Result: Best approximation to common fixed points of a semigroup of nonexpansive operators

Let \(C\) be a nonempty closed convex subset of a Hilbert space \(H,\) \(\Gamma=\{T_{t}:t\in G\},\) \(G\subset\mathbb{R}_{+},\) a semigroup of nonexpansive operators \(T_{t}:C\rightarrow C,\) \(t\in G,\) and \(F=\cap_{t\in G}Fix\left( T_{t}\right) \) the set of common fixed points of the operators from \(\Gamma.\) For a fixed \(u\in C\) one looks for a best approximation element of \(u\) in \(F,\) denoted by \(P_{F}u.\) To solve this problem the author proposes the following algorithm: step \(0:\) \(x_{0}\in C\) is arbitrary; step \(n+1:\) \(x_{n+1}=\alpha_{n}u+\left( 1-\alpha_{n}\right) T_{r_{n}}x_{n},\) for all \(n\geq0,\) where \(0\leq\alpha_{n}\leq1\) and \(\{r_{n}\}_{n\geq0}\subset G\) is a given sequence. The semigroup \(\Gamma\) is called a uniformly asymptotically regular semigroup of nonexpansive operators on \(C\) if \(\lim_{n\rightarrow \infty}\left( \sup_{x\in C}\left\| T_{s}T_{r}x-T_{r}x\right\| \right) =0.\) The sequence \(\{\alpha_{n}\}\) \(_{n\geq0}\) is called a steering sequence if it has the following properties: \(\left( 1\right) \) \(\alpha_{n}\in [0;1],\;n\geq0\) and \(\alpha_{n}\rightarrow0;\) \(\left( 2\right) \) \(\sum_{n=0}^{+\infty}\alpha_{n}=+\infty\) (or, equivalently, \(\prod _{n=0}^{\infty}\left( 1-\alpha_{n}\right) =0);\) \(\left( 3\right) \) \(\sum_{n=0}^{\infty}\left| \alpha_{n+1}-\alpha_{n}\right