Existence, local uniqueness and asymptotic stability of the boundary layer type solution of the Neumann problem for the two equation nonlinear system with different powers of a small parameter

In this paper the author considers the existence, local uniqueness and asymptotic stability of the boundary layer type solution of the nonlinear one-dimensional initial-boundary value problem with nonhomogeneous Neumann conditions. The corresponding theorems are proved for different types of quasimonotone right-hand sides of the equations using the method of upper and lower solutions and its modification --- asymptotical method of differential inequalities.