Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives

In this work, we establish the existence of multiple positive solutions for a general higher order fractional differential equation with derivatives and a sign-changing Caratheodory perturbed term {-D(alpha)x(t) = p(t)f(t,x(t), D(mu 1)x(t), D(mu 2)x(t), ..., D(mu n-1)x(t)) - g(t,x(t), D(mu 1)x(t), D(mu 2)x(t), ..., D(mu n-1)x(t)), D(mu i)x(0) = 0, 1 <= i <= n - 1, D mu n-1+1 x(0) = 0, D(mu n-1)x(1) = Sigma(m-2)(j-1)a(j)D(mu n-1)x(xi(j)), where n - 1 < alpha <= n, n is an element of N and n >= 3 with 0 < mu(1) < mu(2) < ... < mu(n-2) < mu(n-1) and n - 3 < mu(n-1) < alpha - 2, a(j) is an element of R, 0 < xi(1) < xi(2) < ... < xi(m-2) < 1 satisfying 0 < Sigma(m-2)(j-1)a(j)xi(alpha-mu n-1-1)(j) 1, D-alpha is the standard Riemann-Liouville derivative, f is an element of C([0, 1] x R-n, [0, +infinity)). This equation is viewed as a perturbation of a general higher order fractional differential equation, where the perturbed term g : [0, 1] x R-n -> (-infinity, +infinity) only satisfies the global Caratheodory conditions, which implies that the perturbed effect of g on f is quite large so that the nonlinearity can tend to negative infinity at some singular points. (C) 2012 Elsevier Inc. All rights reserved.