Skip to main content

Second-order n-point eigenvalue problems on time scales

Abstract

We discuss conditions for the existence of at least one positive solution to a nonlinear second-order Sturm-Liouville-type multipoint eigenvalue problem on time scales. The results extend previous work on both the continuous case and more general time scales, and are based on the Guo-Krasnosel'skiĭ fixed point theorem.

[1234567891011121314151617181920212212345678910111213141516171819202122]

References

  1. Anderson DR: Extension of a second-order multi-point problem to time scales. Dynamic Systems and Applications 2003,12(3–4):393–403.

    MathSciNet  MATH  Google Scholar 

  2. Anderson DR: Nonlinear triple-point problems on time scales. Electronic Journal of Differential Equations 2004,2004(47):1–12.

    MathSciNet  Google Scholar 

  3. Anderson DR: Twin n -point boundary value problems. Applied Mathematics Letters 2004,17(9):1053–1059. 10.1016/j.aml.2004.07.008

    MathSciNet  Article  MATH  Google Scholar 

  4. Anderson DR, Avery RI: An even-order three-point boundary value problem on time scales. Journal of Mathematical Analysis and Applications 2004,291(2):514–525. 10.1016/j.jmaa.2003.11.013

    MathSciNet  Article  MATH  Google Scholar 

  5. Anderson DR, Avery RI, Henderson J: Existence of solutions for a one dimensional p -Laplacian on time-scales. Journal of Difference Equations and Applications 2004,10(10):889–896. 10.1080/10236190410001731416

    MathSciNet  Article  MATH  Google Scholar 

  6. 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 

  7. Aulbach B, Hilger S: Linear dynamic processes with inhomogeneous time scale. In Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), Math. Res.. Volume 59. Akademie, Berlin; 1990:9–20.

    Google Scholar 

  8. Bohner M, Luo H: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. to appear in Advances in Difference Equations

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  11. 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 

  12. DaCunha JJ, Davis JM, Singh PK: Existence results for singular three point boundary value problems on time scales. Journal of Mathematical Analysis and Applications 2004,295(2):378–391. 10.1016/j.jmaa.2004.02.049

    MathSciNet  Article  MATH  Google Scholar 

  13. Davis JM, Henderson J, Rajendra Prasad K, Yin W: Solvability of a nonlinear second order conjugate eigenvalue problem on a time scale. Abstract and Applied Analysis 2000,5(2):91–99. 10.1155/S108533750000018X

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  15. Kaufmann ER: Positive solutions of a three-point boundary-value problem on a time scale. Electronic Journal of Differential Equations 2003,2003(82):1–11.

    MathSciNet  Google Scholar 

  16. Kaufmann ER, Raffoul YN: Eigenvalue problems for a three-point boundary-value problem on a time scale. Electronic Journal of Qualitative Theory of Differential Equations 2004, (2):1–10.

  17. Kong L, Kong Q: Positive solutions of nonlinear m -point boundary value problems on a measure chain. Journal of Difference Equations and Applications 2003,9(1):121–133. 10.1080/10236100309487539

    MathSciNet  Article  MATH  Google Scholar 

  18. Krasnosel'skiĭ MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen; 1964:381.

    Google Scholar 

  19. Ma R, Thompson B: Positive solutions for nonlinear m -point eigenvalue problems. Journal of Mathematical Analysis and Applications 2004,297(1):24–37. 10.1016/j.jmaa.2003.12.046

    MathSciNet  Article  MATH  Google Scholar 

  20. Peterson AC, Raffoul YN, Tisdell CC: Three point boundary value problems on time scales. Journal of Difference Equations and Applications 2004,10(9):843–849. 10.1080/10236190410001702481

    MathSciNet  Article  MATH  Google Scholar 

  21. Sun HR, 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 

  22. Sun HR, Li WT: Positive solutions for nonlinear $m$-point boundary value problems on time scales. Acta Mathematica Sinica 2006,49(2):369–380.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Douglas R Anderson.

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

Anderson, D.R., Ma, R. Second-order n-point eigenvalue problems on time scales. Adv Differ Equ 2006, 059572 (2006). https://doi.org/10.1155/ADE/2006/59572

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/ADE/2006/59572

Keywords

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