Can Estimation of a Function Determine Exponential Decay for Vector-Functions?

[wrobel]

Let $x(t)\in C^1(\mathbb{R}_+,\mathbb{R}^m)$ be a vector-function such that

1) $\|x(t)\|+\|\dot x(t)\|\to 0$ as $t\to\infty$ and

2) for all $t>0$ one has $\|x(t)\|\le c_1\|\dot x(t)\|$Is it true that $\|x(t)\|\le c_2 e^{-c_3t}$? Here $c_i$ are positive constants.

 

[mfb]

Spoiler

No.$$x(t)= \frac{1}{t^3} \left(sin(t^2),cos(t^2)\right)$$

 

[wrobel]

shame on me. Thanks!