A Relation between formal and Asymptotic Equivalence for linear Systems Containing A Parameter

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ε^Pdy/dx = A(x,ε)y with A holomorphic at (x₀,0), suppose that x₀ is formally not a turning point in the sense that no singularities appear at x₀ during the standard formal solution procedure with formal fractional power series in ε. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x₀. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.