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

Abstract

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 α 0+u(t) + λf(t, u(t), u(t)) = 0, 0 < t < 1, 2 < α ≤ 3, u(0) = u 0 (0) = u 0 (1) = 0, where D α 0+ is the standard Riemann-Liouville fractional derivative, and λ 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.