On Best Proximity Point Theorems without Ordering

Recently, Basha (2013) addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that if 𝐴 and 𝐵 are nonvoid subsets of a partially ordered set that is equipped with a metric and 𝑆 is a non-self-mapping from 𝐴 to 𝐵 , then the mapping 𝑆 has an optimal approximate solution, called a best proximity point of the mapping 𝑆 , to the operator equation 𝑆𝑥 = 𝑥 , when 𝑆 is a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and ordered proximal contraction on 𝑆 .


Introduction
Let be a non-self-mapping from to , where and are nonempty subsets of a metric space . Clearly, the set of fixed points of can be empty. In this case, one often attempts to find an element that is in some sense closest to ( ). Best approximation theory and best proximity point analysis are applicable for solving such problems. The well-known best approximation theorem, due to Fan [1], asserts that if is a nonempty, compact, and convex subset of a normed linear space and is a continuous function from to , then there exists a point in such that the distance of to ( ) is equal to the distance of ( ) to . Such a point is called a best approximation point of in . A point in is said to be a best proximity point for , if the distance of to ( ) is equal to the distance of to . The aim of best proximity point theory is to provide sufficient conditions that assure the existence of best proximity points. Investigation of several variants of contractions for the existence of a best proximity point can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In most of the papers on the best proximity, the ordering, proximal monotonicity, and ordered proximal contraction on the mapping play a key role. A natural question arises that it is possible that we can have other ways that may not require the ordering as well as proximal monotonicity and ordered proximal contraction on the mapping . Very recently, Basha [5] addressed a problem that amalgamates approximation and optimization in the setting of a partially ordered set that is endowed with a metric. He assumed that if and are nonvoid subsets of a partially ordered set that is equipped with a metric and is a non-self-mapping from to , then the mapping has an optimal approximate solution, called a best proximity point of the mapping , to the operator equation = , when is a continuous, proximally monotone, ordered proximal contraction. In this note, we are going to obtain his results by omitting ordering, proximal monotonicity, and ordered proximal contraction on .

Preliminary Results
Let be a nonempty set and let be a metric on . Unless otherwise specified, it is assumed throughout this paper that and are nonempty subsets of . We recollect the following notations and preliminary results: Note that (2) is equivalent to Let us consider  Proof. Suppose that is compact and is closed subset of the Euclidian space Note that for all ∈ N. This means that { } is bounded. It follows from the Bolzano-Weierstrass theorem and the closeness of that Let us consider By employing (5) and (7) and letting → ∞ in (8), we obtain ‖ − ‖ = ( , ). Hence 0 ̸ = ⌀ and 0 ̸ = ⌀. This completes the proof. (i) If is compact and is closed, then 0 is a closed subset of .
(ii) If is compact and is closed, then 0 is a closed subset of .
(iii) If both and are compact, then 0 and 0 are nonempty and closed.
The proof of (ii) is obvious from (i) and also the proof (iii) follows from Proposition 1 and (i) and (ii).
The next result extends Proposition 3.1 of [10] from normed linear spaces to metrizable topological vector spaces.

Proposition 4. Let be real topological vector space whose topology is induced by translation invariant metric with the property
where 0 denotes the zero vector of . Let and be two closed subsets of such that ( , ) > 0. Then where ( ) and ( ) are denoted by the boundary of and , respectively.
The following example shows that there are metrizable topological vector spaces with the properties cited in the previous proposition which are not normable.

Main Results
In this section, we provide an existence result for the best proximity point of the mapping on the metric space by omitting ordering, proximal monotonicity, and ordered proximal contraction on .
We begin with an example which shows that it is possible in the finite dimensional Euclidean space that the proximity points set for even a linear mapping (here projection) be empty.
It is clear that is continuous (since it is projection). It is not hard to verify that (i) both and are closed subset of ; (ii) ( , ) = 0; (iii) there is no * ∈ such that ( * , * ) = ( , ) (i.e., there is no best proximity point).
To achieve understanding in Example 6, let us see Figure 1.
This completes the proof.
The following result establishes an existence result in order to be nonempty best proximity point set for the mapping without assuming any ordering, proximal monotonicity, and ordered proximal contraction on the . It is worth noting that it is only an existence result without applying any iteration method (see Theorem 3.1 of [5]).
(29) If = , then = 0 = 0 = . Then, by Corollary 9, we obtain the following corollary which says that the fixed points set of the mapping is nonempty.
By using the lower semicontinuity of ( ∘ ( ⊗ )), we have that there exists * ∈ such that Proof. It is obvious that the continuity and surjectivity of and imply the lower semicontinuity of ∘ ( ⊗ ), where is the distance function of the metric space ( , ) and ( ⊗ )( ) = × , respectively. Applying Theorem 11, we have the desired result.