Consider the following recurrence:
$$ y_i = y_{i-1} + \frac{\log(p^{y_{i-1} - y_{i-2}} - \frac{\lambda}{y_{i-1}^2})}{\log p} $$ where $0 < p < 1$. We are only interested in values for the constant $\lambda$ such that $y_i \geq 0$ for all $i$. We start with $y_0 = 0, y_1 = 1$.
Some experiments with different parameters show that the sequence $y_j$ approaches $y_j \approx a j + b$ for some constant $a \geq 0, b \leq 0$. It seems like you should take $\lambda = O(p)$ but I cannot prove this.
Are there any relatively simple bounds (upper and lower) for the asymptotic behavior of $y_j$?
