General Iterative Method for Convex Feasibility Problem via the Hierarchical Generalized Variational Inequality Problems

Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Am, Bm : C → H be relaxed cocoercive mappings for each 1 ≤ m ≤ r, where r ≥ 1 is integer. Let f : C → C be a contraction with coefficient k ∈ (0, 1). Let G : C → C be ξ-strongly monotone and L-Lipschitz continuous mappings. Under the assumption ∩m=1GV I(C,Bm, Am) 6= ∅, where GV I(C,Bm, Am) is the solution set of a generalized variational inequality. Consequently, we prove a strong convergence theorem for finding a point x̃ ∈ ∩m=1GV I(C,Bm, Am) which is a unique solution of the hierarchical generalized variational inequality 〈(γf−μG)x̃, x− x̃〉 ≤ 0, ∀x ∈ ∩m=1GV I(C,Bm, Am).