Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2019, Article ID 5923490, 6 pages
https://doi.org/10.1155/2019/5923490
Research Article

The Unique Positive Solution for Singular Hadamard Fractional Boundary Value Problems

1School of Mathematics, Qufu Normal University, Qufu, Shandong, 273165, China
2Department of Mathematics, Jining University, Qufu, Shandong, 273155, China

Correspondence should be addressed to Jinxiu Mao; moc.361@2891uixnijoam

Received 21 February 2019; Accepted 31 May 2019; Published 20 June 2019

Guest Editor: Pedro Garrancho

Copyright © 2019 Jinxiu Mao et al. 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. K. Diethelm and A. Freed, “On the solution of nonlinear fractional order differential equations used in the modeling of viscoelasticity,” in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werther, Eds., pp. 217–224, Springer-Verlag, Heidelberg, Germany, 1999. View at Google Scholar
  2. B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335–1342, 2008. View at Publisher · View at Google Scholar · 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 Publisher · View at Google Scholar · View at Scopus
  4. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Arafa, S. Rida, and M. Khalil, “Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection,” Nonlinear Biomedical Physics, vol. 6, no. 1, pp. 1–7, 2012. View at Google Scholar
  6. J. W. Kirchner, X. Feng, and C. Neal, “Frail chemistry and its implications for contaminant transport in catchments,” Nature, vol. 403, no. 6769, pp. 524–527, 2000. View at Publisher · View at Google Scholar · View at Scopus
  7. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “Application of a fractional advection-dispersion equation,” Water Resources Research, vol. 36, no. 6, pp. 1403–1412, 2000. View at Publisher · View at Google Scholar · View at Scopus
  8. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, “The fractional-order governing equation of Lévy motion,” Water Resources Research, vol. 36, no. 6, pp. 1413–1423, 2000. View at Publisher · View at Google Scholar · View at Scopus
  9. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  10. X. Zhang, L. Liu, and Y. Wu, “Variational structure and multiple solutions for a fractional advection-dispersion equation,” Computers & Mathematics with Applications, vol. 68, no. 12, part A, pp. 1794–1805, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  11. Y. Wang and L. Liu, “Positive solutions for a class of fractional 3-point boundary value problems at resonance,” Advances in Difference Equations, vol. 2017, no. 13, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Z. Yue and Y. Zou, “New uniqueness results for fractional differential equation with dependence on the first order derivative,” Advances in Difference Equations, vol. 2019, no. 1, p. 38, 2019. View at Publisher · View at Google Scholar
  13. K. M. Zhang, “On a sign-changing solution for some fractional differential equations,” Boundary Value Problems, vol. 2017, no. 59, 2017. View at Google Scholar · View at MathSciNet
  14. Y. Guan, Z. Zhao, and X. Lin, “On the existence of positive solutions and negative solutions of singular fractional differential equations via global bifurcation techniques,” Boundary Value Problems, vol. 2016, no. 141, pp. 1–18, 2016. View at Publisher · View at Google Scholar · View at Scopus
  15. Y. L. Guan, Z. Q. Zhao, and X. Lin, “On the existence of solutions for impulsive fractional differential equations,” Advances in Mathematical Physics, vol. 2017, Article ID 1207456, 12 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  16. J. Jiang, W. Liu, and H. Wang, “Positive solutions for higher order nonlocal fractional differential equation with integral boundary conditions,” Journal of Function Spaces, vol. 2018, Article ID 6598351, 12 pages, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. Q. Jiang, W. W. Liu, and H. C. Wang, “Positive solutions to singular dirichlet-type boundary value problems of nonlinear fractional differential equations,” Advances in Difference Equations, vol. 2018, no. 169, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  18. X. S. Du and A. M. Mao, “Existence and multiplicity of nontrivial solutions for a class of semilinear fractional Schrödinger equations,” Journal of Function Spaces, vol. 2017, Article ID 3793872, 7 pages, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  19. J. X. Mao and Z. Q. Zhao, “The existence and uniqueness of positive solutions for integral boundary value problems,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 34, no. 1, pp. 153–164, 2011. View at Google Scholar · View at MathSciNet
  20. J. X. Mao and Z. Q. Zhao, “On existence and uniqueness of positive solutions for integral boundary boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 16, pp. 1–8, 2010. View at Google Scholar
  21. J. X. Mao, Z. Q. Zhao, and C. G. Wang, “The exact iterative solution of fractional differential equation with nonlocal boundary value conditions,” Journal of Function Spaces, vol. 2018, Article ID 8346398, 6 pages, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  22. J. X. Mao, Z. Q. Zhao, and A. X. Qian, “Laplace's equation with concave and convex boundary nonlinearities on an exterior region,” Boundary Value Problems, vol. 51, pp. 1–12, 2019. View at Publisher · View at Google Scholar · View at MathSciNet
  23. K. Y. Zhang and Z. Q. Fu, “Solutions for a class of Hadamard fractional boundary value problems with sign-changing nonlinearity,” Journal of Function Spaces, vol. 2019, Article ID 9046472, 7 pages, 2019. View at Publisher · View at Google Scholar · View at MathSciNet
  24. J. Tariboon, S. K. Ntouyas, S. Asawasamrit, and C. Promsakon, “Positive solutions for Hadamard differential systems with fractional integral conditions on an unbounded domain,” Open Mathematics, vol. 15, no. 1, pp. 645–666, 2017. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. B. Ahmad and S. Ntouyas, “A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations,” Fractional Calculus and Applied Analysis, vol. 17, no. 2, pp. 348–360, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  26. W. G. Yang and Y. P. Qin, “Positive solutions for nonlinear Hadamard fractional differential equations with integral boundary conditions,” ScienceAsia, vol. 43, no. 3, pp. 201–206, 2017. View at Publisher · View at Google Scholar · View at Scopus
  27. Q. Ma, R. Wang, J. Wang, and Y. Ma, “Qualitative analysis for solutions of a certain more generalized two-dimensional fractional differential system with Hadamard derivative,” Applied Mathematics and Computation, vol. 257, pp. 436–445, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. G. Wang, K. Pei, R. P. Agarwal, L. Zhang, and B. Ahmad, “Nonlocal hadamard fractional boundary value problem with hadamard integral and discrete boundary conditions on a half-line,” Journal of Computational and Applied Mathematics, vol. 343, pp. 230–239, 2018. View at Publisher · View at Google Scholar · View at MathSciNet