Nonsmooth variational inequalities on Hadamard manifolds

Nonsmooth variational inequality problem (NVIP) and Minty nonsmooth variational inequality problem (MNVIP) in terms of a bifunction are considered in the setting of Hadamard manifolds. We first establish a Browder type fixed point theorem in the setting of Hadamard spaces, and then by using it we prove some existence results for solutions of NVIP and MNVIP. We give some relations among NVIP, MNVIP and nonsmooth optimization problem. The gap functions for NVIP and MNVIP are also studied in the setting of Hadamard manifolds.