On the rates of asymptotic regularity for some unbounded trajectories

[摘要] Let T be a nonexpansive self-mapping of C where Cis a nonempty closed convex subset of a Banach space E. We define T-lambda for 0<1 by T-lambda = lambda T+(1-lambda) I, where I is the identity operator on C, and denote x(n) = T(lambda)(n)x(0), where x(0) is an element of C. Then the related initial value problem is du/dt = -(I - T) u(t) with u(0) = x(0) is an element of C. The facts that \\x(n) - Tx(n)\\ = O(1/root n) as n --> infinity and \\u'(t)\\ = O(1/root t) as t --> infinity are known when C is bounded. In this paper we look for a rate of asymptotic regularity for \\u'(t)\\ if \\u(t)\\ = O(t(alpha)) where 0 less than or equal to alpha less than or equal to 1. We prove \\u'(t)\\ = O(t(-beta)) as t --> infinity, where alpha + 2 beta = 1 and obtain an estimate on \\u'(t)\\ with the universal constant C-alpha depending only on cc. (C) 1999 Academic Press.

[发布日期] 1999-12-10