Skip to main content
SpringerLink
Log in

Sign-Changing and Multiple Solutions of Impulsive Boundary Value Problems Via Critical Point Methods

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

In this paper, we study the second-order impulsive boundary value problem

$$\left\{\begin{array}{ll} -Lu=f(x, u), \, \, x\in [0, 1]\backslash\{x_{1}, x_{2}, \cdots, x_{l}\}, \\ -{\Delta} (p(x_{i}) u'(x_{i}))=I_{i}(u(x_{i})), \quad i=1, 2, \cdots, l, \\ R_{1}(u)=0, \, \, \, R_{2}(u)=0, \end{array}\right.$$

where Lu = (p(x)u′)′ − q(x)u is a Sturm-Liouville operator, R 1(u) = αu′(0) − βu(0) and R 2(u) = γu′(1) + σu(1). The existence of sign-changing and multiple solutions is obtained. The technical approach is based on minimax methods and invariant sets of descending flow.

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. Agarwal RP, O’Regan D. Multiple nonnegative solutions for second-order impulsive differential equations. Appl Math Comput. 2000;114:51–59.

    Article  MATH  MathSciNet  Google Scholar 

  2. d’Onofrio A. On pulse vaccination strategy in the SIR epidemic model with vertical transmission. Appl Math Lett. 2005;18:729–732.

    Article  MATH  MathSciNet  Google Scholar 

  3. Ahmad B, Sivasundaram S. The monotone iterative technique for impulsive hybrid set valued integro-differential equations. Nonlinear Anal TMA. 2006;65:2260–2276.

    Article  MATH  MathSciNet  Google Scholar 

  4. Dancer EN, Zhang Z. Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity. J Math Anal Appl. 2000;250:449–464.

    Article  MATH  MathSciNet  Google Scholar 

  5. Franco D, Nieto JJ. First-order impulsive ordinary differential equations with antiperiodic and nonlinear boundary conditions. Nonlinear Anal TMA. 2000;42:163–173.

    Article  MATH  MathSciNet  Google Scholar 

  6. Guo D, Sun J, Liu Z. Functional methods in nonlinear ordinary differential equations. Jinan: Shan-dong Science and Technology Press; 1995. (in Chinese).

    Google Scholar 

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

    Article  Google Scholar 

  8. Jiang DQ, Chu J, He Y. Multiple positive solutions of Sturm-Liouville problems for second order impulsive differential equations. Dynam Systems Appl. 2007; 16:611–624.

    MATH  MathSciNet  Google Scholar 

  9. Lakshmikantham V, Bainov DD, Simeonov PS. Theory of impulsive differential equations, Series Modern Appl. Math. Teaneck: World Scientific; 1989. vol. 6.

  10. Lee EK, Lee YH. Multiple positive solutions of singular two point boundary value problems for second-order impulsive differential equations. Appl Math Comput. 2004; 158:745–759.

    Article  MATH  MathSciNet  Google Scholar 

  11. Li W, Huo H. Global attractivity of positive periodic solutions for an impulsive delay periodic model of respiratory dynamics. J Comput Appl Math. 2005;174:227–238.

    Article  MATH  MathSciNet  Google Scholar 

  12. Liu X., Guo D. Periodic boundary-value problems for a class of second-order impulsive integro-differential equations in Banach spaces. Appl Math Comput. 1997; 216:284–302.

    MATH  Google Scholar 

  13. Liu ZL. 1992. Multiple solutions of differential equations, Ph.D thesis, Shandong University, Jinan.

  14. Liu ZL, Sun JX. Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations. J Differ Equ. 2001;172:257–299.

    Article  MATH  Google Scholar 

  15. Nieto JJ. Periodic boundary value problems for first-order impulsive ordinary differential equations. Nonlinear Anal. 2002;51(7):1223–1232.

    Article  MATH  MathSciNet  Google Scholar 

  16. Nieto JJ, O’ Regan D. Variational approach to impulsive differential equations. Nonlinear Anal RWA. 2009;10:680–690.

    Article  MATH  MathSciNet  Google Scholar 

  17. Nieto JJ. Variational formulation of a damped Dirichlet impulsive problem. Appl Math Lett. 2010;23:940–942.

    Article  MATH  MathSciNet  Google Scholar 

  18. Sun J, Chen H. Variational method to the impulsive equation with Neumann boundary conditions. Bound Value Prob. 2009;2009. Article ID 316812.

  19. Sun J, Chen H, Nieto JJ, Otero-Novoa M. The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects. Nonlinear Anal TMA. 2010;72:4575–4586.

    Article  MATH  MathSciNet  Google Scholar 

  20. Sun J, Chen H. Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems. Nonlinear Anal RWA. 2010;11(5):4062– 4071.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Teng K, Zhang C. Existence of solution to boundary value problem for impulsive differential equations. Nonlinear Anal RWA. 2010;11:4431–4441.

    Article  MATH  MathSciNet  Google Scholar 

  23. Tian Y, Ge W. Applications of variational methods to boundary value problem for impulsive differential equations. Proc Edinburgh Math Soc. 2008;51:509–527.

    Article  MATH  MathSciNet  Google Scholar 

  24. Tian Y, Ge W. Variational methods to Sturm-Liouville boundary value problem for impulsive differential equations. Nonlinear Anal. 2010;72(1):277–287.

    Article  MATH  MathSciNet  Google Scholar 

  25. Tian Y, Ge W. Multiple solutions of impulsive Sturm-Liouville boundary value problem via lower and upper solutions and variational methods. J Math Anal Appl. 2012; 387:475–489.

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang ZQ. On a superlinear elliptic equation. Anal Nonlineaire. 1991;8:43–58.

    MATH  Google Scholar 

  27. Wei ZL. Periodic boundary value problems for second order impulsive integrodifferential equations of mixed type in Banach spaces. J Math Anal Appl. 1995; 195:214–229.

    Article  MATH  MathSciNet  Google Scholar 

  28. Yao M, Zhao A, Yun J. Periodic boundary value problems of second-order impulsive differential equations. Nonlinear Anal TMA. 2009;70:262–273.

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Zavalishchin ST, Sesekin AN. Dynamic impulse systems, theory and applications. Math. Appl. Dordrecht: Kluwer Acad. Publ.; 1997. vol. 394.

  31. Zhang H, Li Z. Variational approach to impulsive differential equations with periodic boundary condition. Nonlinear Anal RWA. 2010;11:67–78.

    Article  MATH  Google Scholar 

  32. Zhang W, Fan M. Periodicity in a generalized ecological competition system governed by impulsive differential equations with delays. Math. system governed by impulsive differential equations with delays. Math Comput Model. 2004;39:479–493.

    Article  MATH  MathSciNet  Google Scholar 

  33. Zhang X, Shuai Z, Wang K. Optimal impulsive harvesting policy for single population. Nonlinear Anal RWA. 2003;4:639–651.

    Article  MATH  MathSciNet  Google Scholar 

  34. Mao AM, Zhang ZT. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal. 2009;70(3):1275–1287.

    Article  MATH  MathSciNet  Google Scholar 

  35. Sun JX. 1984. On some problems about nonlinear operators, PhD thesis. Shandong University, Jinan.

  36. Sun JX. The schauder condition in the critical point theory. Kexue Tongbao (English) 1986;31:1157–1162.

  37. Sun JX, Hu SZ. Flow-invariant sets, and critical point theory. Discrete Contin Dyn Syst. 2003;9(2):483–496.

    MATH  MathSciNet  Google Scholar 

  38. Sun JX, Sun JL. Existence theorems of multiple critical points and applications. Nonlinear Anal. 2005;61:1303–1317.

    Article  MATH  MathSciNet  Google Scholar 

  39. Zhang ZT, Perera K. Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J Math Anal Appl. 2006;317:456–463.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

Project 11001028 is supported by the National Science Foundation for Young Scholars, Project 11071014 is supported by the National Science Foundation of P.R. China, and Project YETP0458 is supported by the Beijing Higher Education Young Elite Teacher.

Author information

Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, People’s Republic of China

    Yu Tian

  2. Department of Applied Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Weigao Ge

  3. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

Authors
  1. Yu Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weigao Ge
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yu Tian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tian, Y., Ge, W. & O’Regan, D. Sign-Changing and Multiple Solutions of Impulsive Boundary Value Problems Via Critical Point Methods. J Dyn Control Syst 20, 559–574 (2014). https://doi.org/10.1007/s10883-014-9243-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-014-9243-6

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation