Since Littlewood works in the 1960's, the boundedness of solutions of
Duffing-type equations x¨+g(x)=p(t) has been extensively investigated.
More recently, some researches have focused on the family of non-smooth forced
oscillators x¨+sgn(x)=p(t), mainly because it represents a
simple limit scenario of Duffing-type equations for when g is bounded. Here,
we provide a simple proof for the boundedness of solutions of the non-smooth
forced oscillator in the case that the forcing term p(t) is a T-periodic
Lebesgue integrable function with vanishing average. We reach this result by
constructing a sequence of invariant tori whose union of their interiors covers
all the (t,x,x˙)-space, (t,x,x˙)∈S1×R2