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A periodic solution for a second-order asymptotically linear Hamiltonian system. (English) Zbl 1167.34345

Summary: We study the existence of a periodic solution of the following second-order Hamiltonian system:
\[ \ddot u (t)+\nabla F(t,u(t))=0,\qquad t\in [0,T], \]
where \(F(t,x)= - K(t,x)+W(t,x)\). Assuming that \(K\) satisfies the “pinching” condition and \(W\) is asymptotically linear at infinity, the existence of a nontrivial periodic solution is obtained via the Mountain Pass Theorem.

MSC:

34C25 Periodic solutions to ordinary differential equations
47J30 Variational methods involving nonlinear operators
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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[1] Abbondandolo, A., Morse theory for asymptotically linear Hamiltonian systems, Nonlinear anal., 39, 997-1049, (2000) · Zbl 0942.37004
[2] Bartolo, P.; Benci, V.; Fortunato, D., Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear anal., 7, 241-273, (1983) · Zbl 0522.58012
[3] Cerami, G., An existence criterion for the critical points on unbounded manifolds, Istit. lombardo accad. sci. lett. rend. A, 112, 332-336, (1978), (in Italian) · Zbl 0436.58006
[4] Degiovanni, M.; Olian Fannio, L., Multiple periodic solutions of asymptotically linear Hamiltonian systems, Nonlinear anal., 26, 1437-1446, (1996) · Zbl 0851.34047
[5] Ding, Y., Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear anal., 25, 1095-1113, (1995) · Zbl 0840.34044
[6] Ekeland, I.; Ghoussoub, N., Selected new aspects of the calculus variational in the large, Bull. amer. math. soc., 39, 207-265, (2002) · Zbl 1064.35054
[7] Fei, G., Nontrivial periodic solutions for asymptotically linear Hamiltonian systems, Electron J. differential equations, 2001, 1-17, (2001) · Zbl 1001.37052
[8] Fei, G., On periodic solutions of superquadratic Hamiltonian systems, Electron J. differential equations, 2002, 1-12, (2002) · Zbl 0999.37039
[9] Fei, G., Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity, J. differential equations, 121, 121-133, (1995) · Zbl 0831.34046
[10] Fei, G., Relative Morse index and its application to Hamiltonian systems in the presence of symmetries, J. differential equations, 122, 302-315, (1995) · Zbl 0840.34032
[11] Guo, Y., Nontrivial periodic solutions for asymptotically linear Hamiltonian systems with resonance, J. differential equations, 175, 71-87, (2001) · Zbl 0983.37075
[12] Izydorek, M.; Janczewska, J., Homoclinic solutions for a class of the second order Hamiltonian systems, J. differential equations, 219, 375-389, (2005) · Zbl 1080.37067
[13] Li, S.; Willem, M., Applications of local linking to critical point theory, J. math. anal. appl., 189, 6-32, (1995) · Zbl 0820.58012
[14] Long, Y.M., Multiple solutions of perturbed superquadratic second order Hamiltonian systems, Trans. amer. math. soc., 311, 749-780, (1989) · Zbl 0676.34026
[15] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin, New York · Zbl 0676.58017
[16] Rabinowitz, P.H., Periodic solutions of Hamiltonian systems, Comm. pure appl. math., 31, 157-184, (1978) · Zbl 0358.70014
[17] Rabinowitz, P.H., ()
[18] Su, J., Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity, J. differential equation, 145, 2, 252-273, (1998) · Zbl 0944.34034
[19] Szulkin, A.; Zou, W., Infinite dimensional cohomology groups and periodic solutions for asymptotically linear Hamiltonian systems, J. differential equations, 174, 369-391, (2001) · Zbl 0997.37040
[20] Tao, Z.L.; Tang, C.L., Periodic and subharmonic solutions of second order Hamiltonian systems, J. math. anal. appl., 293, 435-445, (2004) · Zbl 1042.37047
[21] Tang, C.L., Periodic solutions for non-autonomous second order systems with sublinear nonlinearity, Proc. amer. math. soc., 126, 3263-3270, (1998) · Zbl 0902.34036
[22] M. Willem, Oscillations forcées de systèmes hamiltoniens, in: Public. Sémin. Analyse Nonlinéaire, Univ. Besancon, 1981
[23] Zou, W.; Li, S., Infinitely many solutions for Hamiltonian systems, J. differential equations, 186, 141-164, (2002)
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