### Uniqueness and the convergence of successive approximations. II

A. D. Ziebur
1965 Proceedings of the American Mathematical Society
It has long been known that the uniqueness of the solution of an ordinary differential problem of the type of (1) below and the convergence of sequences of successive approximations (Picard sequences) to solutions are logically independent. Thus Brauer and Sternberg [2] list examples (due to Müller and Dieudonné) in which there is uniqueness but not convergence or convergence but not uniqueness. Nevertheless, uniqueness and convergence are closely related, and we have recently shown [lO], for a
more » ... special type of equation, and then only in the case w = l, that we can associate with a given differential problem another problem in such a way that the uniqueness of the solution of the associated problem guarantees the convergence of sequences of successive approximations to the (necessarily unique) solution of the original problem. In this note we remove the restriction to a "special type" of equation and extend the results to systems of n equations, where n is an arbitrary positive integer. If the functions/, in problem (1) satisfy a Lipschitz condition, then the problem has a unique solution, and this solution is the limit of sequences of successive approximations. Over the years, many weaker "Lipschitz-like" conditions which guarantee uniqueness and convergence have been found. Presently known conditions of this kind guarantee the uniqueness of the solution of our associated problem, and in this sense our results are generalizations of previous work. We discuss this point in more detail after we state our main theorem. We consider a differential problem of order n, that is, a system of n differential equations with a given initial condition :