不动点理论及Mazur-Ulam等距定理的一些探讨
|
摘要
硕士学位论文《不动点理论及Mazur-Ulam等距定理的一些探讨》综合运用Banach空间几何理论和算子方面的知识。全文共分如下三个章节:
第一章为绪论。主要介绍本文的研究背景及相关的一些预备知识,并且给出文中所涉及的大部分概念和记号。
第二章中,研究了赋β-范空间中渐近伪压缩和渐近非扩张映象的不动点迭代逼近问题,证明了渐近伪压缩映象T的修改的Ishikawa迭代序列收敛到不动点的充要条件。
第三章中,研究了在赋2-范空间(更一般地,在2-距离空间)框架下有关不动点理论以及相关的问题,运用Picard迭代序列逼近的方法证明了压缩型映象有唯一不动点,进而也讨论了Picard迭代序列的稳定性。
第四章中,探讨了在赋(2,p)范空间中的Mazur-Ulam问题,考虑用其他条件来替代“满射”这个条件。
Master paper "some fixed point theorems and isometric theorem in linear 2-normed space" ,which based on the formers'studies, applies the geometric theory of Banach space and the operator theory comprehensively. The whole paper is divided into three chapters.
The first chapter of the paper starts from the preliminary and some basic theorems and results . Also,we introduce some conceptions and notations needed in the following chapters.
In Chapter 2, we constructed a new Ishikawa iteration precess inβ-normed linear Space, and we prove a sufficient and necessary condition for the new Ishikawa iteration processes with error of asymptotically pseudo-contractive mapping T to converge to fixed points.
In Chapter 3, We first obtain generalizations of the 2-metric space version of some contractive type mappings, and then prove that each such mapping has a unique fixed point. Moreover, this enables us to establish some fixed point theorems in linear 2-normed space. At last we obtain some stability results for Picard iteration in 2-metric space.
In Chapter 4, We focus our attention on the Mazur-Ulam problem in linear (2, p)-normed space, and deal with the following problem :instead of surjectivity,what conditions imply the linearity of isometries?
第一章为绪论。主要介绍本文的研究背景及相关的一些预备知识,并且给出文中所涉及的大部分概念和记号。
第二章中,研究了赋β-范空间中渐近伪压缩和渐近非扩张映象的不动点迭代逼近问题,证明了渐近伪压缩映象T的修改的Ishikawa迭代序列收敛到不动点的充要条件。
第三章中,研究了在赋2-范空间(更一般地,在2-距离空间)框架下有关不动点理论以及相关的问题,运用Picard迭代序列逼近的方法证明了压缩型映象有唯一不动点,进而也讨论了Picard迭代序列的稳定性。
第四章中,探讨了在赋(2,p)范空间中的Mazur-Ulam问题,考虑用其他条件来替代“满射”这个条件。
Master paper "some fixed point theorems and isometric theorem in linear 2-normed space" ,which based on the formers'studies, applies the geometric theory of Banach space and the operator theory comprehensively. The whole paper is divided into three chapters.
The first chapter of the paper starts from the preliminary and some basic theorems and results . Also,we introduce some conceptions and notations needed in the following chapters.
In Chapter 2, we constructed a new Ishikawa iteration precess inβ-normed linear Space, and we prove a sufficient and necessary condition for the new Ishikawa iteration processes with error of asymptotically pseudo-contractive mapping T to converge to fixed points.
In Chapter 3, We first obtain generalizations of the 2-metric space version of some contractive type mappings, and then prove that each such mapping has a unique fixed point. Moreover, this enables us to establish some fixed point theorems in linear 2-normed space. At last we obtain some stability results for Picard iteration in 2-metric space.
In Chapter 4, We focus our attention on the Mazur-Ulam problem in linear (2, p)-normed space, and deal with the following problem :instead of surjectivity,what conditions imply the linearity of isometries?
引文
[1]Goebel.K,Kirk W.A fixed point theorem for asymptotically nonexpansive mappings [J].Pro Amer Math Soc,1972,35(1):171-174.
[2]Schu.Iterative construction of fixed points asymptotically nonexpansive mappings[J].J Math Anal Appl,1991,158:407-413.
[3]Liu.Q.H.Convergence theorems of the sequence of iterates for asymptotically demicontrative and hemicontractive mappings[J].Nonlinear Anal TMA,1996,26(11):1853-1842.
[4]Kato.T.Nonlinear Semi-groups and Evolution Equations[M].J Math Soc Japan,1964,19:508-520.
[5]Rhoades.B.E.Comments on two fixed point iteration methods[J].J Math Anal Appl,1976,56:741-750.
[6]张石生.Banach空间中渐近非扩张映像不动点的迭代逼近问题[J].应用数学学报,2001,4(2):236-241.
[7]唐玉超,刘理蔚.赋范线性空间中渐近伪压缩映像的不动点迭代逼近[J].应用数学学报,2007,30(5):810-815.
[8]Chang.S.S.On some problems and results in study of nonlinear analysis[J].Nonlinear Anal TMA,1997,30(7):4197-4208.
[9]Tank.K,Xu.H.K.The nonlinear ergodic theorem for asymptotically nonexpansive mapping[J].Prco Amer Math Soc,1992,114:399-404.
[10]Qihou Lin.Iterative sequence for asymptotically quasi-nonexpansive mappings with Error Member[J].J Math Anal Appl,2001,259:18-24.
[11]G(a|¨)hler,S.Lineare 2-normierte R(a|¨)ume,Math Nachr,1965,28:1-43
[12]G(a|¨)hler,S.2-metrische R(a|¨)ume und ihre topologische Strucktur,Math Nachr,1963,26:115-148.
[13]G(a|¨)hler,S.(U|¨)ber die uniformisierbarkeit 2-metrische R(a|¨)ume,Math Nachr,1965,28:235-244.
[14]Iseki,K.Fixed point theorems in 2-mertic spaces,Maths Seminar Notes,Kobe Univ,1975,3:133-136.
[15]S.V.R.Naidu,J.Rajendra Prasad,Fixed point theorems in 2-metric spaces,Indian J.Pure Appl Math,1986,17:974-993.
[16]S.V.R.Naidu,Fixed point theorems for self-maps on a 2-metric space,Pure Appl Math Sci,1995,41:73-77.
[17]Q.Liu,A convergence theorem of the sequence of Ishikawa iterates for quasicontractive mappings,J.Math Anal Appl,1990,146:301-305.
[18]B.E.Rhoades,A comparison of various definitions of contractive mappings,Trans.Amer Math Soc,1977,226:257-290.
[19]G(a|¨)hler.S,Uber 2-Banach R(a|¨)ume,Math Nachr,1969,42:335-347.
[20]White.A,2-Banach spaces,Math Nachr,1969,42:43-60.
[21]A.M.Harder and T.L.Hicks,Stability results for fixed point iteration procedures,Mathematica Japonnica,1988,33:693-706.
[22]张石生.不动点理论及其应用.重庆:重庆出版社,1984.
[23]江泽坚,孙善利.泛函分析.北京:高等教育出版社,2005.
[24]张恭庆等.泛函分析讲义.北京:北京大学出版社,1990.
[25]Chang.S.S,ChoY.J,ZhouH.Y,Iterative methods for nonlinear operator equations in Banach spaces,New York:Nova Science Publishers,2002.
[26]Deimling K,Zeroes of accretive operators,Manuscripta Math,1974,13:365-374.
[27]Borwein.D,Borwein J.M,Fixed point iterations for real function,J Math Anal Appl,1991,157:112-126.
[28]Hicks T.L,Kubieck J.R,On the Mann iteration process in Hilbert space,J Math Anal Appl,1979,59:498-504.
[29]Yao Y.H,Chen R.D,Zhou H.Y,Iterative algorithms to fixed point of nonexpansive mapping,Acta Mathematica Sinica,ChineseSeries,2007,50(1):139-144.
[30]Y.Song,R.Chen,Viscosity approximation methods for nonexpansive nonselfmappings,J Math Anal Appl,2006,321:316-326.
[31]Th.M.Rassias,Xiang.S.H,On mappings with conservative distance and the Muzar-Ulam theorem,Univ Beograd Publ Elektrotehn Fak(Ser Mat),2000,11:1-8.
[32]Gao.J.M,On the Alexandrov problem of distance preserving mapping,J Math Anal Appl,2009,352:583-590.
[33]H.Y.Chu,On the Mazur-Ulam problem in linear 2-noremed spaces,J Math Anal Appl,2007,327:1041-1045.
[34]A.P.Bosznay,On a theorem of Mazur and Ulam,Periodica Math Hungarica,1985,16(1):7-13.
[35]H.Y.Chu,C.G.Park,W.G.Park,The Aleksandrov problem in linear 2-normed spaces,J.Math.Anal.Appl.2004,289:666-672
[36]A.D.Aleksandrov,Mappings of families of sets,Soviet Math Dokl,1970,11:116-120.
[37]J.V(a|¨)is(a|¨)l(a|¨),A proof of the Mazur-Ulam theorem,Amer Math Monthly,2005,110:633-635.
[38]J.A.Baker,Isometries in normed spaces,Amer Math Monthly,1971 655-658.
[39]Ren.W.Y,On the Aleksandrov Problem in 2-normed linear spaces,Acta Sci Naturalium Universitatis Nankaiensis,2008,41(3):52-56.
[40]Wang Jian,Some Further Generalizations of the Hyers-Ulam-Rassias Stability of Functional Equations,J Math Anal Appl,2001,263(2):406-423.
[41]Y.J.Cho,P.C.S.Lin,S.S.Kim,A.Misiak,Theory of 2-Inner Product Spaces,Nova Science,New York,2001.
[42]H.Y.Chu,K.H.Lee,C.G.Park,On the Aleksandrov problem in linear n-normed spaces,Nonlinear Anal,2004,59:1001-1011.
[43]S.Mazur,S.Ulam,Sur les transformationes isometriques,espaces vectoriels normes,C.R.Acad.Sci.Paris 1932,94:946-948.
[44]Th.M.Rassias,Properties of isometric mappings,J Math Anal Appl,1999,235:108-121.
[45]Th.M.Rassias,Psemrl,On the Mazur-Ulam problem and the Aleksandrov problem for unit distance preserving mappings,Proc Amer Math Soc,1993,118:919-925.
[46]H.Y.Chu,H.K.Se,D.S.Kang,Characterizations on 2-isometries,J Math Anal Appl,2008,340:621-628.
[47]Wang Jian,The Stability of Approximately Additive mappings on Davison Functional Equations,Nonlinear Funct Anal Appl,2002,7(1):101-114
[48]Wang Jian,On extension of isometries between unit spheres of ALp-spaces(0＜p＜∞),Proc Amer Math Soc,2004,132(10):2899-2909.
[49]Wang Jian,The Isometric Approximation Problem on F~*-spaces,Functional Space Theory and its Application,Proceedings of International Conference 13th Academic Symposium in China.
[50]定光桂.等距算子的延拓、逼近及相关问题,数学进展[J],2003,32(5):529-536.
[51]Wang Jian,On the Generalizations of Mazur-Ulam Isometric Theorem,J Math Anal Appl,2001,263(2):510-521.
[2]Schu.Iterative construction of fixed points asymptotically nonexpansive mappings[J].J Math Anal Appl,1991,158:407-413.
[3]Liu.Q.H.Convergence theorems of the sequence of iterates for asymptotically demicontrative and hemicontractive mappings[J].Nonlinear Anal TMA,1996,26(11):1853-1842.
[4]Kato.T.Nonlinear Semi-groups and Evolution Equations[M].J Math Soc Japan,1964,19:508-520.
[5]Rhoades.B.E.Comments on two fixed point iteration methods[J].J Math Anal Appl,1976,56:741-750.
[6]张石生.Banach空间中渐近非扩张映像不动点的迭代逼近问题[J].应用数学学报,2001,4(2):236-241.
[7]唐玉超,刘理蔚.赋范线性空间中渐近伪压缩映像的不动点迭代逼近[J].应用数学学报,2007,30(5):810-815.
[8]Chang.S.S.On some problems and results in study of nonlinear analysis[J].Nonlinear Anal TMA,1997,30(7):4197-4208.
[9]Tank.K,Xu.H.K.The nonlinear ergodic theorem for asymptotically nonexpansive mapping[J].Prco Amer Math Soc,1992,114:399-404.
[10]Qihou Lin.Iterative sequence for asymptotically quasi-nonexpansive mappings with Error Member[J].J Math Anal Appl,2001,259:18-24.
[11]G(a|¨)hler,S.Lineare 2-normierte R(a|¨)ume,Math Nachr,1965,28:1-43
[12]G(a|¨)hler,S.2-metrische R(a|¨)ume und ihre topologische Strucktur,Math Nachr,1963,26:115-148.
[13]G(a|¨)hler,S.(U|¨)ber die uniformisierbarkeit 2-metrische R(a|¨)ume,Math Nachr,1965,28:235-244.
[14]Iseki,K.Fixed point theorems in 2-mertic spaces,Maths Seminar Notes,Kobe Univ,1975,3:133-136.
[15]S.V.R.Naidu,J.Rajendra Prasad,Fixed point theorems in 2-metric spaces,Indian J.Pure Appl Math,1986,17:974-993.
[16]S.V.R.Naidu,Fixed point theorems for self-maps on a 2-metric space,Pure Appl Math Sci,1995,41:73-77.
[17]Q.Liu,A convergence theorem of the sequence of Ishikawa iterates for quasicontractive mappings,J.Math Anal Appl,1990,146:301-305.
[18]B.E.Rhoades,A comparison of various definitions of contractive mappings,Trans.Amer Math Soc,1977,226:257-290.
[19]G(a|¨)hler.S,Uber 2-Banach R(a|¨)ume,Math Nachr,1969,42:335-347.
[20]White.A,2-Banach spaces,Math Nachr,1969,42:43-60.
[21]A.M.Harder and T.L.Hicks,Stability results for fixed point iteration procedures,Mathematica Japonnica,1988,33:693-706.
[22]张石生.不动点理论及其应用.重庆:重庆出版社,1984.
[23]江泽坚,孙善利.泛函分析.北京:高等教育出版社,2005.
[24]张恭庆等.泛函分析讲义.北京:北京大学出版社,1990.
[25]Chang.S.S,ChoY.J,ZhouH.Y,Iterative methods for nonlinear operator equations in Banach spaces,New York:Nova Science Publishers,2002.
[26]Deimling K,Zeroes of accretive operators,Manuscripta Math,1974,13:365-374.
[27]Borwein.D,Borwein J.M,Fixed point iterations for real function,J Math Anal Appl,1991,157:112-126.
[28]Hicks T.L,Kubieck J.R,On the Mann iteration process in Hilbert space,J Math Anal Appl,1979,59:498-504.
[29]Yao Y.H,Chen R.D,Zhou H.Y,Iterative algorithms to fixed point of nonexpansive mapping,Acta Mathematica Sinica,ChineseSeries,2007,50(1):139-144.
[30]Y.Song,R.Chen,Viscosity approximation methods for nonexpansive nonselfmappings,J Math Anal Appl,2006,321:316-326.
[31]Th.M.Rassias,Xiang.S.H,On mappings with conservative distance and the Muzar-Ulam theorem,Univ Beograd Publ Elektrotehn Fak(Ser Mat),2000,11:1-8.
[32]Gao.J.M,On the Alexandrov problem of distance preserving mapping,J Math Anal Appl,2009,352:583-590.
[33]H.Y.Chu,On the Mazur-Ulam problem in linear 2-noremed spaces,J Math Anal Appl,2007,327:1041-1045.
[34]A.P.Bosznay,On a theorem of Mazur and Ulam,Periodica Math Hungarica,1985,16(1):7-13.
[35]H.Y.Chu,C.G.Park,W.G.Park,The Aleksandrov problem in linear 2-normed spaces,J.Math.Anal.Appl.2004,289:666-672
[36]A.D.Aleksandrov,Mappings of families of sets,Soviet Math Dokl,1970,11:116-120.
[37]J.V(a|¨)is(a|¨)l(a|¨),A proof of the Mazur-Ulam theorem,Amer Math Monthly,2005,110:633-635.
[38]J.A.Baker,Isometries in normed spaces,Amer Math Monthly,1971 655-658.
[39]Ren.W.Y,On the Aleksandrov Problem in 2-normed linear spaces,Acta Sci Naturalium Universitatis Nankaiensis,2008,41(3):52-56.
[40]Wang Jian,Some Further Generalizations of the Hyers-Ulam-Rassias Stability of Functional Equations,J Math Anal Appl,2001,263(2):406-423.
[41]Y.J.Cho,P.C.S.Lin,S.S.Kim,A.Misiak,Theory of 2-Inner Product Spaces,Nova Science,New York,2001.
[42]H.Y.Chu,K.H.Lee,C.G.Park,On the Aleksandrov problem in linear n-normed spaces,Nonlinear Anal,2004,59:1001-1011.
[43]S.Mazur,S.Ulam,Sur les transformationes isometriques,espaces vectoriels normes,C.R.Acad.Sci.Paris 1932,94:946-948.
[44]Th.M.Rassias,Properties of isometric mappings,J Math Anal Appl,1999,235:108-121.
[45]Th.M.Rassias,Psemrl,On the Mazur-Ulam problem and the Aleksandrov problem for unit distance preserving mappings,Proc Amer Math Soc,1993,118:919-925.
[46]H.Y.Chu,H.K.Se,D.S.Kang,Characterizations on 2-isometries,J Math Anal Appl,2008,340:621-628.
[47]Wang Jian,The Stability of Approximately Additive mappings on Davison Functional Equations,Nonlinear Funct Anal Appl,2002,7(1):101-114
[48]Wang Jian,On extension of isometries between unit spheres of ALp-spaces(0＜p＜∞),Proc Amer Math Soc,2004,132(10):2899-2909.
[49]Wang Jian,The Isometric Approximation Problem on F~*-spaces,Functional Space Theory and its Application,Proceedings of International Conference 13th Academic Symposium in China.
[50]定光桂.等距算子的延拓、逼近及相关问题,数学进展[J],2003,32(5):529-536.
[51]Wang Jian,On the Generalizations of Mazur-Ulam Isometric Theorem,J Math Anal Appl,2001,263(2):510-521.