Applications of topological degree theory to generalized ODEs

Abstract

In this work, we present original results concerning the theory of Generalized Ordinary Differential Equations (we write generalized ODEs for short) using tools from the Topological Degree theory. In particular, we proved results on Existence of bifurcation points and we applied the results to measure differential equations; Differentiability of the solution operator of generalized ODEs, including a Fredholm Alternative-type theorem, and we applied the results to measure differential equations; Existence of periodic solutions of linear generalized ODEs to which we applied not only results from the topological degree theory, but also from the Fredholm operator theory; Existence of affine-periodic solutions of generalized ODEs.

It is worth mentioning that the present work generated 3 original articles which are in their final stages of preparation and will be submitted for publication soon.

In addition to the above, we also generalized the results from my Master Thesis which are contained in a submitted article, coauthored by J. Mawhin and M. Federson. While in such article we deal with the existence of periodic solutions of generalized ODEs involving bounded variation functions, in the present work we consider the regulated functions. Such new results are part of a chapter in the book entitled Generalized ODEs in Abstract Spaces and Applications and organized by the editors M. Federson, E. Bonotto and J. Mesquita. The book will be published by Wiley in 2020.