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A case of boundedness in Littlewood's problem on oscillatory differential equations

Published online by Cambridge University Press:  17 April 2009

G.R. Morris
G.R. Morris
Affiliation:
Department of Mathematics, University of New England, Armidale, New South Wales.
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It is shown that all solutions of ẍ + 2x3 = p(t) are bounded, the notation indicating that p is periodic. It is not necessary to have a small parameter multiplying p.

The essential step is to show by appeal to Moser's theorem that, under the mapping (of the initial-value plane) which corresponds to the equation, there are invariant simple closed curves. This implies also that there is an uncountable infinity of almost-periodic solutions and, for each positive integer m, an infinity of periodic solutions of least period 2mπ (2π being taken as the least period of p ).

It is suggested that for a large class of equations the same attack would show all solutions of ẍ + g(x) = p(t) bounded. However, in order to show the method clearly, no generalisation is attempted here.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 1 , February 1976 , pp. 71 - 93
Copyright
Copyright © Australian Mathematical Society 1976

References

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