Noncommutative Semigroups Strong Ergodic Theorems Of Weak Convergence Theorem And Right Reversible Semigroup

The ergodic theory for semitopological semigroups of nonlinear operators is now a focus of nonlinear analysis and it began to be studied in the middle of 1970's .In 1975,J.B.Bai.llon[1] introduced the first ergodic convergence theorem for nonexpansive nonlinear operators acting on closed and convex subset of Hilbert spaces.From then on,this theorem was extended to uniformly convex Banach space with Frechet differentiable norm by Bruck[10],Hirano([23],[18]),Lau-Takahashi[25] and Reich[16]. The analogous results for nonexpansive semigroups were given by Baillon[19], Baillon-Brezis[30],Reich[17]and Hirano-Kido-Takahashi[24]..In addition,by using Bruck's lemma in[7],this theorem was also extended to the almost-orbit of nonexpansive semigroups by Miyadera-Kobayasi[2] and to the almost-orbits of asymptotically nonexpansive mappings by Oka[15].On the other hand, Baillon[19] proved that {yx} is strongly almost convergent as n- to a fixed point of T if X is a Hilbert space and T is odd. Bruck[ll] obtained the same conclusion under the more general assumption that {T(t},t 0} is "asymptotically isometric".At the same time,Bruck's result has been extended by Miyadera-Kobayasi[2] to the case of uniformly convex Banach space. Recently, Kada-Takahashi[22] proved a strongly ergodic theorem for commutative semigroups of nonexpansive mappings.However,in chapter 1 of this paper,we first studied the strong ergodic convergence theorem for general semigroups of asymptotically nonexpansive type semigroups which are of type (r) in reflexive Banach space.In this theorem ,G is a general semigroup and there is no topological structure on it, so this theorem extended many previously known results.We next proved the strong ergodic convergence theorem for orbits of asymptotically nonexpansive type semigroups. Further,we discussed that if G is a right reversible semitological semigroup, the supposition that D has an invariant mean in thecondition of the strong ergodic convergence theorem can be weakened and it is enough to suppose that D has a left invariant mean.In 1982,a weak convergence theorem for nonexpensive semigroups in uniformly convex Banach space was first established by Miyadera and Kobaysi and it was generalized to that for commutative semigroup of asymptotically nonexpansive mappings by Oka[ 15].Feathers and Dotson[16] and Bose[1] gave the weak convergence theorem of asymptotically nonexpansive mappings in a uniformly convex Banach space with weak continuous duality mapping by using Opial's Lemma[17]. By using Bruck's lemma[10],Passty[31] extended the results of[1,16] to uniformly convex Banach space with a Frechet differentiable norm. However, there existed more or less limitations in their methods adopted. By using new techniques, Chapter2 of this paper discussed the weak convergence theorem for right reversible semigroup of asymptotically nonexpansive type semigroup and the corresponding theorem for its almost-orbit in the reflexive Banach space with a Frechet differentiable norm or Opial property. This theorem removed the key supposition that the almost-orbit of the semigroup is "almost asymptotically isometric " in Li Gang [20],and as applications, it embraced all the commutative cases.