Nonlinear Semigroups

For a topological space E and a measurable submarkov semigroup , we consider the restriction on E× + of the kernel associated with the space time semigroup ⊗ . A local Kato-class Kt Loc related to V and for functions φ from E × + × to , notions of locally Kato-bounded, continuous locally Kato-bounded, locally Kato-Lipschitzian and Kato-Lipschitzian, which are not necessarily (locally) bounded and (locally ) Lipschitzian, are introduced. Nonlinear monotone semigroups (Qt)t>0, defined not only for positive but for bounded Borel measurable functions and their monotone limits, are constructed. In contrast to many earlier works, our construction method does not rely on Picard iteration . Introduction Let E be a topological space and Bb the set of bounded and Borel measurable functions on E. Let = (Pt)t≥0 be a measurable submarkov semigroup of linear operators on Bb. Let us consider the kernel V defined on E × + by