Existence and continuous dependence of mild solutions to semilinear functional differential equations in Banach spaces
The authors consider the initial value problem for the semilinear functional-differential equation with infinite delay \[ \frac {d} {dt} u(t) =Au(t)+F(t,u_t), \qquad \sigma < t\leq \sigma +a, \] \[ u(t)=\varphi (t-\sigma), \qquad t\in (-\infty,\sigma), \] where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup on a Banach space \(E\), \(F\) is a continuous mapping defined on a suitably chosen memory function space \(B\), \(\varphi\in B\). They prove some existence and continuous dependence on the data results under the additional assumption that either the semigroup generated by \(A\) is compact, or \(F\) satisfies a certain condition expressed in terms of the measure of compactness.
scientific article; zbMATH DE number 1445642
scientific article