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

Yanan Li, Shurong Sun, Zhenlai Han and Hongling Lu

Abstract and Applied Analysis, 2013, vol. 2013, issue 1

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.

Date: 2013
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https://doi.org/10.1155/2013/301560

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:301560

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