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 $\mathbb {R}$.

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.

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.

$|\Phi _{n}-y_{0}|=|\int _{x_{0}}^{x}f(t,\Phi _{n-1}(t))dt|\leq |\int _{x_{0}}^{x}|f(t,\Phi _{n-1}(t))|dt|\leq |\int _{x_{0}}^{x}Mdt|=M|x_{0}-x|\leq M\delta \leq M\cdot {\frac {b}{M}}=b.$

$\Box$