Corollary | 3.2
If \(c_{1}\neq 0\) and \(c_{2}\) are constants, then,
{\displaystyle u’(t)\leq
\beta (t)\,u(t),\qquad
t\in I^{\circ
},}\(\dot{\Theta}\left(t\right)\leq c_{2}+c_{1}\Theta(t)\Rightarrow\Theta\left(t\right)\leq\Theta\left(0\right)e^{c_{1}t}+\frac{c_{2}}{c_{1}}(e^{c_{1}t}-1)\)
Lemma | 3.3
Consider the triangular system in \(\mathbb{R}^{n}\)
\begin{equation}
\left(\text{TS}\right)\left\{\begin{matrix}\dot{X}=F(X),\\
\dot{Y}=G\left(X,Y\right)\\
\end{matrix}\right.\ \nonumber \\
\end{equation}With \(X\in\mathbb{R}^{n-k}\), \(Y\in\mathbb{R}^{k}\), F and G
C1 functions. Moreover, we assume
A1: \(X=0\) is a globally asymptotically stable fixed
point for \(\dot{X}=F(X)\)
A2: \(Y=Y_{o}\) is a globally asymptotically stable
fixed point for \(\dot{Y}=G\left(0,Y\right)\)
A3: Every forward orbit of (TS) is bounded.
Then, \((0,Y_{o})\) is a globally asymptotically stable point for (TS).