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  • Research Article
  • Open Access

Second-order n-point eigenvalue problems on time scales

Advances in Difference Equations20062006:059572

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

  • Received: 10 December 2004
  • Accepted: 6 November 2005
  • Published:

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.

Keywords

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

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Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562, USA
(2)
Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China

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.MathSciNetMATHGoogle Scholar
  2. Anderson DR: Nonlinear triple-point problems on time scales. Electronic Journal of Differential Equations 2004,2004(47):1–12.MathSciNetGoogle Scholar
  3. Anderson DR: Twin n -point boundary value problems. Applied Mathematics Letters 2004,17(9):1053–1059. 10.1016/j.aml.2004.07.008MathSciNetView ArticleMATHGoogle 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.2003.11.013MathSciNetView ArticleMATHGoogle 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. 10.1080/10236190410001731416MathSciNetView ArticleMATHGoogle 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-XMathSciNetView ArticleMATHGoogle 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 EquationsGoogle Scholar
  9. Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  10. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.MATHGoogle 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.2000.6781MathSciNetView ArticleMATHGoogle 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.2004.02.049MathSciNetView ArticleMATHGoogle 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/S108533750000018XMathSciNetView 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. 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.MathSciNetGoogle 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.Google Scholar
  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. 10.1080/10236100309487539MathSciNetView ArticleMATHGoogle 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.2003.12.046MathSciNetView ArticleMATHGoogle 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. 10.1080/10236190410001702481MathSciNetView ArticleMATHGoogle 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.2004.03.079MathSciNetView ArticleMATHGoogle 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.MathSciNetMATHGoogle Scholar

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