SpringerLink
Log in

Existence of Positive Solutions for an Impulsive Differential Equation with p-Laplacian Operator

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

The aim of this paper is to study the existence and multiplicity of solutions for an impulsive differential equation with Dirichlet boundary conditions. We obtain sufficient conditions for the existence of at least one positive solution, and at least two positive solutions. Our approach is based on variational methods.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Spain)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability Statement

My manuscript has no associate data.

References

  1. Shengda, L., **Rong, W., Yong, Z.: Optimal control of noninstantaneous impulsive differential equations. J. Franklin Inst. 354(17), 7668–7698 (2017). https://doi.org/10.1016/j.jfranklin.2017.09.010

    Article  MathSciNet  MATH  Google Scholar 

  2. Tailei, Z., Zhidong, T.: Dynamic behavior of a delayed impulsive SEIRS model in epidemiology. Rocky Mt. J. Math. 38(5), 1841–1862 (2008). https://doi.org/10.1155/2014/262535

    Article  MathSciNet  MATH  Google Scholar 

  3. Hengguo, Y., Shouming, Z., Ravi, P.A.: Mathematics analysis and chaos in an ecological model with an impulsive control strategy. Commun. Nonlinear Sci. Numer. Simul. 16(2), 776–786 (2011). https://doi.org/10.1016/j.cnsns.2010.04.017

    Article  MathSciNet  MATH  Google Scholar 

  4. Lakshmikantham, V., Bainov, D.D., Simeonov, P.: Theory of Impulsive Differential Equations, vol. 6. World Scientific, Singapore (2011). https://doi.org/10.1142/0906

    Book  Google Scholar 

  5. Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995). https://doi.org/10.1142/2892

    Book  MATH  Google Scholar 

  6. **an, X., O’Regan, D., Agarwal, R.: Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions. Bound. Value Probl. 2008, 1–21 (2008). https://doi.org/10.1155/2008/197205

    Article  MathSciNet  MATH  Google Scholar 

  7. Wei, D., Yu, W.: New result for a class of impulsive differential equation with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 18(5), 1095–1105 (2013). https://doi.org/10.1016/j.cnsns.2012.09.021

    Article  MathSciNet  MATH  Google Scholar 

  8. Ding, Y., O’Regan, D.: Positive solutions for a second-order \(p\)-Laplacian impulsive boundary value problem. Adv. Differ. Equ. 2012(159), 1–12 (2012). https://doi.org/10.1186/1687-1847-2012-159

    Article  MathSciNet  MATH  Google Scholar 

  9. Luo, Y., Wang, W.: Existence for impulsive semilinear functional differential inclusions. Qual. Theory Dyn. Syst. 20(22), 1–8 (2021). https://doi.org/10.1007/s12346-021-00457-x

    Article  MathSciNet  MATH  Google Scholar 

  10. Shi, H., Chen, H., Zhang, Q.: Infinitely many solutions for a \(p\)-Laplacian boundary value problem with impulsive effects. J. Appl. Math. Comput. 46, 93–106 (2014). https://doi.org/10.1007/s12190-013-0739-0

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang, W.: Infinitely many solutions for nonlinear periodic boundary value problem with impulses. RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 111, 1093–1103 (2017). https://doi.org/10.1007/s13398-016-0348-5

    Article  MathSciNet  MATH  Google Scholar 

  12. **g, X., Juan, J.N., Zhiguo, L.: Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods. Commun. Nonlinear Sci. Numer. Simul. 17(1), 426–432 (2012). https://doi.org/10.1016/j.cnsns.2011.05.015

    Article  MathSciNet  MATH  Google Scholar 

  13. Liang, B., Binxiang, D.: Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory. Math. Comput. Model. 53(9–10), 1844–1855 (2011). https://doi.org/10.1016/j.mcm.2011.01.006

    Article  MathSciNet  MATH  Google Scholar 

  14. Peng, C., **anhua, T.: Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems. Appl. Math. Comput. 218(24), 11775–11789 (2012). https://doi.org/10.1016/j.amc.2012.05.027

    Article  MathSciNet  MATH  Google Scholar 

  15. Kaimin, T., Chao, Z.: Existence of solution to boundary value problem for impulsive differential equations. Nonlinear Anal. Real World Appl. 11(5), 4431–4441 (2010). https://doi.org/10.1016/j.nonrwa.2010.05.026

    Article  MathSciNet  MATH  Google Scholar 

  16. Dai, B., Zhang, D.: The existence and multiplicity of solutions for second-order impulsive differential equations on the half-line. Results Math. 63, 135–149 (2013). https://doi.org/10.1007/s00025-011-0178-x

    Article  MathSciNet  MATH  Google Scholar 

  17. Yu, T., Weigao, G.: Applications of variational methods to boundary-value problem for impulsive differential equations. Proc. Edinb. Math. Soc. 51(2), 509–527 (2008). https://doi.org/10.1017/s0013091506001532

    Article  MathSciNet  MATH  Google Scholar 

  18. Weibing, W., Jianhua, S.: Eigenvalue problems of second order impulsive differential equations. Comput. Math. Appl. 62(1), 142–150 (2011). https://doi.org/10.1016/j.camwa.2011.04.061

    Article  MathSciNet  MATH  Google Scholar 

  19. Paul, H.R.: Minimax methods in critical point theory with applications to differential equations, vol. 65. American Mathematical Society, Providence (1986). https://doi.org/10.1090/cbms/065

    Book  Google Scholar 

  20. Kourogenis, N., Papageorgiou, N.S.: Nonsmooth critical point theory and nonlinear elliptic equations at resonance. J. Aust. Math. Soc. 69(2), 245–271 (2000). https://doi.org/10.1017/S1446788700002202

    Article  MathSciNet  MATH  Google Scholar 

  21. Kristály, A., Varga, C., Dontchev, A.L.: Cerami \((C)\) Condition and mountain pass theorem for multivalued map**s. Serdica Math. J. 28(2), 95–108 (2002). http://eudml.org/doc/11549

  22. El Amrouss, A.R.: Variantes du principe variationnel d’Ekeland et applications. Rev. Colomb. Mat. 40(1), 1–14 (2006). https://revistas.unal.edu.co/index.php/recolma/article/view/94631

  23. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47(2), 324–353 (1974). https://doi.org/10.1016/0022-247X(74)90025-0

    Article  MathSciNet  MATH  Google Scholar 

  24. Bouabdallah, M., Chakrone, O., Chehabi, M.: Eigenvalue problems of impulsive differential equations governed by the one-dimensional \(p\)-Laplacian operator. Proyecciones 41(1), 217–247 (2022). https://doi.org/10.22199/issn.0717-6279-4657

    Article  MathSciNet  MATH  Google Scholar 

  25. Coffman, C.V.: A minimum–maximum principle for a class of non-linear integral equations. J. Anal. Math. 22, 391–419 (1969). https://doi.org/10.1007/BF02786802

    Article  MathSciNet  MATH  Google Scholar 

  26. Huiwen, C., Zhimin, H., Zigen, O., Maoxin, L.: New results for some damped Dirichlet problems with impulses. Qual. Theory Dyn. Syst. 21(36), 1–16 (2022). https://doi.org/10.1007/s12346-022-00559-0

    Article  MathSciNet  MATH  Google Scholar 

  27. Liang, B., Juan, J.N., José, M.U.: On a delayed epidemic model with non-instantaneous impulses. Commun. Pure Appl. Anal. 19(4), 1915–1930 (2020). https://doi.org/10.3934/cpaa.2020084

    Article  MathSciNet  MATH  Google Scholar 

  28. Chen, L.: Applications of the Moser’s Twist Theorem to Some Impulsive Differential Equations. Qual. Theory Dyn. Syst. 19(75), 1–20 (2020). https://doi.org/10.1007/s12346-020-00413-1

    Article  MathSciNet  Google Scholar 

  29. Ilkay, Y.K., Fatma, T.F.: Existence of positive solutions for a nonlinear nth-order \(m\)-point \(p\)-Laplacian impulsive boundary value problem. Math. Slovaca 67(2), 467–482 (2017). https://doi.org/10.1515/ms-2016-0282

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the reviewers for attentively reading the paper, and for giving pertinent remarks and precious ideas to enhance the results as well as the presentation of this work.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer, Laboratory of Nonlinear Analysis, Faculty of Science, University Mohammed 1st, Oujda, Morocco

    Mohamed Bouabdallah, Omar Chakrone & Mohammed Chehabi

Authors
  1. Mohamed Bouabdallah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Omar Chakrone
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohammed Chehabi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohamed Bouabdallah.

Ethics declarations

Conflict of Interest

The authors indicate that they have no conflicting interests.

Authors Efforts

The authors declare that the research was carried out in cooperation with the same accountability. All authors have seen and approved the final paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouabdallah, M., Chakrone, O. & Chehabi, M. Existence of Positive Solutions for an Impulsive Differential Equation with p-Laplacian Operator. Mediterr. J. Math. 20, 229 (2023). https://doi.org/10.1007/s00009-023-02426-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-023-02426-x

Keywords

Mathematics Subject Classification

Search

Navigation