Advertisement

The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1999–2013 | Cite as

Infinitely many solutions for a perturbed nonlinear fractional boundary value problems depending on two parameters

  • N. NyamoradiEmail author
  • Y. Zhou
Regular Article

Abstract

In this paper we prove the existence and multiplicity of (weak) solutions for the following fractional boundary value problem:
where \(\alpha \in (\tfrac{1} {2},1]\), 0 D t α−1 and t D T α−1 are the left and right Riemann-Liouville fractional integrals of order 1 − α respectively, λ,μ ∈ [0,+∞), T > 0, F,GC([0,T] × R N ;R)\{0} and A = (a ij (t)) N×N is symmetric. Our approach is based on variational methods.

Keywords

European Physical Journal Special Topic Fractional Derivative Fractional Calculus Singular Integral Operator Fractional Boundary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    C. Bai, Abst. Appl. Anal. 2012 (2012) Article ID 963105, 13 pages, doi: 10.1155/2012/963105
  2. 2.
    C. Bai, Elect. J. Different. Equat. 2012, 1 (2012)Google Scholar
  3. 3.
    D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Series on Complexity, Nonlinearity and Chaos (World Scientific, 2012)Google Scholar
  4. 4.
    G. Bonanno, Journal of Global Optimization 28, 249 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    G. Bonanno, Nonlinear Anal. 75, 2992 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    G. Bonanno, G. Molica Bisci, Bound. Value Probl. 2009, 1 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A.M.A. El-Sayed, Nonlinear Anal. 33, 181 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    J. Feng, Z. Yong, Comput. Math. Appl. 62, 1181 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R. Hilfer (Singapore: World Scientific Publishing Co, 2000)Google Scholar
  10. 10.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, North-Holland Mathematics Studies, Vol. 204 (Elsevier Science B.V., Amsterdam, 2006)Google Scholar
  11. 11.
    A.A. Kilbas, J.J. Trujillo, Appl. Anal. 78, 153 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A.A. Kilbas, J.J. Trujillo, Appl. Anal. 81, 435 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    V. Lakshmikantham, A.S. Vatsala, Nonlinear Anal. TMA 69, 2677 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    K.S. Miller, B. Ross (Wiley, New York, 1993)Google Scholar
  15. 15.
    I. Podlubny (Academic Press, New York, 1999)Google Scholar
  16. 16.
    I. Podlubny, Slovac Academy of Science (Slovak Republic, 1994)Google Scholar
  17. 17.
    B. Ricceri, J. Comput. Appl. Math. 133, 401 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Sabatier, R.P. Agrawal, J.A. Tenreiro Machado (Springer, 2007)Google Scholar
  19. 19.
    G. Samko, A.A. Kilbas, O. Marichev (Gordon and Breach, Amsterdam, 1993)Google Scholar

Copyright information

© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesRazi UniversityKermanshahIran
  2. 2.School of Mathematics and Computational ScienceXiangtan UniversityHunanPR China

Personalised recommendations