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Guglielmo Feltrin

Publications and source records attributed to Guglielmo Feltrin.

At least 19 recordsLinked to original sources
arXiv2026

Periodic orbits with prescribed negative energy for relativistic Keplerian problems

Using a variational approach, we study the existence of periodic solutions with prescribed energy for the relativistic equation \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t}\left(\dfrac{m\dot x}{\sqrt{1-|\dot{x}|^{2}/c^{2}}}\right) = -\alpha \frac{x}{|x|^{3}} + \nabla W(x), \qquad x\in\mathbb{R}^{N}\setminus\{0\}, \end{equation*} where $W$ is a lower-order perturbation of the Kepler potential. The main difficulty stems from the fact that the Kepler singularity is critical for the associated Maupertuis functional, lying exactly at the boundary between the weak force and strong force regimes. To overcome the resulting lack of compactness, we use a penalization procedure and develop a suitable min-max scheme combined with a blow-up analysis of near-collision critical sequences. As a consequence, we establish the existence of periodic solutions on prescribed negative energy levels, obtaining non-perturbative results in every dimension $N\geq 2$.

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arXiv2025

Continuation theorems for periodic systems and applications to problems with nonlinear time-dependent differential operators

In this paper we propose some continuation theorems for the periodic problem \begin{equation*} \begin{cases} \, x_{i}' = g_{i}(t,x_{i+1}), &i=1,\ldots,n-1, \\ \, x_{n}' = h(t,x_{1},\ldots,x_{n}), \\ \, x_{i}(0)=x_{i}(T), &i=1,\ldots,n, \end{cases} \end{equation*} providing a unified framework that improves and extends earlier contributions by Jean Mawhin and collaborators to second-order differential problems governed by nonlinear time-dependent differential operators of the form \begin{equation*} \begin{cases} \, (ϕ(t,x'))'=f(t,x,x'), \\ \, x(0)=x(T),\quad x'(0)=x'(T). \end{cases} \end{equation*} The proof is based on the topological degree theory.

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arXiv2025

Bifurcation from periodic solutions of central force problems in the three-dimensional space

The paper deals with electromagnetic perturbations of a central force problem of the form \begin{equation*} \dfrac{\mathrm{d}}{\mathrm{d}t} \bigl( φ(\dot{x}) \bigr) = V'(|x|) \dfrac{x}{|x|} + E_{\varepsilon}(t,x)+\dot{x} \wedge B_{\varepsilon}(t,x), \qquad x \in \mathbb{R}^3 \setminus \{0\}, \end{equation*} where $V \colon (0,+\infty) \to \mathbb{R}$ is a smooth function, $E_\varepsilon$ and $B_\varepsilon$ are respectively the electric field and the magnetic field, smooth and periodic in time, $\varepsilon\in\mathbb{R}$ is a small parameter. The considered differential operator includes, as special cases, the classical one, $φ(v)=mv$, as well as that of special relativity, $φ(v) = mv/\sqrt{1-\vert v \vert^2/c^2}$. We investigate whether non-circular periodic solutions of the unperturbed problem (i.e., with $\varepsilon=0$) can be continued into periodic solutions for $\varepsilon\neq0$ small, both for the fixed-period problem and, if the perturbation is time-independent, for the fixed-energy problem. The proof is based on an abstract bifurcation theorem of variational nature, which is applied to suitable Hamiltonian action functionals. In checking the required non-degeneracy conditions we take advantage of the existence of partial action-angle coordinates as provided by the Mishchenko--Fomenko theorem for superintegrable systems. Physically relevant problems to which our results can be applied are homogeneous central force problems in classical mechanics and the Kepler problem in special relativity.

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arXiv2025

Atypical bifurcation for periodic solutions of $ϕ$-Laplacian systems

In this paper, we study the $T$-periodic solutions of the parameter-dependent $ϕ$-Laplacian equation \begin{equation*} (ϕ(x'))'=F(λ,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 $λ=0$. Finally, we propose some applications to Liénard-type equations.

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arXiv2025

Uniqueness, non-degeneracy, and exact multiplicity of positive solutions for superlinear elliptic problems

In this paper, we focus our attention on the positive solutions to second-order nonlinear ordinary differential equations of the form $u''+q(t)g(u)=0$, where $q$ is a sign-changing weight and $g$ is a superlinear function. We exploit the classical shooting approach and the comparison theorem to present non-degeneracy and exact multiplicity results for positive solutions. This completes the multiplicity results obtained by Feltrin and Zanolin. Numerical examples and some related open problems are also discussed.

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arXiv2024

A Poincaré-Birkhoff theorem for multivalued successor maps with applications to periodic superlinear Hamiltonian systems

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 $\ddot x + λg(t,x) = 0$, for $λ>0$ sufficiently small, with $g(t,x)$ having a superlinear growth at infinity, without requiring the existence of an equilibrium point.

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arXiv2024

Nearly-circular periodic solutions of perturbed relativistic Kepler problems: the fixed-period and the fixed-energy problems

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) = -α\frac{x}{|x|^{3}} + \varepsilon \, \nabla_{x} U(t,x), \qquad x \in \mathbb{R}^d\setminus\{0\}, \end{equation*} with $d=2$ or $d=3$, bifurcating, for $\varepsilon$ small enough, from the set of circular solutions of the unperturbed system. Both the case of the fixed-period problem (assuming that $U$ is $T$-periodic in time) and the case of the fixed-energy problem (assuming that $U$ is independent of time) are considered.

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arXiv2023

Bifurcation of closed orbits of Hamiltonian systems with application to geodesics of the Schwarzschild metric

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 $\mathcal{T}$ of the unperturbed problem gives rise to infinitely many closed orbits, bifurcating from a family of tori accumulating onto $\mathcal{T}$. The required non-degeneracy condition, which is nothing but a reformulation of the usual non-degeneracy condition in the isoenergetic KAM theory, is expressed in terms of the derivative of the apsidal angle with respect to the angular momentum: in this way, tools from the theory of time-maps of nonlinear oscillators can be used to verify it in concrete problems. Applications are given to perturbations of central force problems in the plane, and to equatorial geodesic dynamics for perturbations of the Schwarzschild metric.

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arXiv2023

Homoclinic and heteroclinic solutions for non-autonomous Minkowski-curvature equations

We deal with the non-autonomous parameter-dependent second-order differential equation \begin{equation*} δ\left( \dfrac{v'}{\sqrt{1-(v')^{2}}} \right)' + q(t) f(v)= 0, \quad t\in\mathbb{R}, \end{equation*} driven by a Minkowski-curvature operator. Here, $δ>0$, $q\in L^{\infty}(\mathbb{R})$, $f\colon\mathopen{[}0,1\mathclose{]}\to\mathbb{R}$ is a continuous function with $f(0)=f(1)=0=f(α)$ for some $α\in \mathopen{]}0,1\mathclose{[}$, $f(s)<0$ for all $s\in\mathopen{]}0,α\mathclose{[}$ and $f(s)>0$ for all $s\in\mathopen{]}α,1\mathclose{[}$. Based on a careful phase-plane analysis, under suitable assumptions on $q$ we prove the existence of strictly increasing heteroclinic solutions and of homoclinic solutions with a unique change of monotonicity. Then, we analyze the asymptotic behaviour of such solutions both for $δ\to 0^{+}$ and for $δ\to+\infty$. Some numerical examples illustrate the stated results.

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arXiv2023

Prescribed energy periodic solutions of Kepler problems with relativistic corrections

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 solutions with prescribed energy bifurcating from periodic invariant tori of the unperturbed problems. Our main tool is an abstract bifurcation theory from periodic manifolds developed by Weinstein, which is applied in the case of nearly integrable Hamiltonian systems satisfying the usual KAM isoenergetic non-degeneracy condition.

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arXiv2022

Periodic solutions to superlinear indefinite planar systems: a topological degree approach

We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - λa(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the second variable, $λ>0$, $a$ is a sign-changing $T$-periodic weight function and $g$ is superlinear. Based on the coincidence degree theory, in dependence of $λ$, we prove the existence of $T$-periodic solutions $(u,v)$ such that $u(t)>0$ for all $t\in\mathbb{R}$. Our results generalize and unify previous contributions about Butler's problem on positive periodic solutions for second-order differential equations (involving linear or $ϕ$-Laplacian-type differential operators).

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arXiv2022

Equilibrium points, periodic solutions and the Brouwer fixed point theorem for convex and non-convex domains

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. We also provide an answer to a question raised in a recent paper of Cid and Mawhin.

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arXiv2021

Periodic perturbations of central force problems and an application to a restricted $3$-body problem

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 $U\colon \mathbb{R} \times (\mathbb{R}^{2} \setminus \{0\}) \to \mathbb{R}$ are smooth functions, and $U$ is $τ$-periodic in the first variable. Based on the introduction of suitable time-maps (the radial period and the apsidal angle) for the unperturbed problem ($\varepsilon=0$) and of an associated non-degeneracy condition, we apply an higher-dimensional version of the Poincaré-Birkhoff fixed point theorem to prove the existence of non-circular $τ$-periodic solutions bifurcating from invariant tori at $\varepsilon=0$. We then prove that this non-degeneracy condition is satisfied for some concrete examples of physical interest (including the homogeneous potential $V(r)=κ/r^α$ for $α\in(-\infty,2)\setminus\{-2,0,1\}$). Finally, an application is given to a restricted $3$-body problem with a non-Newtonian interaction.

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arXiv2021

On the number of positive solutions to an indefinite parameter-dependent Neumann problem

We study the second-order boundary value problem \begin{equation*} \begin{cases} \, -u''=a_{λ,μ}(t) \, u^{2}(1-u), & t\in(0,1), \\ \, u'(0)=0, \quad u'(1)=0, \end{cases} \end{equation*} where $a_{λ,μ}$ is a step-wise indefinite weight function, precisely $a_{λ,μ}\equivλ$ in $[0,σ]\cup[1-σ,1]$ and $a_{λ,μ}\equiv-μ$ in $(σ,1-σ)$, for some $σ\in\left(0,\frac{1}{2}\right)$, with $λ$ and $μ$ positive real parameters. We investigate the topological structure of the set of positive solutions which lie in $(0,1)$ as $λ$ and $μ$ vary. Depending on $λ$ and based on a phase-plane analysis and on time-mapping estimates, our findings lead to three different (from the topological point of view) global bifurcation diagrams of the solutions in terms of the parameter $μ$. Finally, for the first time in the literature, a qualitative bifurcation diagram concerning the number of solutions in the $(λ,μ)$-plane is depicted. The analyzed Neumann problem has an application in the analysis of stationary solutions to reaction-diffusion equations in population genetics driven by migration and selection.

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arXiv2020

Bound sets for a class of $ϕ$-Laplacian operators

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 the case of convex bound sets for which we show some new connections using a characterisation of sublevel sets due to Krantz and Parks. We also extend to the $ϕ$-Laplacian vector case a classical theorem of Reissig for scalar periodically perturbed Liénard equations.

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arXiv2020

Uniqueness of positive solutions for boundary value problems associated with indefinite $ϕ$-Laplacian type equations

The paper provides a uniqueness result for positive solutions of the Neumann and periodic boundary value problems associated with the $ϕ$-Laplacian equation \begin{equation*} \bigl{(} ϕ(u') \bigr{)}' + a(t) g(u) = 0, \end{equation*} where $ϕ$ is a homeomorphism with $ϕ(0)=0$, $a(t)$ is a stepwise indefinite weight and $g(u)$ is a continuous function. When dealing with the $p$-Laplacian differential operator $ϕ(s)=|s|^{p-2}s$ with $p>1$, and the nonlinear term $g(u)=u^γ$ with $γ\in\mathbb{R}$, we prove the existence of a unique positive solution when $γ\in\mathopen{]}-\infty,(1-2p)/(p-1)\mathclose{]} \cup \mathopen{]}p-1,+\infty\mathclose{[}$.

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arXiv2020

Parabolic orbits in Celestial Mechanics: a functional-analytic approach

We prove the existence of half-entire parabolic solutions, asymptotic to a prescribed central configuration, for the equation \begin{equation*} \ddot{x} = \nabla U(x) + \nabla W(t,x), \qquad x \in \mathbb{R}^{d}, \end{equation*} where $d \geq 2$, $U$ is a positive and positively homogeneous potential with homogeneity degree $-α$ with $α\in\mathopen{]}0,2\mathclose{[}$, and $W$ is a (possibly time-dependent) lower order term, for $\vert x \vert \to +\infty$, with respect to $U$. The proof relies on a perturbative argument, after an appropriate formulation of the problem in a suitable functional space. Applications to several problems of Celestial Mechanics (including the $N$-centre problem, the $N$-body problem and the restricted $(N+H)$-body problem) are given.

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