Zhang, Qiongfen; Tang, X. H. Periodic solutions for second order Hamiltonian systems. (English) Zbl 1265.34154 Appl. Math., Praha 57, No. 4, 407-425 (2012). The authors consider the periodic boundary value problem for the Hamiltonian system \[ \begin{aligned} &\ddot {u}(t)=\nabla F(t,u(t))\; \; \text{for a.e. } t\in [0,T], \\ &u(0)-u(T)=\dot {u}(0)-\dot {u}(T=0)\end{aligned} \] and establish some existence theorems for periodic solutions. They use the technique of the least-action principle and the minimax method in critical point theory. The results extend those of Z. Wang and J. Zhang [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 12, 4480–4487 (2010; Zbl 1206.34060)]. Reviewer: Ladislav Adamec (Brno) MSC: 34C25 Periodic solutions to ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34B15 Nonlinear boundary value problems for ordinary differential equations 58E30 Variational principles in infinite-dimensional spaces Keywords:periodic solution; minimax method; second-order Hamiltonian system Citations:Zbl 1206.34060 × Cite Format Result Cite Review PDF Full Text: DOI Link References: [1] M. S. Berger, M. Schechter: On the solvability of semilinear gradient operator equations. Adv. Math. 25 (1977), 97–132. · Zbl 0354.47025 · doi:10.1016/0001-8708(77)90001-9 [2] J. Mawhin: Semi-coercive monotone variational problems. Bull. Cl. Sci., V. Sér., Acad. R. Belg. 73 (1987), 118–130. · Zbl 0647.49007 [3] J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems. Springer, New York, 1989. · Zbl 0676.58017 [4] J. Mawhin, M. Willem: Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance Ann. Inst. Henri. Poincaré, Anal. Non Linéaire 3 (1986), 431–453. · Zbl 0678.35091 [5] P.H. Rabinowitz: On subharmonic solutions of Hamiltonian systems Commun. Pure Appl. Math. 33 (1980), 609–633. · doi:10.1002/cpa.3160330504 [6] P.H. Rabinowitz: Minimax methods in critical point theory with applications to differential equations. In: CBMS Reg. Conf. Ser. Math., Vol. 65. Am. Math. Soc., Providence, 1986. [7] C. L. Tang: Periodic solutions of non-autonomous second order systems with quasisubadditive potential. J. Math. Anal. Appl. 189 (1995), 671–675. · Zbl 0824.34043 · doi:10.1006/jmaa.1995.1044 [8] C. L. Tang: Periodic solutions of nonautonomous second order systems. J. Math. Anal. Appl. 202 (1996), 465–469. · Zbl 0857.34044 · doi:10.1006/jmaa.1996.0327 [9] C. L. Tang: Periodic solutions of nonautonomous second order systems with sublinear nonlinearity. Proc. Am. Math. Soc. 126 (1998), 3263–3270. · Zbl 0902.34036 · doi:10.1090/S0002-9939-98-04706-6 [10] C. L. Tang, X. P. Wu: Periodic solutions for second order systems with not uniformly coercive potential. J. Math. Anal. Appl. 259 (2001), 386–397. · Zbl 0999.34039 · doi:10.1006/jmaa.2000.7401 [11] M. Willem: Oscillations forcees de systémes hamiltoniens. In: Public. Semin. Analyse Non Linéaire. Univ. Besancon, 1981. [12] X. Wu: Saddle point characterization and multiplicity of periodic solutions of nonautonomous second order systems. Nonlinear Anal., Theory Methods Appl. 58 (2004), 899–907. · Zbl 1058.34053 · doi:10.1016/j.na.2004.05.020 [13] X.P. Wu, C. L. Tang: Periodic solutions of a class of nonautonomous second order systems. J. Math. Anal. Appl. 236 (1999), 227–235. · Zbl 0971.34027 · doi:10.1006/jmaa.1999.6408 [14] F.K. Zhao, X. Wu: Periodic solutions for a class of non-autonomous second order systems. J. Math. Anal. Appl. 296 (2004), 422–434. · Zbl 1050.34062 · doi:10.1016/j.jmaa.2004.01.041 [15] F.K. Zhao, X. Wu: Existence and multiplicity of periodic solutions for non-autonomous second-order systems with linear nonlinearity. Nonlinear Anal., Theory Methods Appl. 60 (2005), 325–335. · Zbl 1087.34022 [16] X.H. Tang, Q. Meng: Solutions of a second-order Hamiltonian system with periodic boundary conditions. Nonlinear Anal., Real World Appl. 11 (2010), 3722–3733. · Zbl 1207.34027 · doi:10.1016/j.nonrwa.2010.02.002 [17] Z.Y. Wang, J.H. Zhang: Periodic solutions of a class of second order non-autonomous Hamiltonian systems. Nonlinear Anal., Theory Methods Appl. 72 (2010), 4480–4487. · Zbl 1206.34060 · doi:10.1016/j.na.2010.02.023 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.