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

Let C be a nonempty closed convex subset of real Hilbert space H and S = {T (s) : 0 <= s < infinity} be a nonexpansive semigroup on C such that F(S) not equal empty set. For a contraction f on C, and t is an element of (0, 1), let x(t) is an element of C be the unique fixed point of the contraction x bar right arrow tf(x) + (1 - t)1/lambda(t) integral(lambda t)(0) T(s)xds, where {lambda(t)} is a positive real divergent net. Consider also the iteration process {x(n)}, where x(0) is an element of C is arbitrary and x(n+1) = alpha(n) f(x(n)) + beta(n)x(n) + (1 - alpha(n) - beta(n)) 1/s(n) integral(sn)(0) T(s)x(n)ds for n >= 0, where {alpha(n)}, {beta(n)} subset of (0, 1) with alpha(n) + beta(n) < 1 and {s(n)} are positive real divergent sequences. It is proved that {x(t)} and, under certain appropriate conditions on {alpha(n)} and {beta(n)}, {x(n)} converges strongly to a common fixed point of S. (C) 2007 Elsevier Ltd. All rights reserved.