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On the existence of periodic solutions of a certain third-order differential equation

Published online by Cambridge University Press:  24 October 2008

J. O. C. Ezeilo
J. O. C. Ezeilo
Affiliation:
University CollegeIbadan
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Extract

In this paper we shall be concerned with the differential equation

in which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:

under which every solution x(t) of (1) satisfies

where t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),

or, in the case (3),

In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 56 , Issue 4 , October 1960 , pp. 381 - 389
Copyright
Copyright © Cambridge Philosophical Society 1960

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References

REFERENCES

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