## 一类广义分数阶系统的Hyers-Ulam-Rassias稳定性

1 长治学院数学系 山西长治 046011

2 北京理工大学数学与统计学院 北京 100081

## Hyers-Ulam-Rassias Stability on a Class of Generalized Fractional Systems

Wang Chun,1, Xu Tianzhou2

1 Department of Mathematics, Changzhi University, Shanxi Changzhi 046011

2 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081

 基金资助: 山西省自然科学基金.  201801D121024山西省高等学校科技创新项目.  2019L0903

 Fund supported: the NSF of Shanxi Province.  201801D121024the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi.  2019L0903

Abstract

This paper investigates the stability in the sense of Hyers-Ulam-Rassias for a class of generalized fractional differential systems by the generalized Laplace transform method. Several examples are given to illustrate the theoretical results.

Keywords： Generalized Laplace transform ; Generalized fractional differential systems ; Hyers-Ulam-Rassias stability

Wang Chun, Xu Tianzhou. Hyers-Ulam-Rassias Stability on a Class of Generalized Fractional Systems. Acta Mathematica Scientia[J], 2021, 41(6): 1791-1804 doi:

## 1 引言

Obloza第一个研究了线性微分方程的Hyers-Ulam稳定性, 可参见文献[14, 15]. 从此开始, 很多学者把关注点放在了微分方程的Hyers-Ulam稳定性的研究上, 得到了很多有意义的重要结果. 这方面的研究可参见文献[1, 3, 4, 7-9, 13, 17, 20, 23, 25-33].

$X$是一个赋范空间, $I$是一个开区间. 对某一个$\varepsilon\geq 0$, 如果对任意的满足微分不等式

## 2 定义和引理

$$$\left\|^C_aD_g^\alpha\vec{y}(t)-A\vec{y}(t)-\vec{f}(t)\right\|\leq\varepsilon,$$$

设$\vec{Z}(t) = \ ^C_aD_g^\alpha\vec{y}(t)-A\vec{y}(t)-\vec{f}(t)$, 应用引理2.2, 可得

$\begin{eqnarray} {\cal L}_g\left\{\vec{Z}(t)\right\} & = & {\cal L}_g\left\{^C_aD_g^\alpha\vec{y}(t)-A\vec{y}(t)-\vec{f}(t)\right\} \\ & = &s^\alpha I{\cal L}_g\left\{\vec{y}(t)\right\}-s^{\alpha-1}\vec{y}(a^+)-A{\cal L}_g\left\{\vec{y}(t)\right\}-{\cal L}_g\left\{\vec{f}(t)\right\} \\ & = &\left(s^\alpha I-A\right){\cal L}_g\left\{\vec{y}(t)\right\}-s^{\alpha-1}\vec{y}(a^+)-{\cal L}_g\left\{\vec{f}(t)\right\}, \end{eqnarray}$

$$${\cal L}_g\left\{\vec{y}(t)\right\} = \left(s^\alpha I-A\right)^{-1}\left\{{\cal L}_g\left\{\vec{Z}(t)\right\}+s^{\alpha-1}\vec{y}(a^+)+{\cal L}_g\left\{\vec{f}(t)\right\}\right\}.$$$

$$$\vec{y}^\ast(t) = \int_a^te_\alpha^{A[g(t)-g(\tau)]}\vec{f}(\tau)g'(\tau){\mathrm d}\tau+\int_a^te_\alpha^{A[g(t)-g(\tau)]}A\vec{c}g'(\tau){\mathrm d}\tau+\vec{c}.$$$

$\begin{eqnarray} {\cal L}_g\left\{\vec{y}^\ast(t)\right\} & = & {\cal L}_g\left\{(g(t)-g(a))^{\alpha-1}E_{\alpha, \alpha}\left[A (g(t)-g(a))^\alpha\right]\right\}{\cal L}_g\left\{\vec{f}(t)\right\}\\ &\quad&+ {\cal L}_g\left\{(g(t)-g(a))^{\alpha-1}E_{\alpha, \alpha}\left[A (g(t)-g(a))^\alpha\right]\right\}{\cal L}_g\left\{A\vec{c}\right\}+{\cal L}_g\left\{\vec{c}\right\}\\ & = & (s^\alpha I-A)^{-1}{\cal L}_g\left\{\vec{f}(t)\right\}+(s^\alpha I-A)^{-1}{\cal L}_g\left\{A\vec{c}\right\}+{\cal L}_g\{\vec{c}\} \\ & = &\left(s^\alpha I-A\right)^{-1}{\cal L}_g\left\{\vec{f}(t)\right\}+\left(s^\alpha I-A\right)^{-1}\frac{1}{s}A\vec{c}+\frac{1}{s}\vec{c}. \end{eqnarray}$

$\begin{eqnarray} {\cal L}_g\left\{^C_aD_g^\alpha\vec{y}^\ast(t)-A\vec{y}^\ast(t)\right\} & = & {\cal L}_g\left\{\vec{f}(t)\right\}+\frac{1}{s}A\vec{c}+\left(s^\alpha I-A\right)\frac{1}{s}\vec{c}-s^{\alpha-1}\vec{y}(a^+)\\ & = & {\cal L}_g\left\{\vec{f}(t)\right\}. \end{eqnarray}$

$\begin{eqnarray} {\cal L}_g\left\{\vec{y}(t)-\vec{y}^{\ast}(t)\right\} & = & {\cal L}_g\left\{\vec{y}(t)\right\}-{\cal L}_g\left\{\vec{y}^{\ast}(t)\right\} \\ & = & \left(s^\alpha I-A\right)^{-1}\left\{{\cal L}_g\left\{\vec{Z}(t)\right\}+s^{\alpha-1}\vec{y}(a^+)+{\cal L}_g\left\{\vec{f}(t)\right\}\right\}\\ &\quad&-\left(s^\alpha I-A\right)^{-1}{\cal L}_g\left\{\vec{f}(t)\right\}-\left(s^\alpha I-A\right)^{-1}\frac{1}{s}A\vec{c}-\frac{1}{s}\vec{c} \\ & = & \left(s^\alpha I-A\right)^{-1}{\cal L}_g\left\{\vec{Z}(t)\right\}. \end{eqnarray}$

$\begin{eqnarray} {\cal L}_g\left\{e_\alpha^{A[g(t)-g(a)]}\ast_g\vec{Z}(t)\right\} & = & {\cal L}_g\left\{e_\alpha^{A[g(t)-g(a)]}\right\}{\cal L}_g\left\{\vec{Z}(t)\right\}\\ & = & \left(s^\alpha I-A\right)^{-1}{\cal L}_g\left\{\vec{Z}(t)\right\}. \end{eqnarray}$

$$$\vec{y}(t)-\vec{y}^{\ast}(t) = e_\alpha^{A[g(t)-g(a)]}\ast_g\vec{Z}(t).$$$

$$$\left\|^C_aD_g^\alpha\vec{y}(t)-A\ ^C_aD_g^\beta\vec{y}(t)-\vec{h}(t)\right\|\leq\varepsilon,$$$

为了证明的方便, 设$\vec{Z}(t) = \ ^C_aD_g^\alpha\vec{y}(t)-A\ ^C_aD_g^\beta\vec{y}(t)-\vec{h}(t)$, 这里$t>a,$由引理2.2, 可得

$\begin{eqnarray} {\cal L}_g\left\{\vec{Z}(t)\right\} & = & {\cal L}_g\left\{^C_aD_g^\alpha\vec{y}(t)-A\ ^C_aD_g^\beta\vec{y}(t)-\vec{h}(t)\right\} \\ & = & {\cal L}_g\left\{^C_aD_g^\alpha\vec{y}(t)\right\}-A{\cal L}_g\left\{^C_aD_g^\beta\vec{y}(t)\right\}-{\cal L}_g\left\{\vec{h}(t)\right\} \\ & = &s^\alpha I{\cal L}_g\left\{\vec{y}(t)\right\}-s^{\alpha-1}\vec{y}(a^+)-A\left\{s^\beta I{\cal L}_g\left\{\vec{y}(t)\right\}-s^{\beta-1}\vec{y}(a^+)\right\}-{\cal L}_g\left\{\vec{h}(t)\right\} \\ & = &\left(s^\alpha I-As^\beta \right){\cal L}_g\left\{\vec{y}(t)\right\}-s^{\alpha-1}\vec{y}(a^+)+As^{\beta-1}\vec{y}(a^+)-{\cal L}_g\left\{\vec{h}(t)\right\}, \end{eqnarray}$

$\begin{eqnarray} {\cal L}_g\left\{\vec{y}(t)\right\}& = &\left(s^\alpha I-As^\beta \right)^{-1}\left[s^{\alpha-1}\vec{y}(a^+)-As^{\beta-1}\vec{y}(a^+)+{\cal L}_g\left\{\vec{h}(t)\right\}\right]\\ &\quad&+\left(s^\alpha I-As^\beta \right)^{-1}{\cal L}_g\left\{\vec{Z}(t)\right\}. \end{eqnarray}$

$$$\vec{y}^\ast(t) = y_a(t)\vec{y}(a^+)+\int_a^t(g(t)-g(\tau))^{\alpha-1}E_{\alpha-\beta, \alpha}\left[A(g(t)-g(\tau))^{\alpha-\beta}\right]\vec{h}(\tau)g'(\tau){\mathrm d}\tau,$$$

$$$y_a(t) = E_{\alpha-\beta, 1}(A(g(t)-g(a))^{\alpha-\beta})-A(g(t)-g(a))^{\alpha-\beta}E_{\alpha-\beta, \alpha-\beta+1}(A(g(t)-g(a))^{\alpha-\beta}).$$$

$\begin{eqnarray} {\cal L}_g\{\vec{y}^\ast(t)\} & = &s^{\alpha-\beta-1}\left(s^{\alpha-\beta}I-A\right)^{-1}\vec{y}(a^+)-As^{\alpha-\beta-(\alpha-\beta+1)}\left(s^{\alpha-\beta}I-A\right)^{-1}\vec{y}(a^+) \\ &\quad&+{\cal L}_g\left\{(g(t)-g(a))^{\alpha-1}E_{\alpha-\beta, \alpha}\left[A(g(t)-g(a))^{\alpha-\beta}\right]\right\}{\cal L}_g\left\{\vec{h}(t)\right\}\\ & = &s^{\alpha-\beta-1}\left(s^{\alpha-\beta}I-A\right)^{-1}\vec{y}(a^+)-As^{-1}\left(s^{\alpha-\beta}I-A\right)^{-1}\vec{y}(a^+)\\ &\quad&+s^{(\alpha-\beta)-\alpha}\left(s^{\alpha-\beta}I-A\right)^{-1}{\cal L}_g\left\{\vec{h}(t)\right\}\\ & = &\left(s^{\alpha-\beta}I-A\right)^{-1}\left[s^{\alpha-\beta-1}\vec{y}(a^+)-A\cdot s^{-1}\vec{y}(a^+)+s^{-\beta}{\cal L}_g\left\{\vec{h}(t)\right\}\right]. \end{eqnarray}$

$\begin{eqnarray} &&{\cal L}_g\left\{^C_aD_g^\alpha\vec{y}^\ast(t)-A\ ^C_aD_g^\beta\vec{y}^\ast(t)\right\}\\ & = &\left(s^\alpha I-As^\beta\right)\left(s^{\alpha-\beta}I-A\right)^{-1}\left[s^{\alpha-\beta-1}\vec{y}(a^+)-A\cdot s^{-1}\vec{y}(a^+)+s^{-\beta}{\cal L}_g\left\{\vec{h}(t)\right\}\right]\\ &\quad&-\left(s^{\alpha-1}I-As^{\beta-1}\right)\vec{y}(a^+)\\ & = &s^\beta\left(s^{\alpha-\beta}I-A\right)\left(s^{\alpha-\beta}I-A\right)^{-1}\left[s^{\alpha-\beta-1}\vec{y}(a^+)-A\cdot s^{-1}\vec{y}(a^+)+s^{-\beta}{\cal L}_g\left\{\vec{h}(t)\right\}\right]\\ &\quad&-\left(s^{\alpha-1}I-As^{\beta-1}\right)\vec{y}(a^+)\\ & = &s^\beta\left[s^{\alpha-\beta-1}\vec{y}(a^+)-A\cdot s^{-1}\vec{y}(a^+)+s^{-\beta}{\cal L}_g\left\{\vec{h}(t)\right\}\right]-\left(s^{\alpha-1}I-As^{\beta-1}\right)\vec{y}(a^+)\\ & = &{\cal L}_g\left\{\vec{h}(t)\right\}. \end{eqnarray}$

$\begin{eqnarray} &&{\cal L}_g\left\{\vec{y}(t)-\vec{y}^\ast(t)\right\}\\ & = &\left(s^\alpha I-As^\beta \right)^{-1}\left[s^{\alpha-1}\vec{y}(a^+)-As^{\beta-1}\vec{y}(a^+)+{\cal L}_g\left\{\vec{h}(t)\right\}\right]+\left(s^\alpha I-As^\beta \right)^{-1}{\cal L}_g\left\{\vec{Z}(t)\right\}\\ &\quad&-\left(s^\alpha I-As^\beta\right)^{-1}\left[s^{\alpha-1}\vec{y}(a^+)-A\cdot s^{\beta-1}\vec{y}(a^+)+{\cal L}_g\left\{\vec{h}(t)\right\}\right]\\ & = &\left(s^\alpha I-As^\beta \right)^{-1}{\cal L}_g\left\{\vec{Z}(t)\right\}. \end{eqnarray}$

$\begin{eqnarray} &&{\cal L}_g\left\{\left[(g(t)-g(a))^{\alpha-1}E_{\alpha-\beta, \alpha}\left(A(g(t)-g(a))^{\alpha-\beta}\right)\right]\ast_g \vec{Z}(t)\right\}\\ & = &{\cal L}_g\left[(g(t)-g(a))^{\alpha-1}E_{\alpha-\beta, \alpha}\left(A(g(t)-g(a))^{\alpha-\beta}\right)\right]{\cal L}_g\left\{\vec{Z}(t)\right\}\\ & = &s^{-\beta}(s^{\alpha-\beta}I-A)^{-1}{\cal L}_g\left\{\vec{Z}(t)\right\}\\ & = &(s^\alpha I-As^\beta)^{-1}{\cal L}_g\left\{\vec{Z}(t)\right\}. \end{eqnarray}$

$\begin{eqnarray} \vec{y}(t)-\vec{y}^\ast(t) = \left[(g(t)-g(a))^{\alpha-1}E_{\alpha-\beta, \alpha}\left(A(g(t)-g(a))^{\alpha-\beta}\right)\right]\ast_g \vec{Z}(t). \end{eqnarray}$

应用定理4.1, 容易证明这个结果.

$\begin{eqnarray} ^C_aD_g^{\frac{1}{3}}\vec{y}(t) = A\ ^C_aD_g^{\frac{1}{5}}\vec{y}(t)+\vec{h}(t), \end{eqnarray}$

$\begin{eqnarray} ^C_aD_g^{\frac{4}{5}}\vec{y}(t) = B\ ^C_aD_g^{\frac{3}{4}}\vec{y}(t)+\vec{k}(t), \end{eqnarray}$

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