Error bounds for asymptotic solutions of differential equations.I. Distinct eigenvalue case

The method of Olver for bounding the error term in the asymptotic so lutions of a second-order equation having a n irregular singul arity at infinity is extended to the general system of n first-order equations in the case when the eigenvalu es of the lead coefficient matrix are distinct. Vector and norm bounds are given for the difference between an actual solution vector and a partial su m of a formal so lution vector. Two cases are distinguished geometrica ll y: In one it is possible to express the error vec tor by a s ingle Volterra vec tor integral equat.ion; in the other it is necessary to use a simultaneous pair of Volterra vector integral equations. Some ne w inequalities for integral equations are given in an append ix.