Maximal regularity for generalized boundary conditions in time

We consider autonomous and non-autonomous evolution equations on a time interval $[0,\tau ]$ in a Banach space $X$ with the non-standard time-boundary condition $u(0)=\mathrm{\Phi }u(\tau )$, where $\mathrm{\Phi }$ is a linear map on $X$. If $\mathrm{\Phi }=0$, this is an initial value problem, whereas $\mathrm{\Phi }=I$ corresponds to periodic boundary conditions, and $\mathrm{\Phi }=-I$ to antiperiodic boundary conditions. Our main point is to establish maximal $L^p$-regularity. In the non-autonomous case we consider two situations. The first concerns time-dependent operators with a fixed domain. In the second one we take $X=H$ a Hilbert space and consider evolution equations associated with non-autonomous forms. Of special interest is then maximal regularity in $H$ with a non-standard time-boundary condition.