GEVREY ASYMPTOTIC REPRESENTATION OF THE SOLUTIONS OF EQUATIONS WITH ONE TURNING-POINT

An ordinary differential equation of the type D(t)(m)u+j+/alpha/less t han or equal to m, j<m Sigma a(j,a)(t)xi(alpha)D(t)(j)u=0 with paramet er xi is an element of IR(n) and smooth coefficients a(j,a)is an eleme nt of C-infinity([-T,T]) is studied. It is assumed that all the charac teristic roots of the equation vanish at t=0 while for t not equal 0 t hey are real and distinct. The constructions of real-valued phase func tions phi(kl) (k, l=1,..., m) and of amplitude functions A(jkl) such t hat for a given s is an element of[-T, T] every solution u(t, xi) of t he equation can be represented as u(t,xi)=>(j=0)Sigma(m-1) (l,k=1)Sigm a(m) e(i phi kl(t,s,xi))A(jkl)(t,s,xi)psi(j)(s,xi), where psi(j)(s,xi) =D(t)(j)u(s,xi), j=0,...,m-1, are given.