A novel dual approach to nonlinear semigroups of Lipschitz operators

Peng, Jigen; Xu, Zongben

doi:10.1090/S0002-9947-04-03635-9

A novel dual approach to nonlinear semigroups of Lipschitz operators
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by Jigen Peng and Zongben Xu

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

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

Published electronically: August 11, 2004

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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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