Asymptotics of a steplike contrast structure in a partially dissipative stationary system of equations

A boundary value problem for a system of two ordinary differential equations, one being of the second order, and the other, of the first order, with a small parameter multiplying the derivatives in each equation is considered. The conditions are established under which the problem has a solution involving an internal transition layer near some point within which the solution jumps from a small neighborhood of one root of the corresponding degenerate system to a small neighborhood of its other root. A solution of this type is called a steplike contrast structure (SLCS). An asymptotic approximation of SLCS with respect to the small parameter is constructed and substantiated. It has certain differences from SLCS observed in other singularly perturbed problems. This is concerned primarily with the structure of the solution asymptotics in the transition layer. The constructed asymptotic expansion is substantiated using the asymptotic method of differential inequalities, whose application to the considered problem has a number of qualitative features.