We first recall simple proofs relying on the Schauder Fixed Point Theorem of the Nonlinear Alternative, the Leray-Schauder Alternative and the Coincidence Alternative for compact maps on normed spaces. We present also an alternative for compact maps defined on convex subsets of normed spaces. Those alternatives permit to apply the method of a priori bounds to obtain results establishing the existence of solutions to differential equations. Using those alternatives, we present some new proofs of existence results for first order differential equations.
"Schauder's Theorem and the method of a priori bounds." Topol. Methods Nonlinear Anal. 52 (1) 99 - 109, 2018. https://doi.org/10.12775/TMNA.2017.053