Asymptotic solutions of boundary value problems for third-order ordinary differential equations with turning points

Boundary value problems for third-order ordinary differential equations with turning points are studied as follows: ∈y′″+f(x;∈)y″+g(x;∈)y′+h(x;∈)y=0 (−a<x<b, 0<∈<1), where f(x;0) has several multiple zero points in (−a, b). The necessary conditions for exhibiting resonance is given, and the uniformly valid asymptotic solutions and the estimations of remainder terms are obtained.