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

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 View the MathML sourceγ̄>0. Consider two iterative methods that generate the sequences {xn},{yn}{xn},{yn} by the algorithm, respectively. equation(I) View the MathML sourcexn=(I−αnA)1tn∫0tnT(s)xnds+αnγf(xn) Turn MathJax on equation(II) View the MathML sourceyn+1=(I−αnA)1tn∫0tnT(s)ynds+αnγf(yn) Turn MathJax on 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