Existence and multiplicity of solutions for nonlinear fractional differential equations

In this paper, we consider the following fractional initial value problems: { Mathematical Formulas } 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.