# Solution to autonomous differential equation with locally lipscitz function

As I was learning about the following theorem and its proof from the book Nonlinear Systems by H. K. Khalil, I encountered a difficulty in grasping some parts of the proof.

Theorem: Consider the scalar autonomous differential equation

$$$$\dot{y}=-\alpha(y),\ y(t_0)=y_0,\tag{1}$$$$

where $$\alpha$$ is a locally Lipschitz class $$\kappa$$ function defined on $$[0,a)$$. For all $$0\leq{y_0}, this equation has a unique solution $$y(t)$$ defined for all $$t\geq{t_0}$$. Moreover,

$$$$y(t)=\sigma(y_0,t-t_0),\tag{2}$$$$

where $$\sigma$$ is a class $$\kappa\ell$$ function defined on $$[0,a)\times[0,\infty)$$.

The proof goes as follows.

Since $$\alpha(.)$$ is locally Lipschitz, the equation (1) has a unique solution $$\forall\ {y_0}\geq{0}$$. Because $$\dot{y}(t)<0$$ whenever $$y(t)>0$$, the solution has the property that $$y(t)\leq{y_0}$$ forall $$t\geq{t_0}$$. By integration we have,

$$$$-\int_{y_0}^{y} \dfrac{dx}{\alpha(x)}= \int_{t_0}^{t} d\tau.$$$$

Let b be any positive number less than $$a$$ and define $$\eta(y)=-\int_{b}^{y}\dfrac{dx}{\alpha(x)}$$. The function $$\eta(y)$$ is strictly decreasing differentiable function on $$(0,a)$$. Moreover, $$\lim_{y\to{0}}\eta(y)=\infty$$. This limit follows from two facts.

First, the solution of the differential equation $$y(t)\to{0}$$ as $$t\to\infty$$, since $$\dot{y}(t)<0$$ whenever $$y(t)>0$$.

Second, the limit $$y(t)\to{0}$$ can happen only asymptotically as $$t\to\infty$$; it cannot happen in finite time due to the uniqueness of the solution.

Here I do not quite understand the second fact (in italics) how the uniqueness of solution ensures that $$y(t)$$ goes to $$0$$ asymptotically as $$t\to\infty$$.

Any hints on this are greatly appreciated.

That's not what it's saying. It's saying $$y(t) \to 0$$ can't happen in finite time, i.e. there can't be a solution $$Y(t)$$ of the differential equation with $$Y(t_0) = y_0$$ and $$Y(t_1) = 0$$ for some $$t_1 > t_0$$.
Suppose that did happen. Note that $$y(t) = 0$$ is also a solution of the differential equation, because part of the definition of class $$\kappa$$ is $$\alpha(0)=0$$. So this would contradict the Existence and Uniqueness Theorem, as there would be two different solutions $$Y$$ and $$0$$ having the same value at $$t_1$$.