Editing Picard–Lindelöf theorem (section)

== Detailed proof ==

Let <math>L</math> be the Lipschitz constant of <math>(t, y) \mapsto f(t,y)</math> with respect to <math>y.</math>
The function <math>f</math> is continuous as a function of <math>(t,y)</math>.
In particular, since <math>t \mapsto f(t,y)</math> is a continuous function of <math>t</math>, we have that for any point <math>(t_0, y_0)</math> and <math>\epsilon>0</math>  there exist <math>\delta>0</math>
such that <math>|f(t,y_0)-f(t_0,y_0)| <\epsilon / 2</math> when <math>|t - t_0| < \delta</math>. 
We have
<math display="block"> |f(t,y)-f(t_0,y_0)|\leq |f(t,y)-f(t,y_0)|+|f(t,y_0)-f(t_0,y_0)|<\epsilon, </math>
provided <math>|t-t_0|<\delta</math> and <math>|y-y_0|<\epsilon /2L</math>, which shows that <math>f</math> is continuous at <math>(t_0,y_0)</math>.

Let <math>a := 1/2L</math> and take any <math>b > 0</math> such that 
<math display="block"> C_{a,b} = I_a(t_0) \times B_b(y_0) </math>
is a subset of <math>D,</math> where
<math display="block">\begin{align}
I_a(t_0) &= [t_0-a,t_0+a] \\
B_b(y_0) &= [y_0-b,y_0+b].
\end{align}</math>
Such a set exists because <math>(t_0, y_0)</math> is in the interior of <math>D,</math> by assumption. 
<!--<math>C_{a,b}</math>  is a compact rectangular set where &thinsp;{{math|''f''}}&thinsp; is defined. -->

Let

:<math>M = \sup_{(t,y) \in C_{a,b}}\|f(t,y)\|,</math>

which is the [[supremum]] of (the [[absolute value]]s of) the slopes of the function.
The function <math>f</math> attains a maximum on <math>C_{a,b}</math> because <math>f</math> is continuous and <math>C_{a,b}</math> is compact. 
For a later step in the proof, we need that <math>a < b / M,</math> so if <math>a \geq b / M,</math> then change <math>a</math> to <math>a :=\tfrac{1}{2}\min\{1 / L,\ b / M\},</math> and update <math>I_{a}(t_0),</math> <math>B_{b}(y_0),</math> <math>C_{a,b},</math> and <math>M</math> accordingly (this update will be needed at most once since <math>M</math> cannot increase as a result of restricting <math>C_{a,b}</math>).

Consider <math>\mathcal{C}(I_{a}(t_0),B_b(y_0))</math>, the [[function space]] of continuous functions <math>I_{a}(t_0)\to B_b(y_0).</math>
We will proceed by applying the [[Banach fixed-point theorem]] using the [[metric (mathematics)|metric]] on <math>\mathcal{C}(I_{a}(t_0),B_b(y_0))</math> induced by the [[uniform norm]]. Namely, for each continuous function <math>\varphi :  I_{a}(t_0) \to B_b(y_0),</math> the norm of <math>\varphi</math> is
<math display="block">\| \varphi \|_\infty = \sup_{t \in I_a} \| \varphi(t)\|.</math>
<!--\varphi should be a function of t and x, but it is only written as a function of t!-->
The ''Picard operator'' <math display="block">\Gamma:\mathcal{C}\big(I_{a}(t_0),B_b(y_0)\big) \to \mathcal{C}\big(I_{a}(t_0),B_b(y_0)\big)</math> is defined for each <math>\varphi \in \mathcal{C}(I_{a}(t_0),B_b(y_0))</math> by <math>\Gamma \varphi \in \mathcal{C}(I_{a}(t_0),B_b(y_0))</math> given by
<math display="block">\Gamma \varphi(t) = y_0 + \int_{t_0}^{t} f(s,\varphi(s)) \, ds \quad \forall t \in I_a(t_0).</math>

To apply the Banach fixed-point theorem, we must show that <math>\Gamma</math> maps a complete non-empty [[metric space]] ''X'' into itself and also is a [[contraction mapping]].

We first show that <math>\Gamma</math> takes <math>B_b(y_0)</math> into itself in the space of continuous functions with the uniform norm. 
<!-- The statement "<math>\Gamma</math> takes <math>\overline{B_b(y_0)}</math> into itself" is not technically correct, since \Gamma acts on functions *and* those functions should have \mathcal{C}(I_{a}(t_0),B_b(y_0)) as their domain.-->
Here, <math>B_b(y_0)</math> is a closed ball in the space of continuous (and [[bounded function|bounded]]) functions "centered" at the constant function <math>y_0</math>. 
Hence we need to show that 
<math display="block>\| \varphi -y_0 \|_\infty \le b</math>
implies
<math display="block>\left\| \Gamma\varphi(t)-y_0 \right\| = \left\|\int_{t_0}^t f(s,\varphi(s))\, ds \right\| \leq \int_{t_0}^{t'} \left\|f(s,\varphi(s))\right\| ds \leq \int_{t_0}^{t'} M\, ds = M \left|t'-t_0 \right| \leq M a \leq b</math>

where <math>t'</math> is some number in <math>[t_0-a, t_0 +a]</math> where the maximum is achieved. 
The last inequality in the chain is true since <math>a < b / M.</math>
<!--To-do: This preceding paragraph mixes a statement of what we "want to show" with the actual thing we are showing.--> 

Now let us prove that <math>\Gamma</math> is a contraction mapping as required to apply the [[Banach fixed-point theorem]].
In particular, we want to show that there exists <math>0 \leq q < 1,</math> such that
<math display="block"> \left \| \Gamma \varphi_1 - \Gamma \varphi_2 \right\|_\infty \le q  \left\| \varphi_1 - \varphi_2 \right\|_\infty</math>
for all <math>\varphi_1,\varphi_2\in\mathcal{C}(I_{a}(t_0),B_b(y_0)).</math>

Let <math>q = aL</math> and take any <math>\varphi_1,\varphi_2\in\mathcal{C}(I_{a}(t_0),B_b(y_0)).</math>
Take <math>t</math> such that

:<math>\| \Gamma \varphi_1 - \Gamma \varphi_2 \|_\infty =  \left\| \left(\Gamma\varphi_1 - \Gamma\varphi_2 \right)(t) \right\|.</math>
<!-- To-do: Explain why such a t exists -->

Then, using the definition of <math>\Gamma</math>,

:<math>\begin{align}
\left\|\left(\Gamma\varphi_1 - \Gamma\varphi_2 \right)(t) \right\| &= \left\|\int_{t_0}^t \left( f(s,\varphi_1(s))-f(s,\varphi_2(s)) \right)ds \right\|\\
&\leq \int_{t_0}^t \left\|f \left(s,\varphi_1(s)\right)-f\left(s,\varphi_2(s) \right) \right\| ds \\
&\leq L \int_{t_0}^t \left\|\varphi_1(s)-\varphi_2(s) \right\|ds  && \text{since } f \text{ is Lipschitz-continuous} \\
&\leq L \int_{t_0}^t \left\|\varphi_1-\varphi_2 \right\|_\infty \,ds \\
&\leq La \left\|\varphi_1-\varphi_2 \right\|_\infty,
\end{align}</math>
where <math>t - t_0 \leq a,</math> because the domains of <math>\phi_1,\phi_2</math> are both <math>I_a(t_0) \times B_b(y_0).</math>
By definition, <math>q = aL,</math> and <math>a < 1 / L,</math> so <math>q < 1.</math>
Therefore, <math>\Gamma</math> is a contraction.
<!--To-do: This proof should construct a value of q \in [0, 1) such that |\Gamma \phi_1 - \Gamma \phi_2| \leq q |\phi_1 - \phi_2|_\infty for all \phi_1, \phi_2. -->

We have established that the Picard's operator is a contraction on the [[Banach space]]s with the metric induced by the uniform norm. This allows us to apply  the Banach fixed-point theorem to conclude that the operator has a unique fixed point. In particular, there is a unique function
<math display="bl">\varphi\in \mathcal{C}(I_a (t_0), B_b(y_0))</math>
such that 
<math display="block">\Gamma \varphi = \varphi.</math> 
Thus, <math>\varphi</math> is the unique solution of the initial value problem, valid on the interval <math>I_a.</math>