Hybrid implicit steepest-descent methods for triple hierarchical variational inequalities with hierarchical variational inequality constraints

In this paper, we introduce and analyze a hybrid implicit steepest-descent algorithm for solving the triple hierarchical variational inequality problem with the hierarchical variational inequality constraint for finitely many nonexpansive mappings in a real Hilbert space. The proposed algorithm is based on Korpelevich’s extragradient method, hybrid steepest-descent method, Mann’s implicit iteration method, and Halpern’s iteration method. Under mild conditions, the strong convergence of the iteration sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.