About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 740760, 14 pages
http://dx.doi.org/10.1155/2012/740760
Research Article

Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

1School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, Sichuan, China
2School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, China

Received 22 October 2012; Accepted 19 November 2012

Academic Editor: Dragoş-Pătru Covei

Copyright © 2012 Jing Wu and Xinguang Zhang. 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.

Linked References

  1. H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, New York, NY, USA, John Wiley and Sons, Chichester, UK, 1989. View at Zentralblatt MATH
  2. L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991. View at Scopus
  3. W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995. View at Scopus
  4. K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995. View at Scopus
  6. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH
  7. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH
  9. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. View at Zentralblatt MATH
  10. X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012. View at Publisher · View at Google Scholar
  11. B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1727–1740, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012. View at Publisher · View at Google Scholar
  14. X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012. View at Publisher · View at Google Scholar
  16. H. A. H. Salem, “On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 565–572, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050–1055, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, vol. 2011, 2011. View at Publisher · View at Google Scholar
  20. C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1251–1268, 2011. View at Publisher · View at Google Scholar · View at Scopus
  21. C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. C. S. Goodrich, “Positive solutions to boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 1, pp. 417–432, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. Jia, X. Zhang, and X. Gu, “Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions,” Boundary Value Problems, vol. 2012, 70 pages, 2012. View at Publisher · View at Google Scholar
  24. M. Jia, X. Liu, and X. Gu, “Uniqueness and asymptotic behavior of positive solutions for a fractional-order integral boundary value problem,” Abstract and Applied Analysis, vol. 2012, Article ID 294694, 21 pages, 2012. View at Publisher · View at Google Scholar
  25. P.-L. Lions, “On the existence of positive solutions of semilinear elliptic equations,” SIAM Review, vol. 24, no. 4, pp. 441–467, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. R. Aris, Introduction to the Analysis of Chemical Reactors, Prentice-Hall, Englewood Cliffs, NJ, USA, 1965.
  27. A. Castro, C. Maya, and R. Shivaji, “Nonlinear eigenvalue problems with semipositone,” Electronic Journal of Differential Equations, vol. 5, pp. 33–49, 2000.
  28. V. Anuradha, D. D. Hai, and R. Shivaji, “Existence results for superlinear semipositone BVP'S,” Proceedings of the American Mathematical Society, vol. 124, no. 3, pp. 757–763, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, New York, NY, USA, 1988.