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In the present paper, we consider the existence of solutions with a prescribed l 2 -norm for the following discrete Schrödinger equations, { − Δ 2 u k − 1 − f (u k) = λ u k k ∈ Z , ∑ k ∈ Z | u k | 2 = α 2 , where Δ 2 u k − 1 = u k + 1 + u k − 1 − 2 u k , f ∈ C (R) , α is a fixed constant, and λ ∈ R arises as a Lagrange multiplier. To get the solutions, we investigate the corresponding minimizing problem with the l 2 -norm constraint: E α = inf { 1 2 ∑ | Δ u k − 1 | 2 − ∑ F (u k) : ∑ | u k | 2 = α 2 }. An elaborative analysis on a minimizing sequence with respect to E α is obtained. We prove that there is a constant α 0 ≥ 0 such that there exists a global minimizer if α > α 0 , and there exists no global minimizer if α < α 0 . It seems that it is the first time to consider the solution with a prescribed l 2 -norm of the discrete Schrödinger equations. [ABSTRACT FROM AUTHOR]

By using critical point theory, Lyapunov-Schmidt reduction method, and characterization of the Brouwer degree of critical points, sufficient conditions to guarantee the existence of five or six solutions together with their sign properties to discrete second-order two-point boundary value problem are obtained. An example is also given to demonstrate our main result.