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 Λ . 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 Λ . 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 Λ (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.