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

$$\begin{cases}E\dot{u} = Au, &t>0,\\ u(0)=u_0,\end{cases}$$

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 \(\{U(t) : t \geq 0\}\). 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.