On the approximate solution obtained through Euler's method

[psie]

TL;DR Summary

I'm reading about Euler's method to construct approximate solutions to ODEs in Ordinary Differential Equations by Andersson and Böiers. I have questions about properties of the approximate solution.

This is a bit of a longer post. I have tried to be as brief as possible while still being self-contained. My questions probably do not have much to do with ODEs, but this is the context in which they arose. Grateful for any help.

In what follows $|\cdot|$ denotes either the absolute value of a scalar or the Euclidean norm of a vector (denoted in bold). First a definition of what it means to be an approximate solution:

Definition 1. Let $I$ be an interval on the real axis, and $\Omega$ an open set in $\mathbf R\times\mathbf{R}^n$. Assume that the function $\pmb{f}:\Omega\to\mathbf{R}^n$ is continuous. A continuous function $\pmb{x}(t),\ t\in I$, is called an $\varepsilon$-approximate solution of the system $\pmb{x}'=\pmb{f}(t,\pmb{x})$ if $(t,\pmb{x})\in\Omega$ when $t\in I$ and $$\left|\pmb{x}(t'')-\pmb{x}(t')-\int_{t'}^{t''} \pmb{f}(s,\pmb{x}(s))ds\right|\leq \varepsilon|t''-t'|\quad \text{when } t',t''\in I.$$

Next, a theorem (for the sake of brevity, without proof) on an error estimate on the approximate solution:

Theorem 1. Assume that $\pmb{f}(t,\pmb{x})$ is continuous in $\Omega\subseteq \mathbf{R}\times\mathbf{R}^n$ and satisfies the Lipschitz condition $$|\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})|\leq L|\pmb{x}-\pmb{y}|, \quad (t,\pmb{x}),(t,\pmb{y})\in\Omega.$$ Let $\pmb{\tilde{x}}(t)$ be an $\varepsilon$-approximate and $\pmb{x}(t)$ an exact solution of $\pmb{x}'=\pmb{f}(t,\pmb{x})$ in $\Omega$ when $t\in I$. For an arbitrary point $t_0$ in $I$ we then have $$|\pmb{\tilde{x}}(t)-\pmb{x}|\leq |\pmb{\tilde{x}}(t_0)-\pmb{x}(t_0)|e^{L|t-t_0|}+\frac{\varepsilon}{L}(e^{L|t-t_0|}-1),\quad t\in I.$$

The Euler's method works as follows. Consider the equation $x'=f(t,x)$ and consider an initial value $(t_0,x(t_0))$. We know the slope of the tangent line through $(t_0,x(t_0))$. Follow this tangent a bit to the right, to $t=t_1=t_0+\delta$ and repeat the procedure for the point $(t_1,x(t_1))$. This way, one gets a broken curve of straight line segments resembling the solution.

To make the definition of the broken curve precise, consider the system $\pmb{x}'=\pmb{f}(t,\pmb{x})$ and an initial value $(t_0,\pmb{x}_0)$. Make a division of the interval $[t_0,t_0+a]$ in equally long subintervals: $t_0<t_1<\ldots<t_m=t_0+a$, and put $\delta=t_j-t_{j-1}$. Then define the function $\pmb{x}_{\delta}$ recursively at the step points $t_j$ by \begin{align}
&\pmb{x}_{\delta}(t_0)=\pmb{x}_0, \nonumber \\
&\pmb{x}_{\delta}(t_{j+1})=\pmb{x}_{\delta}(t_j)+(t_{j+1}-t_j)\pmb{f}(t_j,\pmb{x}_{\delta}(t_j)),\quad j=0,1,\ldots,m-1. \tag1
\end{align}
Between the step points the curve of $\pmb{x}_{\delta}$ is supposed to be a straight line, so $$\pmb{x}_{\delta}(t)=\pmb{x}_{\delta}(t_j)+(t-t_j)\pmb{f}(t_j,\pmb{x}_{\delta}(t_j)),\quad t_j\leq t\leq t_{j+1}.\tag2$$ The function $\pmb{x}_{\delta}$ is defined correspondingly in the interval $[t_0-a,t_0]$.

The following theorem shows $\pmb{x}_{\delta}$ is an $\varepsilon$-approximate solution to $\pmb{x}'=\pmb{f}(t,\pmb{x})$ under certain conditions on $\pmb{f}$.

Theorem 2. [Let $B,L$ and $C$ be positive constants.] Assume that

1. the function $\pmb{f}(t,\pmb{x})$ is continuous in $\Omega\subseteq\mathbf{R}\times\mathbf{R}^n$, that $|\pmb{f}(t,\pmb{x})|\leq B$ in $\Omega$, and that $a$ is so small that $\Lambda(a)\subseteq\Omega$, where $\Lambda(a)$ denotes the double cone $$\Lambda(a)=\{(t,\pmb{x})\in \mathbf{R}\times\mathbf{R}^n;|t-t_0|\leq a, \ |\pmb{x}-\pmb{x}_0|\leq B|t-t_0|\},$$
2. $|\pmb{f}(t,\pmb{x})-\pmb{f}(t,\pmb{y})|\leq L|\pmb{x}-\pmb{y}| \qquad \ \ \ (t,\pmb{x}),(t,\pmb{y})\in\Omega$,
3. $|\pmb{f}(t',\pmb{x})-\pmb{f}(t'',\pmb{x})|\leq C|t'-t''| \quad (t',\pmb{x}),(t'',\pmb{x})\in\Omega$

Then $\pmb{x}_{\delta}$ is an $\varepsilon$-approximate solution to the system $\pmb{x}'=\pmb{f}(t,\pmb{x})$ in the interval $[t_0-a,t_0+a]$, with $\varepsilon=\delta(C+LB)$.

Proof. All line segments in the definition of $\pmb{x}_{\delta}$ have a slope at most $B$. Therefor $(t,\pmb{x}_{\delta}(t))\in\Lambda(a)$ when $t\in[t_0-a,t_0+a]$. Moreover, if $|t'-t''|\le\delta$ then \begin{align} |\pmb{f}(t',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t''))|&\leq |\pmb{f}(t',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t'))| \nonumber \\ &+|\pmb{f}(t'',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t''))| \nonumber \\ &\leq C|t'-t''|+L|\pmb{x}_{\delta}(t')-\pmb{x}_{\delta}(t'')| \nonumber \\ &\leq C|t'-t''|+LB|t'-t''| \nonumber \\ &\leq \delta(C+LB).\tag3 \end{align}
[The first inequality is the triangle inequality, the second follows from 2. and 3. in the assumptions and the third by the fact that each line segments of $\pmb{x}_{\delta}$ have a slope at most $B$, so $|\pmb{x}_{\delta}(t')-\pmb{x}_{\delta}(t'')|\leq B|t'-t''|$.] We must prove that $$\left|\pmb{x}_{\delta}(t'')-\pmb{x}_{\delta}(t')-\int_{t'}^{t''} \pmb{f}(s,\pmb{x}_{\delta}(s))ds\right|\leq \delta(C+LB)|t'-t''|.\tag4$$
If $t'$ and $t''$ belong to the same subinterval $[t_j,t_{j+1}]$ then [from $(2)$] $$\pmb{x}_{\delta}(t'')-\pmb{x}_{\delta}(t')=(t''-t')\pmb{f}(t_j,\pmb{x}_{\delta}(t_j))=\int_{t'}^{t''} \pmb{f}(t_j,\pmb{x}_{\delta}(t_j))ds,$$ and $(4)$ follows from $(3)$. (In particular, $(4)$ is valid when $t'=t_j$ and $t''=t_{j+1}$). If $t'$ and $t''$ belong to different subintervals then use $(4)$ on each one of the intervals $[t',t_{j+1}],[t_{j+1},t_{j+2}],\ldots,[t_{k-1},t_k],[t_k,t'']$ and add the results.

Now comes the part I have some questions about. The authors claim that if we drop assumptions 1. and 2. in theorem 2, it is still possible to show that there is a number $\varepsilon(\delta)$, tending to zero as $\delta\to 0$, such that $\pmb{x}_{\delta}$ is an $\varepsilon(\delta)$-approximate solution in the interval $I(a)=[t_0-a,t_0+a]$. They write:

We know that $(t,\pmb{x}_{\delta}(t))\in\Lambda(a)$ when $t\in I(a)$ and that $$|\pmb{x}_{\delta}(t')-\pmb{x}_{\delta}(t'')|\leq B|t'-t''|\quad t',t''\in I(a).\tag5$$ Put $$\varepsilon(\delta)=\sup\limits_{|t'-t''|\leq\delta}|\pmb{f}(t',\pmb{x}_{\delta}(t'))-\pmb{f}(t'',\pmb{x}_{\delta}(t''))|.$$ The computations in the proof of theorem 2 show that $\pmb{x}_{\delta}$ is an $\varepsilon(\delta)$-approximate solution. Furthermore, $\pmb{f}$ is uniformly continuous on $\Lambda(a)$. From $(5)$ it accordingly follows that $$\lim_{\delta\to0}\varepsilon(\delta)=0.\tag6$$

Question 1: I simply do not understand how $(6)$ follow from $(5)$. How does it?

Question 2: The authors go on to claim that when the IVP $\pmb{x}'=\pmb{f}(t,\pmb{x}), \pmb{x}(t_0)=\pmb{x}_0$ has a solution $\pmb{x}(t)$, then it follows from $(6)$ and theorem 1, that $\pmb{x}_{\delta}$ converges uniformly to $\pmb{x}$ as $\delta\to 0$. The definition of uniform convergence I'm used to is $\lVert f_n-f\rVert\to 0$ as $n\to\infty$, where $\lVert\cdot\rVert$ denotes the sup-norm, but here they are claiming that the sup-norm should tend to $0$ as $\delta\to 0$. This makes me wonder; what is $\pmb{x}_{\delta}$? Is it a sequence? If not, what definition of uniform convergence are the authors using?

EDIT: The last quoted passage is just prior to presenting Peano's existence theorem. They note that:

If you only assume that $\pmb f$ is continuous, you can not use [Picard-Lindelöf's theorem]. In fact, you can not even be certain that $\pmb{x}_{\delta}(t)$ converges when $\delta\to 0$. But what you can prove is that there is a sequence $\delta_p$, tending to zero as $p\to\infty$, such that the functions $\pmb {x}_{\delta_p}(t)$ converge uniformly on $I(a)$.

 

[pasmith]

Question 2: The authors go on to claim that when the IVP $\pmb{x}'=\pmb{f}(t,\pmb{x}), \pmb{x}(t_0)=\pmb{x}_0$ has a solution $\pmb{x}(t)$, then it follows from $(6)$ and theorem 1, that $\pmb{x}_{\delta}$ converges uniformly to $\pmb{x}$ as $\delta\to 0$. The definition of uniform convergence I'm used to is $\lVert f_n-f\rVert\to 0$ as $n\to\infty$, where $\lVert\cdot\rVert$ denotes the sup-norm, but here they are claiming that the sup-norm should tend to $0$ as $\delta\to 0$. This makes me wonder; what is $\pmb{x}_{\delta}$? Is it a sequence? If not, what definition of uniform convergence are the authors using?

Let $X$ be a compact space and $f_\alpha : X \to \mathbb{R}^n$ a family of functions defined for $\alpha \in U \subset \mathbb{R}$. Then $f_\alpha \to f$ uniformly on $X$ as $\alpha \to \alpha_0 \in \bar{U}$ if and only if for every $\epsilon > 0$ there exists $\delta > 0$ such that for all $\alpha \in U$, $$|\alpha - \alpha_0| < \delta \quad \Rightarrow\quad \sup_{x \in X} \|f_\alpha(x) - f(x)\| < \epsilon.$$ Note that this has the same relation to the defintiion of uniform convergence of a sequence of functions as the definition of the limit of a function at a point has to the definition of the limit of a sequence.