Existence of multiple positive solutions of a nonlinear arbitrary order boundary value problem with advanced arguments

In this paper, we investigate nonlinear fractional differential equations of arbitrary order with advanced arguments    D 0+u(t) + a(t)f(u(θ(t))) = 0, 0 < t < 1, n− 1 < α ≤ n, u(i)(0) = 0, i = 0, 1, 2, · · · , n− 2, [D 0+ u(t)]t=1 = 0, 1 ≤ β ≤ n− 2, where n > 3 (n ∈ N), D 0+ is the standard Riemann-Liouville fractional derivative of order α, f : [0,∞) → [0,∞), a : [0, 1] → (0,∞) and θ : (0, 1) → (0, 1] are continuous functions. By applying fixed point index theory and Leggett-Williams fixed point theorem, sufficient conditions for the existence of multiple positive solutions to the above boundary value problem are established.