Guglielmo Feltrin

Guglielmo Feltrin
University of Udine | UNIUD · Department of Mathematics, Computer Science and Physics

PhD

About

71
Publications
4,392
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424
Citations
Additional affiliations
May 2019 - April 2022
University of Udine
Position
  • Professor (Assistant)
Description
  • Ricercatore a tempo determinato ai sensi dell'art. 24 comma 3 lettera b) Legge n. 240/2010.
December 2017 - September 2018
University of Turin
Position
  • PostDoc Position
October 2017 - November 2018
University of Mons
Position
  • PostDoc Position
Education
October 2012 - September 2016
International School for Advanced Studies
Field of study
  • Mathematical Analysis
October 2010 - July 2012
Università degli Studi di Udine
Field of study
  • Mathematics
September 2007 - October 2010
Università degli Studi di Udine
Field of study
  • Mathematics

Publications

Publications (71)
Article
Full-text available
The paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{alig...
Preprint
This paper solves in a positive manner a conjecture stated in 2000 by R. G\'omez-Re\~nasco and J. L\'opez-G\'omez regarding the multiplicity of positive solutions of a paradigmatic class of superlinear indefinite boundary value problems.
Article
Full-text available
We provide a new version of the Poincaré–Birkhoff theorem for possibly multivalued successor maps associated with planar non-autonomous Hamiltonian systems. As an application, we prove the existence of periodic and subharmonic solutions of the scalar second order equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usep...
Preprint
We provide a new version of the Poincar\'e-Birkhoff theorem for possibly multivalued successor maps associated with planar non-autonomous Hamiltonian systems. As an application, we prove the existence of periodic and subharmonic solutions of the scalar second order equation $\ddot x + \lambda g(t,x) = 0$, for $\lambda>0$ sufficiently small, with $g...
Preprint
Full-text available
In this paper, we study the $T$-periodic solutions of the parameter-dependent $\phi$-Laplacian equation \begin{equation*} (\phi(x'))'=F(\lambda,t,x,x'). \end{equation*} Based on the topological degree theory, we present some atypical bifurcation results in the sense of Prodi--Ambrosetti, i.e., bifurcation of $T$-periodic solutions from $\lambda=0$....
Chapter
We consider two different relativistic versions of the Kepler problem in the plane: The first one involves the relativistic differential operator, and the second one involves a correction for the usual gravitational potential due to Levi-Civita. When a small external perturbation is added into such equations, we investigate the existence of periodi...
Preprint
Full-text available
The paper studies the existence of periodic solutions of a perturbed relativistic Kepler problem of the type \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\frac{m\dot{x}}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = -\alpha\frac{x}{|x|^{3}} + \varepsilon \, \nabla_{x} U(t,x), \qquad x \in \mathbb{R}^d\setminus\{0\}, \end{equation*} with $d=2$ or...
Preprint
Full-text available
We investigate bifurcation of closed orbits with a fixed energy level for a class of nearly integrable Hamiltonian systems with two degrees of freedom. More precisely, we make a joint use of Moser invariant curve theorem and Poincaré-Birkhoff fixed point theorem to prove that a periodic non-degenerate invariant torus T of the unperturbed problem gi...
Preprint
We consider two different relativistic versions of the Kepler problem in the plane: the first one involves the relativistic differential operator, the second one involves a correction for the usual gravitational potential due to Levi-Civita. When a small external perturbation is added into such equations, we investigate the existence of periodic so...
Preprint
Full-text available
We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - \lambda a(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the second variable, $\lambda>0$, $a$ is a sign-changing $T$-periodic weight function and $g$ is superlinear. Bas...
Article
Full-text available
We show the direct applicability of the Brouwer fixed point theorem for the existence of equilibrium points and periodic solutions for differential systems on general domains satisfying geometric conditions at the boundary. We develop a general approach for arbitrary bound sets and present applications to the case of convex and star-shaped domains....
Article
We investigate the existence of positive solutions for a class of Minkowski-curvature equations with indefinite weight and nonlinear term having superlinear growth at zero and super-exponential growth at infinity. As an example, for the equation [Formula: see text] where [Formula: see text] and [Formula: see text] is a sign-changing function satisf...
Preprint
Full-text available
We show the direct applicability of the Brouwer fixed point theorem for the existence of equilibrium points and periodic solutions for differential systems on general domains satisfying geometric conditions at the boundary. We develop a general approach for arbitrary bound sets and present applications to the case of convex and star-shaped domains....
Preprint
Full-text available
We consider a perturbation of a central force problem of the form \begin{equation*} \ddot x = V'(|x|) \frac{x}{|x|} + \varepsilon \,\nabla_x U(t,x), \quad x \in \mathbb{R}^{2} \setminus \{0\}, \end{equation*} where $\varepsilon \in \mathbb{R}$ is a small parameter, $V\colon (0,+\infty) \to \mathbb{R}$ and $U\colon \mathbb{R} \times (\mathbb{R}^{2}...
Article
We provide an extension of the Hartman–Knobloch theorem for periodic solutions of vector differential systems to a general class of ϕ-Laplacian differential operators. Our main tool is a variant of the Manásevich–Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. Our results concern t...
Article
Full-text available
This paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the ϕ-Laplacian equation (ϕ(u′))′+a(t)g(u)=0,(\phi \left(u^{\prime} ))^{\prime} +a\left(t)g\left(u)=0, where ϕ is a homeomorphism with ϕ(0) = 0, a(t) is a stepwise indefinite weight and g(u) is a continuous function. Wh...
Preprint
Full-text available
We study the second-order boundary value problem \begin{equation*} \begin{cases} \, -u''=a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\ \, u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{\lambda,\mu}$ is a step-wise indefinite weight function, precisely $a_{\lambda,\mu}\equiv\lambda$ in $[0,\sigma]\cup[1-\sigma,1]$ and $a_{\lambda,...
Article
We study the second-order boundary value problem \begin{document}$ \begin{equation*} \begin{cases}\, -u'' = a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\\, u'(0) = 0, \quad u'(1) = 0,\end{cases} \end{equation*} $\end{document} where \begin{document}$ a_{\lambda,\mu} $\end{document} is a step-wise indefinite weight function, precisely \begin{doc...
Preprint
Full-text available
We provide an extension of the Hartman-Knobloch theorem for periodic solutions of vector differential systems to a general class of $\phi$-Laplacian differential operators. Our main tool is a variant of the Man\'{a}sevich-Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. Our results...
Preprint
Full-text available
The paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the $\phi$-Laplacian equation \begin{equation*} \bigl{(} \phi(u') \bigr{)}' + a(t) g(u) = 0, \end{equation*} where $\phi$ is a homeomorphism with $\phi(0)=0$, $a(t)$ is a stepwise indefinite weight and $g(u)$ is a contin...
Article
We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of the Neumann boundary value problem p1′′+...
Article
We prove the existence of a pair of positive radial solutions for the Neumann boundary value problem div(∇u1−|∇u|2)+λa(|x|)up=0,in B,∂νu=0,on ∂B,where B is a ball centered at the origin, a(|x|) is a radial sign-changing function with ∫Ba(|x|)dx<0, p>1 and λ>0 is a large parameter. The proof is based on the Leray–Schauder degree theory and extends t...
Preprint
Full-text available
We investigate the existence of positive solutions for a class of Minkowski-curvature equations with indefinite weight and nonlinear term having superlinear growth at zero and super-exponential growth at infinity. As an example, for the equation \begin{equation*} \Biggl{(} \dfrac{u'}{\sqrt{1-(u')^{2}}}\Biggr{)}' + a(t) \bigl{(}e^{u^{p}}-1\bigr{)} =...
Article
Full-text available
We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation u^{\prime\prime}+cu^{\prime}+\bigl{(}\lambda a^{+}(x)-\mu a^{-}(x)\bigr{)}g(u)% =0, where {\lambda,\mu>0} are parameters, {c\in\mathbb{R}} , {a(x)} is a locally integrable P -periodic sign-changing weight function, a...
Article
Full-text available
We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., T-periodic) and subharmonic (i.e., kT-periodic for some integer k≥2) to the equation(u′1−(u′)2)′+λa(t)g(u)=0, where λ>0 is a parameter, a(t) is a T-periodic sign-changing weight function and g:[0,+∞[→[0,+∞[ is a continuous function having...
Preprint
Full-text available
We consider a perturbed relativistic Kepler problem \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\dfrac{m\dot{x}}{\sqrt{1-|\dot{x}|^2/c^2}}\right)=-\alpha\, \dfrac{x}{|x|^3}+\varepsilon \, \nabla_x U(t,x), \qquad x \in \mathbb{R}^2 \setminus \{0\}, \end{equation*} where $m, \alpha > 0$, $c$ is the speed of light and $U(t,x)$ is a function...
Preprint
Full-text available
We prove the existence of a pair of positive radial solutions for the Neumann boundary value problem \begin{equation*} \begin{cases} \, \mathrm{div}\,\Biggl{(} \dfrac{\nabla u}{\sqrt{1- | \nabla u |^{2}}}\Biggr{)} + \lambda a(|x|)u^p = 0, & \text{in $B$,} \\ \, \partial_{\nu}u=0, & \text{on $\partial B$,} \end{cases} \end{equation*} where $B$ is a...
Article
Full-text available
We study the periodic boundary value problem associated with the \(\phi \)-Laplacian equation of the form \((\phi (u'))'+f(u)u'+g(t,u)=s\), where s is a real parameter, f and g are continuous functions, and g is T-periodic in the variable t. The interest is in Ambrosetti–Prodi type alternatives which provide the existence of zero, one or two soluti...
Preprint
Full-text available
We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of the Neumann boundary value problem \begi...
Preprint
Full-text available
We deal with the periodic boundary value problem associated with the parameter-dependent second-order nonlinear differential equation \begin{equation*} u'' + cu' + \bigr{(} \lambda a^{+}(x) - \mu a^{-}(x) \bigr{)} g(u) = 0, \end{equation*} where $\lambda,\mu>0$ are parameters, $c\in\mathbb{R}$, $a(x)$ is a locally integrable $P$-periodic sign-chang...
Preprint
Full-text available
We prove the existence of parabolic arcs with prescribed asymptotic direction for the equation \begin{equation*} \ddot{x} = - \dfrac{x}{\lvert x \rvert^{3}} + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$ and $W$ is a (possibly time-dependent) lower order term, for $\vert x \vert \to +\infty$, with respect to the Kepl...
Chapter
This appendix is devoted to the coincidence degree. First we recall the classical coincidence degree theory introduced by J. Mawhin for open and bounded sets in a normed linear space.
Chapter
In this chapter we continue the study of the Neumann and periodic boundary value problems associated with indefinite equations, introduced in Chapter 3, dealing with multiplicity results for positive solutions.
Chapter
In this chapter we continue the investigation initiated in Chapter 6 with the aim to provide some multiplicity results for positive solutions to Dirichlet, Neumann and periodic boundary value problems associated with the second-order nonlinear differential equation
Chapter
In this chapter we deal with boundary value problems associated with the nonlinear second-order ordinary differential equation
Chapter
In this chapter we continue the discussion of the previous chapter for the supersublinear indefinite equation
Chapter
In this chapter we study the problem of existence and multiplicity of positive solutions for the nonlinear two-point boundary value problem
Chapter
In the present chapter we study the second-order nonlinear boundary value problem
Chapter
In this final chapter, we collect some recent results that take into consideration classes of indefinite problems which we have not discussed in the present manuscript. Moreover, we illustrate some interesting uninvestigated issues that would be the subject of future researches. Our purpose is to describe a panorama of topics related to the discuss...
Chapter
In this chapter we deal with subharmonic solutions for superlinear indefinite equations.
Chapter
This chapter is devoted to some further investigations on the nonlinear secondorder differential equation
Chapter
In this appendix we present a general version of the Leray–Schauder topological degree for locally compact operators on open possibly unbounded sets in a normed linear space.
Chapter
This appendix is devoted to some technical results which constitute important tools for the discussion in the present manuscript.
Article
Full-text available
Reaction-diffusion equations have several applications in the field of population dynamics and some of them are characterized by the presence of a factor which describes different types of food sources in a heterogeneous habitat. In this context, to study persistence or extinction of populations it is relevant the search of nontrivial steady states...
Preprint
Full-text available
We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., $T$-periodic) and subharmonic (i.e., $kT$-periodic for some integer $k \geq 2$) to the equation \begin{equation*} \Biggl{(} \dfrac{u'}{\sqrt{1-(u')^{2}}} \Biggr{)}' + \lambda a(t) g(u) = 0, \end{equation*} where $\lambda > 0$ is a paramete...
Article
Full-text available
We study the periodic boundary value problem associated with the $\phi$-Laplacian equation of the form $(\phi(u'))'+f(u)u'+g(t,u)=s$, where $s$ is a real parameter, $f$ and $g$ are continuous functions and $g$ is $T$-periodic in the variable $t$. The interest is in Ambrosetti--Prodi type alternatives which provide the existence of zero, one or two...
Preprint
Reaction-diffusion equations have several applications in the field of population dynamics and some of them are characterized by the presence of a factor which describes different types of food sources in a heterogeneous habitat. In this context, to study persistence or extinction of populations it is relevant the search of nontrivial steady states...
Article
Full-text available
We deal with the Neumann boundary value problem \begin{equation*} \begin{cases} \, u" + \bigl{(} \lambda a^{+}(t)-\mu a^{-}(t) \bigr{)}g(u) = 0, \\ \, 0 < u(t) < 1, \quad \forall\, t\in\mathopen{[}0,T\mathclose{]},\\ \, u'(0) = u'(T) = 0, \end{cases} \end{equation*} where the weight term has two positive humps separated by a negative one and $g\col...
Preprint
We deal with the Neumann boundary value problem \begin{equation*} \begin{cases} \, u" + \bigl{(} \lambda a^{+}(t)-\mu a^{-}(t) \bigr{)}g(u) = 0, \\ \, 0 < u(t) < 1, \quad \forall\, t\in\mathopen{[}0,T\mathclose{]},\\ \, u'(0) = u'(T) = 0, \end{cases} \end{equation*} where the weight term has two positive humps separated by a negative one and $g\col...
Article
Full-text available
We study the positive subharmonic solutions to the second order nonlinear ordinary differential equation \begin{equation*} u'' + q(t) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth both at zero and at infinity, and $q(t)$ is a $T$-periodic sign-changing weight. Under the sharp mean value condition $\int_{0}^{T} q(t) ~\!dt < 0$, combi...
Article
Full-text available
Using Mawhin's coincidence degree theory, we obtain some new continuation theorems which are designed to have as a natural application the study of the periodic problem for cyclic feedback type systems. We also discuss some examples of vector ordinary differential equations with a $\phi$-Laplacian operator where our results can be applied. Our main...
Preprint
Using Mawhin's coincidence degree theory, we obtain some new continuation theorems which are designed to have as a natural application the study of the periodic problem for cyclic feedback type systems. We also discuss some examples of vector ordinary differential equations with a $\phi$-Laplacian operator where our results can be applied.
Thesis
The present Ph.D. thesis is devoted to the study of positive solutions to indefinite problems. In particular, we deal with the second order nonlinear differential equation u'' + a(t) g(u) = 0, where g : [0,+∞[→[0,+∞[ is a continuous nonlinearity and a : [0,T]→R is a Lebesgue integrable sign-changing weight. We analyze the Dirichlet, Neumann and per...
Article
Full-text available
We study the second order nonlinear differential equation \begin{equation*} u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j} b_{j}(x)k_{j}(u) = 0, \end{equation*} where $\alpha_{i},\beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0,L\mathclose{]}$, and the nonlinear...
Preprint
We study the second order nonlinear differential equation \begin{equation*} u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j} b_{j}(x)k_{j}(u) = 0, \end{equation*} where $\alpha_{i},\beta_{j}>0$, $a_{i}(x), b_{j}(x)$ are non-negative Lebesgue integrable functions defined in $\mathopen{[}0,L\mathclose{]}$, and the nonlinear...
Article
We prove that the superlinear indefinite equation \begin{equation*} u" + a(t)u^{p} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a $T$-periodic sign-changing function satisfying the (sharp) mean value condition $\int_{0}^{T} a(t)~\!dt < 0$, has positive subharmonic solutions of order $k$ for any large integer $k$, thus providing a further contri...
Preprint
We prove that the superlinear indefinite equation \begin{equation*} u" + a(t)u^{p} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a $T$-periodic sign-changing function satisfying the (sharp) mean value condition $\int_{0}^{T} a(t)~\!dt < 0$, has positive subharmonic solutions of order $k$ for any large integer $k$, thus providing a further contri...
Article
Full-text available
We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( \lambda a^{+}(t) - \mu a^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $\lambda, \mu$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-p...
Preprint
We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation $u''+f(x,u)=0$. We allow $x \mapsto f(x,s)$ to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that $f(x,s)/s$ is below $\lambd...
Article
We study the periodic boundary value problem associated with the second order nonlinear differential equation $$ u" + c u' + \left(a^{+}(t) - \mu \, a^{-}(t)\right) g(u) = 0, $$ where $g(u)$ has superlinear growth at zero and at infinity, $a(t)$ is a periodic sign-changing weight, $c\in\mathbb{R}$ and $\mu>0$ is a real parameter. We prove the exist...
Article
Full-text available
We present a fixed point theorem on topological cylinders in normed linear spaces for maps satisfying a property of stretching a space along paths. This result is a generalization of a similar theorem obtained by D. Papini and F. Zanolin. In view of the main result, we discuss the existence of fixed points for maps defined on different types of dom...
Article
Full-text available
We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon \mathopen{[}0,+\infty\mathclose{[}\to \mathopen{[}0,+\infty\mathclose{[}$ is a sublinear function at infinity having superlinear growth a...
Article
Full-text available
We prove the existence of positive periodic solutions for the second order nonlinear equation $u" + a(x) g(u) = 0$, where $g(u)$ has superlinear growth at zero and at infinity. The weight function $a(x)$ is allowed to change its sign. Necessary and sufficient conditions for the existence of nontrivial solutions are obtained. The proof is based on M...
Article
Full-text available
We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change its sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth cond...
Article
We study the multiplicity of positive solutions for a two-point boundary value problem associated to the nonlinear second order equation $u''+f(x,u)=0$. We allow $x \mapsto f(x,s)$ to change its sign in order to cover the case of scalar equations with indefinite weight. Roughly speaking, our main assumptions require that $f(x,s)/s$ is below $\lambd...

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