Hyperbolicity of solution semigroups for linear neutral differential equations

Abstract

Consider the linear neutral functional differential equation of the form $$\left\{\begin{array}{l@{\quad }l}\frac{\partial}{\partial t}Fu_t=BFu_t+\Phi u_t, & t\ge 0,\\[3pt]u_0(s)=\varphi(s),& s\in [-r,0],\end{array}\right.$$ where the function u(⋅) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ we prove that the solution semigroup for this equation is hyperbolic provided that the differential operator B generates a hyperbolic semigroup on X.