Periodic solutions of 𝑥”+𝑔(𝑥)+𝜇ℎ(𝑥)=0

Butler, G. J.; Freedman, H. I.

doi:10.1090/S0002-9947-1974-0342774-3

Periodic solutions of $x” + g(x) + \mu h(x) = 0$
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by G. J. Butler and H. I. Freedman PDF

Trans. Amer. Math. Soc. 197 (1974), 59-74 Request permission

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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