Functional Differential Equations and its Applications
Organizadores: Pablo Amster (pamster@dm.uba.ar), Gonzalo Robledo (grobledo@uchile.cl)

Wednesday 15
15:00  15:45Periodic positive solutions of superlinear delay equations via topological degree
Pierluigi Benevieri (Universidade de Sao Paulo, Brasil)
We present a joint work with Pablo Amster (Universidad de Buenos Aires, Argentina) and Julián Haddad (Universidade Federal de Minas Gerais, Brazil), which deals with an extension to delay problems of some recent results by G. Feltrin and F. Zanolin. They obtained existence and multiplicity results for positive periodic solutions to nonlinear differential equations of the form \[u''(t)=f(t,u(t),u'(t))\] with periodic or Neumann boundary conditions and special assumptions on $f$. We obtain analogous results for a delay problem of the type \[u''(t)=f(t,u(t),u(t\tau),u'(t))\] Our approach, as well as that of Feltrin and Zanolin, is topological and based on the coincidence degree introduced by J. Mawhin.
15:45  16:30Global bifurcationlike Theorems in presence of nonvanishing Spectral Flow
Julian Haddad (Universidade Federal de Minas Gerais, Brasil)
Given a oneparameter family of functions \(f_t:H \to \mathbb R\) where \(f_t (0) = 0, \nabla f_t (0) = 0\) and \(H\) is a real Hilbert space, the Spectral Flow of the Hessian of \(f\) was related recently to local bifurcation results by Fitzpatrick, Pejsachowicz and Waterstraat. While this invariant is finer than the topological index, the existent results are of local nature, in contrast to the global bifurcation theorem of Krasnoselski and Rabinowitz. We prove a global bifurcation theorem for “target values” of \(f\) under Spectral Flow hypothesis.
16:45  17:30Existence of bifurcation point for impulsive differential equations via generalized ordinary differential equations
Maria Carolina Mesquita (Universidade de Sao Paulo, Brasil)
This is a joint work with Professors M. Federson and J. Mawhin. In this work, we establish conditions for the existence of a bifurcation point with respect to the trivial solution of a generalized ordinary differential equation, whose integral form displays the nonabsolute Kurzweil integral. The main tools employed here are the coincidence degree theory and an ArzelaAscolitype theorem for regulated functions. We also present applications to impulsive differential equations.
17:30  18:15A LazerLeach type result for functionaldifferential equations at resonance
Arturo Sanjuán (Universidad Distrital Francisco José de Caldas, Colombia)
In this talk we will discuss some recent results regarding LazerLeach type conditions for the existence of periodic solutions to systems of functionaldifferential equations at resonance. We consider evendimensional kernels and general delays using Coincidence Degree Theory.

Thursday 16
15:00  15:45Linearized instability for neutral FDEs
Jaqueline Godoy Mesquita (Universidade de Brasilia, Brasil)
In this talk, we will give a brief overview of a class of equations called neutral functional differential equations with statedependent delays, describing some important applications. After this, we will show some recents results in the area, and we will present a principle of linearized instability for these equations.This is a joint work with Professor Bernhard LaniWayda.
15:45  16:30On the nonlinearly determined wavefronts for the MackeyGlass type diffusive equations
Sergei Trofimchuk (Universidad de Talca, Chile)
We study the MackeyGlass type monostable delayed reactiondiffusion equation with a unimodal birth function \(g(u)\). This model, designed to describe evolution of single species populations, is considered in the presence of the weak Allee effect (\(g(u_0)>g'(0)u_0\) for some \(u_0>0\)). We focus our attention on the existence of slow monotonic traveling fronts to the equation: under given assumptions, this problem seems to be rather difficult since the usual positivity and monotonicity arguments are not effective. By considering the particular case of Nicholson's diffusive equation, we also discuss other situation leading to the appearance of nonlinearly determined wavefronts. This is a joint work with Karel Hasík, Jana Kopfová and Petra Nábělková, Mathematical Institute, Silesian University, Czech Republic.
16:45  17:30On the Stability and the Existence and Nonexistence of \(T\)Periodic Solutions for Nonlinear Delayed Differential Equations with Friction and \(\varphi\)Laplacian
Mariel Paula Kuna (Universidad de Buenos Aires, Argentina)
Let us consider the following problem \[(\varphi(x'(t) ))' + h(x(t),x'(t))x'(t)+ g(x(tr)) = p(t), \ \ \ t\in [0,\infty),\] where \(\varphi:\mathbb{R} \rightarrow \mathbb{R}\) is an increasing homeomorphism such that \(\varphi(0)=0\), \(r\) is a positive constant, \(g:\mathbb{R} \rightarrow \mathbb{R}\) is a continuous differentiable function, and \(h:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R}\) and \(p:[0,\infty) \rightarrow \mathbb{R}\) are continuous functions such that \(p\) is \(T\)periodic in \(t\) for \(T\) a positive constant.
Using LyapunovKrasovskii functional and under appropriate assumptions we obtain new results on the global stability, boundedness of solutions, existence and nonexistence of \(T\)periodic solutions. This is a joint work with P. Amster and D.P. Santos.
17:30  18:15Stability and boundedness of solutions of retarded equations
Márcia Federson (Universidade de Sao Paulo, Brasil)
Generalized ODEs were introduced by J. Kurzweil in 1957 and are known to encompass several other types of equations. In this talk, we explore converse Lyapunov theorems for this type of equations. In particular, we derive results for certain integral forms of measure functional differential equations. We also relate Lyapunov stability to the boundedness of solutions of these equations which have nonabsolute integrable righthand sides.

Friday 17
15:00  15:45Coupled reactiondiffusion and difference system with nonlocal dispersal term and implicit time delay
Mostafa Adimy (Institut de la Recherche en Informatique et Automatique, Francia)
We consider a class of biological models represented by a coupled system between reactiondiffusion and difference equations (renewal equation) with nonlocal dispersal term and implicit time delay. The difference equation generally arises, using the characteristic method, from an agestructured partial differential system. We start by studying the existence, uniqueness, positivity, and boundedness of solutions. We then investigate the stability analysis and obtain a threshold condition for the global asymptotic stability of the trivial steady state by using a Lyapunov functional. We also obtain a sufficient condition for the existence and uniqueness of a positive steadystate by using the method of lower and upper solutions. In the case of an unbounded domain, we study the existence of monotonic planar traveling wave fronts connecting the extinction state to the uniform positive state. The corresponding minimum wave speed is also obtained. In addition, we investigate the effect of the parameters on this minimum wave speed and we give a detailed analysis of its asymptotic behavior.
15:45  16:30A Krasnoselskii relatedness principle for delay differential equations
Julian Epstein (Universidad de Buenos Aires, Argentina)
In this talk, we shall present an \textbf{Abstract} formulation of a duality principle established by Krasnoselskii. Under appropriate conditions, it shall be shown that, if the solutions of a nonlinear functional equation can be obtained by finding fixed points of certain operators in possibly different Banach spaces, then these operators have the same topological index. An application to delay differential equations shall be given. This is a joint work with P. Amster.