非线性Lipschitz-α算子半群生成元的存在性
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摘要
Hille和Yosida首先给出了C_0半群的生成元定理,但是人们对于非线性算子半群的研究进展的相当缓慢。本文给出了一种新的非线性算子半群,非线性Lipschitz-α算子半群。这种算子半群是在非线性算子半群的基础上提出的,是对非线性Lipschitz算子半群的进一步扩展,使它的范围更加扩大了。本文仔细研究了这种半群,并给出了它的一种生成元的存在性定理。
本文共分三章,主要内容如下:
在第一章回顾了算子半群的发展历程和非线性算子半群的研究以及发展现状,接着介绍了研究非线性Lipschitz-α算子半群的意义。
在第二章介绍了赋范空间,算子理论知识,算子半群的有关知识。
在第三章中给出了关于非线性Lipschitz-α算子半群的生成元存在性定理。
Hille and Yosida proved the generation theorem of C_0 semigroups firstly, but the course of study of a nonlinear semigroups is very slow. This paper is concerned with the problem of a nonlinear semigroup of Lipschitz-αoperator, based on the nonlinear semigroup of Lipschitz operator. This new semigroup extende the nonlinear semigroup of Lipschitz operator. In this paper, new theorems of the generation of the nonlinear semigroup of Lipschhitz-αoperator are established.
This paper consists of three chapters. The main contents are the following:
In the first chapter, we recall the development of the semigroups. Then we introduce the meaning of doing research on the nonlinear semigroups of Lipschitz -αoperator.
In the second chapter, we present some mathematical preliminaries. These include the theorems in normed spaces, operator theory and operator semigroup. In the third chapter, we established the new theory of the nonlinear semigroup of the Lipschitz-αoperator.
本文共分三章,主要内容如下:
在第一章回顾了算子半群的发展历程和非线性算子半群的研究以及发展现状,接着介绍了研究非线性Lipschitz-α算子半群的意义。
在第二章介绍了赋范空间,算子理论知识,算子半群的有关知识。
在第三章中给出了关于非线性Lipschitz-α算子半群的生成元存在性定理。
Hille and Yosida proved the generation theorem of C_0 semigroups firstly, but the course of study of a nonlinear semigroups is very slow. This paper is concerned with the problem of a nonlinear semigroup of Lipschitz-αoperator, based on the nonlinear semigroup of Lipschitz operator. This new semigroup extende the nonlinear semigroup of Lipschitz operator. In this paper, new theorems of the generation of the nonlinear semigroup of Lipschhitz-αoperator are established.
This paper consists of three chapters. The main contents are the following:
In the first chapter, we recall the development of the semigroups. Then we introduce the meaning of doing research on the nonlinear semigroups of Lipschitz -αoperator.
In the second chapter, we present some mathematical preliminaries. These include the theorems in normed spaces, operator theory and operator semigroup. In the third chapter, we established the new theory of the nonlinear semigroup of the Lipschitz-αoperator.
引文
1 彭济根. 关于非线性 lipschitz 算子半群生成元的存在性. 应用泛函分析学报. 2005,7(1):39~44
2 G. Sorderlind. Bounds on Nonlinear Operators in Finite Dimensional Banach Spaces. Number Math. 1986,50(1):27~44
3 J. B. Conway. A Course in Functional Analysis. New York, Springer-Verlag, 1985
4 S.Y. Shaw. Approximation of Unbound Functions and Applications to Representations of Semigroups. J. Approx, Theory. 1980,28:238~256
5 R.F. Arens, J.J. Eellens. On Embedding Uniform and Topological Spacers. Pacific J. Math. 1956, 6:397~403
6 D.V. Widder. An Introduction to Transform Theory, New York, Academic Press,1971
7 N. Weaver. Order Completeness in Lipschitz Algebras. Funct Anal. 1995, 130:118~130
8 A. Pazy. ”Semigroups of Linear Opeartors and Applications to Partial Differtial Equations”. Spring Verlag, New York,1983
9 Bj. Peng, Z. B. Xu. A Novel Dual Notion of Banach Spaces: Lipschitz dual space. Acta Math. Sinica,1999,42: 61~70(in Chinese)
10 彭济根, 徐宗本. Banach 空间的 lipschitz 对偶空间及其应用. 数学学报. 1999,42(1):61~70
11 V. Barbu. Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoof , Leyden, 1976
12 张恭庆, 郭懋正. 《泛函分析讲义》(上册). 北京大学出版社
13 R. Larsen. Functions Analysis. New York, Marcel Dekker Inc, 1973
14 E. Hille, R.S. Phillips. Functional Analysis and Semigroups. Providence, Amer Math Sor Colloquium Publications, 1957
15 W. Rudin. Functional Analysis(1st edt). New York, John Wiley&Sons, 1974
16 Q. Zheng. Strongly Continuous Semigroups of Linear Operators. Wuhan: Huazhong University of Science and Technology Press, 1994 (in Chinese)
17 I. Miyadera. Some Remarks on Semigroups of Nonlinear Operators. Tohoku Math.J.1971,23:245~258
18 J.A. Goldstein. “Semigroups of Nonlinear Operators and Applications”. Oxford University Press, New York, 1985
19 张恭庆, 郭懋正. 《泛函分析讲义》(下册). 北京大学出版社
20 J.G.. Peng. The Theoretical Researches on Nonliear Lipschitz Opeartors and its Applications. Thesis for Ph.d. at Xi’an jiaotong University. 1998(in Chinese)
21 C.Y. Lin. Cauchy Problems and Applications. Topological Methods in Nonlinear Analysis 2000,15:359~368
22 G.. Morosanu. ”Nonlinear Evolution Equations and Applications”. D.Reidel Publishing Company, Dordrecht,1998
23 T. Takahahi. Convergence of Difference Approximation of Nonlinear Evolution Equations and Generation of Semigroups. J.Math.Soc. Japan, 1976,28:96~113
24 Ph. Benilan. Equations Evolution Dans an Espace Banach Quelconqueet Applications. Ph.D Thesis, Univ.Pairs. rsay,1972
25 M.G., Crandll. “An Introduction to Evolution Equations Governed by Accretive Operators in Dynamical Systems”. Cesari, H.K. Hale and J.P. Lassale(eds.), Academic. New York, 1973:131~165
26 M. G. Cradall and Pazy. Nonlinear Evolution Equations in Banach Spaces. Israel J.Math.1972,11:57~94
27 T. kato. Nonlinear Semigroups and Evolution Equations. J. Math. Soc.Japan,1967,21:508~520
28 I. Miyadera. ”Nonliear Semigroups”. Translations of Mathematical Monographs,vol.109, American Mathematical Society.1992
29 P.M. Nieberg. Banach Lattice. New York, Springer-Verlag, 1990
30 I. Minyadera. Nonlinear Semigroups (transland by Y.C Chong in English). Amer Math. Soc. Providence, 1993
31 L. Chin-Yuan. “On Genaration of Semigroup and Nonlinear Operator Semigroups”. Semigroup Forum . 2003,66: 110~120
32 K. Yosida. Functions Analysis (5th edt.). New York, Springer-Verlag, 1978
33 R. A. Horn, C.R. Johnson. Martx Analysis. London, Cambridge Univ. Press,1985
34 彭济根,徐宗本. 非线性 Lipschitz 算子半群的表示[J]. 数学学报,2004,47(4):723~730
35 C.C. Chen and K.M. Koh . ”Principles and Techniques in Combinatorices”, World Scientific. Singapore,1992
36 M.G. Cradl and T.M. Liggett. Genaration of Semigroups of Nonlinear Transformations on General Banach Spaces. Amer, J.Math.1973,91:256~298
37 G.. Cradall. A Generalized Domain for Semigroup Generators. Proc. Amer.Math.soc.1973,37: 434~440
38 V. Lakshmikantham and S. Leela. “Nonlinear Differential Equations in Abstract Spaces’. Pergaman Press, Oxford,1981
39 C.Y. Lin. “Introduction to Combinatorial Mathematics”. Mc. Graw-Hill, New York,1968
40 R.E. Mickens. ”Difference Equations Theory and Applications”. Second edion, Van Mostrand Reinhold, New York,1990
41 B. H. Pourciau. Analysis and Optimization of Lipschitz Continuous Mapping. J. Optim Theroy and Appl. 1977,22(3):311~351
42 Jigen. Peng, Si Kit. Chung. Lapalace Transforms and Generators of Semigroups of Operators. Proc. Amer. Math. Soc.1998,126(8):2407~2416
43 D. J. Downng, W. O. Ray. Uniformly Lipschitz Semigroups in Hilbert Sparce. Canadian Math Bull. 1982, 25:210~214
2 G. Sorderlind. Bounds on Nonlinear Operators in Finite Dimensional Banach Spaces. Number Math. 1986,50(1):27~44
3 J. B. Conway. A Course in Functional Analysis. New York, Springer-Verlag, 1985
4 S.Y. Shaw. Approximation of Unbound Functions and Applications to Representations of Semigroups. J. Approx, Theory. 1980,28:238~256
5 R.F. Arens, J.J. Eellens. On Embedding Uniform and Topological Spacers. Pacific J. Math. 1956, 6:397~403
6 D.V. Widder. An Introduction to Transform Theory, New York, Academic Press,1971
7 N. Weaver. Order Completeness in Lipschitz Algebras. Funct Anal. 1995, 130:118~130
8 A. Pazy. ”Semigroups of Linear Opeartors and Applications to Partial Differtial Equations”. Spring Verlag, New York,1983
9 Bj. Peng, Z. B. Xu. A Novel Dual Notion of Banach Spaces: Lipschitz dual space. Acta Math. Sinica,1999,42: 61~70(in Chinese)
10 彭济根, 徐宗本. Banach 空间的 lipschitz 对偶空间及其应用. 数学学报. 1999,42(1):61~70
11 V. Barbu. Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoof , Leyden, 1976
12 张恭庆, 郭懋正. 《泛函分析讲义》(上册). 北京大学出版社
13 R. Larsen. Functions Analysis. New York, Marcel Dekker Inc, 1973
14 E. Hille, R.S. Phillips. Functional Analysis and Semigroups. Providence, Amer Math Sor Colloquium Publications, 1957
15 W. Rudin. Functional Analysis(1st edt). New York, John Wiley&Sons, 1974
16 Q. Zheng. Strongly Continuous Semigroups of Linear Operators. Wuhan: Huazhong University of Science and Technology Press, 1994 (in Chinese)
17 I. Miyadera. Some Remarks on Semigroups of Nonlinear Operators. Tohoku Math.J.1971,23:245~258
18 J.A. Goldstein. “Semigroups of Nonlinear Operators and Applications”. Oxford University Press, New York, 1985
19 张恭庆, 郭懋正. 《泛函分析讲义》(下册). 北京大学出版社
20 J.G.. Peng. The Theoretical Researches on Nonliear Lipschitz Opeartors and its Applications. Thesis for Ph.d. at Xi’an jiaotong University. 1998(in Chinese)
21 C.Y. Lin. Cauchy Problems and Applications. Topological Methods in Nonlinear Analysis 2000,15:359~368
22 G.. Morosanu. ”Nonlinear Evolution Equations and Applications”. D.Reidel Publishing Company, Dordrecht,1998
23 T. Takahahi. Convergence of Difference Approximation of Nonlinear Evolution Equations and Generation of Semigroups. J.Math.Soc. Japan, 1976,28:96~113
24 Ph. Benilan. Equations Evolution Dans an Espace Banach Quelconqueet Applications. Ph.D Thesis, Univ.Pairs. rsay,1972
25 M.G., Crandll. “An Introduction to Evolution Equations Governed by Accretive Operators in Dynamical Systems”. Cesari, H.K. Hale and J.P. Lassale(eds.), Academic. New York, 1973:131~165
26 M. G. Cradall and Pazy. Nonlinear Evolution Equations in Banach Spaces. Israel J.Math.1972,11:57~94
27 T. kato. Nonlinear Semigroups and Evolution Equations. J. Math. Soc.Japan,1967,21:508~520
28 I. Miyadera. ”Nonliear Semigroups”. Translations of Mathematical Monographs,vol.109, American Mathematical Society.1992
29 P.M. Nieberg. Banach Lattice. New York, Springer-Verlag, 1990
30 I. Minyadera. Nonlinear Semigroups (transland by Y.C Chong in English). Amer Math. Soc. Providence, 1993
31 L. Chin-Yuan. “On Genaration of Semigroup and Nonlinear Operator Semigroups”. Semigroup Forum . 2003,66: 110~120
32 K. Yosida. Functions Analysis (5th edt.). New York, Springer-Verlag, 1978
33 R. A. Horn, C.R. Johnson. Martx Analysis. London, Cambridge Univ. Press,1985
34 彭济根,徐宗本. 非线性 Lipschitz 算子半群的表示[J]. 数学学报,2004,47(4):723~730
35 C.C. Chen and K.M. Koh . ”Principles and Techniques in Combinatorices”, World Scientific. Singapore,1992
36 M.G. Cradl and T.M. Liggett. Genaration of Semigroups of Nonlinear Transformations on General Banach Spaces. Amer, J.Math.1973,91:256~298
37 G.. Cradall. A Generalized Domain for Semigroup Generators. Proc. Amer.Math.soc.1973,37: 434~440
38 V. Lakshmikantham and S. Leela. “Nonlinear Differential Equations in Abstract Spaces’. Pergaman Press, Oxford,1981
39 C.Y. Lin. “Introduction to Combinatorial Mathematics”. Mc. Graw-Hill, New York,1968
40 R.E. Mickens. ”Difference Equations Theory and Applications”. Second edion, Van Mostrand Reinhold, New York,1990
41 B. H. Pourciau. Analysis and Optimization of Lipschitz Continuous Mapping. J. Optim Theroy and Appl. 1977,22(3):311~351
42 Jigen. Peng, Si Kit. Chung. Lapalace Transforms and Generators of Semigroups of Operators. Proc. Amer. Math. Soc.1998,126(8):2407~2416
43 D. J. Downng, W. O. Ray. Uniformly Lipschitz Semigroups in Hilbert Sparce. Canadian Math Bull. 1982, 25:210~214