Solvability of the Cauchy problem for an evolution equation in a Banach space with a nondensely defined operator coefficient generating a semigroup with a singularity (Q1087698)

Let E be a Banach space, and D a given (generally non-dense) subset of E. Let A(t): \(D\to E\) denote for each \(t\in [0,1]\) a linear operator which generates a semigroup on E, with a singularity at the origin. The authors are concerned with the existence and uniqueness of solutions to the Cauchy problem \(v'(t)+A(t)v(t)=f(t)\) \((0<t\leq 1)\), \(v(0)=v_ 0\), where \(v_ 0\in E\), and f: [0,1]\(\to E\) is continuous.