Permanence for a Generalized Discrete Neural Network System

We prove that the system of difference equationsxn+1(i)=λixn(i)+fi(αixn(i+1)−βixn−1(i+1)),i∈{1,2,…,k},n∈ℕ, (we regard thatxn(k+1)=xn(1)) is permanent, provided thatαi≥βi,λi+1∈[0,βi/αi),i∈{1,2,…,k},fi:ℝ→ℝ,i∈{1,2,…,k}, are nondecreasing functions bounded from below and such that there areδi∈(0,1)andM>0such thatfi(αix)≤δix,i∈{1,2,…,k}, for allx≥M. This result considerably extends the results existing in the literature. The above system is an extension of a two-dimensional discrete neural network system.