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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Periodic solutions of $ x'' + g(x) + \mu h(x) = 0$


Authors: G. J. Butler and H. I. Freedman
Journal: Trans. Amer. Math. Soc. 197 (1974), 59-74
MSC: Primary 34C25
DOI: https://doi.org/10.1090/S0002-9947-1974-0342774-3
MathSciNet review: 0342774
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Abstract: Necessary and sufficient conditions for $ x'' + f(x) = 0$ to admit at least one nontrivial periodic solution are given. The results are applied to $ x'' + g(x) + \mu h(x) = 0,x(0) = A,x'(0) = 0$ in order to characterize those regions of the $ (\mu ,A)$-plane for which nontrivial periodic solutions exist. A converse theorem is given; together with some illustrative examples.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0342774-3
Keywords: Ordinary differential equation, periodic solution, boundary curves, admissible sets
Article copyright: © Copyright 1974 American Mathematical Society