Banach空间非线性算子半群的遍历收敛定理及非交换半群上的弱遍历理论

【作者】 张剑梅；

【导师】 李刚； 庄亚栋；

【作者基本信息】 扬州大学 ， 基础数学， 2003， 硕士

【摘要】 非线性算子理论是非线性理论中的热门话题，它的研究始于上世纪七十年代中期，由于它被广泛的应用于微分方程的数值解、正解的存在性理论、控制论以及最优化等问题中，因而得到了很大的发展。 Miyadera和Kobayasi[15]在非扩张半群上引入了殆轨道的概念。本文第一章在一般半群中引入了广义殆轨道的概念，它包含了半群的殆轨道，并证明了渐近非扩张型半群的广义殆轨道的遍历收敛与渐近非扩张型半群的遍历收敛本质上是等价的。即：定理2．1 X是Banach空间，C是X的非空有界凸闭子集，G是含单位元的一般半群，S={T(t)；t∈G}是C上的渐近非扩张型半群，{μ_α；α∈A}是D上的强正则网，则下列命题等价：(a)对任意的x∈C，存在p_x∈F(S)，使得关于h∈G一致成立。(b)对S任意的广义殆轨道u(·)，有关于h∈G一致成立。定理2．2设X，C，G，S={T(t)；t∈G)，{μ_α；α∈A}同定理2．1，则下列命题等价：(a)对任意的x∈C，存在p_x∈F(S)，使得关于h∈G一致成立。(b)对S任意的广义殆轨道u(·)，有关于h∈G一致成立。上面的定理说明了遍历收敛定理从半群到殆轨道的推广在很多情况下是非本质的。 1975年，J．B．Baillon[1]首先在Hilbert空间的非空凸闭子集上给出了非扩张映照的弱遍历收敛定理。Baillon的定理引起了很多数学家的兴趣，Reich[2]在Hilbert空间中证明了非扩张半群的遍历收敛定理。Takahashi和Zhang[3]，Tan和Xu[4]分别将Baillon的定理推广到渐近非扩张半群及渐近非扩张型半群。近年来，Bruck[5]，Reich[6]，Oka[7]等在具Frechet可微范数的一致凸Banach空间中给出了非扩张及渐近非扩张映射及半群的遍历收敛定理。Li和Ma[13]在具Frechet可微范数的自反Banach空间中给出了一般交换渐近非扩张型拓扑半群的遍历收敛定理，这是一个重大突破。本文第二章用一种新的证明方法在自反Banach空间中，研究了 扬州大学硕士学位论文2一般半群上的(r)类渐近非扩张型半群的弱遍历收敛定理，即:定理3.1设x是具性质(F)的实自反Banach空间，C是X的非空有界闭凸子集，G为含单位元的一般半群，s=仕〔工，。G}是c上犷)类渐近非扩张型半群，D是m(G)的含常值函数的不变子空间，则对D上的任意一族渐近不变平均切。;。。A}，有夕〔卿。(l).已分p oF(s).由本文第一章中的定理2.1易得一般半群上的(r)类渐近非扩张型半群的殆轨道的弱遍历收敛定理.接着我们又利用这种证明方法，给出了右可逆拓扑半群的弱遍历收敛定理，即:定理4.1设X是具性质(F)的实自反Banach空间，C是X的非空有界闭、凸子集，G为右可逆拓扑半群，s二{T(t工，。G}是c上(r)类渐近非扩张型半群.D是m(G)的含常值函数的不变子空间，设D有左不变平均，则对D上的任意强正则网加。;aoA}，有w一lim 口〔A介伽卜咖。(t)二;。F(s殊于”。A阎一致成立再由第一章中的定理2.1得到了半群的殆轨道的弱遍历收敛定理，完全避免了殆渐近等距这一在以往证明中必不可少的假设.它涵盖了所有交换半群的情形.Baillon侈〕，Hiran。和丁hkahashi[91给出了Hilbert空间中非扩张半群的遍历压缩定理.近来Mizoguchi和几kahashi【10〕证明了LIPschitZian半群的遍历压缩定理.Hirano，Kido和丁砍ahashi【川，Hiran。【12]等在具Frechet可微范数的一致凸Banach空间中给出了非扩张映射的遍历压缩定理.1997年，Li和Ma〔161在Hilbert空间中成功地去掉了凸、闭等条件，在一般半群上得到了遍历压缩定理，极大地推广了遍历定理的应用范围.本文第二章在一般半群给出了渐近非扩张型半群的遍历压缩定理:即，定理3.2:设X，C，qs二仕(t工，。G}同定理3.1，那么下列等价:(a)二而份(ts)x;‘任G拍F(s)‘中“任C.(b)存在唯一的非扩张压缩尸:c*F(s)使得pT(t)二T仓护二p，竹。G且Px。而伊(t卜;，。G工vxoc.并给出了半群的殆轨道的遍历压缩定理.以及G为右可逆拓扑半群的殆轨道的遍历压缩定理.更多还原

【Abstract】 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.更多还原