An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces

Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → H be a κ-Lipschitzian and η-strongly monotone operator with constants κ, η > 0, V, T : C → C be nonexpansive mappings with Fix T / ∅ where Fix T denotes the fixed-point set of T , and f : C → H be a ρ-contraction with coefficient ρ ∈ 0, 1 . Let 0 < μ < 2η/κ2 and 0 < γ ≤ τ , where τ 1 − √ 1 − μ 2η − μκ2 . For each s, t ∈ 0, 1 , let xs,t be a unique solution of the fixed-point equation xs,t PC sγf xs,t I − sμF tV 1 − t T xs,t . We derive the following conclusions on the behavior of the net {xs,t} along the curve t t s : i if t s O s , as s → 0, then xs,t s → z∞ strongly, which is the unique solution of the variational inequality of finding z∞ ∈ Fix T such that 〈 μF − γf l I − V z∞, x − z∞〉 ≥ 0, for all x ∈ Fix T and ii if t s /s → ∞, as s → 0, then xs,t s → x∞ strongly, which is the unique solution of some hierarchical variational inequality problem.