In 1986, Dixon and McKee developed a discrete fractional Gr\"{o}nwall inequality [Z. Angew. Math. Mech., 66 (1986), pp. 535--544], which can be seen as a generalization of the classical discrete Gr\"{o}nwall inequality. However, this generalized discrete Gr\"{o}nwall inequality has not been widely applied in the numerical analysis of the time-stepping methods for the time-fractional evolution equations. The main purpose of this paper is to show how to apply the generalized discrete Gr\"{o}nwall inequality to prove the convergence of a class of time-stepping numerical methods for time-fractional nonlinear subdiffusion equations, including the popular fractional backward difference type methods of order one and two, and the second-order fractional Crank-Nicolson type methods. We obtain the optimal $L^2$ error estimate in space discretization. The convergence of the fast time-stepping numerical methods is also proved in a simple manner.

翻译：1986年,狄克逊和麦基开发了一个离散分块 Gr\"{o}nbal不平等[Z. Angew. Math. Mech., 66 (1986), pp.535-544], 这可以被视为古典离散的Gr\"{o}n wall不平等的概括。然而,这种普遍离散的Gr\"{o}n wall不平等在时间偏差进化方程式的时间跨步方法的数值分析中并没有被广泛应用。本文的主要目的是展示如何应用普遍离散的Gr\{o}nball不平等来证明时间偏差非线性非线性亚扩散方程式的时间跨步数字方法的趋同,包括第1和第2级流行的偏差后向型方法,以及第二顺序的分级分数式Crank-Nicolson型方法。我们在空间离散化中获得了最佳的 $L_2美元误差估计值。快速时间跨位数字方法的趋同也以简单的方式证明。