We study the problem of the existence and determining Nash equilibria for stochastic games with finite state and action spaces in the case when a part of players use limiting average reward payoffs criteria and an another part of players use the discounted reward payoffs criteria. Flesh, Thuijsman and Vrieze proved in [1] the existence of ϵ-equilibrium (∀ϵ > 0) in non-stationary strategies for twoplaywer stochastic games, where player 1 uses the limiting average reward payoff ceiterion and player 2 usis the discounted reward payoff criterion. Such an equilibrium in [1] is called average-discounted equilibrium. Note, that for an average stochastic game, in general, a Nash equilibrium in stationary strategies may not exist [2], however for a discounted stochastic game a stationary Nash equilibrium always exists [3]. In this contribution we show that for an m-player stochastic game with finite state and action spaces, where one player uses the limiting average reward payoff criterion and the rest of players use the discounted reward payoffs criteria possesses an ϵ-equilibrium in nonstationary strategies. Additionally we show that for an arbitrary m-player stochastic positional game there exists a Nash equilibrium in stationary strategies. Based on constructive proof of this result we propose an algorithm for determining the optimal stationary strategies of the players.