On Nonexpansive Mappings1

holds for all p, qEX (for all p, q with d(p, q) <e). Mappings, as above, satisfying (1) with the strict inequality sign for all p, qEX, pT^q (for all p, q with 0 <d(p, q) <e) are called contractive (e-contractive). A point yE Y EX is said to belong to the/-closure of Y, yE Y1, if /(F) C F and there is a point r¡E Y and a sequence {re,} of positive integers, («i<re2< • • ■ <Ui< ■ • ■ ), so that lim/n,'(7j) =y. 1.2. In [l] we proved that a point of Xs is fixed if / is contractive, periodic if / is e-contractive. The corresponding statements for nonexpansive and e-nonexpansive mappings do not hold unless some additional assumption is made either on X or on /. We prove (Theorems 1,1') that a weaker conclusion involving suitable generalizations of fixed and periodic points is still true. To this end we introduce the notions of isometric and e-isometric sequences. 1.3. A sequence {x,} EX is said to be an isometric (e-isometric) sequence if the condition