We consider the discrete nonlinear Schr{\o}dinger equation with linear gain and nonlinear loss. We show that for the infinite lattice with nonzero boundary conditions, which describe solutions decaying on a finite background, solutions to the corresponding initial-boundary value problem exist for any initial condition if and only if the background amplitude has a specific value $A_*$, determined by the gain-loss parameters.

For the finite-dimensional dynamical system defined by the periodic lattice, the dynamics for all initial conditions are governed by a global attractor, represented by a plane wave with amplitude $A_*$. Consequently, any instability effects or localized phenomena observed in the finite system are only transient before convergence to this trivial attractor.

Finally, we study the dynamics of localized initial conditions on the constant background and investigate the potential impact of the global asymptotic stability of the background with amplitude $A_*$ on the system`s long-term evolution.