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Periodic solutions of superquadratic Hamiltonian systems with bounded forcing terms. (English) Zbl 0666.34045

We prove the existence of infinitely many distinct T-periodic solutions of the forced Hamiltonian system \(\dot z={\mathcal I}(H_ z(z)+F_ z(t,z))\) under the conditions that H is \(C^ 1\) and superquadratic at infinity and that F is \(C^ 1\), T-periodic in time, bounded in \(C^ 0\), and having a gradient of at most polynomial growth at infinity. The proof of this qualitative results is based on minimax methods.
Reviewer: Yining Long

MSC:

34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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References:

[1] Bahri, A., Berestycki, H.: Forced vibrations of superquadratic Hamiltonian systems. Acta Math.152, 143–197 (1984) · Zbl 0592.70027 · doi:10.1007/BF02392196
[2] Bahri, A., Berestycki, H.: Existence of forced oscillations for some nonlinear differential equations. Commun. Pure Appl. Math.37, 403–442 (1984) · Zbl 0588.34028 · doi:10.1002/cpa.3160370402
[3] Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math.45, 139–174 (1978) · Zbl 0403.57001 · doi:10.1007/BF01390270
[4] Long, Y.: Multiple solutions of perturbed superquadratic second order Hamiltonian systems. Trans. Am. Math. Soc.311, 749–780 (1989) · Zbl 0676.34026 · doi:10.1090/S0002-9947-1989-0978375-4
[5] Long, Y.: Doctoral thesis. University of Wisconsin-Madison (1987)
[6] Long, Y. Periodic solutions of perturbed superquadratic Hamiltonian systems. CMS Technical Summary Report No. 88-23. University of Wisconsin-Madison (1987). Ann. Scuola Norm. Sup. Pisa (to appear)
[7] Long, Y.: On the density of the range for some nonlinear operators. Analyse non linéaire. Ann. Inst. Henri Poincaré6, 139–151 (1989) · Zbl 0757.34034
[8] Rabinowitz, P.H.: Variational methods for nonlinear eigenvalue problems. In: Prodi, G. (ed.) Eigenvalues of nonlinear problems. Roma: Ediz. Cremonese: 1974 · Zbl 0278.35040
[9] Rabinowitz, P.H.: Periodic solutions of Hamiltonian system. Commun. Pure Appl. Math.31, 157–184 (1978) · Zbl 0369.70017 · doi:10.1002/cpa.3160310203
[10] Rabinowitz, P.H.: Multiple critical points of perturbed symmetric functionals. Trans. Am. Math. Soc.272, 753–769 (1982) · Zbl 0589.35004 · doi:10.1090/S0002-9947-1982-0662065-5
[11] Rabinowitz, P.H.: Periodic solutions of large norm of Hamiltonian systems. J. Differ. Equation50, 33–48 (1983) · Zbl 0528.58028 · doi:10.1016/0022-0396(83)90083-9
[12] Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series No. 65 (1986) · Zbl 0609.58002
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