Let A be a resolvent positive (linear) operator (i.e., exists and is positive for on a Banach lattice E. Even though no norm condition on the resolvent is demanded, a theory is developed which - to a large extent - is analogous to the theory of positive -semigroups.
For example, if D(A) is dense or E is reflexive, then for every there exists a unique classical solution of the abstract Cauchy problem
and u(t) (t if . Moreover, A is the generator of a so- called integrated semigroup; i.e. there exists S: [0, strongly continuous s.t. . The solution of (ACP) is given by
A variety of examples is given.