Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 12E (Analysis II)

A local form of the Picard-Lindelof theorem is as follows. Let $F(t,y)$ be continuous on a rectangle about $(t_0,y_0)$ and uniformly Lipschitz in $y$ there. Then some interval about $t_0$ admits a unique continuously differentiable solution ofThe solution is the unique fixed point of the associated integral operator. If $F$ is continuous on $[a,b]\times {\mathbb{R}}^n$ and globally Lipschitz in $y$ with one constant, the solution exists uniquely throughout $[a,b]$.

Since $1\le t\le b$ and $c\ge 0$,Taking suprema givesThus the weighted norm and the usual uniform norm are equivalent norms.

For $f,g\in X$, the Lipschitz condition givesChoose $c$ so that $c+1>Mb$. Then $\varphi$ is a contraction mapping. The space $X$ is complete in the sup norm, and therefore also in the equivalent weighted norm.

For initial value $y_0$, defineIt has the same contraction constant. The Banach fixed-point theorem gives a unique fixed point $f$, which satisfies the integral equation. The fundamental theorem of calculus then givesConversely every solution satisfies that integral equation, proving uniqueness on $[1,b]$. This weighted-norm argument is the Bielecki norm method.