Ordinary Differential Equations/The Existence and Uniqueness Theorem

A major result in the theory of ordinary differential equations is the existence and uniqueness theorem, also known by other names like the Picard–Lindelöf theorem or the Cauchy–Lipschitz theorem. Roughly speaking, it states that initial value problems have unique solutions under certain regularity conditions on the vector field that defines them.

Before we give the precise statement, it is perhaps a good idea to explain why we should care about this theorem.

The main reason is that having an existence and uniqueness theorem allows us to use any method we want to find a solution to an ODE, and as long as we can check that the function we get solves the ODE, we are done. As an example, one method that is commonly used to solve constant coefficient ODEs is to guess a solution in the form of an exponential function. If our guess finds a function that satisfies the ODE, the existence and uniqueness theorem tells us that's the only solution, and that there is no possibility that we found an incorrect or incomplete answer.

Another reason this is important is that the theorem effectively says the solution to an ODE is entirely determined by its initial conditions. This, and generalizations of this, (e.g. to PDEs or cases with boundary conditions) ends up being very important in applications, because it's possible to replace the situation being modeled by a simpler one, as long as the resulting model is the same ODE with the same initial conditions.

Here is the precise statement of the theorem:

Theorem: Let $D$ be an open connected subset of $\mathbb {R} ^{n}$. Let $f:[0,\infty )\times D\rightarrow \mathbb {R} ^{n}$ be continuous in $t$, and Lipschitz continuous in $x$ uniformly in $t$, where $t$ and $x$ are the first and second argument of $f$, respectively. Then, for any $t_{0}\in [0,\infty )$ and $x_{0}\in D$, there exists $\delta >0$ such that the initial value problem

${\begin{cases}{\frac {dx}{dt}}=f(t,x)\\x(t_{0})=x_{0}\end{cases}}$

has a unique solution on $[t_{0},t_{0}+\delta ]$.

By "Lipschitz continuous in $x$ uniformly in $t$", we mean that for any $x,y\in D$, there exists $L>0$ such that ${\frac {|f(t,x)-f(t,y)|}{|x-y|}}\leq L$ for every $t\in [0,\infty )$, which is a slightly stronger condition than continuity. Nonetheless, this condition will be satisfied by most ODEs considered in this course and encountered in applications.