Existence and Exponential Stability of Solutions for Quaternion-Valued Delayed Hopfield Neural Networks by $\xi$ -Norms

Recently, with the development of quaternion applications, quaternion-valued neural networks (QVNNs) have been presented and studied by more and more scholars. In this paper, the existence, uniqueness and exponential stability criteria of solutions for the quaternion-valued delayed Hopfield neural networks (QVDHNNs) are mainly investigated by means of the definitions of ξ-norms. In order to construct a ξ-norm, QVDHNNs system are decomposed into four real-number systems according to Hamilton rules. Then, taking advantage of ξ-norms, inequality technique and Cauchy's test for convergence, time-invariant delays and time-varying delays are considered successively to derive ξ-exponential type sufficient conditions. Based on these, several corollaries about the existence, uniqueness and exponential stability of solutions are obtained. Finally, two numerical examples with time-invariant delays and time-varying delays are given respectively. Their simulated images illustrate the effectiveness of the main theoretical results.