Most of the successive approximations do converge

Abstract

For an example see Coddington-Levinson [l, p. 531. We show here that this situation should be looked at as extraordinary since in reality the successive approximations do converge for most differential equations. For, we prove that “convergence of successive approximations” is a generic property, i.e., that there is a dense second category set M* in the space of continuous functions such that, for all f E M*, the successive approximations converge. Our result has a greater generality since it shows that for f E M* the successive approximations have a unique limit for every starting point y. . Thus we have as corollary that “uniqueness of solution” is a generic property, as proved earlier by Orlicz [3] for f’ d fi s e ne on strips I x R” and by Lasotad Yorke [2] for f’ d fi d s e ne on open subsets of any Banach space.