We study the discrete nonlinear equation
where (the spectrum of L) and is asymptotically linear as for all . We obtain the existence of ground state solitons and the existence of infinitely many pairs of geometrically distinct solitons of this equation. Our method is based on the generalized Nehari manifold method developed recently by Szulkin and Weth. To the best of our knowledge, this technique has not been used for discrete equations with saturable nonlinearities.