Corrigendum to the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings”

Theorem 1.1

(Theorem 2.1 of [1]) Let ( A , B ) be a nonempty, weakly compact and convex pair in a strictly convex Banach space X . Then every cyclic nonexpansive mapping defined on A ∪ B has a best proximity point if and only if every noncyclic nonexpansive mapping defined on A ∪ B has a best proximity pair.

Definition 1.2

Let ( A , B ) be a nonempty pair of subsets of a metric space ( X , d ) such that A 0 is nonempty. We say that the pair ( A , B ) has the diagonal property provided that

for any x 1 , x 2 ∈ A 0 and y 1 , y 2 ∈ B 0 .

Proposition 1.3

Let ( A , B ) be a nonempty, bounded, closed and convex pair in a reflexive and strictly convex Banach space X such that ( A , B ) has the diagonal property. Define P : A 0 ∪ B 0 → A 0 ∪ B 0 as

Then P is a relatively isometry, that is, ∥ P x − P y ∥ = ∥ x − y ∥ for all ( x , y ) ∈ A 0 × B 0 .

Proof

It follows from Proposition 1.8 of [1] that

Since ( A , B ) has the diagonal property, we obtain

that is, P is a relatively isometry on A 0 ∪ B 0 .□