Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications

Abstract

In this paper, we focus on the mild periodic solutions to a class of delayed stochastic reaction-diffusion differential equations. First, the key issues of Markov property in Banach space \begin{document}$ C $\end{document} , \begin{document}$ p $\end{document} -uniformly boundedness, and \begin{document}$ p $\end{document} -point dissipativity of mild solutions \begin{document}$ \boldsymbol{u}_t $\end{document} to the equations are discussed. Then, the theorems of existence-uniqueness and exponential stability in the mean-square sense of the mild periodic solutions are established by using the dissipative theory and the operator semigroup technique, and the relevant results about the existence of mild periodic solutions in the quoted literature are generalized. Next, the given theoretical results are successfully applied to the delayed stochastic reaction-diffusion Hopfield neural networks, and some easy-to-test criteria of exponential stability for the mild periodic solution to the networks are obtained. Finally, some examples are presented to demonstrate the feasibility of our results.