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

Let CC be a nonempty closed convex subset of real Hilbert space HH and S={T(s):0≤s<∞}S={T(s):0≤s<∞} be a nonexpansive semigroup on CC such that F(S)≠0̸F(S)≠0̸. For a contraction ff on CC, and t∈(0,1)t∈(0,1), let xt∈Cxt∈C be the unique fixed point of the contraction x⟼tf(x)+(1−t)1λt∫0λtT(s)xds, where {λt}{λt} is a positive real divergent net. Consider also the iteration process {xn}{xn}, where x0∈Cx0∈C is arbitrary and xn+1=αnf(xn)+βnxn+(1−αn−βn)1sn∫0snT(s)xnds for n≥0n≥0, where {αn},{βn}⊂(0,1){αn},{βn}⊂(0,1) with αn+βn<1αn+βn<1 and {sn}{sn} are positive real divergent sequences. It is proved that {xt}{xt} and, under certain appropriate conditions on {αn}{αn} and {βn}{βn}, {xn}{xn} converges strongly to a common fixed point of SS.