Well-posedness of linear singular evolution equations in Banach spaces: theoretical results

In this work we deal with a singular evolution equation of the form {E-u(center dot)= Au,t>0,<br /> u(0) = u0, where both A and E are linear operators, with E bounded but not necessarily injective, defined in adequate subspaces of a given Banach space X. By using the concept of generalized semigroups, our goal is to prove a Hille-Yosida type theorem for this problem, that is, to find necessary and sufficient conditions under which A is the generator of a generalized semigroup {U(t):t >= 0}. This problem is dealt with by making use of the E-spectral theory and the concept of generalized integrable families. Finally, we present an abstract example that illustrates the theory.