Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter

Abstract

In this paper, we study the existence of positive solutions for the following nonlinear fractional differential equations with integral boundary conditions: D0α+u(t) +h(t)f(t,u(t)) = 0, 0<t<1, u(0) = u′(0) = u″(0) = 0, u(1) = λ∫0ηu(s)ds, where 3 < α ≤ 4,0 < η ≤ 1, 0 -< ληαα <1, D0+α is the standard Riemann–Liouville derivative. h(t) is allowed to be singular at t=0 and t=1. By using the properties of the Green function, u0-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator, we obtain some existence results of positive solution.