Alessandro Fonda

Periodic oscillations for a nonlinear suspension bridge model

Abstract We look for time-periodic solutions of the suspension bridge equation. Lazer and McKenna showed that for a certain configuration of the parameters, one may expect the existence of large-amplitude periodic solutions having the same period as the forcing term. We prove the existence of large-scale subharmonic solutions.

On the use of time-maps for the solvability of nonlinear boundary value problems

(T > 0 is a fixed positive constant). The study of the periodic problem for equation (1.1) (or for some of its generalizations) represents a central subject in the qualitative theory of ordinary differential equations and it has been widely developed by the introduction of powerful tools from nonlinear functional analysis. See e.g. [22, 11, 9, 8, 16] and the references therein, for a source of various different techniques which can be used for this purpose. A classical method to deal with problem (1.1)-(1.2) consists into the search of fixed points of the translation operator (Poincar~-Andronov map) ~ : (x o, Yo) ~ (x (T; Xo, Yo), y (T; xo, Yo)) associated to the equivalent planar system

Periodic solutions of nonlinear differential equations with double resonance

SummaryWe prove the existence of periodic solutions of a second order nonlinear ordinary differential equation whose nonlinearity is at resonance with two successive eigenvalues of the associated linear operator and satisfies some Landesman-Laser type conditions at both of them.

Subharmonic oscillations of forced pendulum-type equations

where f is periodic with minimal period T and mean value zero. We have in mind as a particular case the pendulum equation, where g(x) = A sin x. First results on the existence of subharmonic orbits in a neighborhood of a given periodic motion were obtained by Birkhoff and Lewis (cf. [3] and [ 143) by perturbation-type techniques. Rabinowitz [ 151 was able to prove the existence of subharmonic solutions for Hamiltonian systems by the use of variational methods. His approach is not of local type like the one in [3], and enables one to obtain a sequence of solutions whose minimal period tends toward infinity in the case when the Hamiltonian function has subquadratic or superquadratic growth. These results have been extended in various directions, cf. [2, 5, 6, 8, 13, 16-181. Local results on subharmonies for the forced pendulum equation can be found in [19]. Hamiltonian systems with periodic nonlinearity were studied by Conley and Zehnder [6]. They proved the existence of subharmonic solutions under some assumptions on the nondegenerateness of the solutions, by the use of Morse-Conley theory. In this paper we will prove the existence of subharmonic oscillations of a pendulum-type equation by the use of classical Morse theory together with an iteration formula for the index due to Bott [4] and developed in [7] and [l].

Periodic solutions of radially symmetric perturbations of Newtonian systems

The classical Newton equation for the motion of a body in a gravitational central field is here modified in order to include periodic central forces. We prove that infinitely many periodic solutions still exist in this case. These solutions have periods which are large integer multiples of the period of the forcing, and rotate exactly once around the origin in their period time.

Periodic Orbits of Radially Symmetric Systems with a Singularity: the Repulsive Case

Abstract We study radially symmetric systems with a singularity of repulsive type. In the presence of a radially symmetric periodic forcing, we show the existence of three distinct families of subharmonic solutions: One oscillates radially, one rotates around the origin with small angular momentum, and the third one with both large angular momentum and large amplitude. The proofs are carried out by the use of topological degree theory.

Periodic Solutions of Pendulum-Like Hamiltonian Systems in the Plane

Abstract By the use of a generalized version of the Poincaré-Birkhoff fixed point theorem, we prove the existence of at least two periodic solutions for a class of Hamiltonian systems in the plane, having in mind the forced pendulum equation as a particular case. Our approach is closely related to the one used by Franks in [15], but the proof remains at a more elementary level.

Lower semicontinuous perturbations of maximal monotone differential inclusions

AbstractWe prove existence of solutions to1

A Landesman-Lazer-type condition for asymptotically linear second-order equations with a singularity

\dot x \in - Ax + F\left( {t,x} \right),x\left( a \right) = x^0 ,