Exponential stability for neutral stochastic functional partial differential equations driven by Brownian motion and fractional Brownian motion

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: d[x(t)+g(t,xt)]=[Ax(t)+f(t,xt)]dt+h(t,xt)dW(t)+σ(t)dBH(t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ d \bigl[x(t)+g(t,x_{t}) \bigr]= \bigl[Ax(t)+f(t,x_{t}) \bigr] \,dt+h(t,x_{t})\,dW(t)+\sigma(t)\,dB^{H}(t), $$\end{document} where H∈(1/2,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H\in(1/2,1)$\end{document}. 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).