Rate of Convergence of complicated sequence with interactions I have been working on a problem where the sequence turns out to be so complex that i am unable to find its convergence rate with necessary and sufficient conditions on the parameters.After working through the math I got $$(a+cx_{t-1})x_t \leq cx_{t-1}^2 $$
Assume $a,c$ are variable (but not dependent on $t$) and $x_0$ is the initial point.
It would be great if anyone can help me finding its convergence rate and point me to a good reference.
 A: Suppose $x_t \geq 0$ for all $t$, and that $c > 0$, $a >0$.  The inequality becomes: 
$$ x_t \leq \frac{cx_{t-1}^2}{a+cx_{t-1}} \: \: \mbox{for all $t \in \{0, 1, 2, \ldots\}$}  \: \: \: (\mbox{Equation A})$$ 
Assume $x_0>0$ and define $\rho = \frac{1}{\frac{a}{cx_0} + 1}$. Note that $0< \rho < 1$. 
Claim 1: We have $0 \leq x_t \leq \rho  x_{t-1}$ for all $t \in \{1, 2, 3,  \ldots\}$. 
Proof: It holds for $t=1$ by directly considering equation (A).   Suppose it is true for $t$ (and in particular, $x_t \leq x_0$).  We prove for $t+1$. By equation (A) we get: 
\begin{align*}
x_{t+1} &\leq \frac{cx_{t}^2}{a+cx_{t}}\\
&= \frac{x_{t}}{\frac{a}{cx_{t}}+1}\\
&\leq \frac{x_{t}}{\frac{a}{cx_0} + 1}\\
&=\rho x_t
\end{align*}
$\Box$
Claim 1 proves that $x_t$ converges to 0 with a rate that is at least exponential.  The next claim shows it goes even faster than exponentially once it gets sufficiently close to $0$.
Claim 2:  For all $t$ we have $x_t \leq (c/a)x_{t-1}^2$. 
Proof: By equation (A) we get: 
\begin{align} 
x_t &\leq \frac{cx_{t-1}^2}{a + cx_{t-1}} \\
&\leq \frac{cx_{t-1}^2}{a}
\end{align} 
$\Box$
