On the rates of asymptotic regularity for some unbounded trajectories

Abstract

Let T be a nonexpansive self-mapping of C where C is a nonempty closed convex subset of a Banach space E. We define T λ for 0<λ<1 by T λ=λT+(1-λ)I, where I is the identity operator on C, and denote x n=T n λx 0 where x 0∈C. Then the related initial value problem is du/dt=-(I-T)u(t) with u(0)=x 0∈C. The facts that

x n-Tx n

=O(1/n) as n→∞ and

u′(t)

=O(1/t) as t→∞ are known when C is bounded. In this paper we look for a rate of asymptotic regularity for

if

u(t)

=O(t α) where 0≤α≤1. We prove

=O(t -β) as t→∞, where α+2β=1 and obtain an estimate on

with the universal constant C α depending only on α. © 1999 Academic Press.