A Bolzano's theorem in the new millennium

Abstract

1. IntroductionOurmainobjectiveinthispaperistoestablishacloseconnectionbetweenaclassicaltheorem from real analysis (discovered over two centuries ago) and recent works inMonotone Operator Theory for reexive Banach spaces. Nevertheless, throughout thisexposition,wealsogiveabriefdescriptionofhowtheoriginalproblemhasevolvedintime,passingthroughdierentcategoriesofgeneralizationforthelast30years.Whilewe are progressing toward this goal, we are able to obtain some new results such asTheorem4(seeSection3),wheresomeconvexityconditionisnolongerrequired.Weare also able to obtain a new Invariance of Domain Theorem for monotone operators,as can be seen in Theorem 5 below.The study of the existence of zeros for nonlinear functional equations involvingmonotoneoperatorshasbeenextensivelydiscussedtowardtheveryendofthelastmil-lennium. Concerning the study of the existence of zeros under the boundary condition(2) below, we nd among the contributions, the work of Vainberg and Kachurovskii[19],Minty[12,13],Browder[4–6]andShinbrot[17].Forrelatedwork,wealsomen-tion BrBezis et al. [3], Kachurovskii [9], Leray and Lions [11], and Rockafellar [16].However, our main interest here is to attempt to unify some of the work done in thecontourofthiscondition(2),thatwasperhapsrstobservedbythismathematicianofthe XIX century. In spite of the fact that most of the results presented in Section 2 ofthis paper are known, their proofs share some degree of originality.