Triple positive solutions for a boundary value problem of nonlinear fractional differential equation

In this paper, we investigate the existence of three positive solutions for the nonlinear fractional boundary value problem D(0+)(alpha)u(t) + a(t) f(t, u(t), u ''(t)) = 0, 0 < t < 1, 3 < alpha <= 4, u(0) = u'(0) = u ''(0) = u ''(1) = 0, where D-0+(alpha) is the standard Riemann-Liouville fractional derivative. The method involves applications of a new fixed-point theorem due to Bai and Ge. The interesting point lies in the fact that the nonlinear term is allowed to depend on the second order derivative u ''.