Banach空间中的非Lipschitzian一般半群的非线性遍历定理(英文)

NONLINEAR ERGODIC THEOREMS FOR GENERAL SEMIGROUPS OF NON-LIPSCHITZIAN MAPPINGS IN BANACH SPACES

设G为半群,C为具FrEchet可微范数的一致凸Banach空间X的非空有界闭凸子集.(■)={T_t:t∈G}为C上到自身的渐近非扩张型半群,且F(■)非空.在本文中,我们证明了:对■的任一殆轨道u(·),■co{u(ts),t∈G}∩F(S)至多为单点集.进一步,对x∈C,∩_(s∈G)co{T_(ts)x,t∈G}∩F(■)非空当且仅当存在C到F(■)上非扩张压缩P,使得对任意t∈G,PT_t=T_tP=P,Px∈co{T_tx,t∈G}.这一结果不仅推广了许多已知结果,而且说明它们中的一些关键条件是不必要的.

Abstract Let G be a semigroup. Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space X with a Fr@chet differentiable norm and = {T_t：t∈G} be a semigroup of asymptotically nonexpansive type mappings on C with Fco{u（ts）,t∈G}. We prove in this paper that for every almost orbit u（-） of S, NsEc do co{u（ts）,t∈G}∩F（S） consists of at most one point. Fhrther, [S]sEG do{Ttsx, t C G} N F（S） is nonempty for each x C C if and only if there exists a nonexpansive retraction P of C onto F（~） such that PTt =TtP = P for all t E G and Px E Co{Ttx, t C G}. This result not only gengneralizes some well-known theorems, but also shows that some key conditions in them are not necessary.