A new class of generalized quasi-variational inequalities with applications to Oseen problems under nonsmooth boundary conditions

In this paper, we study a generalized quasi-variational inequality (GQVI for short) with two multivalued operators and two bifunctions in a Banach space setting. A coupling of the Tychonov fixed point principle and the Katutani-Ky Fan theorem for multivalued maps is employed to prove a new existence theorem for the GQVI. We also study a nonlinear optimal control problem driven by the GQVI and give sufficient conditions ensuring the existence of an optimal control. Finally, we illustrate the applicability of the theoretical results in the study of a complicated Oseen problem for non-Newtonian fluids with a nonmonotone and multivalued slip boundary condition (i.e., a generalized friction constitutive law), a generalized leak boundary condition, a unilateral contact condition of Signorini's type and an implicit obstacle effect, in which the multivalued slip boundary condition is described by the generalized Clarke subgradient, and the leak boundary condition is formulated by the convex subdifferential operator for a convex superpotential.