Massimo Tarallo

Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction

A well known theorem says that the forced pendulum equation has periodic solutions if there is no friction and the external force has mean value zero. In this paper we show that this result cannot be extended to the case of linear friction.

Almost periodic upper and lower solutions

Abstract The method of upper and lower solutions is a classical tool in the theory of periodic differential equations of the second order. We show that this method does not have a direct extension to almost periodic equations. To do this we construct equations of this type without almost periodic solutions but having two constants as ordered upper and lower solutions.

On the existence of homoclinic solutions for almost periodic second order systems

Abstract In this paper we prove the existence of at least one homoclinic solution for a second order Lagrangian system, where the potential is an almost periodic function of time. This result generalizes existence theorems known to hold when the dependence on time of the potential is periodic. The method is of a variational nature, solutions being found as critical points of a suitable functional. The absence of a group of symmetries for which the functional is invariant (as in the case of periodic potentials) is replaced by the study of problems “at infinity” and a suitable use of a property introduced by E. Sere.

Almost periodic equations and conditions of Ambrosetti-Prodi type

We discuss the exact number of almost periodic solutions of certain ordinary differential equations of the second order. The class of equations under consideration is inspired by a well-known result in the area of elliptic boundary value problems.

Massera's theorem for quasi-periodic differential equations

For a scalar, first order ordinary differential equation which depends periodically on time, Masseras Theorem says that the existence of a bounded solution implies the existence of a periodic solution. Though the statement is false when periodicity is replaced by quasi-periodicity, solutions with some kind of recurrence are anyway expected when the equation is quasi-periodic in time. Indeed we first prove that the existence of a bounded solution implies the existence of a solution which is quasi-periodic in a weak sense. The partial differential equation, having our original equation as its equation of characteristics, plays a key role in the introduction of this notion of weak quasi-periodicity. Then we compare our approach with others already known in the literature. Finally, we give an explicit example of the weak case, and an extension to higher dimension for a special class of equations.

Taking advantage of petrostructural heterogeneities in subduction-collisional orogens, and effect on the scale of analysis

Since the beginning of the last century, tectonic history of polyphase metamorphic tectonites of orogenic basement complexes is often related to primary links with metasediments, of presumably known origin, and location of their original basins. However such history is worth to be compared with results of an alternative, independent investigation that pursues: i) an objective reconstruction of the evolutionary steps modifying the lithostratigraphic setting and of its deformation-metamorphism interactions during plate-scale events, and ii) a privileged reconstruction of the rock memory for the structural and metamorphic correlation of crystalline basement units. Interpretative merging of data gathered from these affine rock properties made interpretations of orogenic zones more actualistic and based on recognition of tectonic trajectories of units through evolving geodynamic contexts. In this account a refinement of the analytical approach to inferring deformation and metamorphic paths and constructing geological histories of basements in axial zones of orogenic belts is presented and examples are synthesized from the Western Alps and the Canadian Cordillera, based on detailed structural and lithostratigraphic mapping in harmony with macro- and micro- structural techniques of analysis, are reported from the two belts.

Large minimal period orbits of periodic autonomous systems

We prove the existence of periodic orbits with minimal period greater than any prescribed number for a natural Lagrangian autonomous system in several variables that is analytic and periodic in each variable and whose potential is nonconstant.

On the exponential decay for boundary layer problems

Abstract In connection with boundary layers problem, the possible decay of the solution to a Dirichlet problem on the half-space with almost-periodic boundary data is studied.

Multiple periodic solutions for problems at resonance with arbitrary eigenvalues

where Ω is a bounded open set in R , N ≥ 1, g is a bounded (Caratheodory) function and λn is the n–th eigenvalue of the Laplacian with Dirichlet boundary conditions. This and related problems (obtained by changing the boundary conditions), are called resonant and have been the object of much attention and study, as testified by the vast literature concerning the subject. The general aim of papers dealing with problem (1) is the understanding of the conditions on the function g or on the potential G(x, u) = ∫ u 0 g(x, s) ds which ensure existence of one or more solutions to the problem. In this framework, it has long been recognized that the behavior of