Degree Theory for Discontinuous Operators

This unique book contains a generalization of the Leray-Schauder degree theory which applies for wide and meaningful types of discontinuous operators. The discontinuous degree theory introduced in the first section is subsequently used to prove new, applicable, discontinuous versions of many classical fixed-point theorems such as Schauder's. Finally, readers will find in this book several applications of those discontinuous fixed-point theorems in the proofs of new existence results for discontinuous differential problems. Written in a clear, expository style, with the inclusion of many examples in each chapter, this book aims to be useful not only as a self-contained reference for mature researchers in nonlinear analysis but also for graduate students looking for a quick accessible introduction to degree theory techniques for discontinuous differential equations.

This book opens a novel and fruitful line of research based on a generalized degree theory Expands available techniques employed to prove existence of solutions for discontinuous problems Presents many potential applications to other fields

Autorentext

Rubén Figueroa (Salvaterra de Miño, Galicia, 1984) obtained his Ph.D. degree at University of Santiago de Compostela in 2011, with a dissertation about discontinuous differential equations with deviated arguments. In the following years, he focused his research on the study of topological methods for discontinuous operators and its applications to Ordinary Differential Equations. This research gave rise to a score of scientific publications in different journals. At present, he combines his research with teaching at a secondary school.

Rodrigo López Pouso (Ferrol, Galicia, 1973) obtained his Ph.D. degree in 1999 with a thesis on discontinuous differential equations. He currently works as a professor at the University of Santiago de Compostela. His research interests range from discontinuous ordinary differential equations to classical real analysis and he is the author or co-author of more than 60 papers published in high impact mathematical journals.

Jorge Rodríguez-López (A Pontenova, Galicia, 1993) obtained a Ph.D. Degree in Mathematical Analysis at University of Santiago de Compostela in 2020. His line of research focuses on nonlinear analysis, with emphasis in discontinuous differential equations, topological methods and fixed point theory. His results on existence, uniqueness and multiplicity of solutions for ordinary differential equations have been published in several research papers included in the Citation Index Report. Currently, he is Assistant Professor at the University of Santiago de Compostela.

Zusammenfassung

"The book leaves an overall pleasant impression. The contents are very well organized and well written. It is a valuable contribution to the literature on a class of methods of nonlinear analysis, providing an overview of classical techniques while introducing significant new ones." (Luis Fernando Sanchez Rodrigues, Mathematical Reviews, April, 2024)

"The monograph is perfectly organized and very well written. It provides a detailed presentation and insight on this new extension of the degree theory and illustrates the applicability of this tool in several problems. For these reasons, this book is suitable for researchers willing to learn more on this interesting topic." (Guglielmo Feltrin, zbMATH 1505.47003, 2023)

Inhalt

• 1. Introduction. - 2. Degree Theory for a Class of Discontinuous Operators. - 3. Fixed Point Theorems for Some Discontinuous Operators. - 4. First Order Problems. - 5. Second Order Problems and Lower and Upper Solutions. - 6. Positive Solutions for Second and Higher Order Problems.