Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 22G (Linear Analysis)

The sequence $(f_n)$ is equicontinuous on $[0,1]$ when, for every $\epsilon >0$, there is a $\delta >0$ such thatfor every $n$. The Arzela-Ascoli theorem says that a uniformly bounded equicontinuous sequence of continuous real-valued functions on a compact interval has a uniformly convergent subsequence.

Choose $A>0$ and setTakeInductively, if $|f_n(s)|\le A$ on $[0,T_1]$, thenSince $f_0=0$, every iterate is bounded by $A$. Moreover,Thus $(f_n{)}_{n\ge 1}$ is uniformly bounded and uniformly Lipschitz, hence equicontinuous, on $[0,T_1]$.

The Arzela-Ascoli theorem makes the family of iterates relatively compact in $C([0,T_1])$. The standard successive-approximation proof for this scalar autonomous equation shows that, after decreasing to some $T_2\le T_1$, the adjacent differencestend uniformly to zero. One way to finish the existence argument without requiring a Lipschitz hypothesis on $\varphi$ is the following direct scalar construction.

If $\varphi (0)=0$, the function $f(t)=0$ is already a solution. If $\varphi (0)\ne 0$, continuity gives an interval $[-A_0,A_0]$ on which $\varphi$ has constant sign and never vanishes. DefineThen $F$ is continuously differentiable with $F^{\prime }(0)\ne 0$. By the inverse function theorem, it has a continuously differentiable local inverse on $[0,T_2]$ after choosing $T_2>0$ small enough and choosing the appropriate one-sided range. PutThe chain rule givesand henceEquivalently, $f(t)={\int }_0^t\varphi (f(s))ds$, the fixed-point equation approximated by the iterates. This is the local existence for a scalar autonomous ordinary differential equation with continuous vector field.