On the Rates of Asymptotic Regularity for Some Unbounded Trajectories

Let T be a nonexpansive self-mapping of C where C is a nonempty closed convex subset of a Banach space E. We define Tλ for 0<λ<1 by Tλ=λT+(1−λ) I, where I is the identity operator on C, and denote xn=Tnλx0 where x0 C. Then the related initial value problem is du/dt=−(I−T) u(t) with u(0)=x0 C. The facts that xn−Txn =O(1/ ) as n→∞ and u′(t) =O(1/ ) as t→∞ are known when C is bounded. In this paper we look for a rate of asymptotic regularity for u′(t) if u(t) =O(tα) where 0 α 1. We prove u′(t) =O(t−β) as t→∞, where α+2β=1 and obtain an estimate on u′(t) with the universal constant Cα depending only on α.