Periodic solutions of the Brillouin electron beam focusing equation
Maurizio Garrione Manuel Zamora
Quite unexpectedly with respect to the numerical and analytical results found in literature, we establish a new range for the real parameter $b$ for which the existence of $2\pi-$periodic solutions of the Brillouin focusing beam equation \begin{eqnarray} \ddot{x}+b(1+\cos t)x=\frac{1}{x} \end{eqnarray} is guaranteed. This is possible thanks to suitable nonresonance conditions acting on the rotation number of the solutions in the phase plane.
keywords: Brillouin focusing system Poincaré-Bohl theorem singular nonlinearities periodic solutions
Positive solutions to indefinite Neumann problems when the weight has positive average
Alberto Boscaggin Maurizio Garrione
We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$ u'' + q(t)g(u) = 0, \quad t \in [0, T], $$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(u) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$ x' = y, \qquad y' = h(x)y^2 + q(t), $$ with $h(x)$ a continuous function defined on the whole real line.
keywords: average condition Neumann problem Indefinite weight shooting method.

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