Convergence to common fixed points of averaged mappings without commutativity assumption in Hilbert spaces (Q2371169)

Let \(C\) be a closed convex subset of a Hilbert space \(H\). A mapping \(T: C\to C\) is called an averaged mapping on \(C\) if there is a nonexpansive mappng \(G: C\to C\) and a number \(\alpha\in (0,1)\) such that \(T= (1-\alpha)I+\alpha S\). It is clear that an averaged mapping is nonexpansive, but not vice versa. In the present paper, the authors prove the following theorem: Let \(H\) be a real Hilbert space and let \(C\) be a nonempty closed convex subset of \(H\). Let \(S\) and \(T\) be averaged mappings of \(C\) into itself such that \(\text{Fix}(S)\cap \text{Fix}(T)\) is nonempty. Suppose that \(\{\alpha_n\}_{n\geq 0}\) satisfies \[ \text{(i)}\quad \lim_{h\to\infty} \alpha_n= 0,\qquad \text{(ii)}\quad \sum_{u\geq 0} \alpha_n= \infty. \] For arbitrary \(x\in C\), the sequence \(\{x_n\}_{n\geq 0}\) generated by \(x_0= x,\) \[ x_{n+1}= \alpha_n\cdot x+ (1-\alpha_n) {2\over (n+1)(n+2)} \sum^n_{k=0}\;\sum_{i+ j=k} ((ST)^j S^{i-j},v(ST)^i T^{j-i}) x_n, \] converges strongly to a common fixed point \(Px\) of \(S\) and \(T\), where \(P\) is the metric projection of \(H\) onto \(\text{Fix}(S)\cap \text{Fix}(T)\) and \[ (ST)^j S^{i-j}, v(ST)^i T^{j-i}= \begin{cases} (ST)^j S^{i-j},& i\geq j,\\ (ST)^i T^{j-i},& i< j.\end{cases} \] This result extends a result of \textit{T.\,Shimizu} and \textit{W.\,Takahasi} [J.~Math.\ Anal.\ Appl.\ 211, No.\,1, 71--83 (1997; Zbl 0883.47075)].