In this paper, we consider the following fractional initial value problems: Dαu(t) = f(t, u(t), Dβu(t)), t ∈ (0, 1], u (k) (0) = ηk, k = 0, 1, ..., n − 1, where n − 1 < β < α < n, (n ∈ N) are real numbers, Dα and Dβ are the Caputo fractional derivatives and f ∈ C([0, 1] × R). Using the fixed point index theory, we study the existence and multiplicity of positive solutions and obtain some new results. c 2016 All rights reserved.
