The comparison lemma (from Khalil's Nonlinear Systems book) is stated as follows:
Lemma 3.4
Consider the scalar differential equation
$$\dot{u} = f(t,u), \quad u(t_0) = u_0 $$
where $f(t,u)$ is continuous in $t$ and locally Lipschitz in $u$ for all $t \ge0$. Let $[t_0,T)$ be the maximimal interval of existence for the solution $u(t)$. Let $v(t)$ be a continuous function that satisfies the differential inequality
$$\dot{v}(t) \le f(t,v(t)), \quad v(t_0)\le u_0, \quad \forall t\in [t_0,T) $$
Then, $$ v(t) \le u(t), \quad \forall t\in [t_0,T).$$
Question
I have an inequality such as this, but $f(t,u)$ is nontrivial to solve analytically, but I believe I can obtain an asymptotic solution, say
$$ u(t) \sim \tilde{u}(t),\quad t \rightarrow \infty.$$
For, say, $T = \infty$, are there any existing theorems such that it possible to conclude that
$$ v(t) \le \tilde{u}(t), \quad \forall t\in [T_0,\infty),$$
for some $T_0$ sufficiently large?