We study discrete monostable dynamics with general Lipschitz non-linearities. This includes also degenerate non-linearities. In the positive monostable case, we show the existence of a branch of traveling waves solutions for velocities c≥c+, with non existence of solutions for cc∗. This model of discrete dynamics can be seen as a generalized Frenkel-Kontorova model for which we can also add a driving force parameter σ. We show that σ can vary in an interval [σ−,σ+]. For σ∈(σ−,σ+) this corresponds to a bistable case, while for σ=σ+ this is a positive monostable case, and for σ=σ− this is a negative monostable case. We study the velocity function c=c(σ) as σ varies in [σ−,σ+]. In particular for σ=σ+ (resp. σ=σ−), we find vertical branches of traveling waves solutions with c≥c+ (resp. c≤c−). The results have been obtained in collaboration with R. Monneau