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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

\begin{equation} \dot{y}=-\alpha(y),\ y(t_0)=y_0,\tag{1} \end{equation}

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

\begin{equation} y(t)=\sigma(y_0,t-t_0),\tag{2} \end{equation}

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,

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

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.

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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 $t_1$.

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