Open Access

Singular second-order multipoint dynamic boundary value problems with mixed derivatives

Advances in Difference Equations20062006:054989

https://doi.org/10.1155/ADE/2006/54989

Received: 12 September 2005

Accepted: 26 October 2005

Published: 18 June 2006

Abstract

We study a certain singular second-order m-point boundary value problem on a time scale and establish the existence of a solution. The proof of our main result is based upon the Leray-Schauder continuation theorem.

[12345678910111213141516171819202122]

Authors’ Affiliations

(1)
Department of Mathematics, University of Missouri–Rolla
(2)
Department of Mathematics, Northwest Normal University

References

  1. Agarwal RP, O'Regan D: Some new results for singular problems with sign changing nonlinearities. Journal of Computational and Applied Mathematics 2000,113(1–2):1–15. 10.1016/S0377-0427(99)00239-3MathSciNetView ArticleMATHGoogle Scholar
  2. Ahlbrandt CD, Morian C: Partial differential equations on time scales. Journal of Computational and Applied Mathematics 2002,141(1–2):35–55. Special issue on Dynamic Equations on Time Scales, edited by R. P. Agarwal, M. Bohner, and D. O'Regan 10.1016/S0377-0427(01)00434-4MathSciNetView ArticleMATHGoogle Scholar
  3. 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
  4. Asakawa H: Nonresonant singular two-point boundary value problems. Nonlinear Analysis Series A: Theory and Methods 2001,44(6):791–809. 10.1016/S0362-546X(99)00308-9MathSciNetView ArticleMATHGoogle Scholar
  5. 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. Special issue on Dynamic Equations on Time Scales, edited by R. P. Agarwal, M. Bohner, and D. O'Regan 10.1016/S0377-0427(01)00437-XMathSciNetView ArticleMATHGoogle Scholar
  6. Bohner M, Peterson A: Dynamic Equations on Time Scales. An Introduction with Applications. Birkhäuser Boston, Massachusetts; 2001:x+358.View ArticleMATHGoogle Scholar
  7. Bohner M, Peterson A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Massachusetts; 2003:xii+348.MATHGoogle Scholar
  8. 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
  9. Gupta CP, Ntouyas SK, Tsamatos PCh: Solvability of an m -point boundary value problem for second order ordinary differential equations. Journal of Mathematical Analysis and Applications 1995,189(2):575–584. 10.1006/jmaa.1995.1036MathSciNetView ArticleMATHGoogle Scholar
  10. He Z: Existence of two solutions of m -point boundary value problem for second order dynamic equations on time scales. Journal of Mathematical Analysis and Applications 2004,296(1):97–109. 10.1016/j.jmaa.2004.03.051MathSciNetView ArticleMATHGoogle Scholar
  11. Henderson J, Kaufmann ER: Focal boundary value problems for singular difference equations. Computers & Mathematics with Applications 1998,36(10–12):1–10.MathSciNetView ArticleMATHGoogle Scholar
  12. Henderson J, Yin W: Focal boundary-value problems for singular ordinary differential equations. In Advances in Nonlinear Dynamics, Stability Control Theory Methods Appl.. Volume 5. Gordon and Breach, Amsterdam; 1997:283–295.Google Scholar
  13. 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
  14. Ma R: Existence of positive solutions for superlinear semipositone m -point boundary-value problems. Proceedings of the Edinburgh Mathematical Society. Series II 2003,46(2):279–292. 10.1017/S0013091502000391MathSciNetView ArticleMATHGoogle Scholar
  15. Ma R, Luo H: Existence of solutions for a two-point boundary value problem on time scales. Applied Mathematics and Computation 2004,150(1):139–147. 10.1016/S0096-3003(03)00204-2MathSciNetView ArticleMATHGoogle Scholar
  16. Ma R, O'Regan D: Solvability of singular second order m -point boundary value problems. Journal of Mathematical Analysis and Applications 2005,301(1):124–134. 10.1016/j.jmaa.2004.07.009MathSciNetView ArticleMATHGoogle Scholar
  17. Mawhin J: Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics. Volume 40. American Mathematical Society, Rhode Island; 1979:v+122.View ArticleGoogle Scholar
  18. O'Regan D: Theory of Singular Boundary Value Problems. World Scientific, New Jersey; 1994:xii+154.View ArticleMATHGoogle Scholar
  19. Taliaferro SD: A nonlinear singular boundary value problem. Nonlinear Analysis 1979,3(6):897–904. 10.1016/0362-546X(79)90057-9MathSciNetView ArticleMATHGoogle Scholar
  20. Wong PJY, Agarwal RP: On the existence of solutions of singular boundary value problems for higher order difference equations. Nonlinear Analysis 1997,28(2):277–287. 10.1016/0362-546X(95)00151-KMathSciNetView ArticleMATHGoogle Scholar
  21. Zhang Y: Positive solutions of singular sublinear Emden-Fowler boundary value problems. Journal of Mathematical Analysis and Applications 1994,185(1):215–222. 10.1006/jmaa.1994.1243MathSciNetView ArticleMATHGoogle Scholar
  22. Zhang Z, Wang J: The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems. Journal of Computational and Applied Mathematics 2002,147(1):41–52. 10.1016/S0377-0427(02)00390-4MathSciNetView ArticleMATHGoogle Scholar

Copyright

© M. Bohner and H. Luo 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.