Abstracts

• Gabriele Bonanno (University of Messina)
  A local minimizer theorem
  A characterization on the existence of at least one local minimum for a class of functions is presented. Some consequences are highlighted and the relations with the mountain pass theorem are shown. Finally, applications to nonlinear differential problems to obtain existence and multiplicity of solutions are given.


• Alessandro Fonda (University of Trieste)
  Multiplicity results for some boundary value problems associated with Hamiltonian systems in \(\mathbb{R}^N\)
  Different extensions of the Poincaré—Birkhoff Theorem for the periodic problem will be discussed, coupling standard periodic-twist situations with systems involving either lower and upper solutions, or positively homogeneous nonlinearities with or without Landesman—Lazer conditions. Counterparts of these results can be obtained also for two-point boundary value problems, where surprisingly no twist condition is needed.


• Julián López-Gómez (Complutense University of Madrid)
  A class of sublinear elliptic boundary value problems
  Using the global bifurcation theorem from infinity of P. H. Rabinowitz we are going to show the existence of some global continua of positive solutions for a class of sublinear elliptic BVP's, and simultaneously will analyze some interesting one-dimensional prototypes to gain insight into the nature of these problems. Joint work with Fabio Zanolin and Paul H. Rabinowitz.


• Luisa Malaguti (University of Modena e Reggio Emilia)
  A bounding function approach for the investigation of boundary value problems to partial differential equations and systems
  A continuation principle is still successfully used for the investigation of boundary value problems to ordinary differential equations and important contributions in this topic are certainly due to Fabio Zanolin and his co-authors. This method involves, in particular, the existence of a bounding function, i.e. a Lyapunov-like function; it allows, in fact, that the associated vector fields have no fixed points on the boundary of their domains and thus makes possible the use of a degree argument.
  A partial differential equation, when written in its abstract setting, becomes an ordinary differential equation but defined in a function state space. This similarity encourages the employee of a similar topological argument also in this context.
  The talk accounts on some recent results about the study of nonlocal boundary value problems with this method associated both to parabolic and hyperbolic differential equations. The investigation involves some basic properties of the semigroup theory as well as the properties of the duality mapping. For parabolic equations with strongly elliptic differential operators, we can prove the existence of periodic solutions, solutions satisfying the so-called Cauchy-multipoint or the mean value condition. For these equations we can also investigate the existence of solutions satisfying nonlinear nonlocal, possibly multivalued, conditions.
  When less compactness is contained within the model, such as in the case of hyperbolic partial differential equations, the discussion becomes harder and stronger regularity conditions are required on the nonlinear part. We propose some contributions concerning the classical model of age-structured population dynamics.
  The results come from a joint research project with Irene Benedetti (University of Perugia), Stefania Perrotta and Valentina Taddei (University of Modena and Reggio Emilia).


• Alessandro Margheri (University of Lisbon)
  Relative periodic behaviour of simple models of crawling locomotors with a periodically prescribed shape
  In this talk, based on the joint work with Paolo Gidoni (University of Udine), we will address the asymptotic behaviour of a family of dynamic models of crawling locomotors. The locomotors, which can have a discrete or continuous body, move on a line with a periodic prescribed shape change. They are subjected to different classes of friction forces, including multivalued ones, and, possibly, other external forces (e.g., gravity when crawling on a slope). Under mild general assumptions on the forces, we show that the asymptotic motion is relative periodic. However, the asymptotic average velocity of the crawler may still depend on its initial state, and we provide additional assumptions for its uniqueness. In particular, we show that the asymptotic average velocity is unique both for strictly monotone friction forces, and also for dry friction, provided in the latter case that the actuation is sufficiently smooth (for discrete models) or that the friction coefficients are always nonzero (for continuous models). We present several examples and counterexamples illustrating the necessity of our assumptions.


• Jean Mawhin (Catholic University of Louvain)
  Boundary value problems and periodic solutions of nonlinear ordinary differential equations: old and new
  We present a personal (and hence subjective) selection of a few results of Fabio Zanolin in the study of boundary value problems and periodic solutions of nonlinear ordinary differential equations.


• Eduardo Muñoz-Hernández (Complutense University of Madrid)
  Subharmonics and chaos in periodic predator-prey models
  In this talk, we deal with non-autonomous Volterra predator-prey models. First, we analyze the existence and multiplicity of subharmonic solutions through the Poincaré—Birkhoff Theorem and different bifurcation theory methods. Then, by the use of the "stretching along the paths" technique, the existence of chaotic dynamics for a wide class of these predator-prey models will be proved. Joint work with Fabio Zanolin and Julián López-Gómez.


• Pierpaolo Omari (University of Trieste)
  Positive solutions for a class of anisotropic mean curvature problems
  This talk will cover existence, non-existence, and multiplicity of positive solutions for a class of anisotropic mean curvature problems. Various notions of solution will be discussed, and regularity issues will be addressed. A key role in the discussion will be played by lower and upper solutions techniques specifically developed for such problems.


• Rafael Ortega (University of Granada)
  Stability of coexistence states in a periodic prey-predator system
  Consider the system \[ \begin{aligned} & \dot{u}=u(a(t)-b(t)u-c(t)v),\quad \dot{v}=v(d(t)+e(t)u-f(t)v), \\ & u>0, \quad v>0, \end{aligned} \] where all the coefficients are periodic with a common period and the functions \(b,c,e,f\) are positive. The periodic dependence of the coefficients models seasonal effects. This talk will deal with the stability properties of the periodic solutions. The main result is a criterion for the asymptotic stability that is inspired in a classical stability criterion for Hill's equation \(\ddot{x}+a(t)x=0,\) where \(a(t)\) is periodic.


• Maria Patrizia Pera (University of Firenze)
  Periodic perturbations of reducible scalar second order functional differential equations
  In this talk, we will illustrate some results recently obtained with Alessandro Calamai (Università Politecnica delle Marche) and Marco Spadini (University of Firenze) concerning the structure of the set of forced oscillations of a class of reducible second order functional retarded differential equations subject to \(T\)-periodic forcing terms. More precisely, a delay-type functional dependence involving a gamma probability distribution is considered and, by means of a linear chain trick, a first order system of ODEs whose \(T\)-periodic solutions correspond to those of the second order functional equation is formulated. We will also describe the simple notion of ejecting set and present some sufficient conditions yielding multiplicity of the \(T\)-periodic solutions. The approach used is topological and is based on continuation methods together with the computation of the Brouwer degree of a suitable map.


• Alfonso Ruiz-Herrera (University of Oviedo)
  Seasonality in the classical SIR model
  The classical SIR model \[ \label{SIRgeneral} \begin{cases} S' = \lambda-\mu S-\beta S I, \\ I' = \beta SI-(\mu+\gamma) I, \\ R' = \gamma I-\mu R \end{cases} \] describes the evolution of three mutually exclusive compartments: susceptible (\(S\)), infected individuals (\(I\)) and recovered individuals (\(R\)). It is well known that if the basic reproduction number is greater than one in the above model, there is an endemic equilibrium that is globally attracting, otherwise the disease dies out. Thus, the above model always predicts damped oscillations which tend to an equilibrium. However, the evolution of many infectious diseases sometimes exhibits more erratic oscillatory patterns beyond the simple attraction to a steady state. These mismatches between theoretical predictions and real-world patterns are normally related to the absence of seasonal effects.
  In this talk we study model above when the biological parameters depend periodically on time. Since \(R(t)\) does not appear in the first equations, it is enough to analyze \[ \begin{cases} \, S' = \lambda(t)-\mu(t) S-\beta(t) S I, \\ \, I' = \beta(t) SI-(\mu(t)+\gamma(t)) I \end{cases} \] where \(\lambda,\mu,\beta,\gamma:[0,+\infty)\longrightarrow (0,+\infty)\) are continuous and \(T\)-periodic functions. Specifically, we discuss criteria of local stability for a \(T\)-periodic endemic solutions and the existence of unstable solutions. In addition, we provide sufficient conditions for the existence of chaotic dynamics. The main tools employed in our analysis are some criteria of local stability developed by Rafael Ortega and the method of stretching along the paths developed by Fabio Zanolin and co-workers.


• Elisa Sovrano (University of Modena e Reggio Emilia)
  A journey on Ambrosetti-Prodi problems and chaotic dynamics.
  This talk journeys through Ambrosetti—Prodi's periodic problems, inspired by pioneering work [Ambrosetti and Prodi, Ann. Mat. Pura Appl., 1972] and intriguing notes [Ambrosetti, Atti Accad. Naz. Lincei, 2011]. We use topological methods to investigate chaotic dynamics and to relax uniformity assumptions in favor of local coercivity conditions. We also expand our study to include φ-Laplacian operators. These results stem from collaborative research with Fabio Zanolin and Guglielmo Feltrin (University of Udine).


• Andrea Tellini (Polytechnic University of Madrid)
  High multiplicity of positive solutions for superlinear indefinite problems: a review
  In this talk, I will review some results on high multiplicity of positive solutions for superlinear indefinite elliptic problems, starting from the ones obtained with Julián López-Gómez and Fabio Zanolin in CPAA (2013). These results cover several works, in collaboration with Guglielmo Feltrin (University of Udine), Julian López-Gómez (Complutense University of Madrid), Marcela Molina-Meyer (University Carlos III of Madrid), Elisa Sovrano (University of Modena e Reggio Emilia) and Fabio Zanolin.


• Pedro J. Torres (University of Granada)
  Periodic solutions of discontinuous second order differential equations and the porpoising effect
  We construct a mathematical model in order to study the so called porpoising effect in racing cars, and prove that, when adding a small periodic perturbation, large-amplitude subharmonic solutions may arise. This is a work in collaboration with Alessandro Fonda (University of Trieste).


• Gabriele Villari (University of Firenze)
  Seventeen shades of Liénard: 40 years of friendship and mathematical work with my friend Fabio
  Seventeen papers treating the Liénard equation and written together with Fabio Zanolin are shortly presented. However, this is not really a mathematical talk, but instead the story of a joint collaboration which started a long time ago and it is still going on. For this reason, some nice memories are also presented. _ _