Contrasting solutions of singularly perturbed multipoint boundary-value problems with singularities (Q1905298)

In contrast to the known method, we use a new method to study singularly perturbed initial and boundary-value problems. We investigate the existence and construct the asymptotic expansion of the so-called ``contrasting'' solutions of these problems in the case where eigenvalues of the limit operator vanish on a subset of the segment considered. We show that there arise singularities of the power boundary layer type, unconnected with boundary conditions. The solutions of this type are similar to the ``contrasting structures'' studied, for example, by \textit{V. F. Butnzov} and \textit{A. B. Vasil'eva} [Math. Notes 42, No. 6, 956-961 (1987); translation from Mat. Zametki 42, No. 6, 831-841 (1987; Zbl 0725.34061)] in analyzing nonlinear problems, but these solutions are of another nature.