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 D-alpha [(x) over bar (t) - (x) over bar (0)] = A (x) over bar (t), (x) over bar (0) = (x) over bar (0), where D-alpha denotes standard Riemann-Liouville fractional derivative, 0 < alpha < 1, (x) over bar (t) = [x(1)(t),...,x(n)(t)](1) and A is a square matrix. The unique solution to this initial value problem turns out to be Ealpha (t(alpha) A)(x) over bar (0), where Ealpha denotes the Mittag-Leffler function generalized for matrix arguments. Further we analyze the system D-alpha [(x) over bar (t) -(x) over bar (0)] = (f) over bar (t, (x) over bar), (x) over bar (0) = (x) over bar (0), 0 < alpha < 1, and investigate dependence of the solutions on the initial conditions. (C) 2004 Elsevier Inc. All rights reserved.