Analysis of fractional differential equations with multi-orders.

*(English)* Zbl 1176.34008
Summary: We study two kinds of fractional differential systems with multi-orders. One is a system of fractional differential equations with multi-order, ${D}_{*}^{\overline{\alpha}}\overline{x}\left(t\right)=\overline{f}(t,\overline{x}),\overline{x}\left(0\right)={\overline{x}}_{0}$; the other is a multi-order fractional differential equation, ${D}_{*}^{{\beta}^{*}}y\left(t\right)=g(t,y\left(t\right)),{D}_{*}^{{\beta}_{1}}y\left(t\right),\cdots ,{D}_{*}^{{\beta}_{n}}y\left(t\right)$. By the derived technique, such two kinds of fractional differential equations can be changed into equations with the same fractional orders providing that the multi-orders are rational numbers, so the known theorems of existence, uniqueness and dependence upon initial conditions are easily applied. And asymptotic stability theorems for their associate linear systems, ${D}_{*}^{\overline{\alpha}}\overline{x}\left(t\right)=A\overline{x}\left(t\right),\overline{x}\left(0\right)=\overline{x}\left(0\right)$, and ${D}_{*}^{\beta 1*}y\left(t\right)={a}_{0}y\left(t\right)+{\sum}_{i=1}^{n}{a}_{i}{D}_{*}^{{\beta}_{i}}y\left(t\right),{y}^{\left(k\right)}\left(0\right)={y}_{0}^{\left(k\right)},k=0,1,\cdots ,\lceil {\beta}^{*}\rceil -1$, are also derived.

##### MSC:

34A25 | Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) |

34M99 | Differential equations in the complex domain |