Nonhomogeneous boundary value problems for differential-operator equations of mixed type, and their application

Let $A$ and $B$ be symmetric operators in a Hilbert space $H$, such that $B$ is positive and $A$ has an arbitrary spectrum. In this paper nonhomogeneous boundary value problems are considered for an equation of the form
\begin{equation} Au'(t)+Bu(t)=f(t),\qquad t\in(0,T). \end{equation}

An abstract theorem (of the Lax–Milgram type) is proved, which is then used to prove theorems on the weak and strong solvability of boundary value problems for equation (1) in the energy spaces defined by the operators $A$ and $B$, as well as a theorem on the traces of a strong solution.
As an application, nonhomogeneous boundary value problems for partial differential equations are considered.
Bibliography: 16 titles.