Differentiability and growth bounds of solutions of delay equations (Q1766722)

This paper deals with the eventual differentiability of the semigroup solution for the following delay equation in Banach spaces \[ u'(t)=Au(t)+\mu u_t, t\geq0, \quad u_0=\varphi \in C([-1,0];X), \] where \(A\) generates an immediatly norm-continuous semigroup on a Banach space \(X\), \(C([-1,0];X)\) is the space of continuous functions from \([-1,0]\) to \(X\) endowed with the uniform norm topology and \(\mu\) is a bounded linear operator from \(C([-1,0];X)\) to \(X\). The history function is defined by \(u_t(\alpha)=u(t+\alpha),\) \(\alpha \in [-1,0]\). The eventual differentability is obtained by the rate of decay of the resolvent of \(A\) along vertical lines.