## On periodic solutions of subquadratic second order non-autonomous Hamiltonian systems.(English)Zbl 1319.34071

Summary: We are concerned with the existence of periodic solutions for second order non-autonomous Hamiltonian systems under a new subquadratic growth condition. By using the minimax methods in critical point theory, an existence theorem is obtained, which extends and improves some known results in the literature.

### MSC:

 34C25 Periodic solutions to ordinary differential equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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### References:

 [1] Tang, C. L.; Wu, X. P., Notes on periodic solutions of subquadratic second order systems, J. Math. Anal. Appl., 285, 8-16, (2003) · Zbl 1054.34075 [2] Tang, C. L.; Wu, X. P., Periodic solutions for a class of new superquadratic second order Hamiltonian systems, Appl. Math. Lett., 34, 65-71, (2014) · Zbl 1314.34090 [3] Ekeland, I.; Ghoussoub, N., Certain new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc., 39, 207-265, (2002) · Zbl 1064.35054 [4] Mawhin, J.; Wliiem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York [5] Pipan, J.; Schechter, M., Non-autonomous second order Hamiltonian systems, J. Differential Equations, 257, 351-373, (2014) · Zbl 1331.37085 [6] Schechter, M., Periodic nonautonomous second order dynamical systems, J. Differential Equations, 223, 290-302, (2006) · Zbl 1099.34042 [7] Rabinowitz, P., On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 33, 609-633, (1980) · Zbl 0425.34024 [8] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., 7, 981-1012, (1983) · Zbl 0522.58012 [9] Rabinowitz, P., Minimax methods in critical point theory with applications to differential equations, (CBMS Reg. Conf. Ser. Math., vol. 65, (1986), Amer. Math. Soc Providence. RI) [10] Jiang, Q.; Tang, C. L., Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 328, 380-389, (2007) · Zbl 1118.34038 [11] Ma, S.; Zhang, Y., Existence of infinitely many periodic solutions for ordinary $$p$$-Laplacian systems, J. Math. Anal. Appl., 351, 469-479, (2009) · Zbl 1153.37009 [12] Zou, W., Multiple solutions for second-order Hamiltonian systems via computation of the critical groups, Nonlinear Anal., 44, 975-989, (2001) · Zbl 0997.37039 [13] Tang, X. H.; Jiang, J., Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl., 59, 3646-3655, (2010) · Zbl 1206.34059 [14] Wang, Z.; Zhang, J., Periodic solutions of a class of second order non-autonomous Hamiltonian systems, Nonlinear Anal., 72, 4480-4487, (2010) · Zbl 1206.34060 [15] Wang, Z.; Zhang, J.; Zhang, Z., Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential, Nonlinear Anal., 70, 3672-3681, (2009) · Zbl 1179.34037
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