Skip to main content
Erschienen in: Journal of Applied Mathematics and Computing 1-2/2013

01.07.2013 | Applied mathematics

Multiplicity of solutions for second-order Hamiltonian systems with impulses

verfasst von: Huiwen Chen, Zhimin He

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2013

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper, we study the existence of infinitely many solutions for second-order Hamiltonian systems with impulses. By using an infinitely many critical points theorem and Fountain theorem, we obtain some new criteria for guaranteeing that the impulsive Hamiltonian systems have infinitely many solutions. No symmetric condition on the nonlinear term is assumed. Some examples are also given in this paper to illustrate our main results.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Bonanno, G., Livrea, R.: Multiple periodic solutions for Hamiltonian systems with not coercive potential. J. Math. Anal. Appl. 363, 627–638 (2010) MathSciNetMATHCrossRef Bonanno, G., Livrea, R.: Multiple periodic solutions for Hamiltonian systems with not coercive potential. J. Math. Anal. Appl. 363, 627–638 (2010) MathSciNetMATHCrossRef
2.
Zurück zum Zitat Cordaro, G., Rao, G.: Three periodic solutions for perturbed second order Hamiltonian systems. J. Math. Anal. Appl. 359, 780–785 (2009) MathSciNetMATHCrossRef Cordaro, G., Rao, G.: Three periodic solutions for perturbed second order Hamiltonian systems. J. Math. Anal. Appl. 359, 780–785 (2009) MathSciNetMATHCrossRef
3.
Zurück zum Zitat Ding, Y.H.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25, 1095–1113 (1995) MathSciNetMATHCrossRef Ding, Y.H.: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems. Nonlinear Anal. 25, 1095–1113 (1995) MathSciNetMATHCrossRef
4.
Zurück zum Zitat Izydorek, M., Janczewska, J.: Homoclinic solutions for a class of second order Hamiltonian systems. J. Differ. Equ. 219, 375–389 (2005) MathSciNetMATHCrossRef Izydorek, M., Janczewska, J.: Homoclinic solutions for a class of second order Hamiltonian systems. J. Differ. Equ. 219, 375–389 (2005) MathSciNetMATHCrossRef
5.
Zurück zum Zitat Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (1989) MATHCrossRef Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, Berlin (1989) MATHCrossRef
7.
Zurück zum Zitat Rabinowitz, P.H., Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473–499 (1990) MathSciNetCrossRef Rabinowitz, P.H., Tanaka, K.: Some results on connecting orbits for a class of Hamiltonian systems. Math. Z. 206, 473–499 (1990) MathSciNetCrossRef
8.
Zurück zum Zitat Tao, Z., Yan, S., Wu, S.: Periodic solutions for a class of superquadratic Hamiltonian systems. J. Math. Anal. Appl. 331, 152–158 (2007) MathSciNetMATHCrossRef Tao, Z., Yan, S., Wu, S.: Periodic solutions for a class of superquadratic Hamiltonian systems. J. Math. Anal. Appl. 331, 152–158 (2007) MathSciNetMATHCrossRef
9.
Zurück zum Zitat Yang, M., Han, Z.: Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities. Nonlinear Anal. 74, 2635–2646 (2011) MathSciNetMATHCrossRef Yang, M., Han, Z.: Infinitely many homoclinic solutions for second-order Hamiltonian systems with odd nonlinearities. Nonlinear Anal. 74, 2635–2646 (2011) MathSciNetMATHCrossRef
10.
Zurück zum Zitat Yang, J., Zhang, F.: Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials. Nonlinear Anal.: Real World Appl. 10, 1417–1423 (2009) MathSciNetMATHCrossRef Yang, J., Zhang, F.: Infinitely many homoclinic orbits for the second-order Hamiltonian systems with super-quadratic potentials. Nonlinear Anal.: Real World Appl. 10, 1417–1423 (2009) MathSciNetMATHCrossRef
11.
Zurück zum Zitat Yu, J.S.: Subharmonic solutions with prescribed minimal period of a class of nonautonomous Hamiltonian systems. J. Dyn. Differ. Equ. 20, 787–796 (2008) MATHCrossRef Yu, J.S.: Subharmonic solutions with prescribed minimal period of a class of nonautonomous Hamiltonian systems. J. Dyn. Differ. Equ. 20, 787–796 (2008) MATHCrossRef
12.
13.
Zurück zum Zitat He, Z., He, X.: Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions. Comput. Math. Appl. 48, 73–84 (2004) MathSciNetMATHCrossRef He, Z., He, X.: Monotone iterative technique for impulsive integro-differential equations with periodic boundary conditions. Comput. Math. Appl. 48, 73–84 (2004) MathSciNetMATHCrossRef
14.
Zurück zum Zitat Jiang, G., Lu, Q., Qian, L.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fractals 31, 448–461 (2007) MathSciNetMATHCrossRef Jiang, G., Lu, Q., Qian, L.: Complex dynamics of a Holling type II prey-predator system with state feedback control. Chaos Solitons Fractals 31, 448–461 (2007) MathSciNetMATHCrossRef
15.
Zurück zum Zitat Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) MATHCrossRef Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) MATHCrossRef
16.
Zurück zum Zitat Li, J., Shen, J.: Positive solutions for three-point boundary value problems for second-order impulsive differential equations on infinite intervals. J. Comput. Appl. Math. 235, 2372–2379 (2011) MathSciNetMATHCrossRef Li, J., Shen, J.: Positive solutions for three-point boundary value problems for second-order impulsive differential equations on infinite intervals. J. Comput. Appl. Math. 235, 2372–2379 (2011) MathSciNetMATHCrossRef
17.
Zurück zum Zitat Liu, X., Willms, A.R.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 2, 277–299 (1996) MATHCrossRef Liu, X., Willms, A.R.: Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft. Math. Probl. Eng. 2, 277–299 (1996) MATHCrossRef
18.
19.
Zurück zum Zitat Nieto, J.J., Rodriguez-Lopez, R.: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J. Math. Anal. Appl. 318, 593–610 (2006) MathSciNetMATHCrossRef Nieto, J.J., Rodriguez-Lopez, R.: Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations. J. Math. Anal. Appl. 318, 593–610 (2006) MathSciNetMATHCrossRef
20.
Zurück zum Zitat D’Onofrio, A.: On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl. Math. Lett. 18, 729–732 (2005) MathSciNetMATHCrossRef D’Onofrio, A.: On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl. Math. Lett. 18, 729–732 (2005) MathSciNetMATHCrossRef
21.
Zurück zum Zitat Qian, D., Li, X.: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 303, 288–303 (2005) MathSciNetMATHCrossRef Qian, D., Li, X.: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal. Appl. 303, 288–303 (2005) MathSciNetMATHCrossRef
22.
Zurück zum Zitat Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995) MATH Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995) MATH
23.
Zurück zum Zitat Shen, J., Li, J.: Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays. Nonlinear Anal.: Real World Appl. 10, 227–243 (2009) MathSciNetMATHCrossRef Shen, J., Li, J.: Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays. Nonlinear Anal.: Real World Appl. 10, 227–243 (2009) MathSciNetMATHCrossRef
24.
Zurück zum Zitat Chen, H., Li, J.: Variational approach to impulsive differential equations with Dirichlet boundary conditions. Bound. Value Probl. 2010, 325415 (2010) CrossRef Chen, H., Li, J.: Variational approach to impulsive differential equations with Dirichlet boundary conditions. Bound. Value Probl. 2010, 325415 (2010) CrossRef
25.
Zurück zum Zitat Liu, Z., Chen, H., Zhou, T.: Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl. 61, 1687–1699 (2011) MathSciNetMATHCrossRef Liu, Z., Chen, H., Zhou, T.: Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl. 61, 1687–1699 (2011) MathSciNetMATHCrossRef
26.
Zurück zum Zitat Nieto, J.J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal.: Real World Appl. 10, 680–690 (2009) MathSciNetMATHCrossRef Nieto, J.J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal.: Real World Appl. 10, 680–690 (2009) MathSciNetMATHCrossRef
27.
Zurück zum Zitat Sun, J., Chen, H., Nieto, J.J., Otero-Novoa, M.: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 72, 4575–4586 (2010) MathSciNetMATHCrossRef Sun, J., Chen, H., Nieto, J.J., Otero-Novoa, M.: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal. 72, 4575–4586 (2010) MathSciNetMATHCrossRef
28.
Zurück zum Zitat Sun, J., Chen, H., Nieto, J.J.: Infinitely many solutions for second-order Hamiltonian system with impulsive effects. Math. Comput. Model. 54, 544–555 (2011) MathSciNetMATHCrossRef Sun, J., Chen, H., Nieto, J.J.: Infinitely many solutions for second-order Hamiltonian system with impulsive effects. Math. Comput. Model. 54, 544–555 (2011) MathSciNetMATHCrossRef
29.
Zurück zum Zitat Tian, Y., Ge, W.: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509–527 (2008) MathSciNetMATHCrossRef Tian, Y., Ge, W.: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51, 509–527 (2008) MathSciNetMATHCrossRef
30.
Zurück zum Zitat Zhang, D., Dai, B.: Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions. Comput. Math. Appl. 61, 3153–3160 (2011) MathSciNetMATHCrossRef Zhang, D., Dai, B.: Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions. Comput. Math. Appl. 61, 3153–3160 (2011) MathSciNetMATHCrossRef
31.
Zurück zum Zitat Zhou, J., Li, Y.: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal.: Real World Appl. 72, 1594–1603 (2010) MATH Zhou, J., Li, Y.: Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects. Nonlinear Anal.: Real World Appl. 72, 1594–1603 (2010) MATH
32.
Zurück zum Zitat Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983) MathSciNetMATHCrossRef Bartolo, P., Benci, V., Fortunato, D.: Abstract critical point theorems and applications to some nonlinear problems with “strong” resonance at infinity. Nonlinear Anal. 7, 981–1012 (1983) MathSciNetMATHCrossRef
33.
Zurück zum Zitat Bonanno, G., Bisci, G.M.: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009, 670675 (2009) MathSciNetCrossRef Bonanno, G., Bisci, G.M.: Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Bound. Value Probl. 2009, 670675 (2009) MathSciNetCrossRef
34.
Zurück zum Zitat Li, S., Wang, Z.Q.: Ljusternik-Schnirelman theory in partially ordered Hilbert spaces. Trans. Am. Math. Soc. 354, 3207–3227 (2002) MATHCrossRef Li, S., Wang, Z.Q.: Ljusternik-Schnirelman theory in partially ordered Hilbert spaces. Trans. Am. Math. Soc. 354, 3207–3227 (2002) MATHCrossRef
35.
Zurück zum Zitat Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser, Boston (1996) MATHCrossRef Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser, Boston (1996) MATHCrossRef
36.
Zurück zum Zitat Liu, S.B., Li, S.J.: Infinitely many solutions for a superlinear elliptic equation. Acta Math. Sin. Chin. Ser. 46, 625–630 (2003) (in Chinese) MATH Liu, S.B., Li, S.J.: Infinitely many solutions for a superlinear elliptic equation. Acta Math. Sin. Chin. Ser. 46, 625–630 (2003) (in Chinese) MATH
37.
Zurück zum Zitat Qian, A.: Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem. Bound. Value Probl. 2005, 329–335 (2005) MATHCrossRef Qian, A.: Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem. Bound. Value Probl. 2005, 329–335 (2005) MATHCrossRef
38.
Zurück zum Zitat Qian, A., Li, C.: Infinitely many solutions for a Robin boundary value problem. Int. J. Differ. Equ. 2010, 548702 (2010) MathSciNet Qian, A., Li, C.: Infinitely many solutions for a Robin boundary value problem. Int. J. Differ. Equ. 2010, 548702 (2010) MathSciNet
Metadaten
Titel
Multiplicity of solutions for second-order Hamiltonian systems with impulses
verfasst von
Huiwen Chen
Zhimin He
Publikationsdatum
01.07.2013
Verlag
Springer-Verlag
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2013
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-012-0621-5

Weitere Artikel der Ausgabe 1-2/2013

Journal of Applied Mathematics and Computing 1-2/2013 Zur Ausgabe