Multizonal internal layers in the singularly perturbed equation with a discontinuous right-hand side

This paper investigates a two-point boundary value problem for a second-order singularly perturbed ordinary differential equation in the case of multiple roots of the degenerate equation. This is a new class of problems, namely, problems with discontinuous nonlinear terms on the right-hand side of the equation, which leads to the formation of a multizonal interior transitional layer in a neighborhood of the discontinuity point. For sufficiently small parameter values, the existence of a smooth solution is proved, and its asymptotic expansion is constructed, showing that this solution qualitatively differs from the case when the degenerate equation has simple roots.