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

Abstract

In the present manuscript we analyze non-linear multi-order fractional differential equation L(D)u (t)=f(t,u(t)), t ε[O,T], T>0, where L(D)= λncDαn+ λn-1cDαn-1+ ⋯ + λ1cDα1 + λ0c D α0, λ1 ε ℝ (i=0,1, ⋯, n), λn ≠ 0, 0 ≤ α0 < α1 < ⋯ < αn < 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. © 2014 Versita Warsaw and Springer-Verlag Wien. MSC 2010: Primary 26A33 Secondary 33E12, 34A08, 34K37, 35R11.