In Banach Spaces Nonlinear Operator Subsemigroups Ergodic Theorem And The Non-commutative Semigroup Of Weak Ergodic Theory

Nonlinear operator theorem is now a focus in nonlinear theorem.The study of the ergodic theory for semitopologocal semigroups of nonlinear operators began in the middle of 1970's.It got great development because it was widely used in many problems,such as the numerical solution of differentiable equation,the existence theory of positive solution,contral theory and optimization.Miyadera and Kobayasi[15] introduced the concept of almost orbit Chapter 1 of this paper,we give the notion of general almost orbit.It extends the defenition of almost orbit.And we prove two equivalence propositions between orbits and general almost orbits.In 1975,J.B.Baillon[1] introduced the first ergodic convergence theorem for nonexpansive nonlinear operators acting on closed and convex subset of Hilbert spaces.From then on,mathmatics from all over the world have great interests in this" theory. Reich[2] proved the ergodic theorems to nonexpansive semigroups in Hilbert spaces.Takahashi and Zhang[3],Tan and Xu[4] extended Baillon's theorem to asymptotically nonexpansive and asymptotically nonexpansive type semigroups in Hilbert spaces.Recently,Reich[6],Bruck[5],Oka[7] gave the ergodic convergence theorems for nonexpansive,asymptotically nonexpansive mappings and semigroups in uniformly convex Banach spaces with Frechet differentiable norm.Li and Ma[13] obtained the ergodic convergence theorems for general commutative asymptotically nonexpansive type topological semigroups in reflexive Banach space,which is a great breakthrough. Chapter 2 of this paper,by using a new method of proof,we obtain the weak ergodic convergence theorem for general semigroups of asymptotically nonexpansive type semigroups in reflexive Banach space.By theorem 2.1 of chapter 1 we get the weak ergodic convergence theorem of almost orbit for general semigroups of asymptotically nonexpansive type semigroups in reflexive Banach space .By this method of proof ,we give the weak ergodic convergence theorems for right reversible semigroups.By theorem 2.1 of chapter l,we generalize the result to almost orbit case.So we can remove a key supposition that almost orbit is almost asymptotically isometric.It includes all commutative semigroups cases.Baillon[8],Hirano and Takahashi[9] gave nonlinear retraction theorems for nonexpansive semigroups.Recently Mizoguchi and Takahashi[10] proved a nonlinear ergodic retraction theorem for Lipschitzian semigroups.Hirano and Kido and Takahashi[11],Hirano[12] gave nonlinear retraction theorems for nonexpansive mappings in uniformly convex Banach spaces with Frechet differentiable norm..In 1997,Li and Ma[16] proved the ergodic retraction theorem for general semitopological semigroups in Hilbert space without the conditions that the domain is closed and convex,which greatly extended the fields of applications of ergodic theory.Chapter 2 of this paper,we obtain the ergodic retraction theorem for general semigroups and almost orbits of asymptotically nonexpansive type semigroups in reflexive Banach spaces.And we give the ergodic retraction theorem for almost orbits of right reversible semitopological semigroups.