J. Fixed Point Theory Appl. (2018) 20:149 https://doi.org/10.1007/s11784-018-0624-4 c  Springer Nature Switzerland AG 2018 Journal of Fixed Point Theory and Applications A note on best proximity point theory using proximal contractions Moosa Gabeleh and Calogero Vetro Abstract. In this paper, a reduction technique is used to show that some recent results on the existence of best proximity points for various classes of proximal contractions can be concluded from the corresponding re- sults in fixed point theory. Mathematics Subject Classification. 47H09, 46B20, 90C48. Keywords. Best proximity point, fixed point, proximal contraction. 1. Introduction and preliminaries Let (A, B) be a nonempty pair of subsets of a metric space (X, d). In the current paper, we use the following notations and definitions: dist(A, B) := inf {d(x, y):(x, y) ∈ A × B}, A 0 := {x ∈ A : d(x, y) = dist(A, B), for some y ∈ B}, B 0 := {y ∈ B : d(x, y) = dist(A, B), for some x ∈ A}. For a self-mapping T : A → A, we set O(x 0 ) := {T n x 0 : n =0, 1, 2,...}. We say that A is T -orbitally complete if and only if every Cauchy sequence, which is contained in O(x) for some x ∈ A, converges in A (see [1]). Definition 1.1. Let (A, B) be a nonempty pair of subsets of a metric space (X, d) and T : A → B be a non-self-mapping. A point x * ∈ A is called a best proximity point of T provided that d(x * ,Tx * ) = dist(A, B). The concept of best proximity point is a natural generalization of the concept of fixed point, which means a point with zero distance from its image via mapping. Clearly, the interest for best proximity points is real when the mapping under investigation is fixed-point free. There is a wide literature concerning these issues, the key aspects of which are classical subjects in 0123456789().: V,-vol