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Constant-sign solutions of a system of difference equations of Urysohn type

We consider the following system of difference equations where T ≥ N>0. By using fixed point theorems and fixed point index theory, criteria are developed to guarantee the existence of single, double and multiple solutions of the system that are of constant signs, i.e. for each 1 ≤ i ≤ n, u i (k) ≥ 0 or u i (k) ≤ 0 for k ∈ {0,1,…,T}. We also consider the eigenvalue problem and establish criteria for the existence of a constant-sign solution when λ is in some interval.