Periodic Solutions and Exponential Attractivity in Hopfield-Type Neural Networks with Generalized Piecewise Constant Arguments: Impulsive and Nonimpulsive Cases

Abstract

We investigate an impulsive Hopfield-type neural network with a generalized piecewise constant argument, including the nonimpulsive case in which all jump terms vanish. The piecewise constant argument is given by $γ(t)=ζ_k$ on each interval $[t_k,t_{k+1})$, where $t_k\leqζ_k\leq t_{k+1}$, $k\in\mathbb{N}_0$. Depending on the position of $ζ_k$, the argument may be advanced, retarded, or of mixed type. By representing periodic solutions through a Green function and applying a fixed-point argument, we obtain explicit and verifiable sufficient conditions for the existence and uniqueness of an $ω$-periodic solution. We then use a Gronwall-type estimate, together with period averages, to prove that this solution is globally exponentially attractive. These conditions can be checked directly from the system parameters, without requiring explicit knowledge of the periodic solution. In the course of the analysis, we also revisit and correct some estimates presented by Pinto et al. \cite{Pinto_ejqtde}. The case $ζ_k=t_k$ reduces to their setting, whereas our results cover the more general choice of $ζ_k$ within each interval. We also present examples illustrating different types of piecewise constant arguments.