Stability conditions for abstract functional differential equations in Hilbert space (Q1864453)

The author studies some interesting stability properties for a class of functional differential equations of the form \[ \begin{cases} u'(t)= Au(t)+ bAu(t- h)+ \int^0_{-h} a(r) Au(t+ r)\,dr,\;t> 0,\\ u(0)= \phi^0,\quad u(r)= \Phi^1(r),\quad r\in [-h,0).\end{cases} \] Here \(A\) is the infinitesimal generator of an analytic semigroup of linear operators \(S(t)\) on a Hilbert space \(X\), \(a(\cdot)\neq 0\) is a measurable square integrable real function, \(b\neq 0\) a real number and \(h\) is a positive number. The initial value \(\Phi= (\Phi^0,\Phi^1)\) belongs to the product space \(Z= F\times L^2(-h,0; D(A))\) where \(F\) is a suitable intermediate space between \(D(A)\) and \(X\).