Abstract kinetic equations relevant to supercritical media

The abstract Hilbert space equation $\text{(T?)′(x) = ?(A?)(x)}$, x∈$\text{R}$₊, is studied with a partial range boundary condition $\text{(Q}{}_{\mathrm{+}}\text{?)(0) = ?}{}_{\mathrm{+}}\text{?}\text{Ran}\text{Q}{}_{\mathrm{+}}$. Here T is bounded, injective and self-adjoint, A is Fredholm and self-adjoint, with finite-dimensional negative part, and Q₊ is the orthogonal projection onto the maximal T-positive T-invariant subspace. This models half-space stationary transport problems in supercritical media. A complete existence and uniqueness theory is developed.