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Necessary and sufficient conditions for      x + f ( x ) = 0 x + f(x) = 0       to admit at least one nontrivial periodic solution are given. The results are applied to      x + g ( x ) + μ  h ( x ) = 0 , x ( 0 ) = A , x ′ ( 0 ) = 0 x + g(x) + \mu h(x) = 0,x(0) = A,x’(0) = 0       in order to characterize those regions of the      ( μ  , A ) (\mu ,A)      -plane for which nontrivial periodic solutions exist. A converse theorem is given; together with some illustrative examples.

Periodic solutions of 𝑥”+𝑔(𝑥)+𝜇ℎ(𝑥)=0