Comptes Rendus
no. 6
Mathematical analysis/Ordinary differential equations
Existence of periodic solutions for a class of damped vibration problems
[Existence de solutions périodiques pour une classe de problèmes de vibration amortie]

Présenté par : Haïm Brézis

Zhiyong Wang1  ; Jihui Zhang2
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
Comptes Rendus. Mathématique, Volume 356 (2018) no. 6, pp. 597-612.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.014
Zhiyong Wang 1 ; Jihui Zhang 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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Zhiyong Wang; Jihui Zhang. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.014/

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