Existence and Finite-Time Stability of Besicovitch Almost Periodic Solutions of Fractional-Order Quaternion-Valued Neural Networks with Time-Varying Delays

摘要

Almost periodic oscillation is an important dynamics of neural networks. Therefore, there have been many studies on Bohr almost periodic, Stepanov almost periodic and Weyl almost periodic oscillations of neural networks. Besicovitch almost periodicity is a generalization of the above almost periodicity, but there is no research on Besicovitch almost periodicity of neural networks. In this paper, we study the Besicovitch almost periodic oscillation for a class of fractional-order quaternion-valued neural networks with time-varying delays. Firstly, we give a definition of almost periodic functions in the sense of Besicovitch, which we call \(B^{\mathcal {P}}\)-almost periodic functions. Then we introduce and prove some properties of these functions. Since the space composed of these functions is incomplete, we construct a suitable Banach space, and use the contraction fixed point theorem to obtain the existence and uniqueness of \(B^{\mathcal {P}}\)-almost periodic solutions of the considered neural network. Secondly, we use some inequality techniques to study the finite-time stability of the \(B^{\mathcal {P}}\)-almost periodic solution. Even when the system we consider degenerates into a real-valued system, our results are new. Finally, we use a numerical example to illustrate the feasibility and validity of our results.