Let $a_n \geq 0$ for each $n$. I have the relation $$a_n^2 \leq C_1a_na_{n-1} + C_2a_{n-1} + C_3a_n$$ where the constants $C_i > 0$ don't depend on $n$ and in fact they can be made as small as necessary.
I am wondering if I find a bound on $a_n$ that is independent of $n$ (it can depend on $a_0$ or $a_1$). Is it ppossible? Eg. I want $$a_n \leq C.$$
If $C_2=C_3=0$ then this holds if we take $C_1 \leq 1$. Otherwise I don't see it.
Suppose that a sequence $a_0,a_1,a_2, ... $ of nonnegative real numbers satisfies
$$a_n^2 \leq C_1a_na_{n-1} + C_2a_{n-1} + C_3a_n$$
for all positive integers $n$ and all choices of positive real constants $C_1,C_2,C_3$.
Solving the recurrence for $a_{n-1}$ yields
$$a_{n-1} \ge \frac{a_n(a_n - C_3)}{C_1a_n + C_2}$$
For fixed $C_3 > 0$, if we let $C_1,C_2$ approach $0$ from above, the RHS will approach infinity unless $a_n \le C_3$.
Thus, $a_n \le C_3$ for all $n > 0$.
Since $C_3$ can be made arbitrarily small, it follows that $a_n = 0$ for all $n > 0$.
Thus, if I've understood the problem correctly, the only such sequences are those for which $a_0 \ge 0$ and $a_n = 0$ for all $n > 0$.