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)=ζk on each interval [tk,tk+1), where tk≤ζk≤ tk+1, k∈ℕ0. Depending on the position of ζk, 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 ζk=tk reduces to their setting, whereas our results cover the more general choice of ζk within each interval. We also present examples illustrating different types of piecewise constant arguments.
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BibTeX
@misc{naranjo2026periodic,
title = {{Periodic Solutions and Exponential Attractivity in Hopfield-Type Neural Networks with Generalized Piecewise Constant Arguments: Impulsive and Nonimpulsive Cases}},
author = {Ricardo Torres Naranjo and Samuel Castillo Apolonio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36560},
eprint = {2610.36560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36560},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Periodic Solutions and Exponential Attractivity in Hopfield-Type Neural Networks with Generalized Piecewise Constant Arguments: Impulsive and Nonimpulsive Cases
AU - Ricardo Torres Naranjo
AU - Samuel Castillo Apolonio
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.36560
UR - https://arxiv.org/abs/2610.36560
AB - 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≤ζ_k≤ t_k+1, k∈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.
ER -