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On lower and upper solutions without ordering on time scales

Abstract

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.

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References

  1. 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)00004-6

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Akin E: Boundary value problems for a differential equation on a measure chain. Panamerican Mathematical Journal 2000,10(3):17–30.

    MathSciNet  MATH  Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

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

    Google Scholar 

  5. 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.2003.08.018

    MathSciNet  Article  MATH  Google Scholar 

  6. 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. 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/PL00001449

    MathSciNet  Article  MATH  Google Scholar 

  8. 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/BF01455811

    MathSciNet  Article  Google Scholar 

  9. 9.

    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 

  10. 10.

    Peterson A, Thompson HB: The Henstock-Kurzweil delta and nabla integrals. to appear in Journal of Mathematical Analysis and Applications

  11. 11.

    Sattinger DH: Monotone methods in nonlinear elliptic and parabolic boundary value problems. Indiana University Mathematics Journal 1971/1972, 21: 979–1000.

    MathSciNet  Article  Google Scholar 

  12. 12.

    Stehlík P: Periodic boundary value problems on time scales. Advances in Difference Equations 2005,2005(1):81–92. 10.1155/ADE.2005.81

    Article  MATH  Google Scholar 

  13. 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.2002.04.005

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Petr Stehlík.

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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.

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Stehlík, P. On lower and upper solutions without ordering on time scales. Adv Differ Equ 2006, 073860 (2006). https://doi.org/10.1155/ADE/2006/73860

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Keywords

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