In addition to ODE existence theorems, there are also uses for PDE existence/uniqueness theorems. An example of that is constructing weak solutions to the linear Boltzmann equation. I think this example is interesting because it is more of a philosophy, not so much precise "fixed point theorem" that is used here.

The linear Boltzmann equation is:

$\partial_t f + v\cdot \nabla_x f = Kf -af + Q$

where

$Kf = \int k(t,x,v,v') f(t,x,v')dv'$

By Duhamel's principle, we know that a strong solution would satisfy

$ f(t,x,v) = f_0(x-tv,v) + \int_0^t (Kf - af + Q)(s,x-(t-s)v,v)ds$.

We basically use this as our definition of a weak solution. Thus, we can rephrase the search for a weak solution as looking for a fixed point to the operator

$ g \mapsto F[f,Q] + \tau g $

where

$ F[f_0,Q] = f_0(x-vt,v) + \int_0^t Q(s,x-(t-s)v,v)ds$

and

$\tau g = \int_0^t (Kf - af)(s,x-(t-s)v,v)ds$.

Notice that the series

$\sum_{n\geq 0} \tau^n[F[f_0,Q]]$

would be such a fixed point if we had appropriate convergence (just hit it with $\tau$ and see what happens), so basically, we've reduced the problem to bounding the operator $\tau$ in the appropriate space which we would like weak solutions to live. As I mentioned above, this doesn't really use any "fixed point theorems" but is clearly still a fixed point argument.