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Multiple Unbounded Positive Solutions for the Boundary Value Problems of the Singular Fractional Differential Equations

  • Liu, Yuji (Department of Mathematics, Hunan Institute of Science and Technology) ;
  • Shi, Haiping (Basic Courses Department, Guangdong Construction Vocational Technology Institute) ;
  • Liu, Xingyuan (Department of Mathematics, Shaoyang University)
  • Received : 2011.04.22
  • Accepted : 2011.08.01
  • Published : 2013.06.23

Abstract

In this article, we establish the existence of at least three unbounded positive solutions to a boundary-value problem of the nonlinear singular fractional differential equation. Our analysis relies on the well known fixed point theorems in the cones.

Keywords

References

  1. R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl Math, 109(2011), 973-1033.
  2. A. Arara, M. Benchohra, N. Hamidi and J. J. Nieto, Fractional order differential equations on an unbounded domain, Nonlinear Analysis TMA, 72(2010), 580-586, https://doi.org/10.1016/j.na.2009.06.106
  3. Z. Bai, On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Analysis, 72(2010), 916-924. https://doi.org/10.1016/j.na.2009.07.033
  4. R. Dehghant and K. Ghanbari, Triple positive solutions for boundary value problem of a nonlinear fractional differential equation, Bulletin of the Iranian Mathematical Society, 33(2007), 1-14.
  5. A. A. Kilbas and J. J. Trujillo, Differential equations of fractional order: methods, results and problems-I, Applicable Analysis, 78(2001), 153-192. https://doi.org/10.1080/00036810108840931
  6. R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, 28(1979), 673-688. https://doi.org/10.1512/iumj.1979.28.28046
  7. Y. Liu, Positive solutions for singular FDES, U.P.B. Sci. Series A, 73(2011), 89-100.
  8. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley, New York, 1993.
  9. S. Z. Rida, H. M. El-Sherbiny and A. A. M. Arafa, On the solution of the fractional nonlinear Schrodinger equation, Physics Letters A, 372(2008), 553-558. https://doi.org/10.1016/j.physleta.2007.06.071
  10. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integral and Derivative. Theory and Applications, Gordon and Breach, 1993.
  11. X. Xu, D. Jiang and C. Yuan, Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation, Nonlinear Analysis TMA, 71(2009), 4676-4688. https://doi.org/10.1016/j.na.2009.03.030
  12. F. Zhang, Existence results of positive solutions to boundary value problem for fractional differential equation, Positiviyt, 13(2008), 583-599.
  13. S. Q. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252(2000), 804-812. https://doi.org/10.1006/jmaa.2000.7123