General iterative methods for a one-parameter nonexpansive semigroup in Hilbert space

Let HH be a Hilbert space and ff a fixed contractive mapping with coefficient 0<α<10<α<1, AA a strongly positive linear bounded operator with coefficient γ̄>0. Consider two iterative methods that generate the sequences {xn},{yn}{xn},{yn} by the algorithm, respectively. equation(I)xn=(I−αnA)1tn∫0tnT(s)xnds+αnγf(xn)equation(II)yn+1=(I−αnA)1tn∫0tnT(s)ynds+αnγf(yn) where {αn}{αn} and {tn}{tn} are two sequences satisfying certain conditions, and ℑ={T(s):s≥0}ℑ={T(s):s≥0} is a one-parameter nonexpansive semigroup on HH. It is proved that the sequences {xn},{yn}{xn},{yn} generated by the iterative method (I) and (II), respectively, converge strongly to a common fixed point x∗∈F(ℑ)x∗∈F(ℑ) which solves the variational inequality 〈(A−γf)x∗,x∗−z〉≤0z∈F(ℑ).