We consider non-autonomous forced and damped general discrete nonlinear Schrödinger equations on the lattice \({\mathbb {Z}}\) . The forcing is of additive and multiplicative type. We present criteria for the existence and nonexistence of nontrivial travelling wave solutions (TWSs) facilitating fixed point methods where we distinguish between combined additive and multiplicative forcings and only additive or only multiplicative forcing, respectively. We locate a ring in the energy space for nontrivial solutions to exist and energy (norm) thresholds for their existence. We also prove the existence of solitary TWSs and derive an upper bounds on their velocity.
