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Fabio Zanolin

Publications and source records attributed to Fabio Zanolin.

At least 19 recordsLinked to original sources
arXiv2025

Global multiplicity results in a Moore-Nehari type problem with a spectral parameter

This paper analyzes the structure of the set of positive solutions of a Moore-Nehari type problem, where $a\equiv a_h$ is a piece-wise constant function defined for some $h\in (0,1)$. In our analysis, $λ$ is regarded as a bifurcation parameter, whereas $h$ is viewed as a deformation parameter between the autonomous case when $a=1$ and the linear case when $a=0$. In this paper, besides establishing some of the multiplicity results suggested by previous numerical experiments (see Cubillos, López-Gómez and Tellini, 2024), we have analyzed the asymptotic behavior of the positive solutions of the problem as $h\uparrow 1$, when the shadow system of the problem is the linear equation $-u''=π^2 u$. This is the first paper where such a problem has been addressed. Numerics is of no help in analyzing this singular perturbation problem because the positive solutions blow-up point-wise in $(0,1)$ as $h\uparrow 1$ if $λ<π^2$.

math.CA↗
arXiv2024

Non-negative solutions of a sublinear elliptic problem

In this paper the existence of solutions, $(λ,u)$, of the problem $$-Δu=λu -a(x)|u|^{p-1}u \quad \hbox{in }Ω, \qquad u=0 \quad \hbox{on}\;\;\partialΩ,$$ is explored for $0 < p < 1$. When $p>1$, it is known that there is an unbounded component of such solutions bifurcating from $(σ_1, 0)$, where $σ_1$ is the smallest eigenvalue of $-Δ$ in $Ω$ under Dirichlet boundary conditions on $\partialΩ$. These solutions have $u \in P$, the interior of the positive cone. The continuation argument used when $p>1$ to keep $u \in P$ fails if $0 < p < 1$. Nevertheless when $0 < p < 1$, we are still able to show that there is a component of solutions bifurcating from $(σ_1, \infty)$, unbounded outside of a neighborhood of $(σ_1, \infty)$, and having $u \gneq 0$. This non-negativity for $u$ cannot be improved as is shown via a detailed analysis of the simplest autonomous one-dimensional version of the problem: its set of non-negative solutions possesses a countable set of components, each of them consisting of positive solutions with a fixed (arbitrary) number of bumps. Finally, the structure of these components is fully described.

math.AP↗
arXiv2023

Folding Domain Functions (FDF): a Random Variable Transformation technique for the non-invertible case, with applications to RDEs

The Random Variable Transformation (RVT) method is a fundamental tool for determining the probability distribution function associated with a Random Variable (RV) Y=g(X), where X is a RV and g is a suitable transformation. In the usual applications of this method, one has to evaluate the derivative of the inverse of g. This can be a straightforward procedure when g is invertible, while difficulties may arise when g is non-invertible. The RVT method has received a great deal of attention in the recent years, because of its crucial relevance in many applications. In the present work we introduce a new approach which allows to determine the probability density function of the RV Y=g(X), when g is non-invertible due to its non-bijective nature. The main interest of our approach is that it can be easily implemented, from the numerical point of view, but mostly because of its low computational cost, which makes it very competitive. As a proof of concept, we apply our method to some numerical examples related to random differential equations, as well as discrete mappings, all of them of interest in the domain of applied Physics.

math.PR↗
arXiv2022

Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments

This paper deals with the existence, multiplicity, minimal complexity and global structure of the subharmonic solutions to a class of planar Hamiltonian systems with periodic coefficients, being the classical predator-prey model of V. Volterra its most paradigmatic example. By means of a topological approach based on techniques from global bifurcation theory, the first part of the paper ascertains their nature, multiplicity and minimal complexity, as well as their global minimal structure, in terms of the configuration of the function coefficients in the setting of the model. The second part of the paper introduces a dynamical system approach based on the theory of topological horseshoes that permits to detect, besides subharmonic solutions, ``chaotic-type'' solutions. As a byproduct of our analysis, the simplest predator-prey prototype models in periodic environments can provoke chaotic dynamics. This cannot occur in cooperative and quasi-cooperative dynamics, as a consequence of the ordering imposed by the maximum principle.

q-bio.PE↗
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).

math.CA↗
arXiv2022

Rich dynamics in planar systems with heterogeneous nonnegative weights

This paper studies the global structure of the set of nodal solutions of a generalized Sturm--Liouville boundary value problem associated to the quasilinear equation $$ -(ϕ(u'))'= λu + a(t)g(u), \quad λ\in {\mathbb R}, $$ where $a(t)$ is non-negative with some positive humps separated away by intervals of degeneracy where $a\equiv 0$. When $ϕ(s)=s$ this equation includes a generalized prototype of a classical model going back to Moore and Nehari, 1959. This is the first paper where the general case when $λ\in\mathbb{R}$ has been addressed when $a\gneq 0$. The semilinear case with $a\lneq 0$ has been recently treated by López-Gómez and Rabinowitz.

math.CA↗
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.

math.CA↗
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.

math.AP↗
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 $γ\in\mathopen{]}-\infty,(1-2p)/(p-1)\mathclose{]} \cup \mathopen{]}p-1,+\infty\mathclose{[}$.

math.CA↗
arXiv2020

Positive solutions for a Minkowski-curvature equation with indefinite weight and super-exponential nonlinearity

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{)} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a sign-changing function satisfying the mean-value condition $\int_{0}^{T} a(t)\,\mathrm{d}t < 0$, we prove the existence of a positive solution for both periodic and Neumann boundary conditions. The proof relies on a topological degree technique.

math.CA↗
arXiv2019

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

In this paper we prove the existence of multiple periodic solutions (harmonic and subharmonic) for a class of planar Hamiltonian systems which include the case of the second order scalar ODE $x'' + a(t)g(x) = 0$ with $g$ satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. The case of complex dynamics is investigated, too.

math.DS↗
arXiv2018

Periodic solutions to parameter-dependent equations with a $ϕ$-Laplacian type operator

We study the periodic boundary value problem associated with the $ϕ$-Laplacian equation of the form $(ϕ(u'))'+f(u)u'+g(t,u)=s$, 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 solutions depending on the choice of the parameter $s$. We investigate this problem for a broad family of nonlinearities, under non-uniform type conditions on $g(t,u)$ as $u\to \pm\infty$. We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.

math.CA↗
arXiv2017

The Ambrosetti-Prodi periodic problem: Different routes to complex dynamics

We consider a second order nonlinear ordinary differential equation of the form $u'' + f(u) = p(t)$ where the forcing term $p(t)$ is a $T$-periodic function and the nonlinearity $f(u)$ satisfies the properties of Ambrosetti-Prodi problems. We discuss the existence of infinitely many periodic solutions as well as the presence of complex dynamics under different conditions on $p(t)$ and by using different kinds of approaches. On the one hand, we exploit the Melnikov's method and, on the other hand, the concept of "topological horseshoe".

math.DS↗
arXiv2015

Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems

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 Mawhin's coincidence degree and applies also to Neumann boundary conditions. Applications are given to the search of positive solutions for a nonlinear PDE in annular domains and for a periodic problem associated to a non-Hamiltonian equation.

math.CA↗
arXiv2015

Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree

We study the periodic boundary value problem associated with the second order nonlinear equation \begin{equation*} u'' + ( λa^{+}(t) - μa^{-}(t) ) g(u) = 0, \end{equation*} where $g(u)$ has superlinear growth at zero and sublinear growth at infinity. For $λ, μ$ positive and large, we prove the existence of $3^{m}-1$ positive $T$-periodic solutions when the weight function $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a $T$-periodicity interval). As a byproduct of our approach we also provide abundance of positive subharmonic solutions and symbolic dynamics. The proof is based on coincidence degree theory for locally compact operators on open unbounded sets and also applies to Neumann and Dirichlet boundary conditions. Finally, we deal with radially symmetric positive solutions for the Neumann and the Dirichlet problems associated with elliptic PDEs.

math.CA↗
arXiv2015

Multiple positive solutions for a superlinear problem: a topological approach

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 $λ_{1}$ as $s\to 0^{+}$ and above $λ_{1}$ as $s\to +\infty$. In particular, we can deal with the situation in which $f(x,s)$ has a superlinear growth at zero and at infinity. We propose a new approach based on the topological degree which provides the multiplicity of solutions. Applications are given for $u'' + a(x) g(u) = 0$, where we prove the existence of $2^{n}-1$ positive solutions when $a(x)$ has $n$ positive humps and $a^{-}(x)$ is sufficiently large.

math.CA↗
arXiv2015

Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree

We study the periodic boundary value problem associated with the second order nonlinear differential equation $$ u" + c u' + \left(a^{+}(t) - μ\, a^{-}(t)\right) g(u) = 0, $$ where $g(u)$ has superlinear growth at zero and at infinity, $a(t)$ is a periodic sign-changing weight, $c\in\mathbb{R}$ and $μ>0$ is a real parameter. We prove the existence of $2^{m}-1$ positive solutions when $a(t)$ has $m$ positive humps separated by $m$ negative ones (in a periodicity interval) and $μ$ is sufficiently large. The proof is based on the extension of Mawhin's coincidence degree defined in open (possibly unbounded) sets and applies also to Neumann boundary conditions. Our method also provides a topological approach to detect subharmonic solutions.

math.CA↗