Analysis of a system of fractional differential equations

AbstractWe prove existence and uniqueness theorems for the initial value problem for the system of fractional differential equations Dα[x̄(t)−x̄(0)]=Ax̄(t), x̄(0)=x̄0, where Dα denotes standard Riemann–Liouville fractional derivative, 0<α<1, x̄(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)x̄0, where Eα denotes the Mittag–Leffler function generalized for matrix arguments. Further we analyze the system Dα[x̄(t)−x̄(0)]=f̄(t,x̄), x̄(0)=x̄0, 0<α