Triple Hierarchical Variational Inequalities

In this chapter, we give a survey on hierarchical variational inequality problems and triple hierarchical variational inequality problems. By combining hybrid steepest descent method, Mann's iteration method, and projection method, we present a hybrid iterative algorithm for computing a fixed point of a pseudocontractive mapping and for finding a solution of a triple hierarchical variational inequality in the setting of real Hilbert space. We prove that the sequence generated by the proposed algorithm converges strongly to a fixed point which is also a solution of this triple hierarchical variational inequality problem. On the other hand, we also propose another hybrid iterative algorithm for solving a class of triple hierarchical variational inequality problems concerning a finite family of pseudo-contractive mappings in the setting of real Hilbert spaces. Under very appropriate conditions, we derive the strong convergence of the proposed algorithm to the unique solution of this class of problems.