Open Access

On lower and upper solutions without ordering on time scales

Advances in Difference Equations20062006:073860

DOI: 10.1155/ADE/2006/73860

Received: 31 January 2006

Accepted: 16 May 2006

Published: 13 September 2006


In order to enlarge the set of boundary value problems on time scales, for which we can use the lower and upper solutions technique to get existence of solutions, we extend this method to the case when the pair lacks ordering. We use the degree theory and a priori estimates to obtain the existence of solutions for the second-order Dirichlet boundary value problems. To illustrate a wider application of this result, we conclude with an example which shows that a combination of well and non-well ordered pairs can yield the existence of multiple solutions.


Authors’ Affiliations

Department of Mathematics, University of West Bohemia


  1. Agarwal RP, Bohner M, Wong PJY: Sturm-Liouville eigenvalue problems on time scales. Applied Mathematics and Computation 1999,99(2–3):153–166. 10.1016/S0096-3003(98)00004-6MathSciNetView ArticleMATHGoogle Scholar
  2. Akin E: Boundary value problems for a differential equation on a measure chain. Panamerican Mathematical Journal 2000,10(3):17–30.MathSciNetMATHGoogle Scholar
  3. Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  4. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.MATHGoogle Scholar
  5. Cabada A: Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales. Journal of Mathematical Analysis and Applications 2004,290(1):35–54. 10.1016/j.jmaa.2003.08.018MathSciNetView ArticleMATHGoogle Scholar
  6. De Coster C, Habets P: The lower and upper solutions method for boundary value problems. In Handbook of Differential Equations. Edited by: Cañada A, Drábek P, Fonda A. Elsevier/North-Holland, Amsterdam; 2004:69–160.Google Scholar
  7. Drábek P, Girg P, Manásevich R: Generic Fredholm alternative-type results for the one dimensional p -Laplacian. Nonlinear Differential Equations and Applications 2001,8(3):285–298. 10.1007/PL00001449MathSciNetView ArticleMATHGoogle Scholar
  8. Dragoni GS: II problema dei valori ai limiti studiato in grande per le equazioni differenziali del secondo ordine. Mathematische Annalen 1931,105(1):133–143. 10.1007/BF01455811MathSciNetView ArticleGoogle Scholar
  9. 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
  10. Peterson A, Thompson HB: The Henstock-Kurzweil delta and nabla integrals. to appear in Journal of Mathematical Analysis and Applications
  11. Sattinger DH: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana University Mathematics Journal 1971/1972, 21: 979–1000.MathSciNetView ArticleGoogle Scholar
  12. Stehlík P: Periodic boundary value problems on time scales. Advances in Difference Equations 2005,2005(1):81–92. 10.1155/ADE.2005.81View ArticleMATHGoogle Scholar
  13. 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


© Petr Stehlík 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.