Skip to main content
SpringerLink
Log in

Application of Morse Theory for Some Damped Dirichlet Nonlinear Impulsive Differential Equations

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we consider the multiplicity of solutions for some damped nonlinear impulsive differential equations by using Morse theory in combination with the minimax arguments. Under some assumptions, we get some new results on the existence of multiple nontrivial solutions for the problems. Thus, we improve and extend recent results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmad, B., Nieto, J.J.: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. 69, 3291–3298 (2008)

    Article  MathSciNet  Google Scholar 

  2. Agarwal, A.R.P., ORega, D.: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 114, 51–59 (2000)

    MathSciNet  Google Scholar 

  3. Berger, B.M.S., Schechter, M.: On the solvability of semilinear gradient operator equations. Adv. Math. 25, 97–132 (1997)

    Article  MathSciNet  Google Scholar 

  4. Chang, K.: Infinite Dimensional Morse Theory and Multiple Solution Prolems. Birkhëuser, Boston (1993)

    Book  Google Scholar 

  5. Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equation. World Scientific, Singapore (1989)

    Book  Google Scholar 

  6. Nieto, J.J., Rodriguez-Lopez, R.: New comparison results for impulsive integro-differential equations and applications. J. Math. Anal. Appl. 328, 1343–1368 (2007)

    Article  MathSciNet  Google Scholar 

  7. Carter, T.E.: Necessary and sufficient conditions for optimal impulsive rendezvous with linear equations of motion. Dynam. Control. 10, 219–227 (2000)

    Article  MathSciNet  Google Scholar 

  8. Lin, X., Jiang, D.: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 321, 501–514 (2006)

    Article  MathSciNet  Google Scholar 

  9. Yan, J., Zhao, A., Nieto, J.J.: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka–Volterra systems. Math. Comput. Model. 40, 509–518 (2004)

    Article  MathSciNet  Google Scholar 

  10. Li, J., Nieto, J.J., Shen, J.: Impulsive periodic boundary value problems of first-order differential equations. J. Math. Anal. Appl. 325, 226–299 (2007)

    Article  MathSciNet  Google Scholar 

  11. Yang, X., Shen, J.: Nonlinear boundary value problems for first order impulsive differential equations. Appl. Math. Comput. 189, 1943–1952 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Qian, D., Li, X.: Periodic solutions for ordinary differential equations with sublinear impulsive effects. J. Math. Anal Appl. 303, 288–303 (2005)

    Article  MathSciNet  Google Scholar 

  13. De Coster, C., Habets, P.: Upper and lower solutions in the theory ODE boundary value problems: classical and recent results. In: Nonlinear analysis and boundary value problems for ordinary differential equations CISM-ICMS, vol 372, pp 1–78 (1996)

  14. Chen, L., Sun, J.: Nonlinear boundary value problem for first order impulsive functional differential equations. J. Math. Anal. Appl. 318, 726–741 (2006)

    Article  MathSciNet  Google Scholar 

  15. Chu, Nieto, J.J.: Impulsive periodic solutions of first-order singular differential equations. Bull. Lond. Math. Soc. 40(1), 143–150 (2008)

    Article  MathSciNet  Google Scholar 

  16. Nieto, J.J.: Basic theory for nonresonance impulsive periodic problems of first-order. J. Math. Anal. Appl. 205, 423–433 (1997)

    Article  MathSciNet  Google Scholar 

  17. Liu, Z., Chen, H.: Variational methods to the second-order impulsive differential equation with Dirichlet boundary value problem. Comput. Math. Appl. 61, 1687–1699 (1997)

    Article  MathSciNet  Google Scholar 

  18. Nieto, J.J., O’Regan, D.: Variational approach to impulsive differential equations. Nonlinear Anal. 10, 680–690 (2009)

    Article  MathSciNet  Google Scholar 

  19. Tian, Y., Ge, W.G.: Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 72, 277–287 (2010)

    Article  MathSciNet  Google Scholar 

  20. Zhang, Z., Yuan, R.: An application of variational methods to Dirichlet boundary value problem with impulsive. Nonlinear Anal. Real World Appl. 11, 155–162 (2010)

    Article  MathSciNet  Google Scholar 

  21. Zhou, J., Li, Y.: Existence and multiplicity of solutions for some Dirichlet problems with impulsive effects. Nonlinear Anal. 71, 2856–2865 (2009)

    Article  MathSciNet  Google Scholar 

  22. Bai, L., Nieto, J.J.: Variational approach to differential equations with not instantaneous impulses. Appl. Math. Lett. 73, 44–48 (2017)

    Article  MathSciNet  Google Scholar 

  23. Xiao, J., Nieto, J.J.: Variational approach to some damped Dirichlet nonlinear impulsive differential equations. J. Franklin Inst. 348, 369–377 (2011)

    Article  MathSciNet  Google Scholar 

  24. Xiao, J., Nieto, J.J., Luo, Z.: Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Commun. Nonlinear Sci. Numer. Simul.tion 17, 426–432 (2012)

    Article  MathSciNet  Google Scholar 

  25. Chen, S., Wang, C.: Existence of multiple nontrivial solutions for Schrödinger–Poisson system. J. Math. Anal. Appl. 411, 787–793 (2014)

    Article  MathSciNet  Google Scholar 

  26. Jing, M., Sun, M.: Some qualitative results of the critical groups for the p-Laplacian equations. Nonlinear Anal 75, 1778–1786 (2012)

    Article  MathSciNet  Google Scholar 

  27. Sun, M.: Multiplicity of solutions for a class of the quasilinear elliptic equations at resonance. J. Math. Anal. Appl. 17, 661–668 (2012)

    Article  MathSciNet  Google Scholar 

  28. Chang, K.: Infinite Dimensional Morse Theory and Multiple Solutions Problems. Birkhäuser, Boston (1993)

    Book  Google Scholar 

  29. Ayoujil, A., El Amrouss, A.R.: Multiplicity results for quasi-linear prolems. Nonlinear Anal. 68, 1802–1815 (2008)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Information, Xinyang University, Xinyang, 464000, Henan, People’s Republic of China

    Junjun Zhou, Xiao Bie & Yangyang Wang

Authors
  1. Junjun Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao Bie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yangyang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Junjun Zhou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhou, J., Bie, X. & Wang, Y. Application of Morse Theory for Some Damped Dirichlet Nonlinear Impulsive Differential Equations. Differ Equ Dyn Syst 30, 161–171 (2022). https://doi.org/10.1007/s12591-017-0400-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-017-0400-8

Keywords

Search

Navigation