Positive solutions for three-point nonlinear fractional boundary value problems

In this paper, we give sufficient conditions for the existence or the nonexistence of positive solutions of the nonlinear fractional boundary value problem D-0(alpha) + u + a(t)f(u(t)) = 0, 0 < t < 1, 2 < alpha < 3, u(0) - u'(0) - 0, u'(1) - mu u'(eta) - lambda, where D-0+(alpha) is the standard Riemann-Liouville fractional differential operator of order alpha, eta is an element of (0, 1), mu is an element of [0, 1/eta(alpha-2)) are two arbitrary constants and lambda is an element of [0, infinity) is a parameter. The proof uses the Guo-Krasnosel'skii fixed point theorem and Schauder's fixed point theorem.