# zbMATH — the first resource for mathematics

Existence of T-periodic solutions for a class of Lagrangian systems. (English) Zbl 0709.34034
We study the following Lagrangian system of differential equations: $(*)\quad \frac{d}{dt}\frac{\partial {\mathcal L}}{\partial \xi}(q,\dot q,t)- \frac{\partial {\mathcal L}}{\partial q}(q,\dot q,t)=0,\quad q\in C^ 2({\mathbb{R}},{\mathbb{R}}^ N)$ where $${\mathcal L}$$ denotes the Lagrangian function ${\mathcal L}(q,q,t)=\frac{1}{2}\sum^{N}_{i,j=1}a_{ij}(q)\xi_ i\xi_ j-V(q- t),\quad q,\xi \in {\mathbb{R}}^ N,\quad t\in {\mathbb{R}}.$ First we state the existence of multiple periodic solutions of prescribed period when the potential V is unbounded and subquadratic at infinity. Then we prove the existence of at least one nontrivial solution of problem (*) when the potential $$V=V(q,t)$$ is bounded and T-periodic. Finally we consider the following forced Lagrangian system: $(**)\quad \frac{d}{dt}\frac{\partial {\mathcal L}}{\partial \xi}(q,\dot q)- \frac{\partial {\mathcal L}}{\partial q}(q,\dot q)=g(t)$ where g is a T- periodic function; we prove the existence of at least one T-periodic solution of problem (**) when the potential is unbounded and subquadratic at infinity. The results contained in this paper are obtained using variational methods and generalize some well known results referred to the case of the $$a_{ij}'s$$ constant.
Reviewer: E.Mirenghi
##### MSC:
 34C25 Periodic solutions to ordinary differential equations
Full Text:
##### References:
 [1] P. Bartolo - V. BENCI - D. FORTUNATO, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity , Nonlinear Analysis T.M.A. , 7 ( 1983 ), pp. 981 - 1012 . MR 713209 | Zbl 0522.58012 · Zbl 0522.58012 [2] V. Benci , A geometrical index for the group S1 and some applications to the study of periodic solutions of ordinary differential equations , Comm. Pure Appl. Math. , 34 ( 1981 ), pp. 393 - 432 . MR 615624 | Zbl 0447.34040 · Zbl 0447.34040 [3] V. Benci - A. CAPOZZI - D. FORTUNATO, Periodic solutions of Hamiltonian systems with superquadratic potential , Ann. Mat. Pura e Appl. , 143 ( 1986 ), pp. 1 - 46 . MR 859596 | Zbl 0632.34036 · Zbl 0632.34036 [4] H. Berestycki , Solutions périodiques de systèmes Hamiltoniens , Sém. Bourbaki, 35e année ( 1982-83 ), no. 603 . Numdam | MR 728984 | Zbl 0526.58016 · Zbl 0526.58016 [5] A. Capozzi - D. Fortunato - A. Salvatore , Periodic solutions of dynamical systems , Meccanica , 20 ( 1985 ), pp. 281 - 284 . MR 841217 | Zbl 0599.70010 · Zbl 0599.70010 [6] A. Capozzi - D. Fortunato - A. Salvatore , Periodic solutions of Lagrangian systems with a bounded potential , J. Math. Anal. and Appl. , 124 , 2 ( 1987 ), pp. 482 - 494 . MR 887004 | Zbl 0664.34053 · Zbl 0664.34053 [7] A. Capozzi - D. Lupo - S. Solimini , On the existence of a nontrivial solution to nonlinear problems at resonance , Nonlinear Analysis T.M.A. , 13 , 2 ( 1989 ), pp. 151 - 163 . MR 979038 | Zbl 0684.35038 · Zbl 0684.35038 [8] A. Capozzi - A. Salvatore , Periodic solutions for nonlinear problems with strong resonance at infinity , Comm. Math. Univ. Car. , 23 , 3 ( 1982 ), pp. 415 - 425 . MR 677851 | Zbl 0507.34035 · Zbl 0507.34035 [9] V. Coti Zelati , Periodic solutions of Hamiltonian systems and Morse theory, Atti del Convegno Recent advances in Hamiltonian systems , L ’ Aquila ( 1986 ), pp. 155 - 161 . MR 902630 | Zbl 0652.34053 · Zbl 0652.34053 [10] V. Coti Zelati , Periodic solutions of dynamical systems with bounded potential , J. Diff. Eq. , 67 , 3 ( 1987 ), pp. 400 - 413 . MR 884277 | Zbl 0646.34049 · Zbl 0646.34049 [11] F. Giannone , Periodic solutions of dynamical systems by the saddle point theorem of P. H. Rabinowitz , Nonlinear Analysis T.M.A. , 13 , 6 ( 1989 ), pp. 707 - 719 . MR 998515 | Zbl 0729.58044 · Zbl 0729.58044 [12] P.H. Rabinowitz , Some minimax theorems and applications to nonlinear partial differential equations , Nonlinear Analysis (Cesari, Kannan, Wainberger Editors), Academic Press ( 1978 ), pp. 161 - 177 . MR 501092 | Zbl 0466.58015 · Zbl 0466.58015 [13] P.H. Rabinowitz , Periodic solutions of Hamiltonian systems: a survey , SIAM J. Math. Anal. , 13 ( 1982 ), pp. 343 - 352 . MR 653462 | Zbl 0521.58028 · Zbl 0521.58028 [14] A. Salvatore , Periodic solutions of Hamiltonian systems with a sub-quadratic potential , Boll. Un. Mat. Ital. , 3-C ( 1984 ), pp. 393 - 406 . MR 749296 | Zbl 0546.34034 · Zbl 0546.34034
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.