Suppose we have a first order ODE of the form $y^\prime = f(x,y)$ with initial conditions $y(x_0) = y_0$. The existence and uniqueness theorem states that if $f$ and its partial derivative with respect to $y$ are continuous in some rectangular region $\{(x, y); |x – x_0| \leqslant a,|y – y_0| \leqslant b\}$ then there exists a unique solution of the ODE in the closed interval $[x_0 – h, x_0 + h]$ where $h < a$.
I am familiar with the above definition and the various linked topics, such as Picard Iterations, Leibniz integral rule etc but was wondering if anyone could provide a proof of this theorem.
[Math] Proof of uniqueness and existence theorem for first order ordinary differential equations
calculusintegrationordinary differential equations
Best Answer
The fundamental theorem tells you that $$ f(x,y)-f(x,z)=\int_{[y,z]}f_ydy=\int_0^1f_y(x,y+t(z-y))·(z-y)\,dt. $$
The region $\{(x,y);|x−x_0|⩽a,\|y−y_0\|⩽b\}$ is a compact set, $f_y$ is continuous thus bounded, let $L$ be a bound. Then $$ \|f(x,y)-f(x,z)\|\le\int_0^1L·\|z-y\|\,dt=L·\|z-y\| $$ which provides the Lipschitz condition on $f$. Now apply Picard-Lindelöf.