关键词: 分数阶微分方程, 多点边值问题, Schauder不动点定理, Leggett-Williams不动点定理, 多解性

Abstract: In this paper, a class of Riemann-Liouville fractional differential equation with multiple-points boundary conditions are studied: $\left\{\begin{array}{ll} D^{a}_{0+}u(t)+f(t,u(t),u'(t))=0, \\ u(0)=u'(0)=u''(0)=\cdot\cdot\cdot =u^{n-2}(0)=0,u'(1)=\sum^{m-2}_{i=1}\beta_{i}u'(\xi_{i}), \end{array}\right.$其中0≤t≤1,n-1<a≤n, n≥2, 0<β_i<1, 0<ξ_i<1, i=1,2,\cot\cot\cot,m-2。a_i>0,$\sum^{m-2}_{i=1}$β_iξ_i^a-2<1。The existence of solution for the boundary value problem is proved by using the Schauder fixed point theorem. At least three positive solutions of the above fractional differential equation are acquired by using the Leggett-Williams fixed point theorem. An example is given to illustrate the validity of the conclusion obtained in this paper.

Key words: fractional differential equations, multiple-point boundary value problem, Schauder fixed point theorem, Leggett-Williams fixed point theorem, multiple solutions