Solving (non-linear) recurrence relation

Question

I have the following recurrence relation $x_{n+1} \le x_{n-\tau} - a\cdot x_{n-\tau}^2 + b$ where $\tau$ is some positive integer, I'm not sure how to approach it. An upper bound will be enough [my hope is that $x_n \le O(1/n)$].

Any help would be appreciated.

Answer 1

Notice that $x_n \equiv \sqrt{\frac{b}{a}}$ satisfies your relation, so unless there is something you are not telling us your hypothesis does not seem to hold.