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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 391609, 18 pages
http://dx.doi.org/10.1155/2012/391609
Research Article

Nontrivial Solutions for a Class of Fractional Differential Equations with Integral Boundary Conditions and a Parameter in a Banach Space with Lattice

School of Mathematics, Liaocheng University, Liaocheng, Shandong 252059, China

Received 17 September 2012; Accepted 12 December 2012

Academic Editor: Zhanbing Bai

Copyright © 2012 Xingqiu Zhang and Lin Wang. 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. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at Zentralblatt MATH · View at MathSciNet
  2. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, NewYork, NY, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. S. Zhang, “Existence of positive solutions for some class of nonlinear fractional equation,” Journal of Mathematical Analysis and Applications, vol. 278, pp. 136–148, 2003.
  7. H. Jafari and V. Daftardar-Gejji, “Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 700–706, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z. Wei, Q. Li, and J. Che, “Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 260–272, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 11, pp. 5545–5550, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Liang and J. Zhang, “Existence of multiple positive solutions for m-point fractional boundary value problems on an infinite interval,” Mathematical and Computer Modelling, vol. 54, no. 5-6, pp. 1334–1346, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. Liang and J. Zhang, “Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1333–1340, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. Zhang, “Positive solutions for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, pp. 1–12, 2006. View at Zentralblatt MATH · View at MathSciNet
  13. Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950–6958, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. X. Xu, D. Jiang, and C. Yuan, “Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4676–4688, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. B. Ahmad and G. Wang, “A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1341–1349, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Feng, X. Liu, and H. Feng, “The existence of positive solution to a nonlinear fractional differential equation with integral boundary conditions,” Advances in Difference Equations, vol. 2011, Article ID 546038, 14 pages, 2011. View at Zentralblatt MATH · View at MathSciNet
  17. G. Zhang and J. Sun, “Positive solutions of m-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 406–418, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  18. J. Sun and G. Zhang, “Nontrivial solutions of singular sublinear Sturm-Liouville problems,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 242–251, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. X. Hao, L. Liu, Y. Wu, and Q. Sun, “Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 6, pp. 1653–1662, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. B. Liu, L. Liu, and Y. Wu, “Positive solutions for singular second order three-point boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 66, no. 12, pp. 2756–2766, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. F. Xu, H. Su, and X. Zhang, “Positive solutions of fourth-order nonlinear singular boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 5, pp. 1284–1297, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. H. Ma, “Symmetric positive solutions for nonlocal boundary value problems of fourth order,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 3, pp. 645–651, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. S. Wang and J. Liu, “Coexistence of positive solutions of nonlinear three-point boundary value and its conjugate problem,” Journal of Mathematical Analysis and Applications, vol. 330, no. 1, pp. 334–351, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. L. Liu, B. Liu, and Y. Wu, “Nontrivial solutions of m-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term,” Journal of Computational and Applied Mathematics, vol. 224, no. 1, pp. 373–382, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  25. J. Sun and G. Zhang, “Nontrivial solutions of singular superlinear Sturm-Liouville problems,” Journal of Mathematical Analysis and Applications, vol. 313, no. 2, pp. 518–536, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. G. Han and Y. Wu, “Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1327–1338, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. J. Xu, Z. Wei, and W. Dong, “Uniqueness of positive solutions for a class of fractional boundary value problems,” Applied Mathematics Letters, vol. 25, no. 3, pp. 590–593, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. J. Sun and X. Liu, “Computation of topological degree for nonlinear operators and applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 11, pp. 4121–4130, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. J. Sun and X. Liu, “Computation of topological degree in ordered Banach spaces with lattice structure and its application to superlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 348, no. 2, pp. 927–937, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. View at MathSciNet
  32. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, San Diego, Calif, USA, 1988. View at MathSciNet
  33. W. A. J. Lucxemburg and A. C. Zaanen, Riesz Space, vol. 1, North-Holland Publishing, London, UK.
  34. M. G. Krein and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space,” American Mathematical Society Translations, vol. 10, no. 26, pp. 199–325, 1962. View at MathSciNet