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A Note on Uniqueness and Convergence of Successive Approximations

Published online by Cambridge University Press:  20 November 2018

Fred Brauer
Fred Brauer*
Affiliation:
University of Chicago and University of British Columbia
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In a recent note in this Bulletin [3], W.A.J. Luxemburg has shown in two different ways that a condition of Krein and Krasnosel'skii [2] for the uniqueness of solutions of a differential equation also implies the convergence of the successive approximations. Here, a third proof of the uniqueness and the convergence of successive approximations, formulated for systems of differential equations, will be obtained. This third proof is modelled on the methods used in proving general uniqueness and convergence theorems [l]. The approach is suggested by Luxemburg's idea of breaking the argument into two stages and using one of the hypotheses in each stage. Since the proofs given here are hardly shorter than the earlier direct proofs, their main interest lies in the fact that they fit what appeared to be an isolated result into the framework of a general theory.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 2 , Issue 1 , 01 January 1959 , pp. 5 - 8
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Coddington, E.A. and Levinson, Norman, Theory of ordinary differential equations, /(New York, 1955).Google Scholar
2. Krasnosel'skiĭ, M.A. and Krein, S.G., On a class of uniqueness theorems for the equation y'= f(x, y), Uspehi Mat. Nauk. (N.S.) 11 (1956), 209-213 (Russian).Google Scholar
3. Luxemburg, W.A.J., On the convergence of successive approximations in the theory of ordinary differential equations, Canadian Math. Bulletin 1 (1958), 9-20.Google Scholar