Kazuo Kido

Mean ergodic theorems for semigroups of linear operators

converge strongly as n + +a, to a fixed point of T for each x E E. On the other hand, in 1975, Baillon [ 1 ] proved the following nonlinear ergodic theorem: let C be a closed convex subset of a real Hilbert space H and T be a mapping of C into itself such that ]I TX Ty 11 < /Ix y 11 for x, y E C and {Tnx : n = 1,2,...} is bounded for each x E C. Then, the CesPro means S,x converge weakly as IZ -+ +co to a fixed point of T for each x E C. Recently, Rode [5] and Takahashi [7] tried to extend this nonlinear ergodic theorem to semigroups of mappings. Especially, Rode found a sequence of means on the semigroup, generalizing the Cesaro means on N = (1,2,...}, such that the corresponding sequence of mappings converges to a projection onto the set of common fixed points. In this paper, by using Rode’s method, we extend Yosida’s theorem to semigroups of linear operators in Banach spaces. The proofs employ the methods of Greenleaf [3], Rode [5], Takahashi [6], and Yosida [8].

Strong convergence of resolvents of monotone operators in Banach spaces

Let E* be a real strictly convex dual Banach space with a Frechet differentiable norm, and A a maximal monotone operator from E into E* such that A-10 :

Nonexpansive retractions and nonlinear ergodic theorems in banach spaces

0. Fix x E E. Then JAx converges strongly to Px as A -A oo, where JA is the resolvent of A, and P is the nearest point mapping from E onto A-10.