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.

Given a one-parameter 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.

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 Arzela-Ascoli-type theorem for regulated functions. We also present applications to impulsive differential equations.

In this talk we will discuss some recent results regarding Lazer-Leach type conditions for the existence of periodic solutions to systems of functional-differential equations at resonance. We consider even-dimensional kernels and general delays using Coincidence Degree Theory.

In this talk, we will give a brief overview of a class of equations called neutral functional differential equations with state-dependent 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 Lani-Wayda.

We study the Mackey-Glass type monostable delayed reaction-diffusion 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.

Let us consider the following problem \[(\varphi(x'(t) ))' + h(x(t),x'(t))x'(t)+ g(x(t-r)) = p(t), \ \ \ t\in [0,\infty),\] 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 Lyapunov-Krasovskii functional and under appropriate assumptions we obtain new results on the global stability, boundedness of solutions, existence and non-existence of \(T-\)periodic solutions. This is a joint work with P. Amster and D.P. Santos.

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 non--absolute integrable right--hand sides.

We consider a class of biological models represented by a coupled system between reaction-diffusion and difference equations (renewal equation) with nonlocal dispersal term and implicit time delay. The difference equation generally arises, using the characteristic method, from an age-structured 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 steady-state 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.

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.