Exercise 5, § 1

Let $E$ be the complete normed space of sequences $x = (x_n)$ of real numbers such that $\lim_{n \to \infty} x_n = 0$, normed by $\|x\| = \sup_n |x_n|$. For every integer $n \geq 0$ let $e_n$ be the sequence every term of which is zero, apart from the term with index $n$, which is equal to 1; the space $E$ is the direct sum of the subspace $V_n$ of dimension 2 generated by $e_n$ and $e_{n+1}$, and the closed subspace $W_n$ generated by the $e_k$ with indices different from $n$ and $n+1$. Let $f_n$ be a continuous Lipschitz function on $E$, with values in $E$, constant on every equivalence class modulo $W_n$, equal to $e_{n+1} - e_n$ on the line joining $e_n$ to $e_{n+1}$, and equal to 0 on the intersection of $V_n$ and the ball $\|x\| \leq \frac{1}{4}$. On the other hand, for every integer $n > 0$ let $\varphi_n$ be a real function defined and continuous on the interval $\left[ \frac{1}{n+1}, \frac{1}{n} \right]$, equal to 0 at the endpoints of this interval, and such that $\int_{1/(n+1)}^{1/n} \varphi_n(t) dt = 1$. Let $I$ be the interval $[0, 1]$ in $\mathbf{R}$; consider on $I \times E$ the function $f$ with values in $E$, defined as follows: $f(0, x) = 0$ for every $x \in E$; for $\frac{1}{n+1} \leq t \leq \frac{1}{n}$, let $f(t, x) = \varphi_n(t)f_n(x)$. Show that $f$ is continuous and locally Lipschitz on $I \times E$, but that there exists an integral $u$ of the equation $x' = f(t, x)$, defined and bounded on $[0, 1]$, equal to $e_n$ for $t = 1/n$, and consequently having no limit as $t$ tends to 0.