摘要：In this paper, we consider the nonlinear three-point boundary value problem of fractional differential equations D 0 + α u ( t ) + a ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , 2 < α ≤ 3 , $$D^{\alpha}_{0^{+}}u(t)+a(t)f\bigl(t,u(t)\bigr)=0, \quad 0< t< 1, 2< \alpha\leq3, $$ with boundary conditions u ( 0 ) = 0 , D 0 + β u ( 0 ) = 0 , D 0 + β u ( 1 ) = b D 0 + β u ( ξ ) , 1 ≤ β ≤ 2 , $$u(0)=0,\qquad D^{\beta}_{0^{+}}u(0)=0,\qquad D^{\beta}_{0^{+}}u(1)=bD^{\beta}_{0^{+}}u( \xi),\quad 1\leq\beta\leq2, $$ involving Riemann-Liouville fractional derivatives D 0 + α $D^{\alpha}_{0^{+}}$ and D 0 + β $D^{\beta}_{0^{+}}$ , where a ( t ) $a(t)$ maybe singular at t = 0 $t=0$ or t = 1 $t=1$ . We use the Banach contraction mapping principle and the Leggett-Williams fixed point theorem to obtain the existence and uniqueness of positive solutions and the existence of multiple positive solutions. We investigate the above fractional differential equations without many preconditions by the fixed point index theory and obtain the existence of a single positive solution. Some examples are given to show the applicability of our main results.
关键词：fractional differential equations ; three-point boundary value problem ; existence and multiplicity ; fixed point theorem ; positive solution
