A note on some best proximity point theorems proved under P-property

In this article we show that some recent results on the existence of best proximity points can be obtained from the same result in fixed point theory.


Introduction
Let A and B be two nonempty subsets of a metric space (X, d). In this paper, we adopt the following notations and definitions. The notion of best proximity point is defined as follows. Similarly, for a multivalued non-self mapping T : is a nonempty pair of subsets of a metric space (X, d), a point x * ∈ A is a best proximity point of T provided that D(x * , T x * ) = dist(A, B).
Recently, the notion of P-property was introduced in [9] as follows.
where x 1 , x 2 ∈ A 0 and y 1 , y 2 ∈ B 0 . By using this notion, some best proximity point results were proved for various classes of non-self mappings. Here, we state some of them.  Banach space X such that A is compact and A 0 is nonempty. Let T : A → B be a nonexpansive mapping, that is Assume that the pair (A, B) has the P-property and T (A 0 ) ⊆ B 0 . Then T has a best proximity point.
) be a pair of nonempty closed subsets of a complete metric space X such that A 0 is nonempty. Let T : A → B be a Meir-Keeler non-self mapping, that is, for all x, y ∈ A and for any ε > 0, Assume that the pair (A, B) has the P-property and T (A 0 ) ⊆ B 0 . Then T has a unique best proximity point. for all x ∈ A, and T x 0 is included in B 0 for each x 0 ∈ A 0 , then T has a best proximity point in A.

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In this section, we show that the existence of a best proximity point in the main theorems of [1,2,8,9], can be obtained from the existence of the fixed point for a self-map. We begin our argument with the following lemmas. Assume that there exists another pointý ∈ B 0 such that By the fact that (A, B) has the P-property, we conclude that y =ý. Consider the non-self mapping g : A 0 → B 0 such that d(x, gx) = dist (A, B). Clearly, g is well defined. Moreover, g is an isometry. Indeed, if x 1 , x 2 ∈ A 0 then Again, since (A, B) has the P-property, that is, g is an isometry.
Here, we prove that the existence and uniqueness of the best proximity point in Theorem 1.3 is a sample result of the existence of fixed point for a weakly contractive self-mapping.
for all x, y ∈ A 0 which implies that the self-mapping g −1 T is weakly contractive. Note that A 0 is closed by Lemma 2.1. Thus, g −1 T has a unique fixed point ( [7]). Suppose that x * ∈ A 0 is a unique fixed point of the self-mapping g −1 T , that is, g −1 T (x * ) = x * . So, T x * = gx * and then d(x * , T x * ) = d(x * , gx * ) = dist(A, B), 6 from which it follows that x * ∈ A 0 is a unique best proximity point of the non-self weakly contractive mapping T .
Remark 2.1. By a similar argument, using the fact that every nonexpansive self-mapping defined on a nonempty compact and convex subset of a Banach space has a fixed point, we conclude Theorem 1.4. Also, existence and uniqueness of the best proximity point for Meir-Keeler non-self mapping T follows from the Meir-Keeler's fixed point theorem ( [5]). Finally, in Theorem 1.5, the Nadler's fixed point theorem ( [6]), ensures the existence of a best proximity point for multivalued non-self T .