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(D)u(t) = f(t, u(t)), t is an element of [0, T], T > 0, where L(D) = lambda D-c(n)alpha n + lambda(n-1) D-c(alpha n-1) + . . . + lambda(1) D-c(alpha 1) + lambda(0) D-c(alpha 0), lambda(i) is an element of R (i = 0, 1, . . . , n), lambda(n) not equal 0, 0 <= alpha(0) < alpha(1) < . . . < alpha(n-1) < alpha(n) < 1, and D-c(alpha) denotes the Caputo fractional derivative of order alpha. 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.