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The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm‐Liouville Functional Differential Equation

Author

Listed:
  • Yanan Li
  • Shurong Sun
  • Zhenlai Han
  • Hongling Lu

Abstract

We study boundary value problems for the following nonlinear fractional Sturm‐Liouville functional differential equations involving the Caputo fractional derivative: CDβ(p(t) CDαu(t)) + f(t, u(t − τ), u(t + θ)) = 0, t ∈ (0,1), CDαu(010)= CDαu()=( CDαu(0))′′=, au(t) − bu′(t) = η(t), t ∈ [−τ, 0], cu(t) + du′(t) = ξ(t), t ∈ [1,1 + θ], where CDα, CDβ denote the Caputo fractional derivatives, f is a nonnegative continuous functional defined on C([−τ, 1 + θ], ℝ), 1 0, and η ∈ C([−τ, 0], [0, ∞)), ξ ∈ C([1,1 + θ], [0, ∞)). By means of the Guo‐Krasnoselskii fixed point theorem and the fixed point index theorem, some positive solutions are obtained, respectively. As an application, an example is presented to illustrate our main results.

Suggested Citation

  • Yanan Li & Shurong Sun & Zhenlai Han & Hongling Lu, 2013. "The Existence of Positive Solutions for Boundary Value Problem of the Fractional Sturm‐Liouville Functional Differential Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:301560
    DOI: 10.1155/2013/301560
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    References listed on IDEAS

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    1. Yuanyuan Pan & Zhenlai Han & Shurong Sun & Yige Zhao, 2012. "The Existence of Solutions to a System of Discrete Fractional Boundary Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-15, March.
    2. Azizollah Babakhani & Dumitru Baleanu & Ravi P. Agarwal, 2013. "The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, March.
    3. Azizollah Babakhani & Dumitru Baleanu & Ravi P. Agarwal, 2013. "The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. Yige Zhao & Shurong Sun & Zhenlai Han & Qiuping Li, 2011. "Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    5. Yige Zhao & Shurong Sun & Zhenlai Han & Qiuping Li, 2011. "Positive Solutions to Boundary Value Problems of Nonlinear Fractional Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-16, December.
    6. Chuanzhi Bai, 2010. "Existence of Positive Solutions for a Functional Fractional Boundary Value Problem," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-13, June.
    7. Klaus Deimling, 1985. "Nonlinear Functional Analysis," Springer Books, Springer, number 978-3-662-00547-7, March.
    8. Chuanzhi Bai, 2010. "Existence of Positive Solutions for a Functional Fractional Boundary Value Problem," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    9. Yuanyuan Pan & Zhenlai Han & Shurong Sun & Yige Zhao, 2012. "The Existence of Solutions to a System of Discrete Fractional Boundary Value Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
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