Topological Degree Methods in Nonlinear Differential Equations

• Grant number: 25/21527-4
• Support Opportunities: Scholarships abroad - Research Internship - Scientific Initiation
• Start date: January 05, 2026
• End date: March 04, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marta Cilene Gadotti
• Grantee: Gabriela Alves Squaiella
• Supervisor: Francesca Dalbono
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil
• Institution abroad: Università degli Studi di Palermo (UNIPA), Italy
• Associated to the scholarship: 25/08394-5 - A study on delay differential equations and applications, BP.IC

Abstract

Topological degree theory is a fundamental tool in nonlinear analysis and plays a centralrole in the study of differential equations. This theory is used to establish the existenceof solutions without requiring explicit formulas. Its main strength consists in reducinganalytical difficulties to topological invariants: through homotopy invariance, the degreeremains unchanged by deforming, under appropriate conditions, a function into anotherone in a continuous way. Thus, a problem can be trasformed into a simpler one for whichthe degree is easy to calculate, thereby transferring information about solutions to theharder problem.The aim of this project is to introduce the basic concepts of topological degree (Brouwerand Leray-Schauder); study its application to prove existence of solutions for nonlinear ordinarydifferential equations (ODEs).