Blowing-up solutions of multi-order fractional differential equations with the periodic boundary condition

In this paper, we analyze the boundary value problem of a class of multi-order fractional differential equations involving the standard Caputo fractional derivative with the general periodic boundary conditions: { L ( D ) u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , T > 0 , u ( 0 ) = u ( T ) > 0 , u ′ ( 0 ) = u ′ ( T ) > 0 , where L ( D ) = ∑ i = 0 n a i D S i , 1 ≤ S 0 < ⋯ < S n − 1 < S n < 2 , a i ∈ R , a n ≠ 0 , and f : [ 0 , T ] × R → R is a continuous operation. We get the Green’s function in terms of the Laplace transform. We obtain the existence and uniqueness of solution for the class of multi-order fractional differential equations. We investigate the blowing-up solutions to the special case f ( t , u ( t ) ) = | u ( t ) | p , a i ≥ 0 , and give an upper bound on the blow-up time T max .