Regularized algorithms for hierarchical fixed-point problems (Q640212)

Let \(H\) be a real Hilbert space and \(C\) be a closed convex subset of it. Let \(S:C\to C\) and \((T_i:C\to C; i=1,2,\dots)\) be non-expansive maps with \(F:=\bigcap\{\text{Fix}(T_i): i\geq 1\}\neq \emptyset\). For the hierarchical fixed-point problem [Find \(x^*\in F\) such that \(\langle x^*-Sx^*,x^*-x\rangle \leq 0\), \(\forall x\in F\)], the regularized algorithm of the form [\(x_{n+1}=\alpha_nSx_n+(1-\alpha_n)W_n\text{proj}_C[(1-\beta_n)x_n]; \;n\geq 0\)] converges strongly to its maximum norm solution.