Some boundary value problems for the Sobolev-type operator-differential equations

We consider the solvability of boundary value problems for operator-differential equation of the form $Bu_t-Lu=f,$ where $X$ is a Banach space, $B,L\colon X\to X$ are closed operators such that $D(L)\subset D(B)$ ($D(L), D(B)$ are domains of the corresponding operators), with boundary conditions $Bu(0)=Bu_0$ or $\int\limits^T_0 Bu(\tau)\,d\sigma(\tau)=Bu_0,$ where $\sigma$ is a function of bounded variation. Some well-known results on solvability of initial boundary value problems for operator-differential equations of Sobolev type are refined in the case of arbitrary decrease (growth) of the resolvent of the corresponding linear pencil. Existence and uniqueness theorems of solutions to the Cauchy-type problems and general nonlocal boundary value problems are obtained and the maximal regularity of solutions is proven under certain conditions. The results rely on Mikhlin theorems for operator-valued Fourier multipliers. In contrast to the previous results, the function spaces are the Sobolev–Besov spaces.