Generalized solutions for the abstract singular Cauchy problem (Q1012846)

Consider the Cauchy problem \[ \begin{cases} Mv^{\prime}(t)=Av(t)+F(t)x, \quad x\in X,\quad 0\leq t\leq \tau, \\ v(0)=0, \end{cases} \] where \(X\) is a Banach space, \(A\) and \(M\) are closed linear operators in the Banach space \(X\), \(E:[0,\infty)\rightarrow {\mathbb C}\) is a fixed locally integrable function of exponential type \(\omega_1\) and \(F(t)=\int_0^tE(s)ds.\) There are established several conditions on the pair \((M, A)\) to guarantee the existence of a local generalized singular evolution operator.