A. Fonda et al., SUBHARMONIC SOLUTIONS FOR SOME 2ND-ORDER DIFFERENTIAL-EQUATIONS WITH SINGULARITIES, SIAM journal on mathematical analysis, 24(5), 1993, pp. 1294-1311
The existence of infinitely many subharmonic solutions is proved for t
he periodically forced nonlinear scalar equation u'' + g(u) = e(t), wh
ere g is a continuous function that is defined on a open proper interv
al (A, B) subset-of R. The nonlinear restoring field g is supposed to
have some singular behaviour at the boundary of its domain. The follow
ing two main possibilities are analyzed: (a) The domain is unbounded a
nd g is sublinear at infinity. In this case, via critical point theory
, it is possible to prove the existence of a sequence of subharmonics
whose amplitudes and minimal periods tend to infinity. (b) The domain
is bounded and the periodic forcing term e(t) has minimal period T > 0
. In this case, using the generalized Poincare-Birkhoff fixed point th
eorem, it is possible to show that for any m is-an-element-of N, there
are infinitely many periodic solutions having mT as minimal period. A
pplications are given to the dynamics of a charged particle moving on
a line over which one has placed some electric charges of the same sig
n.