Coupled best proximity points in ordered metric spaces
© Kumam et al.; licensee Springer. 2014
Received: 18 January 2014
Accepted: 31 March 2014
Published: 6 May 2014
In this paper, we prove the existence and uniqueness of a coupled best proximity point for mappings satisfying the proximally coupled contraction condition in a complete ordered metric space. Further, our result provides an extension of a result due to Luong and Thuan (Comput. Math. Appl. 62(11):4238-4248, 2011; Nonlinear Anal. 74:983-992, 2011).
MSC:41A65, 90C30, 47H10.
1 Introduction and preliminaries
Let A be a nonempty subset of a metric space . A mapping has a fixed point in A if the fixed point equation has at least one solution. That is, is a fixed point of T if . If the fixed point equation does not possess a solution, then for all . In such a situation, it is our aim to find an element such that is minimum in some sense. The best approximation theory and best proximity pair theorems are studied in this direction. Here we state the following well-known best approximation theorem due to Ky Fan .
Theorem 1.1 ()
Let A be a nonempty compact convex subset of a normed linear space X and be a continuous function. Then there exists such that .
Such an element in Theorem 1.1 is called a best approximant of T in A. Note that if is a best approximant, then need not be the optimum. Best proximity point theorems have been explored to find sufficient conditions so that the minimization problem has at least one solution. To have a concrete lower bound, let us consider two nonempty subsets A, B of a metric space X and a mapping . The natural question is whether one can find an element such that . Since , the optimal solution to the problem of minimizing the real valued function over the domain A of the mapping T will be the one for which the value is attained. A point is called a best proximity point of T if . Note that if , then the best proximity point is nothing but a fixed point of T. Also, best proximity point theory in ordered metric spaces was first studied in .
The existence and convergence of best proximity points is an interesting topic of optimization theory which recently attracted the attention of many authors [3–13]. Also one can find the existence of best proximity point in the setting of partially order metric space in [14–17].
On the other hand, Bhaskar and Lakshmikantham have introduced the concept called mixed monotone mapping and proved coupled fixed point theorems for mappings satisfying the mixed monotone property, which is used to investigate a large class of problems, and they discussed the existence and uniqueness of a solution for a periodic boundary value problem. One can find the existence of coupled fixed points in the setting of partially order metric space in [18–24].
Now we recall the definition of a coupled fixed point which was introduced by Sintunavarat and Kumam in . Let X be a nonempty set and be a given mapping. An element is called a coupled fixed point of the mapping F if and .
ϕ is continuous and nondecreasing,
if and only if ,
Again, let Ψ denote all functions which satisfy for all and .
The main theoretical results of Luong and Thuan, in  is the following.
Theorem 1.2 ()
F is continuous or
X has the following property:
if a nondecreasing sequence , then for all n,
if a nonincreasing sequence , then for all n,
then there exist such that and .
Motivated by the above theorems, we introduce the concept of the proximal mixed monotone property and of a proximally coupled weak contraction on A. We also explore the existence and uniqueness of coupled best proximity points in the setting of partially ordered metric spaces. Further, we attempt to give the generalization of Theorem 1.2.
In , the authors discussed sufficient conditions which guarantee the nonemptiness of and . Also, in , the authors proved that is contained in the boundary of A. Moreover, the authors proved that is contained in the boundary of A in the setting of normed linear spaces.
One can see that, if in the above definition, the notion of the proximal mixed monotone property reduces to that of the mixed monotone property.
From (4) and (5), one can conclude the . Hence the proof. □
Proof The proof is the same as Lemma 1.4. □
where and . It is interesting to note that if the pair considered in the above definition has the P-property, then the mapping F in Theorem 1.2 satisfies the inequality (1).
2 Coupled best proximity point theorems
F is a continuous proximally coupled weak contraction on A having the proximal mixed monotone property on A such that .
- (ii)There exist elements and in such that
Then there exists such that and .
Hence from Lemma 1.4 and Lemma 1.5, we obtain and .
a contradiction. This shows that and are Cauchy sequences. Since A is a closed subset of a complete metric space X, these sequences have limits. Thus, there exist such that and . Therefore in . Since F is continuous, we have and .
Hence the continuity of the metric function d implies that and . But from (9) and (10) we see that the sequences and are constant sequences with the value . Therefore, and . This completes the proof of the theorem. □
F is continuous having the proximal mixed monotone property and proximally coupled weak contraction on A.
There exist and in such that with and with .
Then there exists such that and .
Theorem 2.2 Assume the conditions (25), (26) and is closed in X instead of continuity of F in Theorem 2.1, then the conclusion of Theorem 2.1 holds.
Since and , by taking the limit on the above two inequalities, we get and . Hence, from (29) and (30), we get and . □
Corollary 2.2 Assume the conditions (25) and (26) instead of continuity of F in Corollary 2.1, then the conclusion of Corollary 2.1 holds.
Now, we present an example where it can be appreciated that the hypotheses in Theorem 2.1 and Theorem 2.2 do not guarantee uniqueness of the coupled best proximity point.
Example 2.3 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 a closed subset of X. Then , and . Let be defined as . Then, it can be seen that F is continuous such that . The only comparable pairs of points in A are for , hence the proximal mixed monotone property and the proximally coupled weak contraction on A are satisfied trivially.
It can be shown that the other hypotheses of the theorem are also satisfied. However, F has three coupled best proximity points, , , and .
It is known that this condition is equivalent to the following.
then F has a unique coupled best proximity point.
We distinguish two cases.
this implies that , and using the property of ψ, we get , hence and .
Case 2: Suppose is not comparable. Let be not comparable to , then there exists which is comparable to and .
so that and . Analogously, one can prove that and .
Therefore, and . Hence the proof. □
The following result, due to Theorem 2.4 in Luong and Thuan  follows by taking .
then F has a unique coupled fixed point.
The first author was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU2557). Moreover, Kanokwan Sitthithakerngkiet would like to thank the King Mongkut’s University of Technology North Bangkok for financial support.
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