I have the following delay system: $$x'(t) = g(t,\tau,x,x_\tau)$$ Given that $g(\cdot)$ is smooth and bounded, $x(t)$ is bounded in a non-negative region. What are some possible ways to obtain an upper bound on $\max\frac{x_\tau}{x}$, where $x_\tau = x(t-\tau)$?
I tried using $x_\tau \leq x + \max(|g|)\tau$, but that did not get me very far. Any help would be much appreciated.
I'm mostly interested in the case where $x\rightarrow 0$. I've been trying to involve Lipschitz continuity and some comparison of norms, but I have yet to get anything.