Open Access

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

Advances in Difference Equations20062006:031430

https://doi.org/10.1155/ADE/2006/31430

Received: 6 February 2006

Accepted: 15 May 2006

Published: 13 July 2006

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]

Authors’ Affiliations

(1)
Division of Mathematics, University of Dundee
(2)
Department of Mathematics, Heriot-Watt University

References

  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)00290-4MathSciNetView ArticleMATHGoogle Scholar
  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.2004.11.039MathSciNetView ArticleMATHGoogle Scholar
  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-XMathSciNetView ArticleMATHGoogle Scholar
  4. Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  5. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Massachusetts; 2003:xii+348.MATHGoogle Scholar
  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.MathSciNetView ArticleMATHGoogle Scholar
  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.2000.6781MathSciNetView ArticleMATHGoogle Scholar
  8. Davidson FA, Rynne BP: Global bifurcation on time scales. Journal of Mathematical Analysis and Applications 2002,267(1):345–360. 10.1006/jmaa.2001.7780MathSciNetView ArticleMATHGoogle Scholar
  9. Erbe L, Peterson A: Eigenvalue conditions and positive solutions. Journal of Difference Equations and Applications 2000,6(2):165–191. 10.1080/10236190008808220MathSciNetView ArticleMATHGoogle Scholar
  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)00154-0MathSciNetView ArticleMATHGoogle Scholar
  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)00267-8MathSciNetView ArticleMATHGoogle Scholar
  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/S1687183904308071MathSciNetView ArticleMATHGoogle Scholar
  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.2003.08.030MathSciNetView ArticleMATHGoogle Scholar
  14. Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics 1990,18(1–2):18–56.MathSciNetView ArticleMATHGoogle Scholar
  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)00842-2MathSciNetView ArticleMATHGoogle Scholar
  16. Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic, Dordrecht; 1996:x+285.View ArticleMATHGoogle Scholar
  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.2004.03.079MathSciNetView ArticleMATHGoogle Scholar
  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.2002.04.005MathSciNetView ArticleMATHGoogle Scholar

Copyright

© F. A. Davidson and B. P. Rynne. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.