Continuous and Lipschitz dependence on the initial condition on a Picard-Lindelof cylinder

Given: The open set $U$, the continuous and locally Lipschitz field $F$, the point $a_0\in U$, the numbers $h,R>0$ with $\bar{B}(a_0,R)\subseteq U$, the constants $M,L$ and the radius $\rho$ with $hM\le R-\rho$ and $Lh<1$.

Picard-Lindelöf: if $[t_0-h,t_0+h]\times \bar{B}(x_0,r)\subseteq \mathbb{R}\times U$, if $\mathrm{\parallel }F{\mathrm{\parallel }}_2\le M$ and $F$ has state-Lipschitz constant $L$ on that cylinder, and if $hM\le r$ and $Lh<1$, then exactly one solution of $x^{\mathrm{\prime }}=F(x)$, $x(t_0)=x_0$ exists on $[t_0-h,t_0+h]$ with graph in the cylinder; any two solutions through the same initial data agree on every common subinterval containing the initial time (Picard-Lindelöf local existence and uniqueness for first-order systems, First-order systems, initial value problems, and solutions on intervals).

$F$ is locally Lipschitz in the state variable: every compact time-state cylinder carries a finite state-Lipschitz constant (Local Lipschitz continuity in the state variable, locally uniform in time and parameters); in particular $L$ as chosen in the statement is finite and the inequality $\mathrm{\parallel }F(x)-F(y){\mathrm{\parallel }}_2\le L\mathrm{\parallel }x-y{\mathrm{\parallel }}_2$ holds on $\bar{B}(a_0,R)$.

For $\rho =0$ the notation $\bar{B}(a_0,\rho )$ means the singleton $\{a_0\}$, as follows from the inequality defining a closed ball. $\bar{B}(a_0,R)$ is closed and bounded, hence compact in ${\mathbb{R}}^n$ (Open ball, closed ball and sphere in a metric space, Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line), so the continuous function $\mathrm{\parallel }F{\mathrm{\parallel }}_2$ attains its maximum there and $M<+\mathrm{\infty }$ (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

A curve solves the initial value problem on an interval exactly when it is continuous there, has graph in $U$, and satisfies the Volterra equation $x(t)=a+{\int }_0^{t}F(x(r))dr$; the oriented integral is additive over subintervals (A first-order initial value problem is equivalent to its Volterra integral equation, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$, Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

The Euclidean norm satisfies the triangle inequality (Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation). For an integrable ${\mathbb{R}}^n$-valued curve $g$ on an interval and reals $r\le s$ one has $\parallel {\int }_r^{s}g{\parallel }_2\le \mid {\int }_r^s\mathrm{\parallel }g{\mathrm{\parallel }}_2\mid$, and if a real integrable $u$ satisfies $u\le c$ on a nondegenerate interval then $\int u\le c(\text{length})$ (For $a\le b$ and $f:[a,b]\to {\mathbb{R}}^m$ integrable when $a<b$, $\parallel {\int }_a^{b}f{\parallel }_2\le {\int }_a^b\parallel f{\parallel }_2$; for $a<b$, $\parallel f{\parallel }_2$ is integrable, If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$).

Gronwall: if continuous $u:[0,T]\to \mathbb{R}$ and constants $A\ge 0$, $L\ge 0$ satisfy $u(t)\le A+L{\int }_0^{t}u$ for $t\in [0,T]$, then $u(t)\le A{e}^{Lt}$ there; the reflected form for an integral oriented as ${\int }_t^0$ gives $u(t)\le A{e}^{L\mathrm{\mid }t\mathrm{\mid }}$ for $t\le 0$ (Gronwall's integral inequality with variable and constant coefficients).