Olver’s asymptotic method revisited; Case I

We consider the asymptotic method designed by Olver [F.W.J. Olver, Asymptotics and Special Functions, Academic Press, 1974] for linear differential equations of the second order containing a large (asymptotic) parameter ΛΛ. We only consider here the first case studied by Olver: differential equations without turning or singular points. It is well-known that his method gives the Poincaré-type asymptotic expansion of two independent solutions of the equation in inverse powers of ΛΛ. In this paper we add two initial conditions to the differential equation and consider the corresponding initial value problem. By using the Green’s function of an auxiliary problem and a fixed point theorem, we construct a sequence of functions that converges to the unique solution of the problem. This sequence also has the property of being an asymptotic expansion for large ΛΛ (not of Poincaré type) of the solution of the problem. Moreover, we show that the idea may be applied to nonlinear differential equations with a large parameter.