Resolvents of operators and partial functional differential equations with nonautonomous past.
In this paper, for an abstract differential operator \(G\) corresponding to the integral equation \(u(t)=U(t,s)u(s)+\int_t^sU(t\xi)f(\xi)d\xi\) on a Banach space of \(X\)-valued functions on \({\mathbb{R}_-}\), where \(U(t,s)\) belongs to a backward evolution family, the author computes the resolvent of its restriction to a smaller domain to obtain a generator. By using these results, the author further proves existence, exponential stability and exponential dichotomy of solutions to a partial functional equation with nonautonomous past.
scientific article; zbMATH DE number 2032972