Positive solutions of Neumann problems for a discrete system coming from models of house burglary

We show existence results of positive solutions of Neumann problems for a discrete system: η∆(Ak−1 −Ak−1)−Ak +Ak +NkAk = 0, k ∈ [2, n− 1]Z, ∆ ( ∆Nk−1 − 2Nk ∆Ak−1 Ak ) −NkAk +Ak −Ak = 0, k ∈ [2, n− 1]Z, ∆A1 = 0 = ∆An−1, ∆N1 = 0 = ∆Nn−1, where the assumptions on η, Ak , and Ak are motivated by some mathematics models for house burglary. Our results are based on the topological degree theory.