On the existence and uniqueness of solutions of a class of fractional differential equations

Abstract In this paper, we investigate the existence and uniqueness of solutions for the following class of multi-order fractional differential equations D β 1 γ 1 , δ 1 ⋯ D β n γ n , δ n u ( t ) : = ∏ i = 1 n D β i γ i , δ i u ( t ) : = D β i , n γ i , δ i u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = 0 , ∑ i = 1 n δ i ⩽ 1 , γ i > 0 , β i > 0 , 1 ⩽ i ⩽ n , where D β i , n γ i , δ i denotes the generalized Erdelyi–Kober operator of fractional derivative of order δ i . Moreover, some properties concerning the positive, maximal, minimal, and continuation of solutions are obtained.