ESTIMATES FOR THE GREEN FUNCTIONS OF NONAUTONOMOUS HIGHER ORDER DIFFERENTIAL EQUATIONS

We consider the equation Sigma(n)(k=0) a(k)(t)x((n-k))(t) = 0 (t > 0), where a(0)(t) 1; a(k)(t) (k = 1,...,n) are bounded continuous functions. It is assumed that all the roots r(k)(t) (k = 1,...,n) of the polynomial z(n) + a(1)(t)z(n-1) +...+ a(n)(t) are real for all t >= 0. Sharp estimates for the Green function to the Cauchy problem and their derivatives are derived.