Peano local existence for a continuous first-order system

Since $D$ is open and contains $(t_0,x_0)$, choose $a,r>0$ so that $C=[t_0-a,t_0+a]\times \bar{B}(x_0,r)$ lies in $D$. This cylinder is closed and bounded in ${\mathbb{R}}^{n+1}$, hence compact by [L6]. By [L7], the continuous function $(t,x)\mapsto \mathrm{\parallel }F(t,x){\mathrm{\parallel }}_2$ attains a maximum $M\ge 0$ on $C$. Set $h=a$ if $M=0$, and $h=\min \{a,r\mathrm{/}M\}$ otherwise. Then $h>0$ and $hM\le r$.

Construct forward Euler polygons on $[t_0,t_0+h]$ with mesh tending to zero. Fact [L1] applies with the cylinder from step 1.1, and [L2] gives a subsequence converging uniformly to a continuous curve $x_{+}$.

Apply the same construction to the reflected field $\tilde{F}(s,z)=-F(2t_0-s,z)$ on $[t_0,t_0+h]\times \bar{B}(x_0,r)$. It has the same bound $M$, so [L1] and [L2] give a uniform limit $z$; define $x_{-}(t)=z(2t_0-t)$ on $[t_0-h,t_0]$.

Let ${\eta }_m$ be the forward mesh and let ${\tau }_m(s)$ be the left endpoint of the mesh cell containing $s$. The Euler construction gives $x_m(t)=x_0+{\int }_{t_0}^{t}F({\tau }_m(s),x_m({\tau }_m(s)))ds\mathrm{.}$ By [L1], $\mathrm{\mid }s-{\tau }_m(s)\mathrm{\mid }\le {\eta }_m$ and $\mathrm{\parallel }{x}_m(s)-x_m({\tau }_m(s)){\mathrm{\parallel }}_2\le M{\eta }_m$. Uniform convergence to $x_{+}$ and uniform continuity of $F{\mathrm{\mid }}_C$ from [L4] therefore make the displayed integrands converge uniformly to $F(s,x_{+}(s))$. Applying [L3] componentwise gives the Volterra equation for $x_{+}$. The identical reflected argument gives the oriented Volterra equation for $x_{-}$.

Both half-intervals are nondegenerate and order-convex, both limit graphs lie in $C\subseteq D$, and both limits contain $(t_0,x_0)$, so [L5] makes $x_{+}$ and $x_{-}$ solutions. Their piecewise union is continuous at $t_0$, and the two differential equations give the same one-sided derivative $F(t_0,x_0)$ there. Hence the union is differentiable at $t_0$ and is a solution on $[t_0-h,t_0+h]$.