12-267/Existence And Uniqueness Theorem

Disclamer: This is a student prepared note based on the lecure of Monday September 21st.

Def. $f:\mathbb {R} _{y}\rightarrow \mathbb {R}$ is called Lipschitz if $\exists \epsilon >0,k>0$ (a Lipschitz constant of f) such that $|y_{1}-y_{2}|<\epsilon \implies |f(y_{1})-f(y_{2})|\leq k|y_{1}=y_{2}|$ .

Note that any function that is Lipschitz is uniformly continuous, and that if a function f and its derivative are both continous on a compact set then f is Lipschitz.

Thm. Existence and Uniqueness Theorem for ODEs

Let $f:\mathbb {R} =[x_{0}-a,x_{0}+a]\times [y_{0}-b,y_{0}+b]\rightarrow \mathbb {R}$ be continuous and uniformly Lipschitz relative to y. Then the equation $\Phi '=f(x,\Phi )$ with $\Phi (x_{0})=y_{0}$ has a unique solution $\Phi :[x_{0}-\delta ,x_{0}+\delta ]\rightarrow \mathbb {R}$ where $\delta =min(a,^{b}/_{M})$ where M is a bound of f on $\mathbb {R}$ .

This is proven by showing the equation $\Phi (x)=y_{0}|\int _{x_{0}}^{x}f(t,\Phi (t))dt$ exists, given the noted assumptions.

Let $\Phi _{0}(x)=y_{0}$ and let $\Phi _{n}(x)=y_{0}+\int _{x_{0}}^{x}f(t,\Phi _{n-1}(t))dt$ .

Claim 1: $\Phi _{n}$ is well-defined. More precisely, $\Phi _{n}$ is continuous and $\forall x\in [x_{0}-\delta ,x_{0}|\delta ]$ , $|\Phi _{n}(x)-y_{0}|\leq b$ where b is as referred to above.

Claim 2: For $n\geq 1$ , $|\Phi _{n}(x)-\Phi _{n-1}(x)|\leq {\frac {Mk^{n-1}}{n!}}|x-x_{0}|^{n}$ .

Claim 3: if $\Phi _{n}(x)$ is a series of functions such that $|\Phi _{n}(x)-\Phi _{n-1}(x)|<c_{n}$ , with $\sum _{n=1}^{\infty }c_{n}$ equal to some finite number, then $\Phi _{n}$ converges uniformly to some function $\Phi$

Using these three claims, we have shown that the solution $\Phi (x)$ exists. The proofs of the claims are below.

Proof of Claim 1:

The statement is trivially true for $\Phi _{0}$ . Assume the claim is true for $\Phi _{n-1}$ . $\Phi _{n}$ is continuous, being the integral of a continuous function.