### $(a,k)$-regularized $C$-resolvent families: regularity and local properties.

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We investigate the characterization of almost periodic C-semigroups, via the Hille-Yosida space Z₀, in case of R(C) being non-dense. Analogous results are obtained for C-cosine functions. We also discuss the almost periodicity of integrated semigroups.

A class of C-distribution semigroups unifying the class of (quasi-) distribution semigroups of Wang and Kunstmann (when C = I) is introduced. Relations between C-distribution semigroups and integrated C-semigroups are given. Dense C-distribution semigroups as well as weak solutions of the corresponding Cauchy problems are also considered.

Suppose A is an injective linear operator on a Banach space that generates a uniformly bounded strongly continuous semigroup ${{e}^{tA}}_{t\ge 0}$. It is shown that ${A}^{-1}$ generates an $O(1+\tau )A{(1-A)}^{-1}$-regularized semigroup. Several equivalences for ${A}^{-1}$ generating a strongly continuous semigroup are given. These are used to generate sufficient conditions on the growth of ${{e}^{tA}}_{t\ge 0}$, on subspaces, for ${A}^{-1}$ generating a strongly continuous semigroup, and to show that the inverse of -d/dx on the closure of its image in L¹([0,∞)) does not generate a strongly...

We introduce the notion of a local n-times integrated C-semigroup, which unifies the classes of local C-semigroups, local integrated semigroups and local C-cosine functions. We then study its relations to the C-wellposedness of the (n + 1)-times integrated Cauchy problem and second order abstract Cauchy problem. Finally, a generation theorem for local n-times integrated C-semigroups is given.

This paper is concerned with the problem of real characterization of locally Lipschitz continuous (n + 1)-times integrated semigroups, where n is a nonnegative integer. It is shown that a linear operator is the generator of such an integrated semigroup if and only if it is closed, its resolvent set contains all sufficiently large real numbers, and a stability condition in the spirit of the finite difference approximation theory is satisfied.