P. Gidoni (Czech Academy of Sciences), Existence of a periodic solution for superlinear second order ODEs
Abstract. We prove a necessary and sufficient condition for the existence of a T-periodic solution for the time-periodic second order differential equation x''+f(t,x)+p(t,x,x')=0, where f grows superlinearly in x uniformly in time, while p is bounded. Our method is based on a fixed-point theorem which uses the rotational properties of the dynamics.
September 29, 2021, 17:00 (UTC+2)
E. Affili (University of Deusto), A mathematical model for civil wars: a new Lotka-Volterra competitive system
Abstract. Imagine two populations sharing the same environmental resources in a situation of open hostility. The interactions among these populations are governed not by random encounters but via the strategic decisions of one population that can attack the other according to different levels of aggressiveness. This leads to a non-variational model for the two populations at war, taking into account structural ecological parameters. The analysis of the dynamical properties of the system reveals several equilibria and bifurcation phenomena. Moreover, we present the strategies that may lead to the victory of the aggressive population, i.e., the choices of the aggressiveness parameter, in dependence of the structural constants of the system and possibly varying in time in order to optimize the efficacy of the attacks, which take to the extinction in finite time of the defensive population. The model that we present is flexible enough to include also technological competition models of aggressive companies releasing computer viruses to set a rival companies out of the market. This is a joint work with S. Dipierro, L. Rossi and E. Valdinoci.
October 27, 2021, 17:00 (UTC+2)
J. A. Cid (University of Vigo), Brouwer fixed point theorem and periodic solutions of ODE's
Abstract. In this talk we will present the equivalence between the Brouwer fixed point theorem and the existence of periodic solutions for some second order differential equations defined in convex sets and that satisfy a suitable tangency condition at the boundary. Our approach involves some basic concepts of convex analysis and Stampacchia's method and as an application we will obtain also a very short proof, using the Brouwer fixed point theorem, of a result by Fonda and Gidoni on the existence of zeros of some mappings in convex sets. This is a joint work with J. Mawhin.
November 10, 2021, 17:00 (UTC+1)
Pablo Amster (University of Buenos Aires), On a theorem by Browder and its application to nonlinear boundary value problems
Abstract. In a paper from 1960, Felix Browder established a result concerning the continuation of the fixed points of a family of continuous functions of a compact convex subset of the n-dimensional space depending continuously on a real parameter. In this talk, we shall present a simple extension of Browder's theorem to Banach spaces and a more general family of parameters and show applications to some nonlinear boundary problems.
December 10, 2021, 15:00 (UTC+1)