Consider the function $\dot{x} = f(x,t)$. I want to show that if there exists a function $V(x,t)$ and some positive constants $h,\delta,k_1,k_2,$ and $k_3$ such that for all $x \in B(0,h)$ and for all $t \geq 0$:
$k_1 \|x\|^2 \leq V(x,t) \leq k_2\|x\|^2$,
$\frac{dV}{dt} \leq 0$, and
$\int_t^{t+\delta} \frac{dV}{dt}dt \leq -k_3\|x\|^2$,
then $x(t)$ must exponentially converge to the origin.
I'm confused as to how we can use $V$ to tell us information about $x(t)$. For instance, from the third item in the above list, we know that $V(x,t+\delta) - V(x,t) \leq -k_3\|x\|^2$, which tells us that $V$ strictly decreases with time, but what does this tell us about $x(t)$?