On some nonlocal boundary value problems for evolution equations

In the Sobolev–Besov spaces, we examine the question on solvability of nonlocal boundary value problems for operator-differential equations of the form $u_t-Lu+\gamma u=f$, $u(0)=Bu+u_0$, where $B$ is a linear operator, $L$ is a positive operator, and $\gamma$ is a real parameter. Under certain conditions on the parameter $\gamma$ and the data, the existence and uniqueness theorems for solutions to this boundary value problem are proven. The results are applied to studying nonlocal boundary value problems for parabolic equations and systems.