Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure

Abstract

In this article, we consider the existence of positive solutions of the (n - 1, 1) conjugate-type nonlocal fractional differential equation { D 0α+x(t) + f (t, x(t)) = 0, 0 < t < 1, n - 1 < α ≤ n, x(k)(0) = 0, 0 ≤ k ≤ n - 2, x(1) = ∫ 01 x(s)dA(s), where a ≥ 2, D0α+ is the standard Riemann-Liouville derivative, ∫ 01 x(s)dA(s) is a linear functional given by the Stieltjes integral, A is a function of bounded variation, and dA may be a changing-sign measure, namely the value of the linear functional is not assumed to be positive for all positive x. By constructing upper and lower solutions, some sufficient conditions for the existence of positive solutions to the problem are established utilizing Schauder's fixed point theorem in the case in which the nonlinearities f(t, x) are allowed to have the singularities at t = 0 and (or) 1 and also at x = 0. © 2012 Wu et al.