Abstract semilinear functional differential equations of retarded type (Q2720155)

The authors prove an existence and uniqueness result on integral solutions to a Cauchy problem associated to a functional-differential equation of the form NEWLINE\[NEWLINEu'(t)=Au(t)+F(t,u_t),\ \text{ for } t\leq 0,\tag{1}NEWLINE\]NEWLINEwith \(u_t(\theta)=u(t+\theta)\), for \(\theta\in [-\tau,0]\), \(A\) is a linear operator generating an integrated semigroup on a Banach space \(E\), and \(F:[0,T]\times C([-\tau,0];E)\to E\) is Lipschitz continuous.