\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{Lyupanov theory}
Lyupanov stability
Lyupanov function