Non-convex proximal pairs on hilbert spaces and best proximity points

A sufficient condition is given for a non-convex proximal pair to be a proximal parallel pair on Hilbert spaces. Let (A, B) be a nonempty weakly compact non-convex proximal parallel pair in a Hilbert space X over the real field and T : A ∪ B → X be a, relatively nonexpansive map. We prove that there exists x ε A ∪ B such that

x - Tx

= dist(A,B) whenever A ∪ B is a cyclic T-regular set. We also establish that there exists (x, y) ε A × B such that Tx = x, Ty = y and

x - y

= dist(A, B), if A ∪ B is a T-regular set, T(A) ⊆ A and T(B) ⊆ B. In the above cases, we prove that the Kransnoe'skiǐ's iteration process yields a convergence result under suitable assumption. © 2014.