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In this study, at first we prove that the existence of best proximity points forcyclic nonexpansive mappings is equivalent to the existence of best proximity pairsfor noncyclic nonexpansive mappings in the setting of strictly convex Banach spacesby using the projection operator. In this way, we conclude that the main result ofthe paper [ Proximal normal structure and nonexpansive mappings ,Studia Math.  171  (2005), 283–293] immediately follows. We thendiscuss the convergence of best proximity pairs for noncyclic contractions byapplying the convergence of iterative sequences for cyclic contractions and show thatthe convergence method of a recent paper [ Convergence of Picard's iterationusing projection algorithm for noncyclic contractions , Indag. Math.30  (2019), no. 1, 227–239] is obtained exactly fromPicard’s iteration sequence.

Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings