Some Results on Uniqueness and Successive Approximations

Abstract

In the theory of ordinary differential equations, there is a strange relationship between uniqueness of solutions and convergence of the successive approxi mations. There are examples of differential equations with unique solutions for which the successive approximations do not converge (8) and of differential equations with non-unique solutions for which the successive approximations do converge (2). However, in spite of the known logical independence of these two properties, almost all conditions which assure uniqueness also imply the convergence of the successive approximations. For example, the hypotheses of Kamke's general uniqueness theorem (5), have been shown by Coddington and Levinson to suffice for the convergence of successive approximations, after the addition of one simple monotonicity condition (4). There is one counterexample to this “principle,” a generalization of Kamke's result, to which another condition in addition to a monotonicity assumption must be added before convergence of the successive approximations can be proved (2).