Singular perturbation of boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters

The singularly perturbed boundary value problem for quasilinear third-order ordinary differential equation involving two small parameters has been con sidered. For the three cases epsilon/mu (2) --> 0(mu --> 0), mu (2)/epsilon --> 0(epsilon --> 0) and epsilon = mu (2), the formal asymptotic solutions are constructed by the method of two steps expansions and the existences o f solution are proved by using the differential inequality method. In addit ion, the uniformly valid estimations of the remainder term are given as wel l.