Razumikhin-type theorems of infinite dimensional stochastic functional differential equations

The argument of Razumikhin-type has been well developed and showed significant advantage for the stability of stochastic functional differential equations in finite dimensions. However, so far there have been almost no results of Razun-dkhin-type on the stability of mild solutions of stochastic functional differential equations in infinite dimensions. The main aim of this paper is to establish Razurrtikhin-type stability theorems for stochastic functional differential equations in infinite dimensions. By virtue of these new criteria, we can establish the exponential stability of stochastic delay differential equations and stochastic delay partial differential equations.