We consider the quasi-periodic Fermi–Ulam ping-pong model with no diophantine condition on the frequencies and show that typically the set of initial data which leads to escaping orbits has Lebesgue measure zero.

The theory of twist maps is applied to prove the existence of many harmonic and sub-harmonic solutions for certain Newtonian systems of differential equations. The method of proof leads to very precise information on the oscillatory properties of these solutions.

Let h: 2 → 2 be a dissipative and orientation preserving homeomorphism having an asymptotically stable fixed point. Let U be the region of attraction and assume that it is proper and unbounded. Using Carathéodory's prime ends theory one can associate a rotation number, ρ, to h|U. We prove that any map in the above conditions and with ρ ∉ induces a Denjoy homeomorphism in the circle of prime ends. We also present some explicit examples of maps in this class and we show that, if the infinity point is accessible by an arc in U, ρ ∉ if and only if Per(h) = Fix(h) = {p}.

This study reports the environmental conditions of temperature, moisture and radiation for four years (May 2004 to July 2008) in the area known as Pampas de La Joya in southern Peru, which recently has been considered as a new Mars analogue. The period of evaluation includes the El Niño Southern Oscillation (ENSO) during the months of September 2006 to March 2007, which, despite not having catastrophic effects like its predecessor on 1997–1998, showed an interesting increase in humidity. Our data describe the extreme conditions present in the region and their relationship with the presence of potential habitats that could allow for the survival of micro-organisms. The average environmental temperature was 18.9°C, with a maximum of 35.9°C and a minimum of −4.5°C. The annual average incident solar radiation was 508 W m−2, with high near 1060 W m−2 at noon during the driest period between September and March. The average relative humidity (RH) was 29.5, 20.1 and 20.4% for air, soil and rock, respectively. The RH had higher values at night due to fog during the months of June and August, and during the early morning between December and March. During the months of ENSO event there were four episodes of precipitation (1.1, 1.5, 2.0 and 0.9 mm), of which three increased soil and rock moisture on an average more than 45% and persisted for over 15 days after precipitation, while the atmospheric environment had no significant variations. Finally, quartz rocks and evaporite minerals colonized with micro-organisms were found as the only micro-habitats, in this region, capable of supporting life in this extreme environment.

In this article we consider twist maps that are non-periodic (and hence are defined on the plane rather than on the cylinder) and have small twist at infinity. Under natural assumptions the existence of infinitely many bounded orbits is established, and furthermore it is proved that unbounded orbits follow bounded orbits for long times. An application is given to the Fermi–Ulam ping-pong model with a non-periodic moving wall.

Twist maps (θ1,r1)=f(θ,r) are considered in this paper, with no assumption on the periodicity of the map in θ. Under appropriate assumptions, the existence of infinitely many bounded (in r) complete orbits is proven. In particular, our results apply to the class of maps where λ>0 and no arithmetic condition has to be imposed on ω1/ω2.

In this paper we will show that the optimal bounds for certain static and dynamic bifurcation values of periodic solutions of some superlinear differential equations can be expressed explicitly using Sobolev constants.

We prove that, for every bounded and measurable forcing $p(t)$, the differential equation $\ddot{u}+u^{1/3} =p(t)$ has bounded solutions with arbitrarily large amplitude. In general it is not possible to say that all solutions are bounded, as shown by an example due to Littlewood.

The proof is based on a variational method which can be seen as a dual version of Nehari's method for boundary value problems on compact intervals.

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.

Results of the Landesman–Lazer type provide necessary and sufficient conditions for the existence of periodic solutions of certain nonlinear differential equations with forcing. Typically, they deal with scalar problems. This paper presents a discussion of possible extensions to systems. The emphasis is placed on the new phenomena produced by the increase of the dimension.

Consider the differential equation

$$\ddot{x} +n^2 x+h_L (x) =p(t),$$

where $n=1,2,\dots$ is an integer, $p$ is a $2\pi$-periodic function and $h_L$ is the piecewise linear function

$$ h_L (x)=\begin{cases} L & \text{if $x\geq 1$},\\

Lx & \text{if $|x|\leq 1$},\\

-L & \text{if $x\leq -1$}.\end{cases}$$

A classical result of Lazer and Leach implies that this equation has a $2\pi$-periodic solution if and only if

\begin{equation}\label{ll} |\hat{p}_n |<{2L\over \pi}, \end{equation}

where

$$\hat{p}_n :={1\over 2\pi}\int_0^{2\pi} p(t)e^{-int}\, dt.$$

In this paper I prove that if $p$ is of class $C^5$ then the condition (\ref{ll}) is also necessary and sufficient for the boundedness of all the solutions of the equation.

The proof of this theorem motivates a new variant of Moser's Small Twist Theorem. This variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle.

1991 Mathematics Subject Classification: 34C11, 58F35.

Let $F:{\Bbb R}^2 \to {\Bbb R}^2$ be a mapping that is analytic and area preserving. If $F\neq \hbox{\it identity}$, then every stable fixed point is isolated.

This result can be applied to prove that the number of stable periodic solutions of a fixed period of certain Hamiltonian systems is finite.

We consider two boundary value problems (of Neumann or related type) associated with the equation in Ω. The existence of a solution was previously established assuming that p < N/(N −s2). (N dimension of Ω.) We prove that this exponent is critical for these problems, at least in the radially symmetric case when Ω is a ball. This is understood in the sense that the existence result does not hold when p ≧ N/(N − 2).