Schauder’s fixed-point theorem: new applications and a new version for discontinuous operators

Schauder’s fixed-point theorem, which applies for continuous operators, is used in this paper, perhaps unexpectedly, to prove existence of solutions to discontinuous problems. Moreover, we introduce a new version of Schauder’s theorem for not necessarily continuous operators which implies existence of solutions for wider classes of problems. Leaning on an abstract fixed-point theorem, our approach is not limited to one-dimensional homogeneous Dirichlet problems, the only type of examples worked out in this paper for coherence and simplicity but yet novelty. MSC:47H10, 34A36, 34B15.