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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.

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  • $\begingroup$ 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. $\endgroup$ – kkc Mar 23 at 0:49

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