Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary conditions

In the present manuscript we analyze non-linear multi-order fractional differential equation $$L\left( D \right)u\left( t \right) = f\left( {t,u\left( t \right)} \right), t \in \left[ {0,T} \right], T > 0,$$ where $$L\left( D \right) = \lambda _n ^c D^{\alpha _n } + \lambda _{n - 1} ^c D^{\alpha _{n - 1} } + \cdots + \lambda _1 ^c D^{\alpha _1 } + \lambda _0 ^c D^{\alpha _0 } ,\lambda _i \in \mathbb{R}\left( {i = 0,1, \cdots ,n} \right),\lambda _n \ne 0, 0 \leqslant \alpha _0 < \alpha _1 < \cdots < \alpha _{n - 1} < \alpha _n < 1,$$ and c D α denotes the Caputo fractional derivative of order α. We find the Greens functions for this equation corresponding to periodic/anti-periodic boundary conditions in terms of the two-parametric functions of Mittag-Leffler type. Further we prove existence and uniqueness theorems for these fractional boundary value problems.