The existence of periodic solutions of the quasilinear ordinary differential equation
egin{equation}
-Big(rac{ u'}{sqrt{1-u'^2}}Big)'=f(t,u,u'),
end{equation}
which models an oscillator with relativistic acceleration, is proven in the presence of a couple of lower and upper solutions satisfying or not satisfying the standard ordering condition. The proof is based on a simple trick which allows to reduce the quasilinear singular equation to a semilinear one.
