Existence and uniqueness of positive solutions for boundary value problems of a fractional differential equation with a parameter

In this paper, we are concerned with the existence and uniqueness of positive solutions for the following nonlinear fractional two-point boundary value problem { D-0+(alpha) u(t) + lambda f (t, u (t), u (t)) = 0; 0 < t < 1, 2 < alpha <= 3, u(0) = u'(0) = u'(1) = 0, where D(0+)(alpha)is the standard Riemann-Liouville fractional derivative, and lambda is a positive parameter. Our analysis relies on a fixed point theorem and some properties of eigenvalue problems for a class of general mixed monotone operators. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main results.