A novel dual approach to nonlinear semigroups of Lipschitz operators

Authors:
Jigen Peng and Zongben Xu

Journal:
Trans. Amer. Math. Soc. **357** (2005), 409-424

MSC (2000):
Primary 47H20; Secondary 47D06

DOI:
https://doi.org/10.1090/S0002-9947-04-03635-9

Published electronically:
August 11, 2004

MathSciNet review:
2098102

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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.

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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:
https://doi.org/10.1090/S0002-9947-04-03635-9

Keywords:
Lipschitz operator,
Lipschitzian semigroup,
generator,
Lipschitz dual semigroup,
$C^{*}_{0}$-semigroup

Received by editor(s):
October 8, 2003

Published electronically:
August 11, 2004

Additional Notes:
This work was supported by the Natural Science Foundation of China under contract no. 10101019

Article copyright:
© Copyright 2004
American Mathematical Society