In this paper, new estimates of the sequential Caputo fractional derivatives of a function at its extremum points are obtained.We derive comparison principles for the linear fractional differential equations, then apply these principles to obtain lower and upper bounds of solutions of linear and nonlinear fractional differential equations. The extremum principle is then applied to show that the initial-boundary-value problem for nonlinear anomalous diffusion admits at most one classical solution and this solution depends continuously on the initial and boundary data. This answers positively to the open problem about maximum principle for the space and time-space fractional PDEs posed by Y. Luchko [Fract. Calc. Appl. Anal. 14 (2011)]. The extremum principle for an elliptic equation with a fractional derivative and for the fractional Laplace equation are also proved.

中文翻译：

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空间和时空分数式偏微分方程的极大值原理

在本文中，获得了一个函数在其极值点处的序贯 Caputo 分数阶导数的新估计。我们推导了线性分数阶微分方程的比较原理，然后应用这些原理来获得线性和非线性分数阶解的下界和上限微分方程。然后应用极值原理来证明非线性异常扩散的初始边界值问题最多只允许一个经典解，并且这个解连续依赖于初始和边界数据。这对 Y. Luchko 提出的关于空间和时空分数 PDE 的最大值原理的开放问题做出了肯定的回答 [Fract. 计算。应用程序 肛门。14 (2011)]。