Solution in Banach algebras of differential equations with irregular singular point

Abstract

We have to do with linear first-order differential equations W ' ( z ) = F ( z ) W ( z ) , where z is a complex variable, and F and W are functions taking values in an arb i t rary non-commutat ive Banach algebra 91 with identi ty E. In [4], E. Hille has discussed the existence and nature of analytic solutions when F is holomorphic, near a regular point of F, and near a regular singular point, and has indicated how the theory will go when the equation has an irregular singular point at infinity of rank 1. The methods are adapted from the classical theory in which 91 is the complex field ~. The present paper adds to the discussion with an investigation, for the cases p~> 1, of the equation