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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 274062, 7 pages
The Existence and Uniqueness of Coupled Best Proximity Point for Proximally Coupled Contraction in a Complete Ordered Metric Space
1Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamil Nadu 620 024, India
2Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bang Mod, Thung Khru, Bangkok 10140, Thailand
3Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of North Bangkok (KMUTNB), Wongsawang, Bangsue, Bangkok 10800, Thailand
Received 2 February 2014; Accepted 18 April 2014; Published 14 May 2014
Academic Editor: Salvador Romaguera
Copyright © 2014 V. Pragadeeswarar 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.
We prove the existence and uniqueness of coupled best proximity point for mappings satisfying the proximally coupled contraction in a complete ordered metric space. Further, our result provides an extension of a result due to Bhaskar and Lakshmikantham.
One of the most useful tools in the study of nonlinear functional equation is to describe many problems in physics, chemistry, and engineering. It can be formulated in terms of finding the fixed points of a nonlinear self-mapping. Fixed point theory investigates the techniques for determining a solution of the pattern , where is a self-mapping defined on a subset of a metric space . Noteworthy, a fixed point of on can be written by .
A well-known principle that guarantees a unique fixed point solution is the Banach contraction principle  which states on a complete metric space for a contraction self-mapping (i.e., for all , where is a nonnegative number such that ). Over the years, this principle has been generalized in many ways; see also [2–6]. Recently, an interesting way is to study the extension of Banach contraction principle to the case of non-self-mappings. Certainly, a contraction non-self-mapping does not necessarily have a fixed point, where and are nonempty subsets of a metric space .
Ultimately, one proceeds to find an approximate solution which is closest to in the sense that distance is minimum which implies that and are in close proximity to each other. Indeed, the best approximation theorems and the best proximity point theorem investigate the existence of an approximate solution to fixed point problems for the non-self- mappings. In 1969, Fan  guarantees at least one solution to the minimization problem , where is a nonempty compact convex subset of a normed linear space and is a continuous function. Such an element satisfying the above condition is called a best approximant of in . Note that if is a best approximant, then need not be the optimum. As a matter of fact, the best proximity point theorems have been explored to find sufficient conditions for the existence of an element such that the error is minimum.
To have a concrete lower bound, let us consider two nonempty subsets , of a metric space and a mapping . The natural question is whether one can find an element such that , where . Since for all , the optimal solution to the problem of minimizing the real valued function over the domain of the mapping will be the one for which the value is attained. A point that satisfies the condition is called a best proximity point of . Furthermore, the best proximity theorems also regress to an extension of fixed point theorem; that is, a best proximity point becomes a fixed point if either or the mapping under consideration is a self-mapping.
The existence and convergence of best proximity points are an interesting topic of optimization theory. For more details on this approach, we refer the reader to De la Sen and Agarwal , Kumam et al. [9–11], di Bari et al. , Eldred and Veeramani , Al-Thagafi and Shahzad , Sadiq Basha and Veeramani , Kim and Lee , Kirk et al. , Sankar Raj , Karapınar et al. , and Jleli and Samet . The study of best proximity point in the setting of partially ordered metric space attracted recently the attention of many authors; see [21–33].
Now we recall the definition of coupled fixed point. Let be a nonempty set and a given mapping. An element is called a coupled fixed point of the mapping if and . In 2006, Bhaskar and Lakshmikantham  proved some coupled fixed point theorems for mappings satisfying the mixed monotone mapping. Indeed, let be a partially ordered set; the mapping is said to have the mixed monotone property if Their results investigate a large class of problems and show the existence and uniqueness of a solution for a periodic boundary value problem. For more details on this concept one may go through the references [35–37].
Motivated by the above theorems, we introduce the concept of proximally mixed monotone property and proximally coupled contraction. We also explore the existence and uniqueness of coupled best proximity points in the setting of partially ordered metric spaces, thereby producing optimal approximate solutions for that function with respect to both coordinates. Further, we attempt to give the generalization of the results in .
Let be a nonempty set such that is a metric space. Unless otherwise specified, it is assumed throughout this section that and are nonempty subsets of the metric space ; the following notions are used subsequently: In , the authors discussed sufficient conditions which guarantee the nonemptiness of and . Also, in , the authors proved that is contained in the boundary of . Moreover, the authors proved that is contained in the boundary of in the setting of normed linear spaces.
Definition 1. Let be a partially ordered metric space and , nonempty subsets of . A mapping is said to have proximal mixed monotone property if is proximally nondecreasing in and is proximally nonincreasing in ; that is, for all , where .
One can see that, if in the above definition, the notion of proximal mixed monotone property reduces to that of mixed monotone property.
Lemma 2. Let be a partially ordered metric space and , nonempty subsets of . Assume that is nonempty. A mapping has proximal mixed monotone property with ; then for any , , , , in
Proof. By hypothesis , . Hence there exists such that Using is proximal mixed monotone (in particular is proximally nondecreasing in ) to (4) and (5), we get Analogously, using is proximal mixed monotone (in particular is proximally nonincreasing in ) to (4) and (5), we get From (6) and (7), one can conclude that . Hence the proof is completed.
Lemma 3. Let be a partially ordered metric space and , nonempty subsets of . Assume that is nonempty. A mapping has proximal mixed monotone property with ; then for any , , , , in
Proof. The proof is same as Lemma 2.
Definition 4. Let be a partially ordered metric space and , nonempty subsets of . A mapping is said to be proximally coupled contraction if there exists such that whenever where .
One can see that, if in the above definition, the notion of proximally coupled contraction reduces to that of coupled contraction.
3. Coupled Best Proximity Point Theorems
Let be a partially ordered complete metric space endowed with the product space with the following partial order:
Theorem 5. Let be a partially ordered complete metric space. Let and be nonempty closed subsets of the metric space such that . Let satisfy the following conditions. (i)is a continuous proximally coupled contraction having the proximal mixed monotone property on such that .(ii)There exist elements and in such that Then there exist such that and .
Proof. By hypothesis there exist elements and in such that
Because of the fact that , there exists an element in such that
Hence from Lemmas 2 and 3, we obtain and .
Continuing this process, we can construct the sequences and in such that with and with .
Then , and also we have , , . Now using that is proximally coupled contraction on we get Similarly Adding (16) and (17), we get Finally, we get Now we prove that and are Cauchy sequences.
For , regarding triangle inequality and (19), one can observe that Let be given. Choose a natural number such that for all . Thus, for . Therefore, the sequences and are Cauchy.
Since is closed subset of a complete metric space , these sequences have limits. Thus, there exists such that and . Therefore in . Since is continuous, we have and .
Hence the continuity of the metric function implies that and . But from (14) and (15) we get that the sequences and are constant sequences with the value . Therefore, and . This completes the proof of the theorem.
Corollary 6. Let be a partially ordered complete metric space. Let be nonempty closed subset of the metric space . Let satisfy the following conditions. (i) is continuous having the proximal mixed monotone property and proximally coupled contraction on .(ii)There exist and in such that with and with .Then there exist such that and .
In what follows we prove that Theorem 5 is still valid for not necessarily continuous, assuming the following hypothesis in :
Proof. Following the proof of Theorem 5, there exist sequences and in satisfying the following conditions: Moreover, converges to and converges to in . From (21), we get and . Note that the sequences and are in and is closed. Therefore, . Since , there exists and are in . Therefore, there exists such that Since and . By using is proximally coupled contraction for (22) and (24) also for (25) and (23), we get Since and , by taking limit on the above two inequality, we get and . Consequently the result follows.
Example 9. Let and consider the usual order and .
Thus, is a partially ordered set. Besides, is a complete metric space considering the euclidean metric. Let and be closed subsets of . Then, , and . Let be defined as . Then, it can be seen that is continuous such that . The only comparable pairs of points in are for ; hence proximal mixed monotone property and proximally coupled contraction on are satisfied trivially.
It can be shown that the other hypotheses of the theorem are also satisfied. However, has three coupled best proximity points , and .
One can prove that the coupled best proximity point is in fact unique, provided that the product space endowed with the partial order mentioned earlier has the following property: It is known that this condition is equivalent to the following:
For every pair of , there exists in .
Proof. From Theorem 5 (resp., Theorem 7), the set of coupled best proximity points of is nonempty. Suppose that there exist and in which are coupled best proximity points. That is,
We distinguish two cases.
Case 1. Suppose that is comparable. Let be comparable to with respect to the ordering in . Apply as proximally coupled contraction to and , there exists such that Similarly, one can prove that Adding (31) and (32), we get
This implies that ; hence and .
Case 2. Suppose that is not comparable. Let be noncomparable to ; then there exists which is comparable to and .
Since , there exists such that and . Without loss of generality assume that (i.e., and .) Note that implies that . From Lemmas 2 and 3, we get From the above two inequalities, we obtain . Continuing this process, we get sequences and such that and with . By using that is a proximally coupled contraction, we get Similarly, we can prove that Adding (35) and (36), we obtain As , we get , so that and .
Analogously, one can prove that and . Therefore, and . Hence the proof is completed.
Example 11. Let be endowed with usual metric, and with the usual order in .
Suppose that and . Then and are nonempty closed subsets of and and . Also note that .
Now consider the function defined as Then it can be seen that is continuous and . Hence, . It is easy to see that other hypotheses of the Theorem 10 are also satisfied. Further, it is easy to see that is the unique element satisfying the conclusion of Theorem 10.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research Dr. K. Sitthithakerngkiet was funded by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-GEN-57-19). Moreover, the third author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU2557).
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