Unbounded solutions in asymmetric oscillations

In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator: x ″ + α x + − β x − = f ( t ) is discussed, where α, 3 are positive constants satisfying 1 α + 1 β = 2 ω , , for some ω ∈ R+/Q, ƒ(t) is 2π-periodic and bounded, χ± = max{±χ, 0}.