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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, \(D^{\bar {\alpha }}_* \bar x (t) = \bar f (t,\bar x), \bar x (0) = \bar x_0\); the other is a multi-order fractional differential equation, \(D^{\beta ^*}_* y(t) = g (t,y(t)), D^{\beta _1}_* y(t), \cdots , D^{\beta _n}_* y(t)\). 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, \(D^{\bar {\alpha }}_* \bar x (t) = A\bar x (t),\bar x(0) = \bar x(0)\), and \(D^{\beta 1*}_* y(t) = a_0 y(t) + \sum ^n_{i=1} a_i D^{\beta _i}_* y(t), y^{(k)} (0) = y^{(k)}_0, k = 0, 1, \cdots , \lceil \beta ^* \rceil - 1\), are also derived.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34M99 Ordinary differential equations in the complex domain
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