The Cauchy problem for an inclusion in Banach spaces and distribution spaces

We study well-posedness of degenerate Cauchy problems treated as Cauchy problems for a differential inclusion with a multivalued linear operator. Using a new approach to the definition of degenerate integrated semigroups and their generators in a Banach space, we obtain a well-posedness criterion for the problem. Moreover, we consider the Cauchy problem for a differential inclusion in the space of abstract distributions and give necessary and sufficient conditions for well-posedness in the distribution space.