How can Banach fixed point theorem be used to prove that there is a unique solution to differential equation? I couple of months ago my professor said in class that the Banach fixed Point theorem could be used to prove the uniqueness of a solution to a differential equation. (My memory can be wrong, please correct me if so.).
If the above is correct. How can it be used to prove the uniqueness of the solution to $y=y'$?
 A: You consider an initial value problem (IVP) $y(x_0)=y_0$ of an ordinary differential equation (ODE) $y'=f(x,y)$ of order $1$. $y$ can be a vector, even from a Banach space.
Differentiating is symbolically much more accessible than integrating, but for theory purposes integration is much less demanding of a given situation. Thus transform the differential into an equivalent integral equation
$$
y(x)=y(x_0)+\int_{x_0}^x f(s,y(s))\,ds
$$ 
The right side is valid for any continuous function, not only differentiable functions. It can be interpreted as an operator $P:C([x_0,x_f])\to C([x_0,x_f])$ where $P(y)$ is thus also a function with values $(P(y))(x)$ given by the right side.
Now under various assumptions $P$ can be seen as contractive on a sub-interval $[x_0,x_0+\alpha]$ or the full interval relative to the supremum norm or a modified (and equivalent) supremum norm. As $C([x_0,x_f])$ with these norms is complete one can apply the Banach fixed-point theorem and obtain the existence and uniqueness of a solution as the limit of the recursive sequence $y_0(x)=y_0$, $y_{k+1}=P(y_k)$, which is also called Picard iteration for this operator (mapping of function spaces, function on function spaces etc. means all the same thing).
In total this gives the (or one of the variants of the) theorem of Picard-Lindelöf.
