Loss of energy concentration in nonlinear evolution beam equations

Motivated by the oscillations that were seen at the Tacoma Narrows Bridge, we introduce the notion of solutions with a prevailing mode for the nonlinear evolution beam equation $$u_{tt} + u_{xxxx} + f(u)= g(x, t)$$ in bounded space-time intervals. We give a new definition of instability for these particular solutions, based on the loss of energy concentration on their prevailing mode. We distinguish between two different forms of energy transfer, one physiological (unavoidable and depending on the nonlinearity) and one due to the insurgence of instability. We then prove a theoretical result allowing to reduce the study of this kind of infinite-dimensional stability to that of a finite-dimensional approximation. With this background, we study the occurrence of instability for three different kinds of nonlinearities $f$ and for some forcing terms $g$, highlighting some of their structural properties and performing some numerical simulations

Existence and multiplicity of solutions to boundary value problems associated with nonlinear first order planar systems

The monograph is devoted to the study of nonlinear first order systems in the plane where the principal term is the gradient of a positive and positively 2-homogeneous Hamiltonian (or the convex combination of two of such gradients). After some preliminaries about positively 2-homogeneous autonomous systems, some results of existence and multiplicity of T-periodic solutions are presented in case of bounded or sublinear nonlinear perturbations. Our attention is mainly focused on the occurrence of resonance phenomena, and the corresponding results rely essentially on conditions of Landesman-Lazer or Ahmad-Lazer-Paul type. The techniques used are predominantly topological, exploiting the theory of coincidence degree and the use of the Poincar\ue9-Birkhoff fixed point theorem. At the end, other boundary conditions, including the Sturm-Liouville ones, are taken into account, giving the corresponding existence and multiplicity results in a nonresonant situation via the shooting method and topological arguments

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

We deal with the non-autonomous parameter-dependent second-order differential equation \begin{equation*} \delta \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, $\delta>0$, $q\in L^{\infty}(\mathbb{R})$, $f\colon\mathopen{[}0,1\mathclose{]}\to\mathbb{R}$ is a continuous function with $f(0)=f(1)=0=f(\alpha)$ for some $\alpha \in \mathopen{]}0,1\mathclose{[}$, $f(s)<0$ for all $s\in\mathopen{]}0,\alpha\mathclose{[}$ and $f(s)>0$ for all $s\in\mathopen{]}\alpha,1\mathclose{[}$. 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 $\delta \to 0^{+}$ and for $\delta\to+\infty$. Some numerical examples illustrate the stated results.Comment: 25 pages, 6 figure

Resonance and Landesman-Lazer conditions for first order systems in R^2

The first part of the paper surveys the concept of resonance for T-periodic nonlinear problems. In the second part, some new results about existence conditions for nonlinear planar systems are presented. In particular, the Landesman-Lazer conditions are generalized to systems in R^2 where the nonlinearity interacts with two resonant Hamiltonians. Such results apply to second order equations, generalizing previous theorems by Fabry [4] (for the undamped case), and Frederickson-Lazer [9] (forthe case with friction). The results have been obtained with A. Fonda, and have been published in [8]

A note on a nonresonance condition at zero for first-order planar systems

We introduce a Landesman-Lazer type nonresonance condition at zero for planar systems and discuss its rotational interpretation. We then show an application concerning multiplicity of T-periodic solutions to unforced Hamiltonian systems like (fourmula presented) for which the nonlinearity is resonant both at zero and at infinity, refining and complementing some recent results

An infinite-dimensional version of the Poincar\ue9-Birkhoff theorem on the Hilbert cube

We propose a version of the Poincar\ue9-Birkhoff theorem for infinite-dimensional Hamiltonian systems, which extends a recent result by Fonda and Ure\uf1a. The twist condition, adapted to a Hilbert cube, is spread on a sequence of approximating finite-dimensional systems. Some applications are proposed to pendulum-like systems of infinitely many ODEs. We also extend to the infinite-dimensional setting a celebrated theorem by Conley and Zehnder