We provide a complete solution of the abstract Cauchy problem for operator valued Laplace distributions or hyperfunctions on complete ultrabornological locally convex spaces (like spaces of smooth functions and distributions). This extends results of Komatsu for operators on Banach spaces. Concrete examples are provided. The crucial tools for our solution are a general notion of a resolvent for operators on locally convex spaces and the theory of Laplace transform for Laplace hyperfunctions valued in a complete locally convex space X developed earlier by the authors.