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

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 q∈R, 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 f∈C([0,1]×[0,+∞),[0,+∞)). By using the Leggett-Williams fixed point theorem on a convex cone, some multiplicity results of positive solutions are obtained.