Given the differential equation $dy/dx = f(x,y)$ with initial condition $y(x_0)=y_0$. Let $f$ be a continuous function in $x$ and $y$ and Lipschitz-continuous in $y$ with Lipschitz constant L. Let F be a mapping on the space $C^0(I)$ of continuous functions $u:I\rightarrow \mathbb{R}$. $I = [x_0,M]$. $$Fu(x) = y_0 + \int_{x_0}^x f(s,u(s))ds.$$ Let for $\alpha\geq 0$ the norm $||\cdot||_\alpha$ be given on $C^0(I)$ by $$||u||_\alpha = \max_{x\in I} |u(x)e^{-\alpha x}|,\quad u \in C^0(I).$$
What is an apt value of $\alpha$ such that F is a contraction with respect to the norm $||\cdot||_\alpha$? And how is this $\alpha$ found?