We provide a detailed study of the recursion $y_0 = 1, y_{n+1} = y_n + \tau y_n^m, n = 0, 1, \dots , m > 1$, which arises either as a model discretization of a nonlinear ODE or in the use of the energy method. Sharp bounds and asymptotic estimates are given for the size of the iterates $y_n$.

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Academic Press
Journal of Mathematical Analysis and Applications
Modelling, Analysis and Computation

Sanz-Serna, J. M., & Verwer, J. (1986). A study of the recursion $y_{n + 1} = y_n + \tau y_n^m$. Journal of Mathematical Analysis and Applications, 116(2), 456–464.