Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities

Abstract

Abstract. We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equationThe hypotheses we assume on the nonlinearity g(x) allow us to cover the case b = +∞ (or a = −∞) and g having superlinear growth at infinity, as well as the case b < +∞ (or a > −∞) and g having a singularity in b (respectively in a). Applications are given also to situations like g (−∞) = g (+∞) (including the so-called "jumping nonlinearities"). Our results are connected to the periodic Ambrosetti -Prodi problem and related problems arising from the Lazer -McKenna suspension bridges model.

Multiplicity of periodic solutions for differential equations arising in the study of a nerve fiber model

“…Obviously, also the solutions ũk , ûk ũℓ , ûℓ for ℓ = k belong to different periodicity classes (in fact their rotation numbers are different). We refer also to [13], [18] for a throughout discussion concerning this aspect. Moreover, we observe that the information about the associated rotation numbers permits obtain some conclusions about the minimality of the period.…”

Periodic oscillations of forced pendulums with very small length

“…Ding's theorem then yields the existence of N periodic solutions having period k~T En and making a prime number of rotations around the origin in the time kT En . It is now easy to check that these solutions cannot have a smaller period in the set {T E , 2T E ,..., (ii -l)T E } (see [2,14]). The proof of the theorem can now be easily completed.…”

Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

“…In the study of multiple periodic solutions of the second order equation or planar Hamiltonian systems, Jacobowitz [19] firstly applied the Poincaré-Birkhoff twist theorem to prove the existence of infinitely many periodic solutions for superlinear second order equations (also see Hartman [18] for refined version). Later, many interesting results with the related researches have been obtained, see, for example, [4,25,6,23,22,24] and the references therein.…”