Analysis of a system of fractional differential equations

We prove existence and uniqueness theorems for the initial value problem for the system of fractional differential equations Dα[¯x(t)− ¯x(0)] = A¯x(t), ¯x(0) = ¯x0, where Dα denotes standard Riemann–Liouville fractional derivative, 0 < α < 1, ¯x(t) = [x1(t ), . . . , xn(t )]t and A is a square matrix. The unique solution to this initial value problem turns out to be Eα(tαA) ¯x0, where Eα denotes the Mittag–Leffler function generalized for matrix arguments. Further we analyze the system Dα[x¯(t)−x¯(0)] = f¯(t, x¯), x¯(0)=x¯0, 0<α<1, and investigate dependence of the solutions on the initial conditions.  2004 Elsevier Inc. All rights reserved.