Pythagorean Property and Best-Proximity Point Theorems

In this paper, a notion called proximally complete pair of subsets of a metric space is introduced, which weakens earlier notions in the theory of best-proximity points. By means of this notion, existence and convergence results of best-proximity points are proven for cyclic contraction mappings, which extent other recent results. By observing geometrical properties of Hilbert spaces, the so-called Pythagorean property is introduced. This property is employed to provide sufficient conditions for a cyclic map to be a cyclic contraction.