On a class of forced vibration problems with obstacles

The main purpose of this paper is to study the following damped vibration problemsequation(1.1)−x¨=q(x)|x˙|2+g(t)x˙+f(t) withequation(1.2)x(0)−x(2π)=x˙(0)−x˙(2π)=0,x(t)⩾0,∀t∈R andequation(1.3)x˙(t0−)=−x˙(t0+),ifx(t0)=0, wherex˙(t0−)=limt→t0−0x˙(t),x˙(t0+)=limt→t0+0x˙(t). The variational sets are given and some multiplicity results of periodic solutions satisfying ,  and  are obtained.