An Implicit Hierarchical Fixed-Point Approach to General Variational Inequalities in Hilbert Spaces. (20th March 2011) - Zeng, L. C.; Wen, Ching-Feng; Yao, J. C.
Abstract : LetC be a nonempty closed convex subset of a real Hilbert spaceH . LetF : C → H be aκ -Lipschitzian andη -strongly monotone operator with constantsκ, η > 0, V, T : C → C be nonexpansive mappings withFix ( T ) ≠ ∅ whereFix ( T ) denotes the fixed-point set ofT, andf : C → H be aρ -contraction with coefficientρ ∈ [ 0, 1 ) . Let0 < μ < 2 η / κ 2 and0 < γ ≤ τ, whereτ = 1 - 1 - μ ( 2 η - μ κ 2 ) . For eachs, t ∈ ( 0, 1 ), letx s, t be a unique solution of the fixed-point equationx s, t = P C [ s γ f ( x s, t ) + ( I - s μ F ) ( t V + ( 1 - t ) T ) x s, t ] . We derive the following conclusions on the behavior of the net x s, t along the curvet = t ( s ) : (i) ift ( s ) = O ( s ), ass → 0, thenx s, t ( s ) → z ∞ strongly, which is the unique solution of the variational inequality of findingz ∞ ∈ Fix ( T ) such that〈 [ ( μ F - γ f ) + l ( I - V ) ] z ∞, x - z ∞ 〉 ≥ 0, for all x ∈ Fix ( T ) and (ii) ift ( s ) / s → ∞, ass → 0, thenx s, t ( s ) → x ∞ strongly, which is the unique solution of some hierarchical variational inequality problem.