In this paper, we study the exponential stability in the pth moment of mild solutions to neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion: where . Our method for investigating the stability of solutions is based on the Banach fixed point theorem. The obtained results generalize and improve the results due to Boufoussi and Hajji (Stat. Probab. Lett. 82:1549–1558, 2012), Caraballo et al. (Nonlinear Anal. 74:3671–3684, 2011), and Luo (J. Math. Anal. Appl. 355:414–425, 2009).

We focus on a class of neutral stochastic delay partial differential equations perturbed by a standard Brownian motion and a fractional Brownian motion. Under some suitable assumptions, the existence, uniqueness, and controllability results for these equations are investigated by means of the Banach fixed point method. Moreover, an example is presented to illustrate our main results.

This work investigates the stability of Halanay inequality. Some novel results are obtained by means of constructing an auxiliary differential equation. Some previous works are improved and extended. After that, the obtained results are applied to investigate the stability of neural networks with time-varying and distributed delays. At last, some examples along with numerical simulations are presented to illustrate the validity of the theoretical results. INDEX TERMS Stability, Halanay inequality, delay neural networks.

No abstract