# Vector function going to zero faster than the norm of its variable uniformly in time

I need to find some conditions on a function $$h:\mathbb{R}^+\times \mathbb{R}^n\rightarrow\mathbb{R}^n$$ to ensure that it goes, in norm, to zero faster than $$\|z\|$$ uniformly in time, i.e. $$\lim_{\|z\|\to 0}\sup_{t\geq 0}\frac{\|h(t,z)\|}{\|z\|}=0.$$

I know that for any fixed time $$t\in\mathbb{R}^+$$ the limit goes to zero, but in general this convergence is not uniform in time.

My function $$h$$ has even the property that $$h(t,0)=0$$ for any $$t\in\mathbb{R}^+$$. Therefore the first property I thought for $$h$$ in order to satisfy this condition is that it is so that $$\|h(t,x)\|\leq \|x\|^p$$ where $$p>1$$, which however does not give me relevant information about it.

Do you suggest other possible conditions to ensure this kind of behaviour?

If it might help, I need this condition in order to define the linearization of a non-autonomous dynamical system, where $$h(t,z)$$ is the nonlinear part of my ODE.

• Do you know about moduli of continuity? Maybe they're what you're looking for. en.wikipedia.org/wiki/Modulus_of_continuity Mar 27, 2021 at 9:16
• Thank you, I knew about uniform continuity and the modulus, but I did not think about it to solve my problem Mar 30, 2021 at 8:33

There is a function $$F\colon\mathbb R^n\to\mathbb R_+$$ satisfying $$F(0)=0$$ and $$F'(0)=0$$ such that $$\lVert h(t,z)\rVert\le F(z)$$ for all $$t\ge0$$ and all $$z$$ in a neighborhood $$U\subseteq\mathbb R^n$$ of $$0$$.
(Indeed, if your property is satisfied, then $$F(z)=\min\{1,\sup_{t\ge0}\lVert h(t,z)\rVert\}$$ is such a function with the set $$U=\{z\in\mathbb R^n:F(z)<1\}$$ (which is a neighborhood of $$0$$ by the mere definition of the limit). Conversely, given such a pair $$(F,U)$$, there holds $$0\le\sup_{t\ge0}\frac{\lVert h(t,z)\rVert}{\lVert z\rVert}\le\underbrace{\frac{\lVert F(z)\rVert}{\lVert z\rVert}}_{\to0\quad\text{as z\to0}}\quad\text{for all t\ge0, z\in U,}$$ and so the middle term tends to $$0$$ as $$z\to0$$.)
The choices $$F(z)=C\lVert z\rVert^p$$ are of course the simplest examples of such functions as you observed. You can do finer choices like $$F(z)=f(\lVert z\rVert)$$ with $$f(0)=f'(0)=0$$. However, as proved above, you cannot do much better, in principle.