Persistence for a class of nonautonomous systems of differential equations with unbounded delays Teresa Faria, José J. Oliveira Mathematical Biosciences, 2026 • Criteria for the uniform persistence of some nonautonomous systems of differential equations with time-dependent unbounded delays are established. • An original method to deal with the unbounded delays is proposed, as the usual techniques for systems with finite delays do not apply to the systems considered here. • This method is applied to study the persistence of non-cooperative delay differential equations with unbounded delays. • This method has the potential to be adapted to other classes of differential equations with unbounded delays. • Even for the situation of delay differential equations with finite delays, the results on persistence presented here are in general better than the ones in recent literature, as less constraints on the boundedness of the coeffcients are imposed. • The class of differential equations under study is broad enough to encompass important models with unbounded delays used in mathematical biology and other natural sciences. For a family of nonautonomous systems of differential equations with unbounded delays, sufficient conditions for their persistence are established. In general, these systems are non-cooperative, so the usual techniques for monotone systems do not apply. The class of differential equations studied here is broad enough to encompass important models with unbounded delays used in mathematical biology. Our results extend and improve recent criteria for persistence of systems of differential equations with finite delays, and are illustrated with some selected examples given here.
Global attractivity criteria for a discrete-time Hopfield neural network model with unbounded delays via singular M—matrices José J. Oliveira, Ana Sofia Teixeira Neurocomputing, 2025 In this work, we establish two global attractivity criteria for a multidimensional discrete-time non-autonomous Hopfield neural network model with infinite delays and delays in the leakage terms. The first criterion, which applies when the activation functions are bounded, is based on 𝑀-matrices that are not necessarily invertible. The second criterion, relevant for unbounded activation functions, requires that a related singular 𝑀-matrix be irreducible. We contrast our findings with existing results in the literature and present numerical simulations to illustrate the novelty of the proposed criteria.
Existence and exponential stability of a periodic solution of an infinite delay differential system with applications to Cohen–Grossberg neural networks A. Elmwafy, José J. Oliveira, César M. Silva Communications in Nonlinear Science and Numerical Simulation, 2024 In the present paper, we investigate both the global exponential stability and the existence of a periodic solution of a general differential equation with unbounded distributed delays. The main stability criterion depends on the dominance of the non-delay terms over the delay terms. The criterion for the existence of a periodic solution is obtained with the application of the coincidence degree theorem. We use the main results to get criteria for the existence and global exponential stability of periodic solutions of a generalized higher-order periodic Cohen–Grossberg neural network model with discrete-time varying delays and infinite distributed delays. Additionally, we provide a comparison with the results in the literature and two numerical examples to illustrate the effectiveness of some of our results.
Existence and stability of a periodic solution of a general difference equation with applications to neural networks with a delay in the leakage terms António J.G. Bento, José J. Oliveira, César M. Silva Communications in Nonlinear Science and Numerical Simulation, 2023 In this paper, a new global exponential stability criterion is obtained for a general multidimensional delay difference equation using induction arguments. In the cases that the difference equation is periodic, we prove the existence of a periodic solution by constructing a type of Poincaré map. The results are used to obtain stability criteria for a general discrete-time neural network model with a delay in the leakage terms. As particular cases, we obtain new stability criteria for neural network models recently studied in the literature, in particular for low-order and high-order Hopfield and Bidirectional Associative Memory (BAM).
Global stability criteria for nonlinear differential systems with infinite delay and applications to BAM neural networks José J. Oliveira Chaos Solitons and Fractals, 2022 For a general n-dimensional nonautonomous and nonlinear differential equation with infinite delay, we give sufficient conditions for its global asymptotic stability. The main stability criterion depends on the size of the delay on the linear part and the dominance of the linear terms over the nonlinear terms. We apply our main result to answer several open problems left by Berezansky et al. (2014). Using the obtained theoretical stability results, we get sufficient conditions for both the global asymptotic and global exponential stability of a bidirectional associative memory neural network model with delays which generalizes models recently studied. Finally, a numerical example is given to illustrate the novelty of our results.
Global exponential stability of discrete-time Hopfield neural network models with unbounded delays José J. Oliveira Journal of Difference Equations and Applications, 2022 In this paper, a general setting is presented to study the exponential stability of discrete-time systems with bounded or unbounded delays. Based on the M-matrix theory, we establish sufficient conditions to ensure the global exponential stability of the zero equilibrium of low-order, and high-order, discrete-time Hopfield neural network models with unbounded delays and delay in the leakage terms. A comparison of the literature shows that our results generalize and improve some in recent publications.
