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COURSE | Degree Theory with applications to Ordinary Differential Equations

Lectured by
Doctor Ruben Figueroa, PhD by the University of Santiago de Compostela.

The course will occur on the
11th, 12th, 18h and
19th of March at ISEG.


Degree Theory has demonstrated to be a powerful tool in order to look for roots of continuous functions and, nowadays, it is applied to a very large class of analytical and geometrical problems. This course includes some topics on degree theory in Banach spaces. More concretely, we will speak about Brouwer degree in Rn and Leray-Schauder degree in innitedimensional spaces. Finally, we will show how this techniques can be applied when looking for solutions for some kind of Ordinary Dierential Equations, and how it can be coupled with another standard techniques, with particular emphasis in lower and upper solutions method.


1. Motivation: a periodic problem with constant lower and upper solutions.

2. Brouwer degree in Rn: construction of Brouwer degree; properties; typical applications

Brouwer's xed-point theorem and hairy ball theorem.

3. Leray-Schauder degree in innite-dimensional spaces: construction of L-S degree; properties;

typical applications Schauder's xed-point theorem.

4. Application of degree theory to boundary value problems.

Timetable and Location:

>1st lecture:
11 March, Monday |
Sala BES, 4o piso, Ed. do Quelhas

>2nd lecture:
12 March, Tuesday |
Anteatro 1, 4o piso, Ed. Quelhas

>3rd lecture:
18 March, Monday |
Sala BES, 4o piso, Ed. Quelhas

>2nd lecture:
19 March, Tuesday |
Anteatro 1, 4o piso, Ed. Quelhas

*Class schedule:


For registration please send an email message to Prof. Maria do Rosário Grossinho at

There is no registration fee.