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| We will deal with the scalar second order equation (known as Brillouin electron beam focusing equation)
\begin{equation}\label{Brillouin}
x'' + b(1+\cos t) x = \frac{1}{x},
\end{equation}
where $b$ is a positive constant.
A classical problem related to this equation is to determine the values of $b$ for which there exists a $2\pi$-periodic positive solution. This gave rise to an unproven conjecture, motivated by some numerical experiments, saying that this always happens if $b \in (0, 1/4)$. In this talk, we present a new range of values of $b$ for which the Brillouin equation has a $2\pi$-periodic solution, disjoint from the conjectured one $(0, 1/4)$. Precisely, we will prove $2\pi$-periodic solvability for $b \in [0.4705, 0.59165]$. The technique of proof relies on careful estimates of the rotation numbers of the solutions in the phase-plane, under suitable nonresonance assumptions which will be briefly commented. |
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