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Pierluigi Benevieri

Publications and source records attributed to Pierluigi Benevieri.

10 recordsLinked to original sources
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.

math.CA↗
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.

math.CA↗
arXiv2023

An introduction to topological degree in Euclidean spaces

This paper aims to provide a careful and self-contained introduction to the theory of topological degree in Euclidean spaces. It is intended for people mostly interested in analysis and, in general, a heavy background in algebraic or differential topology is not required.

math.FA↗
arXiv2021

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces: the odd multiplicity case

We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds.

math.SP↗
arXiv2021

A degree associated to linear eigenvalue problems in Hilbert spaces and applications to nonlinear spectral theory

We extend to the infinite dimensional context the link between two completely different topics recently highlighted by the authors: the classical eigenvalue problem for real square matrices and the Brouwer degree for maps between oriented finite dimensional real manifolds. Thanks to this extension, we solve a conjecture regarding global continuation in nonlinear spectral theory that we have formulated in a recent article. Our result (the ex conjecture) is applied to prove a Rabinowitz type global continuation property of the solutions to a perturbed motion equation containing an air resistance frictional force.

math.SP↗
arXiv2019

Global persistence of the unit eigenvectors of perturbed eigenvalue problems in Hilbert spaces

We consider the nonlinear eigenvalue problem $Lx + \varepsilon N(x) = λCx$, $\|x\|=1$, where $\varepsilon,λ$ are real parameters, $L, C\colon G \to H$ are bounded linear operators between separable real Hilbert spaces, and $N\colon S \to H$ is a continuous map defined on the unit sphere of $G$. We prove a global persistence result regarding the set $Σ$ of the solutions $(x,\varepsilon,λ) \in S \times \mathbb R\times \mathbb R$ of this problem. Namely, if the operators $N$ and $C$ are compact, under suitable assumptions on a solution $p_*=(x_*,0,λ_*)$ of the unperturbed problem, we prove that the connected component of $Σ$ containing $p_*$ is either unbounded or meets a triple $p^*=(x^*,0,λ^*)$ with $p^* \not= p_*$. When $C$ is the identity and $G=H$ is finite dimensional, the assumptions on $(x_*,0,λ_*)$ mean that $x_*$ is an eigenvector of $L$ whose corresponding eigenvalue $λ_*$ is simple. Therefore, we extend a previous result obtained by the authors in the finite dimensional setting. Our work is inspired by a paper of R. Chiappinelli concerning the local persistence property of the unit eigenvectors of perturbed self-adjoint operators in a real Hilbert space.

math.SP↗
arXiv2019

The Brouwer degree associated to classical eigenvalue problems and applications to nonlinear spectral theory

Thanks to a connection between two completely different topics, the classical eigenvalue problem in a finite dimensional real vector space and the Brouwer degree for maps between oriented differentiable real manifolds, we were able to solve, at least in the finite dimensional context, a conjecture regarding global continuation in nonlinear spectral theory that we formulated in some recent papers. The infinite dimensional case seems nontrivial, and is still unsolved.

math.SP↗
arXiv2018

Eigenvalue problems for Fredholm operators with set-valued perturbations

By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

math.AP↗
arXiv2008

On a formula for the spectral flow and its applications

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index.

math.FA↗