Analysis of a System for Linear Fractional Differential Equations

Abstract

The main purpose of this paper is to obtain the unique solution of the constant coefficient homogeneous linear fractional differential equations $D_{t_0}^{q}X(t)=PX(t),X(a)=B$ and the constant coefficient nonhomogeneous linear fractional differential equations $D_{t_0}^{q}X(t)=PX(t)+D,X(a)=B$ if $P$ is a diagonal matrix and $X(t)\in C_{1-q}[t_0,T]\times C_{1-q}[t_0,T]\times \cdots \times C_{1-q}[t_0,T]$ and prove the existence and uniqueness of these two kinds of equations for any $P\in L(R^m)$ and $X(t)\in C_{1-q}[t_0,T]\times C_{1-q}[t_0,T]\times \cdots \times C_{1-q}[t_0,T]$. Then we give two examples to demonstrate the main results.