Composite implicit general iterative process for a nonexpansive semigroup in Hilbert space (Q1008507)

Let \(C\) be nonempty closed convex subset of a real Hilbert space \(H\). Consider on \(C\) a nonexpansive semigroup \({\mathfrak I}=\{T(s):s\geq 0\}\) with a common fixed point, a contraction \(f\) with coefficient \(0< \alpha<1\), and a strongly positive linear bounded operator \(A\) with coefficient \(\overline\gamma>0\). Let \(0<\gamma< \overline\gamma/ \alpha\). It is proved that the sequence \(\{x_n\}\) generated iteratively by \(x_n=(I-\alpha_nA)(1/t_n)\int_0^{t_n}T(s)y_n\,ds+\alpha_n\gamma f(x_n)\), \(y_n=(I-\beta_nA)x_n+\beta_n\gamma f(x_n)\), converges strongly to a common fixed point \(x^*\in F({\mathfrak J})\) which solves the variational inequality \(\langle(\gamma f-A)x^*,z-x^*\rangle\leq 0\) for all \(z\in F({\mathfrak J})\).