I am teaching a course in functional analysis and I would like to illustrate the Schauder fixed point theorem (just for Banach spaces) with some nice applications. One that comes to my mind is the existence of solutions of ordinary differential equations. Applications to nonlinear PDEs would require too much knowledge.
Question What are the elementary and nice applications of the Schauder fixed point theorem?
Check out the Chapter 3 of the Methods in Nonlinear Integral Equations
I am not an expert on integral equations but the authors uses Schauder's theorem to prove existence theorems for continuous solutions of the integral equation associated with the Fredholm integral operator, the Volterra integral operator, and in particular integral operator with delay.
Obviously you are right about the most elementary application which comes to mind. Peano's existence theorem for ODE in Jake Hale's ODE book on page 15 is proved by Schauder's theorem.