On the Existence of the Exact Solution of Quaternion-Valued Neural Networks Based on a Sequence of Approximate Solutions

In many practical applications, it is difficult or impossible to obtain the exact solution of the mathematical model due to the limitations of solving methods and the complexity of the neural network itself. A natural problem is given as follows: does the exact solution of quaternion-valued neural networks (QVNNs) exist when successively improved approximate solutions can be obtained? Fortunately, the Hyers-Ulam stability happens to be one of the important means to deal with this problem. In this article, the issue of Hyers-Ulam stability of QVNNs with time-varying delays is addressed. First, inspired by the Hyers-Ulam stability of general functional equations, the concept of the Hyers-Ulam stability of QVNNs is proposed along with the QVNNs model. Then, by utilizing the successive approximation method, both delay-dependent and delay-independent Hyers-Ulam stability criteria are obtained to ensure the Hyers-Ulam stability of the QVNNs considered. Finally, a simulation example is given to verify the effectiveness of the derived results.

Automated summary

This article addresses the issue of Hyers-Ulam stability of quaternion-valued neural networks (QVNNs) with time-varying delays. The concept of Hyers-Ulam stability for QVNNs is proposed and a model is presented. Both delay-dependent and delay-independent stability criteria are obtained using the successive approximation method. The effectiveness of the derived results is verified through a simulation example.