Abstract kinetic equations with positive collision operators

We consider “forward-backward” parabolic equations in the abstract form $Jd \psi / d x + L \psi = 0$,$ 0< x < \tau \leq \infty$,where$J$and$L$ are operators in a Hilbert space $H$ such that $J=J^{\ast }=J^{-1}$, $L=L^{\ast }\ge 0$, and $\ker L=0$. The following theorem is proved: if the operator $B=JL$ is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation $ \mu \frac {\partial \psi}{\partial x} (x,\mu) = b(\mu) \frac {\partial^2 \psi}{\partial \mu^2} (x, \mu)$,$ 0<x<\tau$,$ \mu \in \R$, as well as to other parabolic equations of the “forward-backward” type. The abstract kinetic equation $Td\psi \mathrm{/}dx=-A\psi (x)+f(x)$, where $T=T^{\ast }$ is injective and $A$ satisfies a certain positivity assumption, is considered also.