Eventual differentiability of functional differential equations in Banach spaces (Q860614)

Let us consider the functional differential equation \[ \left\{\begin{matrix} \dot{u}(t)=Au(t)+B_1u(t-r)+\int_{-r}^0a(s)B_2u(t+s)\,ds,\quad t>0,\\ u(0)=x,\\ u(\theta)=g(\theta),\quad \theta \in [-r,0], \end{matrix} \right. \tag{1} \] where \(A\) generates an analytic semigroup on a Banach space \(X\), \(B_1,\,B_2\) are bounded linear operators from \(X_\alpha\) into \(X\) with \(X_\alpha =D((\gamma -A)^\alpha)\) endowed with the norm \(\| x\| _\alpha =\| (\gamma -A)^\alpha x\| ,\,0<\alpha <1,\,\gamma >\omega_0(A)\) and \(a\in L^q(-r,0)\) such that \(\frac{1}{p}+\frac{1}{q}=1\). It is known that the semigroup solution \({\mathcal T}(\cdot)\) of (1) exists in \(X\times L^p([-r,0],X_\alpha)\). Based on these results, the authors show the differentiability of \({\mathcal T}(t)\) for \(t>\frac{3r}{1-\alpha}\) provided that \(1<p<\frac{1}{\alpha}\).