In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. We have established that $q(x_0) < \alpha \Rightarrow \forall t \geqslant 0,\ q(f_{x_0})(t) \leqslant e^{-\beta t} q(x_0)$. Deduce the existence of three strictly positive constants $\tilde { \alpha } , C$ and $\beta$ such that: $$\forall x _ { 0 } \in B ( 0 , \tilde { \alpha } ) , \quad \forall t \in \mathbf { R } _ { + } , \quad \left\| f _ { x _ { 0 } } ( t ) \right\| \leqslant C e ^ { - \beta t } \left\| x _ { 0 } \right\| ,$$ where $B ( 0 , \tilde { \alpha } )$ denotes the open ball, for the norm $\|.\|$, with center 0 and radius $\tilde { \alpha }$.
In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ and $q(x) = b(x,x)$. Let $x_0 \in \mathbf{R}^n$ and $f_{x_0}$ the solution of $y' = \varphi(y),\ y(0) = x_0$. We have established that $q(x_0) < \alpha \Rightarrow \forall t \geqslant 0,\ q(f_{x_0})(t) \leqslant e^{-\beta t} q(x_0)$.
Deduce the existence of three strictly positive constants $\tilde { \alpha } , C$ and $\beta$ such that:
$$\forall x _ { 0 } \in B ( 0 , \tilde { \alpha } ) , \quad \forall t \in \mathbf { R } _ { + } , \quad \left\| f _ { x _ { 0 } } ( t ) \right\| \leqslant C e ^ { - \beta t } \left\| x _ { 0 } \right\| ,$$
where $B ( 0 , \tilde { \alpha } )$ denotes the open ball, for the norm $\|.\|$, with center 0 and radius $\tilde { \alpha }$.