Quartz 5

Picard-Lindelof Theorem

1 min read

Let and . Suppose . Let be continuous and for some we have There there is a s.t. : has a unique solution in .

Proof

Firstly, is cts on a closed set hence bounded, . Pick . Let . The metric space (of continuous functions) is complete in uniform metric. Hence define Note that is indeed cts whenever is cts (by FTA). We do need to check that : Now we check that is a contraction mapping on .

Misplaced &&\leq \sup_t\int_{t_0}^t|\phi(x,g(x))-\phi(x,h(x))|dx\\ &\leq \epsilon \sup_x|\phi(x,g(x))-\phi(x,h(x))|\\ &\leq \epsilon K|g(x)-h(x)|\\ &\leq |g(x)-h(x)|/2 \end{align}$$ Hence, we are done by [[Analysis/Contraction mapping\|contraction mapping theorem]] and FTA (i.e. $(Tg)=g\iff g$ is the solution to original problem).

Graph View

Backlinks