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Multiplicity of periodic solutions for a class of second-order perturbed Hamiltonian systems. (English) Zbl 1477.34066

The authors consider the following second-order Hamiltonian systems with non-autonomous perturbed term
\[\left\{\begin{array}{l} \ddot{u}(t)+\nabla F(u) = \nabla_u G(t,u), ~~t \in \mathbb R,\\ u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0,~~ T>0, \end{array}\right.\] where \(F(u)=-K(u)+W(u),~K,~W \in C^2(\mathbb R^N,\mathbb R),~G \in C^2(\mathbb R \times \mathbb R^N,\mathbb R)\) with \(G(t+T,u)=G(t,u)\), \(K\) and \(W\) are even functions.
It is assumed that \(G(t,u)\) has no parity in \(u\) and is subquadratic at infinity, \(W\) is superquadratic at infinity and \(K(u)/|u|^2\) is bounded by two constants. By using Bolle’s perturbation method, the authors prove the existence of infinitely many periodic solutions. This extends some results in the current literature.

MSC:

34C25 Periodic solutions to ordinary differential equations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34D10 Perturbations of ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
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[1] Bahri, A.; Berestycki, H., A perturbation method in critical point theory and applications, Trans. Am. Math. Soc., 267, 1, 1-32 (1981) · Zbl 0476.35030
[2] Bahri, A.; Berestycki, H., Existence of forced oscillations for some nonlinear differential equations, Commun. Pure Appl. Math., 37, 4, 403-442 (1984) · Zbl 0588.34028
[3] Bolle, P., On the Bolza problem, J. Differ. Equ., 152, 2, 274-288 (1999) · Zbl 0923.34025
[4] Bolle, P.; Ghoussoub, N.; Tehrani, H., The multiplicity of solutions in non-homogeneous boundary value problems, Manuscr. Math., 201, 3, 325-350 (2000) · Zbl 0963.35001
[5] Bonanno, G.; Livrea, R., Periodic solutions for a class of second-order Hamiltonian systems, Electron. J. Differ. Equ., 2005, 115, 357-370 (2005)
[6] Ekeland, I.; Ghoussoub, N.; Tehrani, H., Multiple solutions for a classical problem in the calculus of variations, J. Differ. Equ., 131, 2, 229-243 (1996) · Zbl 0872.49001
[7] Gu, H.; An, T., Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems, Bound. Value Probl., 2013, 1, 1-8 (2013) · Zbl 1294.34047
[8] Long, Y., Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. Am. Math. Soc., 311, 2, 749-780 (1989) · Zbl 0676.34026
[9] Long, Y., Index Theory for Symplectic Paths with Applications, vol. 207 (2002), Birkhäuser: Birkhäuser Basel · Zbl 1012.37012
[10] Mawhin, J.; Willem, M., Critical Point Theory and Hamiltonian Systems, vol. 74 (1989), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0676.58017
[11] Rabinowitz, P., Periodic solutions of Hamiltonian systems, Commun. Pure Appl. Math., 31, 2, 157-184 (1978) · Zbl 0358.70014
[12] Rabinowitz, P., Multiple critical points of perturbed symmetric functionals, Trans. Am. Math. Soc., 272, 2, 753-769 (1982) · Zbl 0589.35004
[13] Salvatore, A., Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems, Discrete Contin. Dyn. Syst., 2003, 1, 778-787 (2003) · Zbl 1073.34046
[14] Salvatore, A., Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Stud., 3, 1, 1-23 (2003) · Zbl 1221.35400
[15] Struwe, M., Infinitely many critical points for functionals which are not even and applications to nonlinear boundary value problems, Manuscr. Math., 32, 3, 335-364 (1980) · Zbl 0456.35031
[16] Tang, C., Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Am. Math. Soc., 126, 11, 3263-3270 (1998) · Zbl 0902.34036
[17] Tang, X.; 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
[18] Tang, C.; Wu, X., Periodic solutions for a class of nonautonomous subquadratic second order Hamiltonian systems, J. Math. Anal. Appl., 275, 2, 870-882 (2002) · Zbl 1043.34045
[19] Wang, Z.; Zhang, J., New existence results on periodic solutions of non-autonomous second order Hamiltonian systems, Appl. Math. Lett., 79, 43-50 (2018)
[20] Yin, Q.; Liu, D., Periodic solutions of a class of superquadratic second order Hamiltonian systems, Appl. Math. J. Chin. Univ. Ser. B, 15, 3, 259-266 (2000) · Zbl 0974.34039
[21] Zhang, Q.; Liu, C., Infinitely many periodic solutions for second order Hamiltonian systems, J. Differ. Equ., 251, 4-5, 816-833 (2011) · Zbl 1230.37081
[22] Zhang, L.; Tang, X.; Chen, Y., Infinitely many homoclinic solutions for a class of indefinite perturbed second-order Hamiltonian systems, Mediterr. J. Math., 13, 5, 3673-3690 (2016) · Zbl 1367.34052
[23] Zhao, F.; Chen, J.; Yang, M., A periodic solution for a second-order asymptotically linear Hamiltonian system, Nonlinear Anal., 70, 11, 4021-4026 (2009) · Zbl 1167.34345
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