Permanence for a generalized discrete neural network system

Abstract

We prove that the system of difference equations xn+1(i) = λi xn(i) + fi(αi xn(i+1) - βi xn-1(i+1)), i ∈ {1, 2, ..., k}, n ∈ ℕ, (we regard that xn(k+1) = xn(1)) is permanent, provided that αi ≥ 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 are δi ∈ (0, 1) and M > 0 such that fi(αix) ≤ δix, i ∈ {1, 2, ..., k}, for all x ≥ 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.