Multi-valued perturbations on stochastic evolution equations driven by fractional Brownian motions

Abstract

We consider a stochastic evolution inclusion having deterministic multi-valued nonlinearity and fractional Brownian motion with nonlinear diffusion. We establish the nonemptiness and compactness of its solution set. After that, the upper semicontinuity with respect to random parameters and initial values of the corresponding solution map is proved. In particular, the results on nonemptiness and upper semicontinuity imply that the inclusion under consideration defines a multi-valued random dynamical system. Moreover, under an extra smooth assumption on the diffusion, it is demonstrated that the solution set has the topological structure of R δ -type.