We study the discrete nonlinear equation

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where inline image (the spectrum of L) and inline image is asymptotically linear as inline image for all inline image. 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.