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We consider the completely positive discretizations of fractional ordinarydifferential equations (FODEs) on nonuniform meshes. Making use of theresolvents for nonuniform meshes, we first establish comparison principles forthe discretizations. Then we prove some discrete Gr\"onwall inequalities usingthe comparison principles and careful analysis of the solutions to the timecontinuous FODEs. Our results do not have any restrictions on the step sizeratio. The Gr\"onwall inequalities for dissipative equations can be used toobtain the uniform-in-time error control and decay estimates of the numericalsolutions. The Gr\"onwall inequalities are then applied to subdiffusionproblems and the time fractional Allen-Cahn equations for illustration.

Some Gr\"onwall inequalities for a class of discretizations of time fractional equations on nonuniform meshes