On the complex differential equation Y ''+G(z)Y=0 in Banach algebras

Asymptotic representations, as z --> infinity, are presented as a basis of solutions to linear complex differential equations in the framework of Bana ch algebras, such as d(k)Y/dz(k) + G(z)Y = 0, k = 1,2, z is an element of O mega subset of or equal to C. Here Omega is open, unbounded, and simply con nected, and the coefficient G(z) is assumed to be "asymptotically negligibl e," in the sense that suitable "moments" of parallel to G(z)parallel to are finite on certain paths in Omega. Precise pathwise as well as uniform boun ds for the asymptotic error terms are obtained by exploiting the geometric properties of the paths via the successive approximations method. Such resu lts extend to the complex domain in previous work on matrix and abstract di fferential equations on the real domain, and also appear new for scalar and matrix differential equations on complex domains other than sectors.