Nonlinear ergodic theorems

LEMMA 1. Let {xk} and {yk} be two sequences in H, F a nonempty subset of H, Cm the convex closure of{Jj>m {x-}. Suppose that (a) ForeachfinF, \xf ƒ i 2 + p ( ƒ) < + ~; (b) dist(.yA:, Cm) —> 0 as k —> °° for each m\ (c) Any weak limit of an infinite subsequence of {yk} lies in F. Then yk converges weakly to a point of F. PROOF OF LEMMA 1. Since {yk} is bounded, it suffices to show that if ƒ and g in F are weak limits of infinite subsequences of {yk }, then ƒ = g. For each ƒ,