On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak ω-limit set

LetT be a nonexpansive self-mapping of a closed convex subsetC of a real Hilbert space. In this paper we deal with the structure of the weak ω-limit set of iterates {T ⁿx}, establish conditions under which it is invariant underT, and show that {T ⁿx} converges weakly iffT has a fixed-point andT ⁿx-Tⁿ⁺¹x→0 weakly. Dedicated to the memory of my father Supported by NSF Grant MCS 76-08217.