Infinitely many solutions for the discrete Schrodinger equations with a nonlocal term

In the present paper, we consider the following discrete Schrodinger equations -(a+b Sigma(k)(is an element of Z)vertical bar Delta u(k-1)vertical bar(2)) Delta(2)u(k-1) + V(k)u(k) = f(k)(u(k)) k is an element of Z, where a, b are two positive constants and V = {V-k} is a positive potential. Delta u(k)(-1) = u(k)-u(k-1) and Delta(2) = Delta(Delta) is the one-dimensional discrete Laplacian operator. Infinitely many high-energy solutions are obtained by the Symmetric Mountain Pass Theorem when the nonlinearities {f (k)} satisfy 4-superlinear growth conditions. Moreover, if the nonlinearities are sublinear at infinity, we obtain infinitely many small solutions by the new version of the Symmetric Mountain Pass Theorem of Kajikiya.