On the Complex Differential Equation Y"+G(z)Y=0 in Banach Algebras

Asymptotic representations, as z →∞, are presented as a basis of solutions to linear complex differential equations in the framework of Banach algebras, such as d^kY / dz^k + G ( z ) Y =0, k =1, 2, z ∈Ω⊆ C . Here Ω is open, unbounded, and simply connected, and the coefficient G ( z ) is assumed to be "asymptotically negligible," in the sense that suitable "moments" of ‖ G ( z )‖ are finite on certain paths in Ω. Precise pathwise as well as uniform bounds for the asymptotic error terms are obtained by exploiting the geometric properties of the paths via the successive approximations method. Such results extend to the complex domain in previous work on matrix and abstract differential equations on the real domain, and also appear new for scalar and matrix differential equations on complex domains other than sectors.