ON THE STUDY TO A TYPE OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEM WITH TWO DOUBLE ROOTS OF THE DEGENERATE EQUATION

This paper addresses a singularly perturbed boundary value problem where the degenerate equation has three distinct roots: two double roots and one simple root. It is shown that for a sufficiently small parameter, the solution of the problem switches between the two double roots in a neighbor-hood of the transition point. As a result, the inner layer can be divided into multiple regions. An asymptotic expansion is constructed, and the existence of smooth solutions is established. Additionally, an estimate for the remainder term is provided.