## Periodic solutions for a class of new superquadratic second order Hamiltonian systems.(English)Zbl 1314.34090

Summary: A new superquadratic growth condition is introduced, which is an extension of the well-known superquadratic growth condition due to P.H. Rabinowitz and the nonquadratic growth condition due to Gui-Hua Fei. An existence theorem is obtained for periodic solutions of a class of new superquadratic second order Hamiltonian systems by the minimax methods in critical point theory, specially, a local linking theorem.

### MSC:

 34C25 Periodic solutions to ordinary differential equations
Full Text:

### References:

 [1] Berger, M. S.; Schechter, M., On the solvability of semilinear gradient operator equations, Adv. Math., 25, 2, 97-132, (1977) · Zbl 0354.47025 [2] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (Applied Mathematical Sciences, vol. 74, (1989), Springer-Verlag New York), xiv+277 pp · Zbl 0676.58017 [3] Tang, C.-L.; Wu, X.-P., Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl., 259, 2, 386-397, (2001) · Zbl 0999.34039 [4] Mawhin, J., Semicoercive monotone variational problems, Acad. Roy. Belg. Bull. Cl. Sci.(5), 73, 3-4, 118-130, (1987) · Zbl 0647.49007 [5] Tang, C.-L.; Wu, X.-P., Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems, J. Differential Equations, 4, 660-692, (2010) · Zbl 1191.34053 [6] Tang, C.-L., Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc., 126, 11, 3263-3270, (1998) · Zbl 0902.34036 [7] Jiang, Q.; Tang, C.-L., Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., 328, 1, 380-389, (2007) · Zbl 1118.34038 [8] Rabinowitz, P. H., On subharmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 33, 5, 609-633, (1980) · Zbl 0425.34024 [9] Rabinowitz, P. H., On a class of functionals invariant under a $$Z^n$$ action, Trans. Amer. Math. Soc., 310, 1, 303-311, (1988) · Zbl 0718.34057 [10] Tang, C.-L.; Wu, X.-P., A note on periodic solutions of nonautonomous second-order systems, Proc. Amer. Math. Soc., 132, 5, 1295-1303, (2004) · Zbl 1055.34084 [11] Willem, M., Periodic oscillations of odd second order Hamiltonian systems, Boll. Unione Mat. Ital. B (6), 3, 2, 293-304, (1984) · Zbl 0582.58014 [12] Wu, X.-P.; Tang, C.-L., Periodic solutions of nonautonomous second-order Hamiltonian systems with even-typed potentials, Nonlinear Anal., 55, 6, 759-769, (2003) · Zbl 1030.37043 [13] Rabinowitz, P. H., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31, 2, 157-184, (1978) · Zbl 0358.70014 [14] Benci, V., Some critical point theorems and applications, Comm. Pure Appl. Math., 33, 2, 147-172, (1980) · Zbl 0472.58009 [15] Li, S. J.; Willem, M., Applications of local linking to critical point theory, J. Math. Anal. Appl., 189, 1, 6-32, (1995) · Zbl 0820.58012 [16] Fei, G., On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differential Equations, 8, 12, (2002) · Zbl 0999.37039 [17] Luan, S. X.; Mao, A. M., Periodic solutions of nonautonomous second order Hamiltonian systems, Acta Math. Sin. (Engl. Ser.), 21, 4, 685-690, (2005) · Zbl 1081.37009 [18] Schechter, M., Periodic non-autonomous second-order dynamical systems, J. Differential Equations, 2, 290-302, (2006) · Zbl 1099.34042 [19] Tao, Z.-L.; Yan, S.-A.; Wu, S.-L., Periodic solutions for a class of superquadratic Hamiltonian systems, J. Math. Anal. Appl., 331, 1, 152-158, (2007) · Zbl 1123.34311 [20] He, X.; Wu, X., Periodic solutions for a class of nonautonomous second order Hamiltonian systems, J. Math. Anal. Appl., 341, 2, 1354-1364, (2008) · Zbl 1133.37025 [21] Ye, Y.-W.; Tang, C.-L., Periodic solutions for some nonautonomous second order Hamiltonian systems, J. Math. Anal. Appl., 344, 1, 462-471, (2008) · Zbl 1142.37023 [22] Chen, G.; Ma, S., Ground state periodic solutions of second order Hamiltonian systems without spectrum 0, Israel J. Math., 198, 1, 111-127, (2013) · Zbl 1280.37053 [23] Li, X.; Su, J.; Tian, R., Multiple periodic solutions of the second order Hamiltonian systems with superlinear terms, J. Math. Anal. Appl., 385, 1, 1-11, (2012) · Zbl 1237.37042 [24] Chen, P.; Tang, X. H., New existence of homoclinic orbits for a second-order Hamiltonian system, Comput. Math. Appl., 62, 1, 131-141, (2011) · Zbl 1228.34064 [25] Zhang, Q.; Liu, C., Infinitely many periodic solutions for second order Hamiltonian systems, J. Differential Equations, 4-5, 816-833, (2011) · Zbl 1230.37081 [26] Aizmahin, N.; An, T., The existence of periodic solutions of non-autonomous second-order Hamiltonian systems, Nonlinear Anal., 74, 14, 4862-4867, (2011) · Zbl 1221.34174 [27] Tang, X. H.; Jiang, J., Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl., 59, 12, 3646-3655, (2010) · Zbl 1206.34059 [28] Luan, S.; Mao, A., Periodic solutions for a class of non-autonomous Hamiltonian systems, Nonlinear Anal., 61, 8, 1413-1426, (2005) · Zbl 1071.34039
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.