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A boundedness theorem for non-linear differential equations of the second order

Published online by Cambridge University Press:  24 October 2008

G. E. H. Reuter
G. E. H. Reuter
Affiliation:
The UniversityManchester
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Extract

1. This paper deals with the differential equation

(dots denoting derivatives with respect to t), where for large x the ‘restoring force’ term g(x) has the sign of x and the ‘damping factor’ kf(x) is positive on the average. It will be shown that every solution of (1) ultimately (for sufficiently large t) satisfies

with B independent of k. The conditions on f(x), g(x) and p(t) (stated in §§ 2, 3) are rather milder than those assumed by Cartwright and Littlewood (1, 2) and Newman (3) in proving similar results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 1 , January 1951 , pp. 49 - 54
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

REFERENCES

(1)Cartwright, M. L. and Littlewood, J. E.Ann. Math. 48 (1947), 472–94.Google Scholar
(2)Cartwright, M. L.Ann. Math. Stud. no. 20 (Princeton, 1950), 142–241.Google Scholar
(3)Levinson, N.J. Math. Phys. 22 (1943), 181–7.CrossRefGoogle Scholar
(4)Levinson, N.Ann. Math. 45 (1944), 723–37.CrossRefGoogle Scholar
(5)Newman, M. H. A.Compositio Math. 8 (1950), 142–56.Google Scholar