A Schauder-type theorem for discontinuous operators with applications to second-order BVPs

Rubén Figueroa,Gennaro Infante

doi:10.1186/s13663-016-0547-y

Abstract

We prove a new fixed point theorem of Schauder-type which applies to discontinuous operators in non-compact domains. In order to do so, we present a modification of a recent Schauder-type theorem due to Pouso. We apply our result to second-order boundary value problems with discontinuous nonlinearities. We include an example to illustrate our theory.

Highlights

In the recent and interesting paper [ ], Pouso proved a novel version of Schauder’s theorem for discontinuous operators in compact sets
We apply our new result to prove the existence of solutions of a large class of discontinuous second-order ordinary differential equation (ODE) subject to separated boundary conditions (BCs), complementing the results of [ ] and improving them in the special case of Dirichlet BCs
We introduce the main result which is an extension of Theorem . to the case of discontinuous operators

Summary

Introduction

In the recent and interesting paper [ ], Pouso proved a novel version of Schauder’s theorem for discontinuous operators in compact sets. ([ ], Theorem .A) Let K be a nonempty, closed, bounded, convex subset of a Banach space X and suppose that T : K −→ K is a compact operator (that is, T is continuous and maps bounded sets into precompact ones).

Results

Conclusion