The problem of steady-state oscillations of a transversally isotropic half-cylinder

The authors study the problem of solvability on the semiaxis of the equation
$$ \mathscr P(u)=-A\frac{d^2u}{dy^2}+iB\frac{du}{dy}+(C-\omega^2R)u=0, $$
where $\omega\in\mathbf R$, and $A$, $B$, $C$, and $R$ are unbounded symmetric operators in a Hilbert space $\mathfrak H$. Models of this equation are problems of steady-state oscillations of an elastic half-cylinder with various boundary conditions. The main results are theorems on factorization of a pencil related to this problem and solvability theorems.