Topological methods for discontinuous operators and applications

Abstract

Topological methods are crucial in nonlinear analysis, especially in the study of existence of solutions to diverse boundary value problems. As a well–known fact, continuity is a basic assumption in the classical theory and the clearest limitation of its applicability. That is why most discontinuous differential equations fall outside its scope because the corresponding fixed point operators are not continuous.

The main goal of this thesis is to introduce a new definition of topological degree which applies for a wide class of non necessarily continuous operators. This generalization is based on the degree theory for upper semicontinuous multivalued mappings. As a consequence, new fixed point theorems for this class of discontinuous operators are derived.

This new theory for discontinuous operators is combined with classical techniques in nonlinear analysis in order to obtain existence, localization and multiplicity results for discontinuous differential equations.