Variational inequality and proximal scheme for enriched nonexpansive mappings in CAT(0) spaces

This paper presents the variational inequality problem in CAT(0) spaces, with the underlying operator being an enriched nonexpansive mapping. The existence of a solution to the problem is discussed with fixed points of the mapping. The paper presents substantial properties of the problem, including convergence analyses of certain bounded nets. The regularized version of the problem is also analyzed, leading to the proposal of an inexact proximal point scheme for approximating fixed points of the mapping that also solves the problem. Under different mild conditions, the sequences generated from the proposed algorithm are shown to Δ-converge to a fixed point which solves the problem.