Permanence for a generalized discrete neural network system (Q2478368)

Summary: We prove that the system of difference equations \(x_{n+1}^{(i)}= \lambda_ix_n^{(i)}+ f_i(\alpha_ix_n^{(i+1)}- \beta_ix_{n-1}^{(i+1)})\), \(i\in\{1,2,\dots,k\}\), \(n\in\mathbb N\) (we regard that \(x_n^{(k+1)}= x_n^{(1)}\)) is permanent, provided that \(\alpha_i\geq\beta_i\), \(\lambda_{i+1}\in [0,\beta_i/\alpha_i)\), \(i\in\{1,2,\dots,k\}\), \(f_i:\mathbb R\to\mathbb R\), \(i\in\{1,2,\dots,k\}\), are nondecreasing functions bounded from below and such that there are \(\delta_i\in(0,1)\) and \(M>0\) such that \(f_i(\alpha_ix)\leq\delta_ix\), \(i\in\{1,2,\dots,k\}\), for all \(x\geq M\). This result considerably extends the results existing in the literature. The above system is an extension of a two-dimensional discrete neural network system.