I have a function $s(x,y)$ where $x,y,s$ are positive integers. $s$ is a bit strange, in that its growth depends on itself, namely $s=O(x^2\log(y+s))$. I would like to find an asymptotic upper bound for $s$ that does not depend on itself but I am unsure of how to do this. More generally, I am also curious if there is a general way to solve other "self referential" bound problems. All Google searches assumed that I wanted to bound functions satisfying recurrence relations.
I know that there exists an $M$ and a $C>0$ such that, when $\max(s(x,y),x,y) > M$, $$s(x,y) < Cx^2log(y+s(x,y)),$$ however I can't figure out how to isolate $s$. I am okay with a weaker upper bound, but I don't know if or how, using the information that I have, I can prove any upper bound not dependent on $s$. Also, I have some control over the definition of $s$, so if it would be easier to bound, $s$ can be monotonically increasing in both variables.
Playing naively.
$s \lt c x^2\log(y+s)) $.
Suppose $s \approx x^ay^b$. Then $x^ay^b \lt c x^2\log(y+x^ay^b)) $ or $x^{a-2}y^b \lt c \log(y+x^ay^b)) $.
If $a > 2$ or $y > 0$ the left side will be too large, so $a=2$ and $y=0$.
This gives $s = \Theta(x^2)$ which does work.