Existence and Multiplicity Results for Periodic Solutions of Semilinear Duffing Equations

We obtain existence an multiplicity of harmonic and subharmonic solutions to the periodically forced nonlinear Duffing′s equation x″ + g(x) = p(t) in the semi-linear case: 0 < A ≤ g(x)/x ≤ B < + ∞, for x large. We introduce new conditions of nonresonance with respect to the spectrum of the associated linear differential operator x −x″ and show, by some nonlinear counterexamples, that the results are sharp.