The solutions of forced pendulum equation with small damping (Q2780920)

Consider the differential equation NEWLINE\[NEWLINE\ddot x+\varepsilon\dot x+ f(x)=\varepsilon g(t,x,\varepsilon),\tag{\(*\)}NEWLINE\]NEWLINE where \(f: \mathbb{R}\to \mathbb{R}\) and \(g: \mathbb{R}\times\mathbb{R}\times (0,1)\to\mathbb{R}\) are continuous, \(g\), is periodic in \(t\) with period \(T_\varepsilon\) satisfying NEWLINE\[NEWLINE\lim_{\varepsilon\to 0}\, T_\varepsilon=+\infty,\quad \lim_{\varepsilon\to 0}\,\varepsilon T_\varepsilon= 0,NEWLINE\]NEWLINE and is uniformly bounded. The author derives conditions on \(f\) such that to any given positive integers \(M\) and \(N\) equation \((*)\) has for each \(k= 1,\dots, M\) at least \(N\) periodic solutions with period \(kT_\varepsilon\) for sufficiently small \(\varepsilon\).