OPTIMAL UNIFORM ESTIMATES AND RIGOROUS ASYMPTOTICS BEYOND ALL ORDERS FOR A CLASS OF ORDINARY DIFFERENTIAL-EQUATIONS

For first-order differential equations of the form y' = Sigma(p=0)(P) F-p(x)y(p) and second-order homogeneous linear differential equations y'' + a(x)y' + b(x)y = 0 with locally integrable coefficients having a symptotic (possibly divergent) power series when \x\ --> infinity on a ray arg(x) = const., under some further assumptions, it is shown that , on the given ray, there is a one-to-one correspondence between true solutions and (complete) formal solutions. The correspondence is based on asymptotic inequalities which are required to be uniform in a: and optimal with respect to certain weights.