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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
Published electronically: August 11, 2004
MathSciNet review: 2098102
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Abstract: Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains $C_{0}$-semigroup, nonlinear semigroup of contractions and uniformly $k$-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 $C_{0}$-semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of $C_{0}$-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

Zongben Xu
Affiliation: Research Center for Applied Mathematics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China

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

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