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Updated: Dec 20, 2025

Author Spotlight: Modular Neuronal Networks for Analyzing Brain Functions
Published on: June 7, 2024
Generalized norm for existence, uniqueness and stability of Hopfield neural networks with discrete and distributed
Huamin Wang1, Guoliang Wei2, Shiping Wen3
1Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471934, China.
Abstract:
In this paper, the existence, uniqueness and stability criteria of solutions for Hopfield neural networks with discrete and distributed delays (DDD HNNs) are investigated by the definitions of three kinds of generalized norm (ξ-norm). A general DDD HNN model is firstly introduced, where the discrete delays τpq(t) are asynchronous time-varying delays. Then, {ξ,1}-norm, {ξ,2}-norm and {ξ,∞}-norm are successively used to derive the existence, uniqueness and stability criteria of solutions for the DDD HNNs. In the proof of theorems, special functions and assumptions are given to deal with discrete and distributed delays. Furthermore, a corollary is concluded for the existence and stability criteria of solutions. The methods given in this paper can also be used to study the synchronization and μ-stability of different DDD NNs. Finally, two numerical examples and their simulation figures are given to illustrate the effectiveness of these results.
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