Degenerate operator equations

The differential-operator equation
$$ [-D_tt^\alpha D_t-D_tA-P]u=f $$
is studied, where $D_t\equiv\frac d{dt}$, $t\in[0,b]$, $\alpha\geqslant0$, and the operators $A,P\colon\mathscr H\to\mathscr H$, which commute with $D_t$, act in a Hilbert space $\mathscr H$ and satisfy appropriate (quite strong) requirements formulated in terms of resolvent, or spectral, properties. The character of the boundary conditions with respect to $t$ (at $t=0,b$), which are imposed on the equation and ensure existence and uniqueness of the solution, is elucidated, and properties of the solution depending on $\alpha$ and on the properties of the operators $A$ and $P$ are investigated, as well.
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