Stationary waves for the discrete Boltzmann equation in the half space with reflective boundaries

where each F = (Fi)i∈Λ is an unknown function representing the mass density of gas particles; Λ is a finite set {1, 2, . . .m} and Λ± := {i ∈ Λ : vi ≷ 0}; each νi is a positive integer; each vi is a constant representing the x-component of the i-th velocity and vi’s are not necessarily distinct and not necessarily non-zero; each Qi(F ) is a given function called the collision term; each Bij is a nonnegative constant. We assume that the initial data satisfies the spatial asymptotic condition,