Asymptotics of the fundamental system of solutions of a linear functional-differential equation with respect to a parameter

Abstract

We study a functional-differential equation, where F is a linear operator acting from the Hölder space Hγ into the Sobolev space W p s [0, 1] and ρ is a complex parameter. For large absolute values of ρ, we construct a one-to-one correspondence between the solutions x(ρ;t) and y(ρ;t) of the equations and y(n)+ρyn=0. We also establish conditions that should be imposed on the operatorF in order that specially selected fundamental systems of solutions of these equationsx j (ρ;t) andy j (ρ;t), j=1,...,n, satisfy the estimate with constantsc, κ>0 for the functional space.