|
A novel dual approach to nonlinear semigroups of Lipschitz operators
Author(s):
Jigen
Peng;
Zongben
Xu
Journal:
Trans. Amer. Math. Soc.
357
(2005),
409-424.
MSC (2000):
Primary 47H20;
Secondary 47D06
Posted:
August 11, 2004
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains  -semigroup, nonlinear semigroup of contractions and uniformly  -Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lipschitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semigroup can be isometrically embedded into a certain  -semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of  -semigroup are generalized to Lipschitzian semigroup.
References:
-
- 1.
- R. F. Arens, J. J. Eells, On embedding uniform and topological spaces. Pacific J. Math., 6(1956): 397-403. MR 18:406e
- 2.
- V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Groningen, 1976 MR 52:11666
- 3.
- O. Bratteli, W. Robinson, Operator Algebras and Quantum Statistical Mechanics I:
 -and  -Algebras, Algebras, Symmetry Groups, Decomposition of States. Springer-Verlag, New York, 1979. MR 81a:46070 - 4.
- J. Czipszer and L. Geher, Extension of function satisfying a Lipschitz condition, Acta. Math. Acad. Sci. Hunger., 6(1955) 213-220. MR 17:136b
- 5.
- E. B. Davies, One parameter Semigroups, London Math. Soc. Mono., Vol.15, 1980. MR 82i:47060
- 6.
- J. R. Dorroh, J. W. Neuberger, A theory of strongly continuous semigroups in terms of Lie generators, J. Funct. Anal., 136 (1996): 114-126. MR 96m:47072
- 7.
- D. J. Downing, W. O. Ray, Uniformly Lipschitz semigroup in Hilbert space, Cana. Math. Bull., 25(1982): 210-214. MR 84e:47066
- 8.
- E. Hille and B. S. Phillips, Functional Analysis and Semigroups (2nd Edit.). Amer. Math. Soc., 1957 MR 19:664d
- 9.
- R. Larsen, Functional Analysis. Marcel Dekker Inc., New York, 1973. MR 57:1055
- 10.
- I. Miyadera, Nonlinear Semigroups (translated by Y. C. Choong in English). Amer. Math. Soc., Providence, 1992. MR 93j:47093
- 11.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983. MR 85g:47061
- 12.
- J. G. Peng, Z. B. Xu, A novel dual notion of Banach space: Lipschitz dual space (in Chinese), Acta Math. Sinica, 42(2): 61-70, 1999 MR 2000d:46017
- 13.
- J. G. Peng, K. S. Chung, Laplace transforms and generators of semigroups of operators. Proc. Amer. Math. Soc., 126(8): 2407-2416, 1998. MR 2000a:47088
- 14.
- Q. Zheng, Strongly Continuous Semigroups of Linear Operators (in Chinese). The Press of Huazhong University of Science and Technology, Wuhan, China, 1994.
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
47H20,
47D06
Retrieve articles in all Journals with MSC
(2000):
47H20,
47D06
Additional Information:
Jigen
Peng
Affiliation:
Research Center for Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
Email:
jgpeng@mail.xjtu.edu.cn
Zongben
Xu
Affiliation:
Research Center for Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
Email:
zbxu@mail.xjtu.edu.cn
DOI:
10.1090/S0002-9947-04-03635-9
PII:
S 0002-9947(04)03635-9
Keywords:
Lipschitz operator,
Lipschitzian semigroup,
generator,
Lipschitz dual semigroup,
$C^{*}_{0}$-semigroup
Received by editor(s):
October 8, 2003
Posted:
August 11, 2004
Additional Notes:
This work was supported by the Natural Science Foundation of China under contract no. 10101019
Copyright of article:
Copyright
2004,
American Mathematical Society
|
Comments: webmaster@ams.org