Unbounded Solutions of Asymmetric Oscillator

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation(φp(x′))+(p−1)[αφp(x+)−βφp(x−)]=(p−1)f(t,x,x′), whereφp(u)=|u|p−2u, p>1, x+=max{x,0}, x−=max{−x,0},α,βare positive constants, andfis continuous, bounded, andT-periodic intfor someT>0.

Nonlinear Resonance: a Comparison Between Landesman-Lazer and Ahmad-Lazer-Paul Conditions

We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

Exponential Growth Rates of Periodic Asymmetric Oscillators

In this paper we will study the dynamics of the periodic asymmetric oscillator xʺ + qdoes exist for each non-zero solution x(t) of the oscillator. The properties of these rates, or the Lyapunov exponents, will be given using the induced circle di®eomorphism of the oscillator. The proof is extensively based on the Denjoy theorem in topological dynamics and the unique ergodicity theorem in ergodic theory.