Date: 20 Nov 2013

Positive solutions to m-point boundary value problem of fractional differential equation

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Abstract

In this paper, we study the multiplicity of positive solutions to the following m-point boundary value problem of nonlinear fractional differential equations: $\left\{ \begin{gathered} D^q u(t) + f(t,u(t)) = 0,0 < t < 1, \hfill \\ u(0) = 0,u(1) = \sum\limits_{i = 1}^{m - 2} {\mu _i D^p u(t)|_{t = \xi _i } } \hfill \\ \end{gathered} \right. $ , where qR, 1<q≤2, 0<ξ 1<ξ 2<⋯< \(\xi _{m - 2} \leqslant \tfrac{1} {2} \) , µ i ∈[0,+∞) and \(p = \tfrac{{q - 1}} {2},\Gamma (q)\sum\limits_{i = 1}^{m - 2} {\mu _i \xi _i^{\tfrac{{q - 1}} {2}} } < \Gamma (\tfrac{{q + 1}} {2}) \) , D q is the standard Riemann-Liouville differentiation, and fC([0,1]×[0,+∞),[0,+∞)). By using the Leggett-Williams fixed point theorem on a convex cone, some multiplicity results of positive solutions are obtained.

Project supported by Hunan Provincial Natural Science Foundation of China (11JJ3009); was also supported by the Scientific Research Foundation of Hunan Provincial Education Department(11C1187) and the Construct Program of the Key Discipline in Hunan Province.