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This paper is devoted to a system of stochastic partial differentialequations (SPDEs) that have a slow component driven by fractional Brownianmotion (fBm) with the Hurst parameter $H >1/2$ and a fast component driven byfast-varying diffusion. It improves previous work in two aspects: Firstly,using a stopping time technique and an approximation of the fBm, we prove anexistence and uniqueness theorem for a class of mixed SPDEs driven by both fBmand Brownian motion; Secondly, an averaging principle in the mean square sensefor SPDEs driven by fBm subject to an additional fast-varying diffusion processis established. To carry out these improvements, we combine the pathwiseapproach based on the generalized Stieltjes integration theory with the It\^ostochastic calculus. Then, we obtain a desired limit process of the slowcomponent which strongly relies on an invariant measure of the fast-varyingdiffusion process.

Pathwise unique solutions and stochastic averaging for mixed stochastic  partial differential equations driven by fractional Brownian motion and  Brownian motion