非交换半群上的强遍历收敛定理及右可逆半群上的弱收敛定理
摘要
非线性算子半群的遍历理论是非线性分析方面的热点话题，它的研究开始于上世纪七十年代中期。1975年，J．B．Baillon[1]首先在Hilbert空间的凸闭集上提出了非扩张映照的遍历收敛定理。之后，此定理被Bruck[10]，Hirano([23]，[18])，Lau-Takahashi[25]和Reich[16]推广到具F-可微范数的一致凸Banach空间，又由Baillon[19]，Baillon-Brezis[30]，Reich[17]和Hirano-Kido-Takahashi[24]给出了非扩张半群的类似结果。除此以外，通过使用Bruck引理[7]，Miyadera-Kobayasi[2]和Oka[15]分别把此定理推广到非扩张半群和渐近非扩张映照的殆轨道。另一方面，Baillon[19]证明了如果X是一个Hilbert空间且T是奇映照，则当n→∞时，{T~n X}殆强收敛于F(T)中一点；Bruck[11]在{T(t)；t≥0}是渐近等距这一更一般性的假设下得到了相同的结论。同时，Bruck的结果又被Miyadera-Kobayasi[2]推广到一致凸Banach空间的情形。目前，Kada-Takahashi[22]就交换半群上的非扩张映照证明了一个强遍历定理。而本文第一章首先研究了自反Banach空间中，一般半群上的(Γ)类渐近非扩张型半群的强遍历收敛定理，即：设C是自反Banach空间X的有界凸闭子集，G是有单位元的一般半群，S={T(t)；t∈G)是C上(Γ)类渐近非扩张型半群，D是m(G)的含常值函数且关于左、右平移不变的子空间，对任意x∈C及x~*∈X~*，函数t→含于D中，{μ_α；α∈A}是D上强正则网，假设S的轨道T(·)x是渐近等距的且D有不变平均，则有关于h∈G一致成立，这里μ为D的任一个不变平均。在此定理的证明中G为一般的代数半群，其上没有涉及到任何的拓扑结构，使以前的一些结果得到了推广；本章紧接着又证明了一般半群上的(Γ)类渐近非扩张型半群的殆轨道的强遍历收敛定理；最后，进一步讨论了当G为右可逆半群时，上述定理条件中D上有不变平均的假设可以减弱为D上有一个左不变平均，此时定理如下：设X是自反Banach空间，C是X的非空有界凸闭子集，D是m(G)的
扬州大学硕士学位论文
含常值函数且关于左、右平移不变的子空间。s=仕(t)，。G}是右可逆半群上的犷)
类渐近非扩张型半群，S的殆轨道。(.)满足对任意x’ oX’，函数‘*(。摊:’)含于
D中，设加。;a oA}是D上强正则网，且D有左不变平均，。0是渐近等距的，则
介(nt知·(t)关于”6G一致强收敛于二而扣(t)t全:}门F(s)中某点:，进一步，有
息而位摊‘七‘}nF(s)为单点集树.
也正是由Miyadera和Kobayasi于1 982年首次在一致凸Banach空中给出了非
扩张半群的弱收敛定理，随后，由Okaf15]把此弱收敛定理推广到交换半群的渐近
非扩张映照。Feattier and Dotson【16]和Bose[l]通过使用Opial引理【17}在具弱连续
对偶映照的一致凸B~h空间中证明了渐近非扩张映照的弱收敛定理，Passty【31】
通过使用Bruck引理【10]把〔1，16]的结果推广到具Freehet可微范数的一致凸Banach
空间，然而，他们的证明存在着种种局限性。本文第二章就针对这些局限性，通
过采用新的证明方法，在具Freehet可微范数或满足Opial条件的自反Banach空间
中证明了右可逆拓扑半群上的仍类渐近非扩张型半群及其殆轨道的弱收敛定理，
即:设X是实自反Banach空间，C是X的非空有界凸闭子集，S=价令工，。G}是
c上的犷)类渐近非扩张型半群，G是右可逆半群，u(.)是s的殆轨道且满足
”一瞥(u(nt)一u(t))二0饰任G，则叽(u)〔F(s)，特别地，如果X具‘性质冈或满
足opial条件，则有‘一鲁u(t)=，任F(s)。在此关于半群的弱收敛定理从略。
上述定理去掉了Li Gang[20]文章弱收敛定理中u(.)是殆渐近等距的这一在其证明
中起着关键性作用的假设，并且作为此定理的应用，它涵盖了所有交换半群的情
况。
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 sp
ace 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 the
condition 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 al
most-orbit of the semigroup is "almost asymptotically isometric " in Li Gang [20],and as applications, it embraced all the commutative cases.
扬州大学硕士学位论文
含常值函数且关于左、右平移不变的子空间。s=仕(t)，。G}是右可逆半群上的犷)
类渐近非扩张型半群，S的殆轨道。(.)满足对任意x’ oX’，函数‘*(。摊:’)含于
D中，设加。;a oA}是D上强正则网，且D有左不变平均，。0是渐近等距的，则
介(nt知·(t)关于”6G一致强收敛于二而扣(t)t全:}门F(s)中某点:，进一步，有
息而位摊‘七‘}nF(s)为单点集树.
也正是由Miyadera和Kobayasi于1 982年首次在一致凸Banach空中给出了非
扩张半群的弱收敛定理，随后，由Okaf15]把此弱收敛定理推广到交换半群的渐近
非扩张映照。Feattier and Dotson【16]和Bose[l]通过使用Opial引理【17}在具弱连续
对偶映照的一致凸B~h空间中证明了渐近非扩张映照的弱收敛定理，Passty【31】
通过使用Bruck引理【10]把〔1，16]的结果推广到具Freehet可微范数的一致凸Banach
空间，然而，他们的证明存在着种种局限性。本文第二章就针对这些局限性，通
过采用新的证明方法，在具Freehet可微范数或满足Opial条件的自反Banach空间
中证明了右可逆拓扑半群上的仍类渐近非扩张型半群及其殆轨道的弱收敛定理，
即:设X是实自反Banach空间，C是X的非空有界凸闭子集，S=价令工，。G}是
c上的犷)类渐近非扩张型半群，G是右可逆半群，u(.)是s的殆轨道且满足
”一瞥(u(nt)一u(t))二0饰任G，则叽(u)〔F(s)，特别地，如果X具‘性质冈或满
足opial条件，则有‘一鲁u(t)=，任F(s)。在此关于半群的弱收敛定理从略。
上述定理去掉了Li Gang[20]文章弱收敛定理中u(.)是殆渐近等距的这一在其证明
中起着关键性作用的假设，并且作为此定理的应用，它涵盖了所有交换半群的情
况。
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 sp
ace 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 the
condition 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 al
most-orbit of the semigroup is "almost asymptotically isometric " in Li Gang [20],and as applications, it embraced all the commutative cases.
引文
[1] Baillon,J.B.Un theorem de type ergodic pour les contractions non lineares darts un espace de Hibert ,C.K.A. cad.Sci.paris 1975 280:1511—1514
[2] Miyadera J. and Kobayasi K. On the asymptotic behavior of almost-orbits of
nonlinear contraction semigroups in Banach space . Nonlinear Analysis1982,6: 349-365
[3] Oka H. On the strong ergodic theorems for commutative semigroups in Banach spaces. TOKOYO,math. 1993,2:385—398
[4] Li Gang Weak convergence and nonlinear ergodic theorem for reversible semigroups of non-lipschitzian mappings. J.Math.Anal&Appl, 1997 Vol 206,No. 13—14:451—463
[5] Li Gang Nonlinear ergodic heorems for reversible smigroups of Lipschitzian mapping in Banach spaces . Dynamic System and Applications 2000, 9:255—268
[6] Day M.M. Amenable semigroups. Illions J.Math. 1957,1:509—544
[7] R.E.Bruck On the convex approximation property and the asymptotically behavior of nonlinear contractions in Banach Space. Isreal J.Math. 1981,34:304—314
[8] Li Gang and J.K.Kim Recent results on nonlinear ergodic theorems and asymptotic behavior for nonlinear operators. Nonlinear Funct. And Appl. 1998 Vol.3 :221—240
[9] Li Gang Strong ergodic theorems for asymptotically nonexpansive type semigroups in Banach spaces. Journal of Nanjing University Mathematical Bquarerly 1997 ,Vol. 14, No. 1:187—193
[10] Bruck,R.E.,A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isreal J.Math.,1979,32:107—116
[11] Bruck,R.E.,On the almost convergence of iterates of a nonexpansive mappings in a Hilbert space and the structure of the weak w-limit set ,Isreal J.Math., 1978,29:1—17
[12] Takahashi,W.,Fixde point theorem and nonlinear ergodic theorems for nonexpansive semigroups without convicity, Can.J.Math. 1992,44:880—887
[13] Tan K K,Xu H K.,An ergodic theorem for nonlinear semigroups of Lipschitzian appings in Banach space Nonlinear Analy., 1992,9:805—813
[14] Li Gang and Ma Jipu, Ergodic theorem for non-Lipschitzian semigroups without convexity, Chin.Ann, of Math., 1998,19B,2:209—216
[15] H.Oka,Nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappings,Nonlinear Anal. 18(1992) 619—635
[16] Reich S.Weak convergence theorems for nonexpansive mapping in Banach Space.J Math Analy Appl,1979,67:274—276
[17] Reich S.A note on the mean ergodic theorem for nonlinear semigroups J Math Analy Appl, 1983,91:547—551
[18] N:Hirano,Nonlinear ergodic theorems and weak convergence theorems, J.Math.Soc. Japan, 34(1982),35—46
[19] Baillon,J.B.,Quelques properietes de convergence asymptotique pour les semigroups de contractions impaires,C.R. Acad.Sci .Pairs 283(1976),57—58
[20] Jong Kyu Kim and Li Gang Nonlinear ergodic theorems for reversible of Lipschizian mappings in Banach spaces ,Dynamic Systems and
Applications ,2000,9:255—268
[21] Li.Ma, Nonlinear ergodic theorems for commutative semigroups in Banach spaces ,Acta Math. Sinic, 1997,40:191—201
[22] Kada, O.and Takahashi,W.,Strong convergence and nonlinear ergodic theorems for commutative semigroups of nonexpansive mappings, Nonlinear Analysis TMA,28(1997),495—511
[23] N.Hirano,A proof of the mean ergodic theorems for nonexpansive mapping in Banach spaces,proc. Amer. Math. Soc.,78(1980),361—365
[24] N.Hirano,K.Kido&W.Takahashi,Nonexpansive retractions and nonlinear ergodic theorems of nonexpansive mappings in Banach spaces,Nonlinear Analysis TMA., 12(1988), 1269—1281
[25] A.T.Lau,W. Takahashi, weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mapping, Pacific J.Math. 12(1987)277—294
[26] G.Li,J.P.Ma, Nonlinear ergodic theorems for commutative semigroups of non-lipschitzian mappings in Banach spaces, Acta Math.Sinica39(1996)803—809
[27] G.Li,Weak convergence and nonlinear ergodic theorems for reversible topological semigroups of non-lipschitzian mappings, J.Math.Anal.Appl.206(1997) 1411—1428
[28] P.K.Lin,H.K.Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal.24(1995)929—946
[29] P.K.Lin,Asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal.26(1996)1137—1141
[30] J.B.Baillon and H.Brezis, Une remarque sur le comportement asymptotique des semigroups nonlineaires, Houston J.Math.,2(1976),5—7
[31] Reish S.Almost convergence and nonlinear ergodic theorems J App Theory, 1978,24:269—272
[2] Miyadera J. and Kobayasi K. On the asymptotic behavior of almost-orbits of
nonlinear contraction semigroups in Banach space . Nonlinear Analysis1982,6: 349-365
[3] Oka H. On the strong ergodic theorems for commutative semigroups in Banach spaces. TOKOYO,math. 1993,2:385—398
[4] Li Gang Weak convergence and nonlinear ergodic theorem for reversible semigroups of non-lipschitzian mappings. J.Math.Anal&Appl, 1997 Vol 206,No. 13—14:451—463
[5] Li Gang Nonlinear ergodic heorems for reversible smigroups of Lipschitzian mapping in Banach spaces . Dynamic System and Applications 2000, 9:255—268
[6] Day M.M. Amenable semigroups. Illions J.Math. 1957,1:509—544
[7] R.E.Bruck On the convex approximation property and the asymptotically behavior of nonlinear contractions in Banach Space. Isreal J.Math. 1981,34:304—314
[8] Li Gang and J.K.Kim Recent results on nonlinear ergodic theorems and asymptotic behavior for nonlinear operators. Nonlinear Funct. And Appl. 1998 Vol.3 :221—240
[9] Li Gang Strong ergodic theorems for asymptotically nonexpansive type semigroups in Banach spaces. Journal of Nanjing University Mathematical Bquarerly 1997 ,Vol. 14, No. 1:187—193
[10] Bruck,R.E.,A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces. Isreal J.Math.,1979,32:107—116
[11] Bruck,R.E.,On the almost convergence of iterates of a nonexpansive mappings in a Hilbert space and the structure of the weak w-limit set ,Isreal J.Math., 1978,29:1—17
[12] Takahashi,W.,Fixde point theorem and nonlinear ergodic theorems for nonexpansive semigroups without convicity, Can.J.Math. 1992,44:880—887
[13] Tan K K,Xu H K.,An ergodic theorem for nonlinear semigroups of Lipschitzian appings in Banach space Nonlinear Analy., 1992,9:805—813
[14] Li Gang and Ma Jipu, Ergodic theorem for non-Lipschitzian semigroups without convexity, Chin.Ann, of Math., 1998,19B,2:209—216
[15] H.Oka,Nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappings,Nonlinear Anal. 18(1992) 619—635
[16] Reich S.Weak convergence theorems for nonexpansive mapping in Banach Space.J Math Analy Appl,1979,67:274—276
[17] Reich S.A note on the mean ergodic theorem for nonlinear semigroups J Math Analy Appl, 1983,91:547—551
[18] N:Hirano,Nonlinear ergodic theorems and weak convergence theorems, J.Math.Soc. Japan, 34(1982),35—46
[19] Baillon,J.B.,Quelques properietes de convergence asymptotique pour les semigroups de contractions impaires,C.R. Acad.Sci .Pairs 283(1976),57—58
[20] Jong Kyu Kim and Li Gang Nonlinear ergodic theorems for reversible of Lipschizian mappings in Banach spaces ,Dynamic Systems and
Applications ,2000,9:255—268
[21] Li.Ma, Nonlinear ergodic theorems for commutative semigroups in Banach spaces ,Acta Math. Sinic, 1997,40:191—201
[22] Kada, O.and Takahashi,W.,Strong convergence and nonlinear ergodic theorems for commutative semigroups of nonexpansive mappings, Nonlinear Analysis TMA,28(1997),495—511
[23] N.Hirano,A proof of the mean ergodic theorems for nonexpansive mapping in Banach spaces,proc. Amer. Math. Soc.,78(1980),361—365
[24] N.Hirano,K.Kido&W.Takahashi,Nonexpansive retractions and nonlinear ergodic theorems of nonexpansive mappings in Banach spaces,Nonlinear Analysis TMA., 12(1988), 1269—1281
[25] A.T.Lau,W. Takahashi, weak convergence and nonlinear ergodic theorems for reversible semigroups of nonexpansive mapping, Pacific J.Math. 12(1987)277—294
[26] G.Li,J.P.Ma, Nonlinear ergodic theorems for commutative semigroups of non-lipschitzian mappings in Banach spaces, Acta Math.Sinica39(1996)803—809
[27] G.Li,Weak convergence and nonlinear ergodic theorems for reversible topological semigroups of non-lipschitzian mappings, J.Math.Anal.Appl.206(1997) 1411—1428
[28] P.K.Lin,H.K.Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal.24(1995)929—946
[29] P.K.Lin,Asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal.26(1996)1137—1141
[30] J.B.Baillon and H.Brezis, Une remarque sur le comportement asymptotique des semigroups nonlineaires, Houston J.Math.,2(1976),5—7
[31] Reish S.Almost convergence and nonlinear ergodic theorems J App Theory, 1978,24:269—272