Second order periodic problems in the presence of dry friction

We prove, via an approach by ordinary differential equations, the existence of oscillations for second order differential inclusions of the form u″+u∈φ(t)−μ(u)S(u′),u″+u∈φ(t)−μ(u)S(u′), where φφ is 2π2π-periodic, μμ is allowed to satisfy the at most linear growth condition of the form μ0≤μ(u)≤μ0+μ1|u|μ0≤μ(u)≤μ0+μ1|u| with some restrictions on μ1μ1, SS is bounded and continuous in R∖{0}R∖{0} with a jump discontinuity at 0 and S(0−)

► We prove the existence of periodic solutions of a generalized friction oscillator.
► The second order differential inclusion is at resonance at the second eigenvalue.
► The friction coefficient is allowed to be at most linear growth.