On asymptotic integration of a differential equation with slowly varying and oscillating coefficients in Banach space (Q1110707)

The author and \textit{N. A. Sotnichinko} [Asymptotic representation of solution of differential equations with slowly varying and oscillating coefficients in a Banach space, Kiev, Ukr. VINITI, No.93, UK-85 (1984)] considered the problem of decomposing the first order system of differential equations (1) \(dx/dt=A(t,\tau,\epsilon)x\) where x(t,\(\epsilon)\) is a vector-function on [0,L/\(\epsilon\) ] taking values in a Banach space B and A(t,\(\tau\),\(\epsilon)\) is an operator function. In this paper the author constructs a solution for the case of a multiple eigenvalue of the principal part of \[ A(t,\epsilon,\tau)=A_ 0(\tau)+\sum^{\infty}_{s=1}\sum^{+\infty}_{m=-\infty}\epsilon^ se^{im\omega t}A_{sm}(\tau). \]