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We consider infinite-horizon difference games where the dynamics are governed by difference equations or difference inequalities and present various notions of Nash- equilibrium. Pontryagin maximum principles in both strong and weak forms are provided as necessary conditions for Nash equilibrium by using tools of reduction to the finite horizon and conditions related to the infinite horizon. Some special cases are then examined: the case of bounded processes and the case of asymptotical constraints for which tools of optimization in Banach spaces are used to provide necessary conditions for NE. Sufficient conditions are also studied