Mathematical analysis/Ordinary differential equations
Existence of periodic solutions for a class of damped vibration problems
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 597-612.

In this paper, we are concerned with the existence of periodic solutions for a class of damped vibration problems. By introducing some new kinds of superquadratic and asymptotically quadratic conditions, and making use of the generalized mountain pass theorem in critical point theory, we propose a unified approach when the potential function F(t,x) exhibits either an asymptotically quadratic or a superquadratic behavior at infinity, and establish some sufficient conditions on periodic solutions, which extend and improve some recent results in the literature, even without damped vibration term.

Nous nous intéressons ici à l'existence de solutions périodiques pour une classe de problèmes de vibration amortie. Nous introduisons de nouvelles conditions de quadraticité asymptotique et de super-quadraticité, et nous utilisons un théorème du col généralisé de la théorie des points critiques. Ainsi, nous proposons une approche unifiée lorsque la fonction potentiel F(t,x) présente un comportement quadratique asymptotique ou super-quadratique à l'infini, et nous établissons des conditions suffisantes pour l'existence de solutions périodiques, ce qui étend et améliore plusieurs résultats récents, même en l'absence du terme de vibration amortie.

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DOI: 10.1016/j.crma.2018.04.014
Wang, Zhiyong 1; Zhang, Jihui 2

1 Department of Mathematics, Nanjing University of Information Science & Technology, Nanjing 210044, PR China
2 Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210023, PR China
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Wang, Zhiyong; Zhang, Jihui. Existence of periodic solutions for a class of damped vibration problems. Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 597-612. doi : 10.1016/j.crma.2018.04.014. http://www.numdam.org/articles/10.1016/j.crma.2018.04.014/

[1] Bartolo, P.; Benci, V.; Fortunato, D. Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlinear Anal., Volume 7 (1983), pp. 981-1012

[2] Bonanno, G.; Livrea, R.; Schechter, M. Some notes on a superlinear second order Hamiltonian system, Manuscr. Math., Volume 154 (2017), pp. 59-77

[3] Ekeland, I.; Ghoussoub, N. Certain new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc., Volume 39 (2002), pp. 207-265

[4] Fei, G. On periodic solutions of superquadratic Hamiltonian systems, Electron. J. Differ. Equ., Volume 8 (2002), pp. 1-12

[5] Jiang, Q.; Tang, C.L. Periodic and subharmonic solutions of a class of subquadratic second-order Hamiltonian systems, J. Math. Anal. Appl., Volume 328 (2007), pp. 380-389

[6] Li, L.; Schechter, M. Existence solutions for second order Hamiltonian systems, Nonlinear Anal., Real World Appl., Volume 27 (2016), pp. 283-296

[7] Ma, S.; Zhang, Y. Existence of infinitely many periodic solutions for ordinary p-Laplacian systems, J. Math. Anal. Appl., Volume 351 (2009), pp. 469-479

[8] Mawhin, J.; Willem, M. Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989

[9] Pipan, J.; Schechter, M. Non-autonomous second order Hamiltonian systems, J. Differ. Equ., Volume 257 (2014), pp. 351-373

[10] Rabinowitz, P.H. Periodic solutions of Hamiltonian systems, Commun. Pure Appl. Math., Volume 31 (1978), pp. 157-184

[11] Rabinowitz, P.H. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., vol. 65, American Mathematical Society, Providence, RI, USA, 1986

[12] Schechter, M. New linking theorems, Rend. Semin. Mat. Univ. Padova, Volume 99 (1998), pp. 255-269

[13] Schechter, M. Periodic solutions of second-order nonautonomous dynamical systems, Bound. Value Probl., Volume 1 (2006)

[14] Tang, C.L.; Wu, X.P. Periodic solutions for a class of new superquadratic second order Hamiltonian systems, Appl. Math. Lett., Volume 34 (2014), pp. 65-71

[15] Tao, Z.; Tang, C.L. Periodic solutions of second-order Hamiltonian systems, J. Math. Anal. Appl., Volume 293 (2004), pp. 435-445

[16] Tao, Z.; Yan, S.; Wu, S. Periodic solutions for a class of superquadratic Hamiltonian systems, J. Math. Anal. Appl., Volume 331 (2007), pp. 152-158

[17] Wang, Z.; Xiao, J. On periodic solutions of subquadratic second order non-autonomous Hamiltonian systems, Appl. Math. Lett., Volume 40 (2015), pp. 72-77

[18] Wang, Z.; Zhang, J. Periodic solutions of a class of second order non-autonomous Hamiltonian systems, Nonlinear Anal., Volume 72 (2010), pp. 4480-4487

[19] Wang, Z.; Zhang, J. New existence results on periodic solutions of non-autonomous second order Hamiltonian systems, Appl. Math. Lett., Volume 79 (2018), pp. 43-50

[20] Wang, Z.; Zhang, J.; Zhang, Z. Periodic solutions of second order non-autonomous Hamiltonian systems with local superquadratic potential, Nonlinear Anal., Volume 70 (2009), pp. 3672-3681

[21] Ye, Y.; Tang, C.L. Periodic and subharmonic solutions for a class of superquadratic second order Hamiltonian systems, Nonlinear Anal., Volume 71 (2009), pp. 2298-2307

[22] Ye, Y.; Tang, C.L. Infinitely many periodic solutions of non-autonomous second-order Hamiltonian systems, Proc. R. Soc. Edinb., Sect. A, Volume 144 (2014), pp. 205-223

[23] Zhang, Q.; Liu, C. Infinitely many periodic solutions for second-order Hamiltonian systems, J. Differ. Equ., Volume 251 (2011), pp. 816-833

[24] Zhang, X.; Tang, X.H. Subharmonic solutions for a class of non-quadratic second order Hamiltonian systems, Nonlinear Anal., Real World Appl., Volume 13 (2012), pp. 113-130

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