# Approximating $\max \big\{\frac{x_\tau}{x}\big\}$ as $x$ and $x_\tau \rightarrow 0^+$

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.

• For any differentiable function, if the derivative is bounded, you can show that the function is Lipschitz continuous (using mean value theorem). From there, I believe you could reach a useful inequality. – kkc Mar 23 at 0:49