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Periodic Solutions by Picard's Approximations

Published online by Cambridge University Press:  20 November 2018

T.A. Burton
T.A. Burton*
Affiliation:
Southern Illinois University
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In [1] Demidovic considered a system of linear differential equations

with A(t) continuous, T-periodic, odd, and skew symmetric. He proved that all solutions of (1) are either T-periodic or 2T-periodic0 In [2] Epstein used Floquet theory to prove that all solutions of (1) are T-periodic without the skew symmetric hypothesis. Epstein's results were then generalized by Muldowney in [7] using Floquet theory. Much of the above work can also be interpreted as being part of the general framework of autosynartetic systems discussed by Lewis in [5] and [6]. According to private correspondence with Lewis it seems that he was aware of these results well before they were published. However, it appears that these theorems were neither stated nor suggested in the papers by Lewis.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 5 , December 1968 , pp. 743 - 745
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Demidovic, B.P., Moskov. Gos. Univ. UČ. Zap. 6 (1952) 123-132.Google Scholar
2. Epstein, I.J., Periodic solutions of systems of differential equations. Proc. Amer. Math. Soc. 13 (1962) 690-694.Google Scholar
3. Hurewicz, W., Lectures on ordinary differential equations. (Wiley, New York, 1958).Google Scholar
4. Lefschetz, S., Differential equations: geometric theory. (Interscience, New York, 1957).Google Scholar
5. Lewis, D.C., Autosynartetic solutions of differential equations Amer. J. Math. 83 (1961) 1-32.Google Scholar
6. Lewis, D. C., Int. Sym. nonlinear differential equations and nonlinear mechanics (Edited by LaSalle, J. P. and Lefschetz, S.). (Academic Press, New York, 1963) 99-104.Google Scholar
7. Muldowney, J. S., Linear systems of differential equations with periodic solutions. Proc. Amer. Math. Soc. 18 (1967) 22-27.Google Scholar