Singularly Perturbed Boundary Value Problems in Case of Exchange of StabilitiesU

We consider a mixed boundary value problem for a system of two second-order nonlinear differential equations where one equation is singularly perturbed. We assume that the associated equation has two intersecting families of equilibria. This property excludes the application of standard results. By means of the method of upper and lower solutions, we prove the existence of a solution of the boundary value problem and determine its asymptotic behavior with respect to the small parameter. The results can be used to study differential systems modeling bimolecular reactions with fast reaction rates.