Harmonic solutions of a dry friction system

In this paper we study harmonic solutions of the second order differential inclusion ẍ+g(x,ẋ)+u(x)Sgnẋ∋ϕ(t), which models a mechanical system with dry friction, viscous damping and T-periodic external force. Under a weaker assumption than published works, we give sufficient conditions for the existence of harmonic solutions not only by the Degree Theory when g(x,ẋ) depends on ẋ but also by the Schauder's Fixed Point Theorem when g(x,ẋ) is independent of ẋ. After obtaining the necessary and sufficient condition for the existence of constant solutions, we find the number of harmonic solutions based on the number of constant solutions when u(x) is a positive constant.