Norm continuity and stability for a functional differential equation in Hilbert space (Q1614689)

The author considers stability properties for the functional-differential equation \[ u'(t)= Au(t)+ \int^0_{-h} a(r) Au(t+ r) dr,\quad t> 0,\tag{1} \] \[ u(0)= \phi^0,\quad u(r)= \phi^1(r),\quad r\in [-h,0). \] He proves that when \(a(\cdot)\) is a measurable square integrable complex function then the solution semigroup is norm continuous for \(g> h\). The norm continuity of the solution semigroup is used to consider the stability of the solution to (1) in the case when \(a(\cdot)\) is a real function, more precisely, the author proves that the solution is asymptotically stable.