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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 375604, 8 pages

http://dx.doi.org/10.1155/2013/375604

## The Existence Theorems of an Optimal Approximate Solution for Generalized Proximal Contraction Mappings

^{1}Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand^{2}Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), Bangkok 10140, Thailand

Received 3 June 2013; Accepted 31 July 2013

Academic Editor: Calogero Vetro

Copyright © 2013 Wutiphol Sintunavarat and Poom Kumam. 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

Recently, Basha (2011) established the best proximity point theorems for proximal contractions of the first and second kinds which are extension of Banach's contraction principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the notions of proximal contractions of the first and second kinds which are more general than the notion of self-contractions, establish the existence of an optimal approximate solution theorems for these non-self-mappings, and also give examples to validate our main results.

#### 1. Introduction

Since Banach’s contraction principle [1] first appeared, several authors have generalized this principle in different directions. However, they have shown the existence of a fixed point for self-mappings. One of the most interesting results on Banach’s contraction principle is the case of non-self-mappings. In fact, for any nonempty closed subsets and of a complete metric space , a contractive non-self-mapping does not necessarily have a fixed point . In this case, a best proximity point, that is, a point for which represents an optimal approximate solution to the equation . It is well known that a best proximity point reduces to a fixed point if the underlying mapping is assumed to be a self-mapping. Consequently, best proximity point theorems are improvement of Banach’s contraction principle in case of non-self-mappings.

A classical best approximation theorem was introduced by Fan [2]. Afterward, several authors including Prolla [3], Reich [4], and Sehgal and Singh [5, 6] have derived extensions of Fan’s Theorem in many directions. Other works of the existence of a best proximity point for contractive mappings can be found in [7–13]. On the other hand, many best proximity point theorems for set-valued mappings have been established in [14–19]. In particular, Eldred et al. [20] have obtained best proximity point theorems for relatively nonexpansive mappings.

Recently, Basha [21] gave necessary and sufficient conditions to claim the existence of best proximity point for proximal contraction of first and second kinds which are non-self-mapping analogues of contraction self-mappings, and they also established some best proximity theorems. Afterward, several mathematicians extended and improved these results in many ways (see in [22–25]).

The purpose of this paper is to extend and generalize the class of proximal contraction of first and second kinds which are different from another type in the literature. For such mappings, we seek the necessary condition for these classes to have best proximity points and also give some examples to illustrate our main results. The results of this paper are generalizations of results of Basha in [21] and some results of the fundamental metrical fixed point and best proximity point theorems in the literature.

#### 2. Preliminaries

Throughout this paper, suppose that and are nonempty subsets of a metric space . We use the following notations:

*Remark 1. * It is easy to see that and are nonempty whenever . Further, if and are closed subsets of a normed linear space such that , then and , where is a boundary of .

*Definition 2 (see [21]). *A mapping is called a proximal contraction of the first kind if there exists such that, for all ,

*Remark 3. * If is self-mapping, then is a proximal contraction of the first kind deduced to which is a contraction mapping. But a non-self-proximal contraction is not necessarily a contraction.

*Definition 4 (see [21]). *A mapping is said to be a proximal contraction of the second kind if there exists such that, for all ,

The necessary condition for a self-mapping to be a proximal contraction of the second kind is that
for all , in the domain of . Therefore, every contraction self-mapping is a proximal contraction of the second kind, but the converse is not true (see Example 5).

*Example 5. * Consider endowed with the Euclidean metric. Let the self-mapping be defined as follows:
It is easy to prove that is a proximal contraction of the second kind. However, is not a contraction mapping.

The above example also exhibits that a self-mapping, that is, a proximal contraction of the second kind, is not necessarily continuous.

*Definition 6. *Let and be mappings. The pair is said to be(1)a cyclic contractive pair if for all and ;(2)a cyclic expansive pair if for all and ;(3)a cyclic inequality pair if for all and .

*Definition 7. *Let and be mappings. The pair is said to satisfy min-max condition if, for all and ,
where and are defined by

We observe that the cyclic contractive pairs, cyclic expansive pairs, and cyclic inequality pairs satisfy the min-max condition.

*Definition 8. *Let a mapping and be an isometry. The mapping is said to preserve isometric distance with respect to if
for all .

*Definition 9. *A point is said to be a best proximity point of a mapping if it satisfies the condition that

Observe that a best proximity reduces to a fixed point if the underlying mapping is a self-mapping.

*Definition 10. * is said to be approximatively compact with respect to if every sequence in satisfies the condition that for some has a convergent subsequence.

*Remark 11. *Any nonempty subset of metric space is approximatively compact with respect to itself.

#### 3. Main Results

In this section, we introduce the notions of generalized proximal contraction mappings of the first and second kinds which are different from another type in the literature. We also give the existence theorems of an optimal approximate solution for these mappings.

*Definition 12. *Let , be nonempty subset of metric space , and . A mapping is said to be a generalized proximal contraction of the first kind with respect to if
for all .

*Remark 13. *If we take for all , where , then a generalized proximal contraction of the first kind with respect to reduces to a proximal contraction of the first kind (Definition 2). In case of a self-mapping, it is apparent that the class of contraction mapping is contained in the class of generalized proximal contraction of the first kind with respect to mapping.

Now, we give an example to claim that the class of proximal contraction mapping of the first kind is a proper subclass of the class of generalized proximal contractions of the first kind with respect to mapping.

*Example 14. *Consider the metric space with Euclidean metric. Let and . Define a mapping as follows:
for all .

It is easy to check that there is no satisfing
for all . Therefore, is not a proximal contraction of the first kind.

Consider a function defined by
Next, we claim that is a generalized proximal contraction of the first kind with respect to .

If such that
for all , then we have
Therefore, it follows that
This implies that is a generalized proximal contraction of the first kind with respect to .

*Definition 15. *Let , be nonempty subset of metric space , and . A mapping is said to be a generalized proximal contraction of the second kind with respect to if
for all .

Clearly, a proximal contraction of the second kind (Definition 4) is a generalized proximal contraction of the second kind.

Next, we extend the results of Basha [21] and many results in the literature.

Theorem 16. *Let a complete metric space and , be nonempty closed subsets of such that and are nonempty. Suppose that , , and are mappings satisfying the following conditions:*(a)* is a continuous generalized proximal contraction of first kind with respect to ;*(b)* and ;*(c)* is an isometry;*(d)*, whenever .**
Then there exists a unique point such that . *

*Proof. *Let be a fixed element in . From and , it follows that there exists a point such that
Again, since and , there exists a point such that
Continuing this process, we can construct the sequence in such that
for all . Since is a generalized proximal contraction of the first kind with respect to , it follows that
for all . Also, since is an isometry, we have
for all . By using (20) and (d), we have
for all . By repeating (23), we get
for all . Now, we let . For positive integers and with , it follows from (24) that
Since , we have as , which implies that is a Cauchy sequence in . Since is complete, it follows that the sequence converges to point . Since and are continuous, we get
Next, we suppose that is another point in such that
Since is a generalized proximal contraction of the first kind with respect to , by using (26) and (27), we get
Since is an isometry, it follows that
which implies that . This completes the proof.

Now, we give an example to illustrate Theorem 16.

*Example 17. *Consider the complete metric space with Euclidean metric. Let and . Define two mappings and as follows:
for all . Then it is easy to see that , , , and the mapping is an isometry.

Consider a function defined by

Next, we claim that is a generalized proximal contraction of the first kind with respect to . If such that
for all , then we have
Therefore, it follows that
This implies that the non-self-mapping is a generalized proximal contraction of the first kind with respect to . It is easy to see that whenever . Moreover, since is continuous and is an isometry, all the conditions of Theorem 16 are satisfied, and so has a unique element such that

Corollary 18 (see [21, Theorem 3.3]). *Let be a complete metric space and , nonempty closed subsets of such that and are nonempty. Suppose that and are mappings satisfying the following conditions:*(a)* is a continuous proximal contraction of the first kind;*(b)* and ;*(c)* is an isometry.**
Then there exists a unique element such that . *

*Proof. *Since a proximal contraction of the first kind is a special case of a generalized proximal contraction of the first kind, we can prove this result by applying Theorem 16.

In Theorem 16, if is the identity mapping, then it yields the following best proximity point theorem.

Corollary 19. *Let a complete metric space and , be nonempty closed subsets of such that and are nonempty. Suppose that and are mappings satisfying the following conditions:*(a)* is a continuous generalized proximal contraction of first kind with respect to ;*(b)*;*(c)*, whenever .**
Then has a unique best proximity point in .*

Corollary 20 (see [21, Corollary 3.4]). * Let be a complete metric space and , nonempty closed subsets of such that and are nonempty. Let be a mapping satisfying the following conditions:*(a)* is a continuous proximal contraction of the first kind;*(b)*.**
Then has a unique best proximity point in . *

*Proof. *Since a proximal contraction of the first kind is a special case of a generalized proximal contraction of the first kind with respect to , we can prove this result by applying Corollary 19.

Next, we prove the second main result for generalized proximal contraction of the second kind with respect to mapping.

Theorem 21. *Let a complete metric space and , be nonempty closed subsets of such that is approximatively compact with respect to . Suppose that and are nonempty and , , and are mappings satisfying the following conditions:*(a)* is a continuous generalized proximal contraction of the second kind with respect to ;*(b)* and ;*(c)* is an isometry;*(d)* preserves isometric distance with respect to ;*(e)*, whenever . **
Then there exists a point such that . Moreover, if is another point in for which , then . *

*Proof. *As in the proof of Theorem 16, for fixed , we can define a sequence in such that
for all . Since is a generalized proximal contraction of the second kind with respect to , it follows that
Since preserves isometric distance with respect to , we have
for all . By using (36) and (e), we have
for all . By repeating (39), we get
for all . Now, we let . For positive integers and with , it follows from (40) that
Since , we have as , which implies that is a Cauchy sequence in . By completeness of , there exists a point such that as . By (36) and the triangle inequality, we have
Letting in (42), we get . Since is approximatively compact with respect to , it follows that has a convergence subsequence ; say as . Thus we have
which implies that . Since , we have for some . Therefore, as . Since is an isometry, we get as . By the continuity of , we have as and then . From (43), we can conclude that

Next, we suppose that is another point in such that
Since is a generalized proximal contraction of the second kind with respect to , by the virtue of (44) and (45), we get
Since preserves isometric distance with respect to , it follows that
which implies that . This completes the proof.

Corollary 22 (see [21, Theorem 3.1]). *Let be a complete metric space and , nonempty closed subsets of such that is approximatively compact with respect to . Suppose that and are nonempty and and are mappings satisfying the following conditions:*(a)* is a continuous proximal contraction of the second kind;*(b)* and ;*(c)* is an isometry;*(d)* preserves isometric distance with respect to . **
Then there exists a point such that . Moreover, if is another point in for which , then . *

*Proof. *Since a proximal contraction of the second kind is a special case of a generalized proximal contraction of the second kind with respect to , we can prove this result by applying Theorem 21.

Corollary 23. * Let be a complete metric space and , nonempty closed subsets of such that is approximatively compact with respect to . Suppose that and are nonempty and and are mappings satisfying the following conditions:*(a)* is a continuous generalized proximal contraction of the second kind with respect to ;*(b)*;*(c)*, whenever . **
Then has a best proximity point. Moreover, if is another best proximity point of , then . *

*Proof. *We can prove this result by applying Theorem 21 with , where is an identity mapping on .

Corollary 24 (see [21, Corollary 3.2]). *Let be a complete metric space and , nonempty closed subsets of such that is approximatively compact with respect to . Suppose that and are nonempty and is mapping satisfying the following conditions:*(a)* is a continuous generalized proximal contraction of the second kind;*(b)*.**
Then has a best proximity point. Moreover, if is another best proximity point of , then . *

*Proof. *Since a proximal contraction of the second kind is a special case of a generalized proximal contraction of the second kind, we can prove this result by applying Corollary 23.

Here, we give the last result in this work.

Theorem 25. *Let be a complete metric space, and nonempty closed subsets of , and . Suppose that is a mapping satisfying
**
for all . Then the following holds.***(A)*** There exists a nonexpansive mapping such that satisfies the min-max condition whenever has a best proximity point. ***(B)*** If there exists a nonexpansive mapping such that satisfies the min-max condition and and for all , then has a best proximity point. ***(C)*** For two any best proximity points and of , we have
*

*Proof. ***(A)** Let has a best proximity point . We define a mapping by for all . Clearly, is a nonexpansive mapping. It follows from the definition of that
for all . Thus we can conclude that for all and .

Next, we show that satisfies the min-max condition. Suppose that and such that . Then we have
which implies that the pair satisfies the min-max condition. Therefore, we can find a nonexpansive mapping such that satisfies the min-max condition.**(B)** Fix and define a sequence in by
for all . Since is nonexpansive, it follows from (48) that
for all . By repeating the above argument, we have
for all , which implies that the sequence is a Cauchy sequence in . A similar argument asserts that the sequence is a Cauchy sequence in . By the completeness of , we conclude that converges to a point and converges to a point . Since is continuous, converges to , which implies that converges to . Thus .

Similarly, it is easy to check that . Therefore, we have
Now, we can conclude that
By the virtue of the min-max condition of , we get . Since , we have . Therefore, we have
which implies that has a best proximity point in .**(C)** Let and be best proximity points of . Then and . Using the triangle inequality and (48), we have
This implies that . This completes the proof.

Corollary 26 (see [21, Theorem 3.6]). * Let be a complete metric space and and nonempty closed subsets of . Suppose that is a contraction mapping. Then has a best proximity point if and only if there exists a nonexpansive mapping such that satisfies the min-max condition.**Moreover, for some and any two best proximity points and of . *

*Proof. *Since is a contraction mapping, we have for some and all . Now, we can prove this result by applying Theorem 25 with a function defined by for all .

#### Acknowledgments

The second author would like to thank the Commission on Higher Education, the Thailand Research Fund, and the King Mongkut’s University of Technology Thonburi (Grant no. MRG5580213) for financial support during the preparation of this paper.

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