Skip to main content

The formulation of second-order boundary value problems on time scales

Abstract

We reconsider the basic formulation of second-order, two-point, Sturm-Liouville-type boundary value problems on time scales. Although this topic has received extensive attention in recent years, we present some simple examples which show that there are certain difficulties with the formulation of the problem as usually used in the literature. These difficulties can be avoided by some additional conditions on the structure of the time scale, but we show that these conditions are unnecessary, since in fact, a simple, amended formulation of the problem avoids the difficulties.

[123456789101112131415161718]

References

  1. 1.

    Agarwal RP, O'Regan D: Nonlinear boundary value problems on time scales. Nonlinear Analysis 2001,44(4):527–535. 10.1016/S0362-546X(99)00290-4

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Amster P, Rogers C, Tisdell CC: Existence of solutions to boundary value problems for dynamic systems on time scales. Journal of Mathematical Analysis and Applications 2005,308(2):565–577. 10.1016/j.jmaa.2004.11.039

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Atici FM, Guseinov GSh: On Green's functions and positive solutions for boundary value problems on time scales. Journal of Computational and Applied Mathematics 2002,141(1–2):75–99. 10.1016/S0377-0427(01)00437-X

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2001:x+358.

    Book  MATH  Google Scholar 

  5. 5.

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

    MATH  Google Scholar 

  6. 6.

    Chyan CJ, Davis JM, Henderson J, Yin WKC: Eigenvalue comparisons for differential equations on a measure chain. Electronic Journal of Differential Equations 1998,1998(35):1–7.

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Chyan CJ, Henderson J: Eigenvalue problems for nonlinear differential equations on a measure chain. Journal of Mathematical Analysis and Applications 2000,245(2):547–559. 10.1006/jmaa.2000.6781

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Davidson FA, Rynne BP: Global bifurcation on time scales. Journal of Mathematical Analysis and Applications 2002,267(1):345–360. 10.1006/jmaa.2001.7780

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Erbe L, Peterson A: Eigenvalue conditions and positive solutions. Journal of Difference Equations and Applications 2000,6(2):165–191. 10.1080/10236190008808220

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Erbe L, Peterson A: Positive solutions for a nonlinear differential equation on a measure chain. Mathematical and Computer Modelling 2000,32(5–6):571–585. 10.1016/S0895-7177(00)00154-0

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Erbe L, Peterson A, Mathsen R: Existence, multiplicity, and nonexistence of positive solutions to a differential equation on a measure chain. Journal of Computational and Applied Mathematics 2000,113(1–2):365–380. 10.1016/S0377-0427(99)00267-8

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Henderson J, Peterson A, Tisdell CC: On the existence and uniqueness of solutions to boundary value problems on time scales. Advances in Difference Equations 2004,2004(2):93–109. 10.1155/S1687183904308071

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Henderson J, Tisdell CC: Topological transversality and boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,289(1):110–125. 10.1016/j.jmaa.2003.08.030

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

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

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Hong C-H, Yeh C-C: Positive solutions for eigenvalue problems on a measure chain. Nonlinear Analysis 2002,51(3):499–507. 10.1016/S0362-546X(01)00842-2

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic, Dordrecht; 1996:x+285.

    Book  MATH  Google Scholar 

  17. 17.

    Sun H-R, Li W-T: Positive solutions for nonlinear three-point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,299(2):508–524. 10.1016/j.jmaa.2004.03.079

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Topal SG: Second-order periodic boundary value problems on time scales. Computers & Mathematics with Applications 2004,48(3–4):637–648. 10.1016/j.camwa.2002.04.005

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fordyce A Davidson.

Rights and permissions

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

Reprints and Permissions

About this article

Cite this article

Davidson, F.A., Rynne, B.P. The formulation of second-order boundary value problems on time scales. Adv Differ Equ 2006, 031430 (2006). https://doi.org/10.1155/ADE/2006/31430

Download citation

Keywords

  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Analysis
  • Functional Equation