On Contrast Structures with a Multizonal Interior Layer

Abstract

A boundary value  problem  for a singularly  perturbed differential  equation  of second order  is considered  in two  cases, when one root  of the  degenerate equation  is two-tuple.   It  is proved that in the  first  case the  problem  has  a solution  with  the  transition from  the  two-tuple root  of the degenerate equation  to one-tuple  root  in the  small neighbourhood of an internal  point of the  interval, and in the second case the problem has a solution which has the spike in the interior  layer.  Such solutions are named,  correspondingly, a contrast structure of step-type and a contrast structure of spike-type.  In each case the  asymptotic expansion  of the  contrast structure is constructed.  It distinguishes from the known expansion  in the case, when all the roots of the degenerate equation  are one-tuple,  in particular, the interior  layer is multizonal.