Picard's Existence Theorem/Proof 2

== Theorem ==
== Proof ==
Let us define the following series of functions:
Denote this sequence by $\sequence {y_k}_{k \mathop \in \N_0}$.
What we are going to do is prove that $\ds \map y x = \lim_{n \mathop \to \infty} \map {y_n} x$ is the required solution.
=== The curve lies in the rectangle ===
We will show that for $a - h \le x \le a + h$, the curve $y = \map {y_n}x$ lies in the rectangle $R$.
That is, that $b - k < y < b + k$.
Suppose $y = \map {y_{n - 1} } x$ lies in $R$.
Then:
Clearly $y_0$ lies in $R$, and the argument holds for $y_1$.
So by induction, $y = \map {y_n} x$ lies in $R$ for all $n \in \N$.
=== Existence ===
The sequence $\sequence {y_k}_{k \mathop \in N_0}$ can be expressed as a telescoping series:
:$\ds y_{n + 1} = y_0 + \sum_{k \mathop = 0}^n \paren {y_{k + 1} - y_k}$
The theorem contains more variables $\paren {\set {x, y_1, y_2} }$ and parameters $\paren {\set{h, k, M, A} }$ than inequality constraints.
Thus, more relations between them can be chosen without affecting the constraints.
Choose $h = \dfrac A 2$.
For $a \le x \le h$ we have:
By taking supremum norm of both sides, we get:
:$\norm {y_{n + 1} - y_n}_\infty \le \dfrac 1 2 \norm {y_n - y_{n - 1} }_\infty$
By induction, the inequality can be extended:
:$(1): \quad \norm {y_{n + 1} - y_n} \le \dfrac 1 {2^n} \norm {y_1 - y_0}$
Therefore:
Same argument applies to $-h \le x \le a$.
Hence, $\sequence {y_k}_{k \mathop \in \N_0}$ converges in $\struct {\map {C^1} {\size{x - a} \le h}, \norm {\cdot}_\infty}$ to $y \in \map {C^1} {\size {x - a} \le h}$ absolutely.
Therefore, the sequence is convergent:
:$\ds \map y x = \lim_{n \mathop \to \infty} \map {y_{n + 1} } x = x_0 + \lim_{n \mathop \to \infty} \int_a^x \map f {x, \map {y_n} x} \rd x$
To find the limit, consider the following sequence:
:$\map {g_n} x = \map f {x, \map {y_n} x}$
The sequence $\sequence {g_n}_{n \mathop \in \N_0}$ is a sequence of partial sums $\ds g_0 + \sum_{k \mathop = 0}^n \paren {g_{k + 1} - g_k}$.
It follows that:
So $\sequence {g_n}_{n \mathop \in \N_0}$ converges to some $g$ in $\struct {\map C {\size {x - a} \le h}, \norm {\, \cdot \,}_\infty}$ absolutely.
It follows that:
On the other hand, a Riemann integral is a continuous mapping.
From Continuous Map Preserves Convergent Sequences:
We conclude that:
:$\ds \map y x = y_0 + \int_a^x \map f {t, \map y t} \rd t$
where:
:$\map y a = y_0 + 0 = b$
and, by Fundamental Theorem of Calculus:
:$\map {y'} x = 0 + \map f {x, \map y x}$
for all $x \in \R : \size {x - a} \le h$.
=== Uniqueness ===
 that the solution to IVP is not unique.
Then, for the same initial conditions there exists a non-empty subset of $R$ where solutions differ.
Let $y_1, y_2$ be solutions to IVP for $x \in \R : \size {x - a} \le h$.
Let $x_* := \max \set {x \in \R : \size {x - a} \le h : \map {y_1} t = \map {y_2} t, \forall t \le x }$
Then:
:$\ds \map {y_1} x - \map {y_1} {x_*} = \int_{x_*}^x \map {y_1'} t \rd t = \int_{x_*}^x \map {f_1} {t, \map {y_1} t} \rd t$
:$\ds \map {y_2} x - \map {y_2} {x_*} = \int_{x_*}^x \map {y_2'} t \rd t = \int_{x_*}^x \map {f_2} {t, \map {y_2} t} \rd t$
After taking the difference:
:$\ds \map {y_1} x - \map {y_2} x = \int_{x_*}^x \paren {\map {f_1} {t, \map {y_1} t} - \map {f_2} {t, \map {y_2} t}} \rd t$
Let $N \in \R$ be such that:
:$N > \max \set {1, \dfrac 1 A, \dfrac 1 {A \paren {a \mathop + h \mathop - x_*} } }$
For all cases it holds that:
:$x_* + \dfrac 1 {A N} < a + h$
Let:
:$\ds B = \max_{t \mathop \in \closedint {x_*} {x_* \mathop + \frac 1 {A N} } } \size {\map {x_2} t - \map {x_1} t} \le 2 k$
Then $\forall x \in \closedint {x_*} {x_* + \dfrac 1 {AN} }$ we have:
Thus: 
:$\forall t \in \closedint {x_*} {x_* + \dfrac 1 {A N} } : \size {\map {x_1} t - \map {x_2} t} \le \dfrac B N$
and $B \le \dfrac B N$ or $N \le 1$. 
This brings us to a contradiction.
Hence our assumption that the solution to IVP is not unique was false.
Hence the result, by Proof by Contradiction.
== Sources ==
* : $\S 1.4$: Normed and Banach spaces. Sequences in a normed space; Banach spaces
Category:Picard's Existence Theorem