A novel dual approach to nonlinear semigroups of lipschitz operators

Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains Co-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 Co-semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of Co-semigroup are generalized to Lipschitzian semigroup.