Infinitely many periodic solutions for a second-order nonautonomous system. (English) Zbl 1055.34082

The existence of solutions \(u\in C^1([0,T],\mathbb{R}^k)\) to the system \[ \ddot u= A(t)u+ b(t)\nabla G(u)\quad\text{a.e. in }[0,T],\quad u(0)- u(T)= \dot u(0)-\dot u(T)= 0,\tag{1} \] is investigated. Roughly speaking, it is shown that if \(G\) has a suitable oscillation behavior at infinity (or at zero), then (1) has an unbounded sequence of solutions (or a sequence of nonzero solutions tending to zero). The proofs are based on a recent local minima result by R. Ricceri.


34C25 Periodic solutions to ordinary differential equations
Full Text: DOI


[1] Antonacci, F.; Magrone, P., Second order nonautonomous systems with symmetric potential changing sign, Rend. mat. appl., 18, 2, 367-379, (1998) · Zbl 0917.58006
[2] Bahri, A.; Berestycki, H., Existence of forced oscillations for some nonlinear differential equations, Commun. pure appl. math., 37, 4, 403-442, (1984) · Zbl 0588.34028
[3] Benci, V., Some critical point theorems and applications, Commun. pure appl. math., 33, 2, 147-172, (1980) · Zbl 0472.58009
[4] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer New York · Zbl 0676.58017
[5] Ricceri, B., A general variational principle and some of its applications, J. comput. appl. math., 113, 401-410, (2000) · Zbl 0946.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.