Sumários
Lecture 3
20 Novembro 2017, 15:30 • Maria Teresa Faria da Paz Pereira
Criteria
for positivity and comparison of solutions for DDEs with (both discrete and
distributed) delays: positivity and Smith's quasimonotone conditions. Statements of the general results about existence and existence
and uniqueness of local solutions for an abstract DDE x'(t)=f(t,x_t); proof of
the existence and uniqueness theorem only in the case of f(t,\phi) continuous
and Lipschitzian relative to \phi in each bounded set of $\R\times C([-r,0],\R^n), by using the
contraction principle (rather than Schauder's fixed point theorem).
(This class took place on Wednesday Nov. 15, 2017, from 18h00 to 19h30.)
Lecture 2
13 Novembro 2017, 15:30 • Maria Teresa Faria da Paz Pereira
The classical example
x'(t)=-ax(t-r) (r>0,a in R) (continuation): pure imaginary roots and complex
roots with positive real part; stability; necessary and sufficient condition
for oscillation of all solutions. Criteria for positivity and comparison of
solutions for DDEs with one (or more) discrete delays.
(This class took place on Wednesday Nov. 15, 2017, from 18h15 to 20h.)
Brower degree exposition
13 Novembro 2017, 12:00 • Maria Carlota da Rocha Xavier Rebelo Gonçalves
The student João Casimiro did an exposition on Brower's degree theory (existence and uniqueness of the degree) (the lesson took place on the 17/11/2017)
Aula 1 de EDF
6 Novembro 2017, 15:30 • Maria Teresa Faria da Paz Pereira
Presentation of the course and main references.
Delay differential equations (DDEs): motivation and examples; phase space and notation; similar aspects and main differences between ODEs and DDEs. The method of steps. The classical example of de equation x′(t) = −βx(t − r) (r > 0, β ∈ IR): solutions of exponential form and characteristic equation. Real characteristic roots.
(This class took place on 8/11/2017, 18h30-20h.)
Degree theory and fixed point index
30 Outubro 2017, 15:30 • Maria Carlota da Rocha Xavier Rebelo Gonçalves
Degree in R and R^2.
Fixed point index in an open bounded set of a Banach space: axioms.