The author examined non-zero $T$-periodic (in time) solutions for a semilinear beam equation under the condition that the period $T$ is an irrational multiple of the length. It is shown that for a.e. $T \in R^1$ (in the sense of the Lebesgue measure on $R^1$) the solutions do exist provided the right-hand side of the equation is sublinear.
@article{104291,
author = {Eduard Feireisl},
title = {On the existence of free vibrations for a beam equation when the period is an irrational multiple of the length},
journal = {Applications of Mathematics},
volume = {33},
year = {1988},
pages = {94-102},
zbl = {0684.35057},
mrnumber = {0940709},
language = {en},
url = {http://dml.mathdoc.fr/item/104291}
}
Feireisl, Eduard. On the existence of free vibrations for a beam equation when the period is an irrational multiple of the length. Applications of Mathematics, Tome 33 (1988) pp. 94-102. http://gdmltest.u-ga.fr/item/104291/
Periodic solutions of a nonlinear wave equation without assumption of monotonicity, Math. Ann. 262 (1983), 273-285. (1983) | Article | MR 0690201 | Zbl 0489.35061
Multiple critical points of invariant functional and applications, Séminaire de Mathématique 2-éme Semestre Université Catholique de Louvain.
Convex analysis and variational problems, North-Holland Publishing Company 1976. (1976) | MR 0463994
A linear and weakly nonlinear equation of a beam: the boundary value problem for free extremities and its periodic solutions, Czechoslovak Math. J. 21 (1971), 535-566. (1971) | MR 0289918