Skip to main content
SpringerLink
Log in

Positive solutions for a second order boundary value problem on time scales

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper using fixed point index theory we study the existence of positive solutions for a second order boundary value problem on time scales. Our existence theorems will be expressed under some conditions concerning the first eigenvalue of a relevant linear operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales, An Introduction with Applications. Birkhauser, Boston (2001)

    Book  MATH  Google Scholar 

  2. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003)

    Book  MATH  Google Scholar 

  3. Hilger, S.: Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atici, F., Biles, D., Lebedinsky, A.: An application of time scales to economics. Math. Comput. Model. 43, 718–726 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Keller, S.: Asymptotisches Verhalten invarianter Faserbündel bei Diskretisierung und Mittelwertbildung im Rahmen der Analysis auf Zeitskalen. Ph.D. thesis, Universität Augsburg (1999)

  6. Spedding, V.: Taming nature’s numbers. New Sci.: Global Sci. Technol. Wkly. 2404, 28–31 (2003)

    Google Scholar 

  7. Thomas, D., Vandemuelebroeke, L., Yamaguchi, K.: A mathematical evolution model for phytoremediation of metals. Discrete Contin. Dyn. Syst. Ser. B 5, 411–422 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, Y., Shu, J.: Multiple positive solutions for first-order impulsive integral boundary value problems on time scales. Bound. Value Prob. 2011, 1–19 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Li, H., Sun, J., Cui, Y.: Positive solutions of nonlinear differential equations on a measure chain. Chin. Ann. Math. 30A, 97–106 (2009). (Chinese)

    MathSciNet  MATH  Google Scholar 

  10. Sang, Y., Wei, Z.: Existence of solutions to a semipositone third-order three-point BVP on time scales. Acta Math. Sci. 31A, 455–465 (2011). (Chinese)

    MathSciNet  MATH  Google Scholar 

  11. Denk, A., Topal, S.: Existence of positive solutions for the second order semipositone \(m\)-point boundary value problem on time scales. Differ. Equ. Dyn. Syst. 22, 265–280 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hamal, N., Yoruk, F.: Positive solutions of nonlinear \(m\)-point boundary value problems on time scales. J. Comput. Appl. Math. 231, 92–105 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Karaca, I.: Multiple positive solutions for dynamic \(m\)-point boundary value problems. Dyn. Syst. Appl. 17, 25–42 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Karaca, I.: Positive solutions for boundary value problems of second-order functional dynamic equations on time scales. Adv. Differ. Equ. 2009, Article ID 829735, p. 21 (2009)

  15. Karaca, I., Ozen, O., Tokmak, F.: Multiple positive solutions of boundary value problems for \(p\)-Laplacian impulsive dynamic equations on time scales. Fixed Point Theo. 15, 475–486 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Han, W., Kang, S.: Multiple positive solutions of nonlinear third-order BVP for a class of \(p\)-Laplacian dynamic equations on time scales. Math. Comput. Model. 49, 527–535 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dogan, A.: Eigenvalue problems for nonlinear third-order \(m\)-point \(p\)-Laplacian dynamic equations on time scales. Math. Meth. Appl. Sci. (2014). doi:10.1002/mma.3258

  18. Dogan, A., Graef, J., Kong, L.: Higher order semipositone multi-point boundary value problems on time scales. Comput. Math. Appl. 60, 23–35 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goodrich, C.: Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale. Comment. Math. Univ. Carolin. 54, 509–525 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Zhang, C., Wang, L., Fan, Y.: Existence of positive solutions for a dynamic equation on measure chains. Appl. Math. Lett. 35, 24–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deimling, K.: Nonlinear Functional Analysis. Springer-Verlag, Berlin (1985)

    Book  MATH  Google Scholar 

  22. Zeidler, E.: Nonlinear Functional Analysis and its Applications, I. Fixed-Point Theorems. Springer-Verlag, New-York (1985)

    Book  MATH  Google Scholar 

  23. Zhang, Q., Lin, Q.: Lectures on Functional Analysis. Peking University Press, Beijing (1987). (in Chinese)

    Google Scholar 

  24. Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, Orlando (1988)

    MATH  Google Scholar 

Download references

Acknowledgments

Research supported by the NNSF-China (11371117), Shandong Provincial Natural Science Foundation (ZR2013AM009).

Author information

Authors and Affiliations

  1. School of Mathematics, Chongqing Normal University, Chongqing, 401331, China

    Jiafa Xu

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

  3. Nonlinear Analysis and Applied Mathematics (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

    Donal O’Regan

Authors
  1. Jiafa Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Donal O’Regan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiafa Xu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, J., O’Regan, D. Positive solutions for a second order boundary value problem on time scales. J. Appl. Math. Comput. 51, 127–144 (2016). https://doi.org/10.1007/s12190-015-0895-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-015-0895-5

Keywords

Mathematics Subject Classification

Search

Navigation