contraction of the first kind and is an isometry, we have Notice also that Taking in above inequality, by (3.13), (3.16), and property of , we get . Therefore, , which is a contradiction. So we obtain the claim and hence converge to some element . Similarly, in view of the fact that and , we can conclude that there is a sequence such that and converge to some element . Since the pair is a proximal cyclic contraction and is an isometry, we have We take limit in (3.20) as ; it follows that so, we concluded that and . Since and , there is and such that From (3.9), (3.22), and the notion of generalized proximal -contraction of first kind of , we get Letting , we get and thus . Therefore Similarly, we can show that and then From (3.21), (3.25), and (3.26), we get
Next, to prove the uniqueness, let us suppose that there exist and with such that Since is an isometry, and are generalized proximal -contractions of the first kind and the property of ; it follows that which is a contradiction, so we have and . On the other hand, let be a sequence in and let be a sequence of positive real numbers such that where satisfies the condition that . Since is a generalized proximal -contraction of first kind and is an isometry, we have Given , we choose a positive integer such that for all ; we obtain that Therefore, we get We claim that as ; supposing the contrary, by inequality (3.33) and property of , we get which is a contradiction, so we have is convergent and it converges to . This completes the proof of the theorem.

If is assumed to be the identity mapping, then by Theorem 3.3, we obtain the following corollary.

Corollary 3.4. Let be a complete metric space and let and be nonempty, closed subsets of . Further, suppose that and are nonempty. Let , and satisfy the following conditions:(a) and are generalized proximal -contraction of the first kind;(b);(c) the pair is a proximal cyclic contraction.Then, there exists a unique point and there exists a unique point such that

If we take , where , we obtain following corollary.

Corollary 3.5 (see [37, Theorem 3.1]). Let be a complete metric space and and be non-empty, closed subsets of . Further, suppose that and are non-empty. Let , and satisfy the following conditions:(a) and are proximal contractions of first kind;(b) is an isometry;(c) the pair is a proximal cyclic contraction;(d);(e) and .Then, there exists a unique point and there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the element . For any fixed , the sequence , defined by converges to the element .

If is assumed to be the identity mapping in Corollary 3.5, we obtain the following corollary.

Corollary 3.6. Let be a complete metric space and let and be nonempty, closed subsets of . Further, suppose that and are nonempty. Let , , and satisfy the following conditions:(a) and are proximal contractions of first kind;(b);(c) the pair is a proximal cyclic contraction.Then, there exists a unique point and there exists a unique point such that

For a self-mapping, Theorem 3.3 includes the Boy and Wong' s fixed point theorem [3] as follows.

Corollary 3.7. Let be a complete metric space and let be a mapping that satisfies for all , where is an upper semicontinuous function from the right such that for all . Then has a unique fixed point . Moreover, for each converges to .

Next, we give an example to show that Definition 3.1 is different form Definition 2.1; moreover we give an example which supports Theorem 3.3.

Example 3.8. Consider the complete metric space with metric defined by for all . Let Then . Define the mappings as follows: First, we show that is generalized proximal -contraction of the first kind with the function defined by Let and be elements in satisfying It follows that Without loss of generality, we may assume that , so we have Thus is a generalized proximal -contraction of the first kind.
Next, we prove that is not a proximal contraction. Suppose is proximal contraction then for each satisfying there exists such that From (3.47), we get and thus Letting with , we get which is a contradiction. Therefore is not a proximal contraction and Definition 3.1 is different form Definition 2.1.

Example 3.9. Consider the complete metric space with Euclidean metric. Let Define two mappings , and as follows: Then it is easy to see that , , and the mapping is an isometry.
Next, we claim that and are generalized proximal -contractions of the first kind. Consider a function defined by for all . If such that for all , then we have Because, Hence is a generalized proximal -contraction of the first kind. If such that for all , then we get In the same way, we can see that is a generalized proximal -contraction of the first kind. Moreover, the pair forms a proximal cyclic contraction and other hypotheses of Theorem 3.3 are also satisfied. Further, it is easy to see that the unique element and such that
Next, we establish a best proximity point theorem for non-self-mappings which are generalized proximal -contractions of the first kind and the second kind.

Theorem 3.10. Let be a complete metric space and let and be non-empty, closed subsets of . Further, suppose that and are non-empty. Let and satisfy the following conditions:(a) is a generalized proximal -contraction of first and second kinds;(b) is an isometry;(c) preserves isometric distance with respect to ;(d);(e).Then, there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the element .
On the other hand, a sequence in converges to if there is a sequence of positive numbers such that where satisfies the condition that .

Proof. Since and , similarly in the proof of Theorem 3.3, we can construct the sequence of element in such that for nonnegative number . It follows from that is an isometry and the virtue of a generalized proximal -contraction of the first kind of ; we see that for all . Similarly to the proof of Theorem 3.3, we can conclude that the sequence is a Cauchy sequence and converges to some . Since is a generalized proximal -contraction of the second kind and preserves isometric distance with respect to that this means that the sequence is nonincreasing and bounded below. Hence, there exists such that If , then which is a contradiction, unless . Therefore We claim that is a Cauchy sequence. Suppose that is not a Cauchy sequence. Then there exists and subsequence of such that with for . Thus it follows from (3.68) that Notice also that Taking in previous inequality, by (3.68), (3.71), and property of , we get . Hence, , which is a contradiction. So we obtain the claim and then it converges to some . Therefore, we can conclude that That is . Since , we have for some and then . By the fact that is an isometry, we have . Hence and so becomes to a point in . As that for some . It follows from (3.63) and (3.74) that is a generalized proximal -contraction of the first kind that for all . Taking limit as , we get the sequence converging to a point . By the fact that is continuous, we have By the uniqueness of limit of the sequence, we conclude that . Therefore, it results that . The uniqueness and the remaining part of the proof follow as in Theorem 3.3. This completes the proof of the theorem.

If is assumed to be the identity mapping, then by Theorem 3.10, we obtain the following corollary.

Corollary 3.11. Let be a complete metric space and let and be nonempty, closed subsets of . Further, suppose that and are nonempty. Let satisfy the following conditions:(a) is a generalized proximal -contraction of first and second kinds;(b).Then, there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the best proximity point of .

If we take , where in Theorem 3.10, we obtain following corollary.

Corollary 3.12 (see [37, Theorem 3.4]). Let be a complete metric space and let and be non-empty, closed subsets of . Further, suppose that and are non-empty. Let and satisfy the following conditions:(a) is a proximal contraction of first and second kinds;(b) is an isometry;(c) preserves isometric distance with respect to ;(d);(e).Then, there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the element .

If is assumed to be the identity mapping in Corollary 3.12, we obtain the following corollary.

Corollary 3.13. Let be a complete metric space and let and be non-empty, closed subsets of . Further, suppose that and are non-empty. Let satisfy the following conditions:(a) is a proximal contraction of first and second kinds;(b).Then, there exists a unique point such that Moreover, for any fixed , the sequence , defined by converges to the best proximity point of .

Acknowledgments

W. Sanhan would like to thank Faculty of Liberal Arts and Science, Kasetsart University, Kamphaeng-Saen Campus and Centre of Excellence in Mathematics, CHE, Sriayudthaya Rd., Bangkok, Thailand. C. Mongkolkeha was supported from the Thailand Research Fund through the the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0029/2553). P. Kumam was supported by the Commission on Higher Education, the Thailand Research Fund, and the King Mongkuts University of Technology Thonburi (Grant no. MRG5580213).

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