We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Gr\"onwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have any restrictions on the step size ratio. The Gr\"onwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Gr\"onwall inequalities are then applied to subdiffusion problems and the time fractional Allen-Cahn equations for illustration.

翻译：本文研究分数阶常微分方程在非均匀网格上的完全正离散格式。首先利用非均匀网格上的预解式建立离散格式的比较原理，然后通过比较原理并结合对时间连续分数阶常微分方程解的精细分析，证明若干离散Grönwall不等式。所得结果对步长比无任何限制。对于耗散型方程，这些Grönwall不等式可用于获得数值解的一致时间误差控制和衰减估计。最后将所证Grönwall不等式应用于次扩散问题及时间分数阶Allen-Cahn方程以作说明。