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.

In this paper, we consider and study a system of generalized variational inclusions involving Cayley operators and an XOR-operation in q-uniformly smooth Banach spaces. To obtain the solution of the system of generalized variational inclusions involving Cayley operators and an XOR-operation, we use some properties of Cayley operators as well as an XOR-operation. We also discuss the convergence criterion. In support of our main result, we provide an example.

In this paper, we study the existence of solutions of equilibrium problems in the setting of Hadamard manifolds under the pseudomonotonicity and geodesic up

We devise a relaxed-inertial subgradient extragradient method for solving pseudomonotone equilibrium problems on Hadamard manifolds. The method uses a self-adaptive step size that does not depend on the Lipschitz constant. Under suitable conditions, we demonstrate that every sequence produced by the proposed algorithms converges to a solution of the equilibrium problem. We also prove linear convergence of the proposed method when the bifunction satisfies strong pseudomonotonicity and Lipschitz continuity. We then provide an application of our main result to variational inequality and optimization problems using the proposed algorithms. We conclude our paper with a numerical example to verify our convergence result.

This is the first paper dealing with the study of minimum and maximum principle sufficiency properties for nonsmooth variational inequalities by using gap functions in the setting of Hadamard manifolds. We also provide some characterizations of these two sufficiency properties. We conclude the paper with a discussion of the error bounds for nonsmooth variational inequalities in the setting of Hadamard manifolds.