\part{Advanced}

\chapter{Existence and uniqueness of differential equations}

We know that the solutions of a differential equation can be represented as a (possibly empty, possibly infinite, etc.) solution set.

For algebraic equations, we understand the nature of these sets quite well due to the fundamental theorem of algebra; algebraic equations of degree $n$ always have $n$ solutions in $\mathbb{C}$ up to multiplicity. We also know other neat facts like that algebraic equations of odd degree always have at least 1 real solution, and so forth.

The nature of differential equations and their solutions is much more difficult to pin down, however ideas from topology and analysis can be used to make some remarks about certain classes of differential equations. Mostly, these theorems reveal classes of differential equations for which at least 1 solution exists, or even better, when a unique solution exists.

\section{Peano existence theorem}

Among the simplest existence theorems is the Peano existence theorem.

\begin{theorem}[Peano existence theorem]
Let $D \subseteq \mathbb{R}\times \mathbb{R}$ be a set open in $\mathbb{R}^2$ (Euclidean topology) with $(t_0,y_0) \in  (D)$. If $f : D \to \mathbb{R}$ is continuous then there exists some interval $t_0 \in I$ on which the IVP $y'(t) = f(t,y(t)), y(t_0) = \mathbf{y}_0$ has a solution.
\end{theorem}

\section{Carathéodory existence theorem}
To those familiar with measure theory, there is a generalization of the theorem that weakens the conditions.

\begin{theorem}[Carathéodory existence theorem]
\end{theorem}


\section{Picard–Lindelöf theorem}

As nice as these theorems are, they don't say anything about the uniqueness of a solution, indeed there are examples of such ODEs having multiple solutions.

The following theorem employs further conditions that show uniqueness for certain well-behaved IVPs on a neighborhoood of the initial value.

\begin{theorem}[Picard–Lindelöf theorem]
Let $D \subseteq \mathbb{R}\times \mathbb{R}^n$ be a closed rectangle $(t_0,\mathbf{y}_0) \in \mathrm{int} (D)$. If $f : D \to \mathbb{R}^n$ is $t$-continuous and $\mathbf{y}$-Lipschitz continuous then there exists some closed interval $t_0 \in I$ on which the IVP $y'(t) = f(t,y(t)), y(t_0) = \mathbf{y}_0$ has a unique solution.
\end{theorem}

Proof is based on applying the Banach fixed-point theorem to the integral equation form of $f$.

Due to the constructive nature of the Banach fixed-point theorem, the proof of the Picard-Lindelöf theorem inspires a solution method called \emph{Picard iteration}.

\subsection{Picard iteration}

\section{Cauchy-Kovalevskaya theorem}

\begin{theorem}[Cauchy-Kovalevskaya theorem]
\end{theorem}

\section{Okamura's theorem}

\begin{theorem}[Okamura's theorem]
\end{theorem}

\chapter{Sturm-Liouville theory}


Operator are an object from functional analysis; familiar with differential operators is required for this chapter.

Sturm-Liouville studies differential equations of specific type, appropriately called Sturm-Liouville equations.

\begin{definition}[Sturm-Liouville equation]
A \emph{Sturm-Liouville equation} is a second order ODE in terms of $p,q,w \in C^1(\mathbb{R})$ and $\lambda \in \mathbb{R}$
\[[p(x)y']' q(x)y = -\lambda w(x) y \]
\end{definition}

In modern times, it is studied with a 'functional analysis' flavour, where the Sturm-Liouville equation is constructed by means of an operator, called (what a surprise) Sturm-Liouville operators, so function solutions of the Sturm-Liouville equation are considered as eigenfunctions of some Sturm-Liouville operator.

\begin{definition}[Sturm-Liouville operator]
A \emph{Sturm-Liouville operator} is an operator in terms of $p,q,w \in C^1(\mathbb{R})$
\[\mathcal{L}(y) =-\frac{1}{w(x)}[p(x)y']' q(x)y \]
\end{definition}

From this perspective, the Sturm-Liouville equation is the eigenequation of the Sturm-liouville operator $\mathcal{L}y = \lambda y$.

\chapter{Fuchsian theory}
Perhaps the most powerful method familiar to us so far is the power series method and method of Frobenius. The Fuchsian theory studies in depth the extent to which these methods can be used.


\chapter{Lie theory}

Lie's theory employs a structure called Lie groups to study the algebraic sy. His ultimate goal was that Lie theory for differential equations would compare with Galois theory for algebraic equations. A
-Lie point symmetry


- Holonomic function
- D-module