×

Saddle point reduction method for some non-autonomous second order systems. (English) Zbl 1053.34039

Nonautonomous second-order systems of the form \(\ddot u(t)=\nabla F(t,u(t))\) are considered. It is assumed that \(F(t,x)\) is measurable in \(t\in [0,T]\) for every fixed vector \(x\) and is continuously differentiable in \(x\) for almost every fixed \(t\). The potential is allowed to be nonconvex and noncompact. By means of the technique of critical points, sufficient conditions are obtained for the existence of periodic solutions that possess saddle point character with respect to the corresponding functional.

MSC:

34C25 Periodic solutions to ordinary differential equations
47J30 Variational methods involving nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Amann, H., Saddle points and multiple solutions of differential equations, Math. Z., 169, 127-166 (1979) · Zbl 0414.47042
[2] Berger, M. S.; Schechter, M., On the solvability of semi-linear gradient operator equations, Adv. Math., 25, 97-132 (1977) · Zbl 0354.47025
[3] Ekeland, I.; Ghoussoub, N., Selected new aspects of the calculus variational in the large, Bull. Amer. Math. Soc., 39, 207-265 (2002) · Zbl 1064.35054
[4] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0676.58017
[5] Mawhin, J., Semi-coercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci. (5), 73, 118-130 (1987) · Zbl 0647.49007
[6] Tang, C.-L., Periodic solutions for non-autonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126, 3263-3270 (1998) · Zbl 0902.34036
[7] Thews, K., A reduction method for some nonlinear Dirichlet problems, Nonlinear. Anal., 3, 795-813 (1979) · Zbl 0419.35027
[8] Willem, M., Oscillations forcées de systèmes hamiltoniens, (Publ. Sémin. Analyse Nonlinéaire (1981), Univ. Besancon) · Zbl 0482.70020
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.