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 - µ(2η - µκ²). For each s, t ∈ (0, 1), let x_s,t be a unique solution of the fixed-point equation x_s,t PC[sγf(x_s,t) + (I - sµF)(tV + (1 - t)T)x_s,t].We derive the following conclusions on the behavior of the net {x_s,t} along the curve t = t(s): (i) if t(s) = O(s), as s → 0, then x_s,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 x_s,t(s) → x_∞ strongly, which is the unique solution of some hierarchical variational inequality problem. [ABSTRACT FROM AUTHOR]