Fixed-point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces

Abstract

Let C be a nonempty closed convex subset of real Hilbert space H and S = {T (s) : 0 ≤ s < ∞} be a nonexpansive semigroup on C such that F (S) ≠ 0{combining long solidus overlay}. For a contraction f on C, and t ∈ (0, 1), let xt ∈ C be the unique fixed point of the contraction x {long rightwards arrow from bar} t f (x) + (1 - t) frac(1, λt) ∫0λt T (s) x d s, where {λt} is a positive real divergent net. Consider also the iteration process {xn}, where x0 ∈ C is arbitrary and xn + 1 = αn f (xn) + βn xn + (1 - αn - βn) frac(1, sn) ∫0sn T (s) xn d s for n ≥ 0, where {αn}, {βn} ⊂ (0, 1) with αn + βn < 1 and {sn} are positive real divergent sequences. It is proved that {xt} and, under certain appropriate conditions on {αn} and {βn}, {xn} converges strongly to a common fixed point of S. © 2007 Elsevier Ltd. All rights reserved.