Existence results and implicit viscosity methods for hierarchical problems on hadamard manifolds

In this paper, we consider the variational inequality problems in¬volving two nonexpansive mappings or a nonexpansive mapping and a weakly contraction mapping and defined over a set of fixed points of another nonex¬pansive mapping in the setting of Hadamard manifolds. We also consider some special cases of our problems. We first establish the existence of a solution of the considered problem. Then we propose implicit type viscosity approximation methods for solving considered problems and study their convergence results. A numerical example is given at the end of this paper in support of proposed algorithms and results. The results of this paper not only extend the results from linear structure to manifolds but also improve several known results.