Asymptotic Solutions of Boundary Value Problems for Third-Order Ordinary Differential Equations with Turning Points

Abstract

Boundary value problems for third-order ordinary differential equations with turning points are studied as follows: $\varepsilon y\prime \prime \prime + f\left( {x;\varepsilon } \right)y\prime \prime + g\left( {x;\varepsilon } \right)y\prime + h\left( {x;\varepsilon } \right)y = 0{\text{ }}\left( {{\text{ - }}a < x < b,0 < \varepsilon \ll 1} \right),$, 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.