The authors consider the nonlinear discrete periodic system
where is continuous in and with saturable nonlinearity for each , , are real valued -periodic sequences. They are interested in the existence of nontrivial homoclinic solutions for this equation; this problem appears when one looks for the discrete solitons of the periodic discrete nonlinear Schrödinger equations. A new sufficient condition guaranteeing the existence of homoclinic solutions is obtained by using critical point theory. It is proved that it is also necessary in some special cases. Moreover, the rate of decay is established.