Skip to main content


Extremal Solutions of Periodic Boundary Value Problems for First-Order Impulsive Integrodifferential Equations of Mixed-Type on Time Scales

Article metrics

  • 1025 Accesses

  • 2 Citations


We consider the existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive integrodifferential equations of mixed-type on time scales by establishing a comparison result and using the monotone iterative technique.



  1. 1.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.

  2. 2.

    Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.

  3. 3.

    Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.

  4. 4.

    Henderson J: Double solutions of impulsive dynamic boundary value problems on a time scale. Journal of Difference Equations and Applications 2002,8(4):345–356. 10.1080/1026190290017405

  5. 5.

    Benchohra M, Henderson J, Ntouyas SK, Ouahab A: On first order impulsive dynamic equations on time scales. Journal of Difference Equations and Applications 2004,10(6):541–548. 10.1080/10236190410001667986

  6. 6.

    Atici FM, Biles DC: First- and second-order dynamic equations with impulse. Advances in Difference Equations 2005,2005(2):119–132. 10.1155/ADE.2005.119

  7. 7.

    Liu X, Guo D: Periodic boundary value problems for impulsive integro-differential equations of mixed type in Banach spaces. Chinese Annals of Mathematics. Series B 1998,19(4):517–528.

  8. 8.

    Ladde GS, Sathananthan S: Periodic boundary value problem for impulsive integro-differential equations of Volterra type. Journal of Mathematical and Physical Sciences 1991,25(2):119–129.

  9. 9.

    Erbe LH, Guo DJ: Periodic boundary value problems for second order integrodifferential equations of mixed type. Applicable Analysis 1992,46(3–4):249–258. 10.1080/00036819208840124

  10. 10.

    Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.

  11. 11.

    Kelley WG, Peterson AC: Difference Equations. An Introduction with Applications. Academic Press, Boston, Mass, USA; 1991:xii+455.

  12. 12.

    Xing Y, Han M, Zheng G: Initial value problem for first-order integro-differential equation of Volterra type on time scales. Nonlinear Analysis: Theory, Methods & Applications 2005,60(3):429–442.

Download references

Author information

Correspondence to Yongkun Li.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Comparison Result