I have a sequence $\{x_n\}_{n\ge 0}$ with $x_0>0$, controlled by the difference inequality: $$x_{n+1}\le ax_n^2+b$$ where, $a,b>0$. Had $b$ been $0$ and $a<1$, I would find $x_n\to 0$ as $n\to \infty$.

However, the presence of $b$ makes finding closed form next to impossible, except maybe for some specialized values of $b$. But I am not interested in closed forms, I am interested only in the necessary conditions on $a, b$ for the convergence or divergence of the sequence, or **at least** finding an upper bound for $\lim_{n\to \infty}x_n$, if the limit exists. It seems that if $a,b<1$, the sequence becomes bounded, and an upper bound is possible (although not sure if a closed form of the upper bound exists), and that if $a>1$, the sequence might diverge to infinity, at least the right hand side of the inequality seems to do so; but what about the following cases:

1) $a<1,b>1$,

2) $a=1,b>0$

Please direct me to references. I think probably the literature of nonlinear dynamics would be helpful in answering questions like this, but it would be really helpful to get pointers for specific topics in that field that might help answering this question. Thanks in advance.