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 c < c+ . We also give certain sufficient conditions to insure that c+ ≥ 0 and we give an example when c+ c* . 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− ).
