Skip to main content
SpringerLink
Log in

Existence and Multiplicity of Mild Solutions for First-Order Hamilton Random Impulsive Differential Equations with Dirichlet Boundary Conditions

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper, we study the sufficient conditions for multiple solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. For systems with random impulses, the variational principle with random impulses is constructed, and the energy function is treated by its conjugate action. Then, the generalized saddle point theorem is used to show that the energy functional has multiple critical points, that is, the first-order Hamiltonian random impulses differential equation has multiple weak solutions. Finally, we give an example to illustrate the feasibility and effectiveness of this method.

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. Smith, R.J., Wahl, L.M.: Distinct effects of protease and reverse transcriptase inhibition in an immuno- logical model of HIV-1 infection with impulsive drug effects. Bull. Math. Biol. 66, 1259–1283 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Smith, R.J., Wahl, L.M.: Drug resistance in an immunological model of HIV-1 infection with impulsive drug effects. Bull. Math. Biol. 67, 783–813 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Tang, S., Chen, L.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. Math. Biol. 44, 185–199 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lakmeche, A., Aqino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynamics of continuous, discrete and impulsive systems, Series A. Mathematical analysis, vol. 7 (2000)

  5. Liu, X.: Stability results for impulsive differential systems with applications to population growth models. Dyn. Stabil. Syst. 9(2), 163–174 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Guo, Y., Shu, X., Yin, Q.: Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions. Discrete Contin. Dyn. Syst. Ser. B 27, 4455–4471 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  7. Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    Book  MATH  Google Scholar 

  8. Choisy, M., Guegan, J.F., Rohani, P.: Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects. Phys. D: Nonlinear Phenom. 223, 26–35 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gao, S., Chen, L., Nieto, J.J., Torres, A.: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24, 6037–6045 (2006)

    Article  Google Scholar 

  10. Jiao, J., Yang, X., Chen, L., Cai, S.: Effect of delayed response in growth on the dynamics of a chemostat model with impulsive input. Chaos Solitons Fract. 42, 2280–2287 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ekeland, I., Hofer, H.: Subharmonics for convex nonautonomous Hamiltonian systems. Commun. Pure Appl. Math. 40, 1–36 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Li, Z., Soh, Y., Wen, C.: Analysis and design of impulsive control systems. IEEE Trans. Autom. Control 46, 894 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, X.: Impulsive stabilization and control of chaotic system. Nonlinear Anal.: Theory,Methods Appl. 47, 1081–1092 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cortés, J.C., Delgadillo-Aleman, S.E., Kú-Carrillo, R.A., Villanueva, R.J.: Full probabilistic analysis of random first-order linear differential equations with Dirac delta impulses appearing in control. Math. Methods Appl. Sci. (2021). https://doi.org/10.1002/mma.7715

    Article  MATH  Google Scholar 

  15. Cortés, J.C., Delgadillo-Alemán, S.E., Kú-Carrillo, R.A., Villanueva, R.J.: Probabilistic analysis of a class of impulsive linear random differential equations via density functions. Appl. Math. Lett. 121, 107519 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, Z., Shu, X., Xu, F.: The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem. AIMS Math. 5(6), 6189–6210 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shu, X., Xu, Y., Huang, L.: Infinite periodic solutions to a class of second-order Sturm–Liouville neutral differential equations. Nonlinear Anal. Theory Methods Appl. 68, 905–911 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shu, X., Xu, Y.: Multiple periodic solutions for a class of second-order nonlinear neutral delay equations. Abstr. Appl. Anal. (2006). https://doi.org/10.1155/AAA/2006/10252

    Article  MathSciNet  MATH  Google Scholar 

  19. Shu, X., Shi, Y.: A study on the mild solution of impulsive fractional evolution equations. Appl. Math. Comput. 273, 465–476 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Guo, Y., Shu, X., Li, Y., Xu, F.: The existence and Hyers–Ulam stability of solution for an impulsive Riemann–Liouville fractional neutral functional stochastic differential equation with infinite delay of order 1\(<\beta <\)2. Boundary Value Problems, volume 2019, Article number: 59 (2019)

  21. Guo, Y., Shu, X.-B., Fei, X., Yang, C.: HJB equation for optimal control system with random impulses. Optimization (2022). https://doi.org/10.1080/02331934.2022.2154607

    Article  Google Scholar 

  22. Vinodkumar, A., Senthilkumar, T., Hariharan, S., Alzabut, J.: Exponential stabilization of fixed and random time impulsive delay differential system with applications. Math. Biosci. Eng. 18(3), 2384–2400 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kawakubo, T., Kobayashi, T., Sakamoto, S.: Drift motion of granules in chara cells induced by random impulses due to the myosin-actin interaction. Physica A 248(1–2), 21–27 (1998)

    Article  Google Scholar 

  24. Long, Y., Xu, X.: Periodic solutions for a class of nonautonomous Hamiltionian systems. Nonlinear Anal.: Theory Methods Appl. 41, 455–463 (2000)

    Article  MATH  Google Scholar 

  25. Timoumi, M.: Subharmonie oscillations of a class of Hamiltonian systems. Nonlinear Anal. Theory Methods Appl. 68, 2697–2708 (2008)

    Article  MATH  Google Scholar 

  26. Liu, P., Guo, F.: Multiplicity of periodic solutions for second order Hamiltonian systems with asymptotically quadratic conditions. Acta Math. Sin. (English Series) 36, 55–65 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu, Y., Guo, F.: Multiplicity of periodic solutions for a class of second-order perturbed Hamiltonian systems. J. Math. Anal. Appl. 491, 124386 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Silva, E.A.: Subharmonic solutions for subquadratic Hamiltonian systems. J. Differ. Equ. 115, 120–145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sun, J., Chen, H., Nieto, J.J., Mario, O.N.: The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal.: Theory Methods Appl. 72, 4575–4586 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang, S., Shu, X., Shu, L.: Existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions. AIMS Math. 7(5), 7685–7705 (2022)

    Article  MathSciNet  Google Scholar 

  32. Mawhin, J., Willem, M.: Critical Point Theory and Hamiltonian Systems. Springer, New York (1989)

    Book  MATH  Google Scholar 

  33. Liu, J.Q.: A generalized saddle point theorem. J. Differ. Equ. 82(2), 372–385 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Timoumi, Mohsen: Subharmonic oscillations of a class of Hamiltonian systems. Nonlinear Anal.: Theory Methods Appl. 68(9), 2697–2708 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Tian, Y., Ge, W.: Multiple periodic solutions for a class of non-autonomous Hamiltonian systems with even-typed potentials. J. Dyn. Control Syst. 18, 339–354 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics, Hunan University, Changsha, 410082, Hunan, People’s Republic of China

    Qian-Bao Yin, Yu Guo, Dan Wu & Xiao-Bao Shu

Authors
  1. Qian-Bao Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dan Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xiao-Bao Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Bao Shu.

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

Yin, QB., Guo, Y., Wu, D. et al. Existence and Multiplicity of Mild Solutions for First-Order Hamilton Random Impulsive Differential Equations with Dirichlet Boundary Conditions. Qual. Theory Dyn. Syst. 22, 47 (2023). https://doi.org/10.1007/s12346-023-00748-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00748-5

Keywords

Search

Navigation