Professor's Keita Yokoyama (Japan Advanced Institute of Science and Technology) report will be held on November 22 at 16:30 in aud. 610 of the II building of KFU.
Название доклада: Ekeland's variational principle in reverse mathematics.
Аннотация: Ekeland's variational principle is a key theorem used in various areas of analysis such as continuous optimization, fixed point theory and functional analysis. It guarantees the existence of pseudo minimal solutions of optimization problems on complete metric spaces. Let f be a positive real valued continuous (or lower semi-continuous) function on a complete metric space (X,d). Then, a point x in X is said to be a pseudo minimum if f(x)=f(y)+d(x,y) implies x=y. Now, Ekeland's variational principle says that for any point a in X, there exists a pseudo minimum x such that f(x)<=f(a)-d(a,x). In reverse mathematics, it is observed that many theorems for continuous optimization problems are provable within the system of arithmetical comprehension (ACA_0), and thus most such problems have arithmetical solutions. However, this is not the case for pseudo minima. We will see that Ekeland's variational principle or its restriction to the space of continuous functions C([0,1]) are both equivalent to Pi^1_1-comprehenstion. This is a joint work with Paul Shafer and David Fernandez-Duque.