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 ($\xi$-norm). A general DDD HNN model is firstly introduced, where the discrete delays ${\tau }_{pq}\left(t\right)$ are asynchronous time-varying delays. Then, $\{\xi ,1\}$-norm, $\{\xi ,2\}$-norm and $\{\xi ,\infty \}$-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 $\mu$-stability of different DDD NNs. Finally, two numerical examples and their simulation figures are given to illustrate the effectiveness of these results.

本文通过三种广义范数的定义来研究具有离散和分布时滞（DDD HNN）的Hopfield神经网络解的存在性，唯一性和稳定性准则。$\xi$-规范）。首先介绍了通用DDD HNN模型，其中离散延迟${\tau }_{pq}（Ť）$是异步时变延迟。然后，$\{\xi ，\mathrm{1个}\}$-规范， $\{\xi ，2\}$-规范和 $\{\xi ，\infty \}$-norm连续用于推导DDD HNN的解的存在性，唯一性和稳定性准则。在定理的证明中，给出了特殊的函数和假设来处理离散和分布式延迟。此外，得出了解决方案的存在性和稳定性标准的推论。本文给出的方法也可用于研究同步和$\mu$-不同DDD NN的稳定性。最后，给出了两个数值示例及其仿真图，以说明这些结果的有效性。