There are three questions that this work tries to answer. First, we investigate some spectral of fine properties of a 3 x 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. An application to transport equations that describes the neutron transport in a plane-parallel domain with width 2a, or the transfer of unpolarized light in a plane-parallel atmosphere of optical thickness 2a is given. Second, we discuss under which conditions a 2 x 2 operator matrix with nonlinear entries, acting on a product of convex closed subsets of Banach spaces have a fixed point. As application, we give some existence results for a mixed stationary problem on Lp-spaces (1